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the prisoner's dilemma</span> </div> </a> <ul id="toc-Strategy_for_the_prisoner's_dilemma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_form" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalized_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generalized form</span> </div> </a> <ul id="toc-Generalized_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_iterated_prisoner's_dilemma" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#The_iterated_prisoner's_dilemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>The iterated prisoner's dilemma</span> </div> </a> <button aria-controls="toc-The_iterated_prisoner's_dilemma-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle The iterated prisoner's dilemma subsection</span> </button> <ul id="toc-The_iterated_prisoner's_dilemma-sublist" class="vector-toc-list"> <li id="toc-General_strategy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_strategy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>General strategy</span> </div> </a> <ul id="toc-General_strategy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Axelrod's_tournament_and_successful_strategy_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Axelrod's_tournament_and_successful_strategy_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Axelrod's tournament and successful strategy conditions</span> </div> </a> <ul id="toc-Axelrod's_tournament_and_successful_strategy_conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_strategies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_strategies"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Other strategies</span> </div> </a> <ul id="toc-Other_strategies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stochastic_iterated_prisoner's_dilemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stochastic_iterated_prisoner's_dilemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Stochastic iterated prisoner's dilemma</span> </div> </a> <ul id="toc-Stochastic_iterated_prisoner's_dilemma-sublist" class="vector-toc-list"> <li id="toc-Zero-determinant_strategies" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Zero-determinant_strategies"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4.1</span> <span>Zero-determinant strategies</span> </div> </a> <ul id="toc-Zero-determinant_strategies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Continuous_iterated_prisoner's_dilemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_iterated_prisoner's_dilemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Continuous iterated prisoner's dilemma</span> </div> </a> <ul id="toc-Continuous_iterated_prisoner's_dilemma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Real-life_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Real-life_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Real-life examples</span> </div> </a> <button aria-controls="toc-Real-life_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Real-life examples subsection</span> </button> <ul id="toc-Real-life_examples-sublist" class="vector-toc-list"> <li id="toc-Environmental_studies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Environmental_studies"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Environmental studies</span> </div> </a> <ul id="toc-Environmental_studies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Animals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Animals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Animals</span> </div> </a> <ul id="toc-Animals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Psychology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Psychology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Psychology</span> </div> </a> <ul id="toc-Psychology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Economics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Economics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Economics</span> </div> </a> <ul id="toc-Economics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sport" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sport"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Sport</span> </div> </a> <ul id="toc-Sport-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-International_politics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#International_politics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>International politics</span> </div> </a> <ul id="toc-International_politics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplayer_dilemmas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplayer_dilemmas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Multiplayer dilemmas</span> </div> </a> <ul id="toc-Multiplayer_dilemmas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Academic_settings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Academic_settings"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Academic settings</span> </div> </a> <ul id="toc-Academic_settings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_games" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_games"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Related games</span> </div> </a> <button aria-controls="toc-Related_games-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Related games subsection</span> </button> <ul id="toc-Related_games-sublist" class="vector-toc-list"> <li id="toc-Closed-bag_exchange" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed-bag_exchange"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Closed-bag exchange</span> </div> </a> <ul id="toc-Closed-bag_exchange-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Friend_or_Foe?" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Friend_or_Foe?"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span><i>Friend or Foe?</i></span> </div> </a> <ul id="toc-Friend_or_Foe?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Iterated_snowdrift" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Iterated_snowdrift"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Iterated snowdrift</span> </div> </a> <ul id="toc-Iterated_snowdrift-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coordination_games" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coordination_games"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Coordination games</span> </div> </a> <ul id="toc-Coordination_games-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymmetric_prisoner's_dilemmas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymmetric_prisoner's_dilemmas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Asymmetric prisoner's dilemmas</span> </div> </a> <ul id="toc-Asymmetric_prisoner's_dilemmas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Software" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Software"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Software</span> </div> </a> <ul id="toc-Software-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_fiction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_fiction"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>In fiction</span> </div> </a> <ul id="toc-In_fiction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_moral_philosophy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_moral_philosophy"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>In moral philosophy</span> </div> </a> <button aria-controls="toc-In_moral_philosophy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle In moral philosophy subsection</span> </button> <ul id="toc-In_moral_philosophy-sublist" class="vector-toc-list"> <li id="toc-Pure_and_impure_prisoner's_dilemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pure_and_impure_prisoner's_dilemma"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Pure and impure prisoner's dilemma</span> </div> </a> <ul id="toc-Pure_and_impure_prisoner's_dilemma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">12.1</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Prisoner's dilemma</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 57 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-57" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">57 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%B6%D9%84%D8%A9_%D8%A7%D9%84%D8%B3%D8%AC%D9%8A%D9%86%D9%8A%D9%86" title="معضلة السجينين – Arabic" lang="ar" hreflang="ar" data-title="معضلة السجينين" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D0%BB%D0%B5%D0%BC%D0%B0_%D0%BD%D0%B0_%D0%B7%D0%B0%D1%82%D0%B2%D0%BE%D1%80%D0%BD%D0%B8%D0%BA%D0%B0" title="Дилема на затворника – Bulgarian" lang="bg" hreflang="bg" data-title="Дилема на затворника" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Zatvorenikova_dilema" title="Zatvorenikova dilema – Bosnian" lang="bs" hreflang="bs" data-title="Zatvorenikova dilema" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Dilema_del_presoner" title="Dilema del presoner – Catalan" lang="ca" hreflang="ca" data-title="Dilema del presoner" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C4%9Bz%C5%88ovo_dilema" title="Vězňovo dilema – Czech" lang="cs" hreflang="cs" data-title="Vězňovo dilema" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Dilema%27r_carcharorion" title="Dilema'r carcharorion – Welsh" lang="cy" hreflang="cy" data-title="Dilema'r carcharorion" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Fangernes_dilemma" title="Fangernes dilemma – Danish" lang="da" hreflang="da" data-title="Fangernes dilemma" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%B9%D8%B6%D9%84%D8%A9_%D8%AF%D9%8A%D8%A7%D9%84_%D9%84%D8%AD%D8%A8%D8%A7%D8%B3%D8%A7" title="معضلة ديال لحباسا – Moroccan Arabic" lang="ary" hreflang="ary" data-title="معضلة ديال لحباسا" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gefangenendilemma" title="Gefangenendilemma – German" lang="de" hreflang="de" data-title="Gefangenendilemma" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Vangi_dilemma" title="Vangi dilemma – Estonian" lang="et" hreflang="et" data-title="Vangi dilemma" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%AF%CE%BB%CE%B7%CE%BC%CE%BC%CE%B1_%CF%84%CE%BF%CF%85_%CF%86%CF%85%CE%BB%CE%B1%CE%BA%CE%B9%CF%83%CE%BC%CE%AD%CE%BD%CE%BF%CF%85" title="Δίλημμα του φυλακισμένου – Greek" lang="el" hreflang="el" data-title="Δίλημμα του φυλακισμένου" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Dilema_del_prisionero" title="Dilema del prisionero – Spanish" lang="es" hreflang="es" data-title="Dilema del prisionero" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Prizonula_Dilemo" title="Prizonula Dilemo – Esperanto" lang="eo" hreflang="eo" data-title="Prizonula Dilemo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Presoaren_dilema" title="Presoaren dilema – Basque" lang="eu" hreflang="eu" data-title="Presoaren dilema" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D9%88%D8%B1%D8%A7%D9%87%DB%8C_%D8%B2%D9%86%D8%AF%D8%A7%D9%86%DB%8C" title="دوراهی زندانی – Persian" lang="fa" hreflang="fa" data-title="دوراهی زندانی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Dilemme_du_prisonnier" title="Dilemme du prisonnier – French" lang="fr" hreflang="fr" data-title="Dilemme du prisonnier" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Dilema_do_Prisioneiro" title="Dilema do Prisioneiro – Galician" lang="gl" hreflang="gl" data-title="Dilema do Prisioneiro" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A3%84%EC%88%98%EC%9D%98_%EB%94%9C%EB%A0%88%EB%A7%88" title="죄수의 딜레마 – Korean" lang="ko" hreflang="ko" data-title="죄수의 딜레마" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%B6%D5%BF%D5%A1%D6%80%D5%AF%D5%B5%D5%A1%D5%AC%D5%AB_%D5%A4%D5%AB%D5%AC%D5%A5%D5%B4%D5%A1" title="Բանտարկյալի դիլեմա – Armenian" lang="hy" hreflang="hy" data-title="Բանտարկյալի դիլեմա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%82%E0%A4%A6%E0%A5%80_%E0%A4%95%E0%A5%80_%E0%A4%A6%E0%A5%81%E0%A4%B5%E0%A4%BF%E0%A4%A7%E0%A4%BE" title="बंदी की दुविधा – Hindi" lang="hi" hreflang="hi" data-title="बंदी की दुविधा" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Zatvorenikova_dilema" title="Zatvorenikova dilema – Croatian" lang="hr" hreflang="hr" data-title="Zatvorenikova dilema" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Dilemo_di_la_karcerano" title="Dilemo di la karcerano – Ido" lang="io" hreflang="io" data-title="Dilemo di la karcerano" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Dilema_tahanan" title="Dilema tahanan – Indonesian" lang="id" hreflang="id" data-title="Dilema tahanan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Vandam%C3%A1l_fangans" title="Vandamál fangans – Icelandic" lang="is" hreflang="is" data-title="Vandamál fangans" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Dilemma_del_prigioniero" title="Dilemma del prigioniero – Italian" lang="it" hreflang="it" data-title="Dilemma del prigioniero" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%9C%D7%9E%D7%AA_%D7%94%D7%90%D7%A1%D7%99%D7%A8" title="דילמת האסיר – Hebrew" lang="he" hreflang="he" data-title="דילמת האסיר" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9E%E1%83%90%E1%83%A2%E1%83%98%E1%83%9B%E1%83%A0%E1%83%98%E1%83%A1_%E1%83%93%E1%83%98%E1%83%9A%E1%83%94%E1%83%9B%E1%83%90" title="პატიმრის დილემა – Georgian" lang="ka" hreflang="ka" data-title="პატიმრის დილემა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Dilemma_captivi" title="Dilemma captivi – Latin" lang="la" hreflang="la" data-title="Dilemma captivi" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Cietumnieka_dilemma" title="Cietumnieka dilemma – Latvian" lang="lv" hreflang="lv" data-title="Cietumnieka dilemma" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Kalinio_dilema" title="Kalinio dilema – Lithuanian" lang="lt" hreflang="lt" data-title="Kalinio dilema" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fogolydilemma" title="Fogolydilemma – Hungarian" lang="hu" hreflang="hu" data-title="Fogolydilemma" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D0%BE%D1%80%D0%B8%D0%B3%D0%B4%D0%BB%D1%8B%D0%BD_%D0%B4%D0%B8%D0%BB%D0%B5%D0%BC%D0%BC%D0%B0" title="Хоригдлын дилемма – Mongolian" lang="mn" hreflang="mn" data-title="Хоригдлын дилемма" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma – Dutch" lang="nl" hreflang="nl" data-title="Prisoner's dilemma" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%9A%E4%BA%BA%E3%81%AE%E3%82%B8%E3%83%AC%E3%83%B3%E3%83%9E" title="囚人のジレンマ – Japanese" lang="ja" hreflang="ja" data-title="囚人のジレンマ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fangens_dilemma" title="Fangens dilemma – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fangens dilemma" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Dil%C3%A8ma_del_presoni%C3%A8r" title="Dilèma del presonièr – Occitan" lang="oc" hreflang="oc" data-title="Dilèma del presonièr" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%95%E0%A9%88%E0%A8%A6%E0%A9%80_%E0%A8%A6%E0%A9%80_%E0%A8%A6%E0%A9%81%E0%A8%B5%E0%A8%BF%E0%A8%A7%E0%A8%BE" title="ਕੈਦੀ ਦੀ ਦੁਵਿਧਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਕੈਦੀ ਦੀ ਦੁਵਿਧਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dylemat_wi%C4%99%C5%BAnia" title="Dylemat więźnia – Polish" lang="pl" hreflang="pl" data-title="Dylemat więźnia" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Dilema_do_prisioneiro" title="Dilema do prisioneiro – Portuguese" lang="pt" hreflang="pt" data-title="Dilema do prisioneiro" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Dilema_prizonierului" title="Dilema prizonierului – Romanian" lang="ro" hreflang="ro" data-title="Dilema prizonierului" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D0%BB%D0%B5%D0%BC%D0%BC%D0%B0_%D0%B7%D0%B0%D0%BA%D0%BB%D1%8E%D1%87%D1%91%D0%BD%D0%BD%D0%BE%D0%B3%D0%BE" title="Дилемма заключённого – Russian" lang="ru" hreflang="ru" data-title="Дилемма заключённого" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma – Simple English" lang="en-simple" hreflang="en-simple" data-title="Prisoner's dilemma" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/V%C3%A4z%C5%88ova_dilema" title="Väzňova dilema – Slovak" lang="sk" hreflang="sk" data-title="Väzňova dilema" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Zapornikova_dilema" title="Zapornikova dilema – Slovenian" lang="sl" hreflang="sl" data-title="Zapornikova dilema" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%97%D0%B0%D1%82%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%B8%D0%BA%D0%BE%D0%B2%D0%B0_%D0%B4%D0%B8%D0%BB%D0%B5%D0%BC%D0%B0" title="Затвореникова дилема – Serbian" lang="sr" hreflang="sr" data-title="Затвореникова дилема" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Dilema_zatvorenika" title="Dilema zatvorenika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Dilema zatvorenika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vangin_dilemma" title="Vangin dilemma – Finnish" lang="fi" hreflang="fi" data-title="Vangin dilemma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/F%C3%A5ngarnas_dilemma" title="Fångarnas dilemma – Swedish" lang="sv" hreflang="sv" data-title="Fångarnas dilemma" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Mahirap_na_kalagayan_ng_bilanggo" title="Mahirap na kalagayan ng bilanggo – Tagalog" lang="tl" hreflang="tl" data-title="Mahirap na kalagayan ng bilanggo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%88%E0%AE%A4%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B4%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%AE%E0%AF%8D" title="கைதியின் குழப்பம் – Tamil" lang="ta" hreflang="ta" data-title="கைதியின் குழப்பம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B8%A5%E0%B8%B3%E0%B8%9A%E0%B8%B2%E0%B8%81%E0%B9%83%E0%B8%88%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%99%E0%B8%B1%E0%B8%81%E0%B9%82%E0%B8%97%E0%B8%A9" title="ความลำบากใจของนักโทษ – Thai" lang="th" hreflang="th" data-title="ความลำบากใจของนักโทษ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Tutsak_ikilemi" title="Tutsak ikilemi – Turkish" lang="tr" hreflang="tr" data-title="Tutsak ikilemi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D0%BB%D0%B5%D0%BC%D0%B0_%D0%B2%27%D1%8F%D0%B7%D0%BD%D1%8F" title="Дилема в'язня – Ukrainian" lang="uk" hreflang="uk" data-title="Дилема в'язня" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D9%88_%D9%82%DB%8C%D8%AF%DB%8C%D9%88%DA%BA_%DA%A9%DB%8C_%D8%AF%DB%81%D8%B1%DB%8C_%D9%85%D8%B4%DA%A9%D9%84" title="دو قیدیوں کی دہری مشکل – Urdu" lang="ur" hreflang="ur" data-title="دو قیدیوں کی دہری مشکل" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Song_%C4%91%E1%BB%81_t%C3%B9_nh%C3%A2n" title="Song đề tù nhân – Vietnamese" lang="vi" hreflang="vi" data-title="Song đề tù nhân" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%9B%A3%E7%8A%AF%E5%9B%B0%E5%A2%83" title="監犯困境 – Cantonese" lang="yue" hreflang="yue" data-title="監犯困境" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9B%9A%E5%BE%92%E5%9B%B0%E5%A2%83" title="囚徒困境 – Chinese" lang="zh" hreflang="zh" data-title="囚徒困境" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q41693#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Prisoner%27s_dilemma" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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.hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Prisoner%27s_dilemma_(disambiguation)" class="mw-disambig" title="Prisoner's dilemma (disambiguation)">Prisoner's dilemma (disambiguation)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Three_prisoners_problem" title="Three prisoners problem">Three prisoners problem</a>, <a href="/wiki/Unexpected_hanging_paradox" title="Unexpected hanging paradox">Unexpected hanging paradox</a>, <a href="/wiki/100_prisoners_problem" title="100 prisoners problem">100 prisoners problem</a>, or <a href="/wiki/Innocent_prisoner%27s_dilemma" title="Innocent prisoner's dilemma">Innocent prisoner's dilemma</a>.</div> <p>The <b>prisoner's dilemma</b> is a <a href="/wiki/Game_theory" title="Game theory">game theory</a> thought experiment involving two <a href="/wiki/Rational_agent" title="Rational agent">rational agents</a>, each of whom can either cooperate for mutual benefit or betray their partner ("defect") for individual gain. The dilemma arises from the fact that while defecting is rational for each agent, cooperation yields a higher payoff for each. The puzzle was designed by <a href="/wiki/Merrill_Flood" class="mw-redirect" title="Merrill Flood">Merrill Flood</a> and <a href="/wiki/Melvin_Dresher" title="Melvin Dresher">Melvin Dresher</a> in 1950 during their work at the <a href="/wiki/RAND_Corporation" title="RAND Corporation">RAND Corporation</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> They invited economist <a href="/wiki/Armen_Alchian" title="Armen Alchian">Armen Alchian</a> and mathematician John Williams to play a hundred rounds of the game, observing that Alchian and Williams often chose to cooperate. When asked about the results, <a href="/wiki/John_Forbes_Nash_Jr." title="John Forbes Nash Jr.">John Nash</a> remarked that rational behavior in the <a class="mw-selflink-fragment" href="#The_iterated_prisoner's_dilemma">iterated version of the game</a> can differ from that in a single-round version. This insight anticipated a <a href="/wiki/Folk_theorem_(game_theory)" title="Folk theorem (game theory)">key result in game theory</a>: cooperation can emerge in repeated interactions, even in situations where it is not rational in a one-off interaction. </p><p><a href="/wiki/Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a> later named the game the "prisoner's dilemma" by framing the rewards in terms of prison sentences.<sup id="cite_ref-FOOTNOTEPoundstone19938,_117_2-0" class="reference"><a href="#cite_note-FOOTNOTEPoundstone19938,_117-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The prisoner's dilemma models many <a href="#Real-life_examples">real-world situations</a> involving strategic behavior. In casual usage, the label "prisoner's dilemma" is applied to any situation in which two entities can gain important benefits by cooperating or suffer by failing to do so, but find it difficult or expensive to coordinate their choices. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Premise">Premise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=1" title="Edit section: Premise"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Prisoners_dilemma.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Prisoners_dilemma.svg/280px-Prisoners_dilemma.svg.png" decoding="async" width="280" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Prisoners_dilemma.svg/420px-Prisoners_dilemma.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Prisoners_dilemma.svg/560px-Prisoners_dilemma.svg.png 2x" data-file-width="512" data-file-height="448" /></a><figcaption>An example prisoner's dilemma payoff matrix</figcaption></figure> <p><a href="/wiki/William_Poundstone" title="William Poundstone">William Poundstone</a> described this "typical contemporary version" of the game in his 1993 book <i>Prisoner's Dilemma</i>: </p> <blockquote><p>Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a <a href="/wiki/Faustian_bargain" class="mw-redirect" title="Faustian bargain">Faustian bargain</a>. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh, yes, there is a catch ... If <i>both</i> prisoners testify against each other, both will be sentenced to two years in jail. The prisoners are given a little time to think this over, but in no case may either learn what the other has decided until he has irrevocably made his decision. Each is informed that the other prisoner is being offered the very same deal. Each prisoner is concerned only with his own welfare—with minimizing his own prison sentence.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></p></blockquote> <p>This leads to four different possible outcomes for prisoners A and B: </p> <ol><li>If A and B both remain silent, they will each serve one year in prison.</li> <li>If A testifies against B but B remains silent, A will be set free while B serves three years in prison.</li> <li>If A remains silent but B testifies against A, A will serve three years in prison and B will be set free.</li> <li>If A and B testify against each other, they will each serve two years.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Strategy_for_the_prisoner's_dilemma"><span id="Strategy_for_the_prisoner.27s_dilemma"></span>Strategy for the prisoner's dilemma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=2" title="Edit section: Strategy for the prisoner's dilemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Prisoner%27s_dilemma" title="Special:EditPage/Prisoner's dilemma">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">January 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>Two prisoners are separated into individual rooms and cannot communicate with each other. It is assumed that both prisoners understand the nature of the game, have no loyalty to each other, and will have no opportunity for retribution or reward outside of the game. The normal game is shown below:<sup id="cite_ref-FOOTNOTEPoundstone1993118_4-0" class="reference"><a href="#cite_note-FOOTNOTEPoundstone1993118-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">Prisoner B</div><div style="margin-right:2em;text-align:left"><br />Prisoner A</div></th> <th>Prisoner B stays silent<br />(<i>cooperates</i>)</th> <th>Prisoner B testifies<br />(<i>defects</i>) </th></tr> <tr> <th>Prisoner A stays silent<br />(<i>cooperates</i>) </th> <td>Each serves 1 year</td> <td>Prisoner A: 3 years<br />Prisoner B: goes free </td></tr> <tr> <th>Prisoner A testifies<br />(<i>defects</i>) </th> <td>Prisoner A: goes free<br />Prisoner B: 3 years</td> <td>Each serves 2 years </td></tr></tbody></table> <p>Regardless of what the other decides, each prisoner gets a higher reward by betraying the other ("defecting"). The reasoning involves analyzing both players' <a href="/wiki/Best_response" title="Best response">best responses</a>: B will either cooperate or defect. If B cooperates, A should defect, because going free is better than serving 1 year. If B defects, A should also defect, because serving 2 years is better than serving 3. So, either way, A should defect since defecting is A's best response regardless of B's strategy. Parallel reasoning will show that B should defect. </p><p>Defection always results in a better payoff than cooperation, so it is a strictly dominant strategy for both players. Mutual defection is the only strong <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a> in the game. Since the collectively ideal result of mutual cooperation is irrational from a self-interested standpoint, this Nash equilibrium is not <a href="/wiki/Pareto_efficient" class="mw-redirect" title="Pareto efficient">Pareto efficient</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Generalized_form">Generalized form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=3" title="Edit section: Generalized form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Prisoner%27s_dilemma" title="Special:EditPage/Prisoner's dilemma">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">January 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The structure of the traditional prisoner's dilemma can be generalized from its original prisoner setting. Suppose that the two players are represented by the colors red and blue and that each player chooses to either "cooperate" or "defect". </p><p>If both players cooperate, they both receive the reward <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> for cooperating. If both players defect, they both receive the punishment payoff <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. If Blue defects while Red cooperates, then Blue receives the temptation payoff <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, while Red receives the "sucker's" payoff, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>. Similarly, if Blue cooperates while Red defects, then Blue receives the sucker's payoff <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, while Red receives the temptation payoff <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>. </p><p>This can be expressed in <a href="/wiki/Normal-form_game" title="Normal-form game">normal form</a>: </p> <table class="wikitable" style="text-align:center"> <caption>Canonical prisoner's dilemma payoff matrix </caption> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:#900">Red</span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">Blue</span></div> </th> <th scope="col" style="width:60px;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">Cooperate</span> </th> <th scope="col" style="width:60px;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">Defect</span> </th></tr> <tr> <th scope="row" style="width:60px;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">Cooperate</span> </th> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900"><i>R</i></span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009"><i>R</i></span></div> </td> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900"><i>T</i></span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009"><i>S</i></span></div> </td></tr> <tr> <th scope="row"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">Defect</span> </th> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900"><i>S</i></span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009"><i>T</i></span></div> </td> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900"><i>P</i></span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009"><i>P</i></span></div> </td></tr></tbody></table> <p>and to be a prisoner's dilemma game in the strong sense, the following condition must hold for the payoffs: </p> <dl><dd><span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>R>P>S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>R</mi> <mo>></mo> <mi>P</mi> <mo>></mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>R>P>S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6fac90a83474d2b977ae4ff2075c146534cb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.94ex; height:2.176ex;" alt="{\displaystyle T>R>P>S}"></span>⁠</span></dd></dl> <p>The payoff relationship <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R>P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>></mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R>P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37afbcb755b88dab03bced457fbdbcd90d55a904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.176ex;" alt="{\displaystyle R>P}"></span>⁠</span> implies that mutual cooperation is superior to mutual defection, while the payoff relationships <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T>R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>></mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T>R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1fa60c9da906b73422a9f79364c30dda045601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.499ex; height:2.176ex;" alt="{\displaystyle T>R}"></span>⁠</span> and <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P>S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>></mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P>S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7edfe25ef9f2cf28786fe0526e7104453dbda43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.343ex; height:2.176ex;" alt="{\displaystyle P>S}"></span>⁠</span> imply that defection is the dominant strategy for both agents. </p> <div class="mw-heading mw-heading2"><h2 id="The_iterated_prisoner's_dilemma"><span id="The_iterated_prisoner.27s_dilemma"></span>The iterated prisoner's dilemma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=4" title="Edit section: The iterated prisoner's dilemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-More_citations_needed_section plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Prisoner%27s_dilemma" title="Special:EditPage/Prisoner's dilemma">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a> in this section. Unsourced material may be challenged and removed.</span> <span class="date-container"><i>(<span class="date">November 2012</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>If two players play the prisoner's dilemma more than once in succession, remember their opponent's previous actions, and are allowed to change their strategy accordingly, the game is called the iterated prisoner's dilemma. </p><p>In addition to the general form above, the iterative version also requires that <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2R>T+S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>R</mi> <mo>></mo> <mi>T</mi> <mo>+</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2R>T+S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b75e44a0fe5e91b7367b6fe318f0aa3e84066e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.001ex; height:2.343ex;" alt="{\displaystyle 2R>T+S}"></span>⁠</span>, to prevent alternating cooperation and defection giving a greater reward than mutual cooperation. </p><p>The iterated prisoner's dilemma is fundamental to some theories of human cooperation and trust. Assuming that the game effectively models transactions between two people that require trust, cooperative behavior in populations can be modeled by a multi-player iterated version of the game. In 1975, <a href="/wiki/Bernard_Grofman" title="Bernard Grofman">Grofman</a> and <a href="/wiki/Jonathan_Pool" title="Jonathan Pool">Pool</a> estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoner's dilemma is also called the "<a href="/wiki/Peace_war_game" title="Peace war game">peace-war game</a>".<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Shy_6-0" class="reference"><a href="#cite_note-Shy-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="General_strategy">General strategy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=5" title="Edit section: General strategy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the iterated prisoner's dilemma is played a finite number of times and both players know this, then the dominant strategy and Nash equilibrium is to defect in all rounds. The proof is <a href="/wiki/Mathematical_induction" title="Mathematical induction">inductive</a>: one might as well defect on the last turn, since the opponent will not have a chance to later retaliate. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on. The same applies if the game length is unknown but has a known upper limit.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (May 2023)">citation needed</span></a></i>]</sup> </p><p>For <a href="/wiki/Cooperation" title="Cooperation">cooperation</a> to emerge between rational players, the number of rounds must be unknown or infinite. In that case, "always defect" may no longer be a dominant strategy. As shown by <a href="/wiki/Robert_Aumann" title="Robert Aumann">Robert Aumann</a> in a 1959 paper,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> rational players repeatedly interacting for indefinitely long games can sustain cooperation. Specifically, a player may be less willing to cooperate if their counterpart did not cooperate many times, which causes disappointment. Conversely, as time elapses, the likelihood of cooperation tends to rise, owing to the establishment of a "tacit agreement" among participating players. In experimental situations, cooperation can occur even when both participants know how many iterations will be played.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>According to a 2019 experimental study in the <i>American Economic Review</i> that tested what strategies real-life subjects used in iterated prisoner's dilemma situations with perfect monitoring, the majority of chosen strategies were always to defect, <a href="/wiki/Tit_for_tat" title="Tit for tat">tit-for-tat</a>, and <a href="/wiki/Grim_trigger" title="Grim trigger">grim trigger</a>. Which strategy the subjects chose depended on the parameters of the game.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Axelrod's_tournament_and_successful_strategy_conditions"><span id="Axelrod.27s_tournament_and_successful_strategy_conditions"></span>Axelrod's tournament and successful strategy conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=6" title="Edit section: Axelrod's tournament and successful strategy conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Interest in the iterated prisoner's dilemma was kindled by <a href="/wiki/Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a> in his 1984 book <i><a href="/wiki/The_Evolution_of_Cooperation" title="The Evolution of Cooperation">The Evolution of Cooperation</a></i>, in which he reports on a tournament that he organized of the <i>N</i>-step prisoner's dilemma (with <i>N</i> fixed) in which participants have to choose their strategy repeatedly and remember their previous encounters. Axelrod invited academic colleagues from around the world to devise computer strategies to compete in an iterated prisoner's dilemma tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth. </p><p>Axelrod discovered that when these encounters were repeated over a long period of time with many players, each with different strategies, greedy strategies tended to do very poorly in the long run while more <a href="/wiki/Altruism" title="Altruism">altruistic</a> strategies did better, as judged purely by self-interest. He used this to show a possible mechanism for the evolution of altruistic behavior from mechanisms that are initially purely selfish, by <a href="/wiki/Natural_selection" title="Natural selection">natural selection</a>. </p><p>The winning <a href="/wiki/Deterministic_algorithm" title="Deterministic algorithm">deterministic</a> strategy was <a href="/wiki/Tit_for_tat" title="Tit for tat">tit for tat</a>, developed and entered into the tournament by <a href="/wiki/Anatol_Rapoport" title="Anatol Rapoport">Anatol Rapoport</a>. It was the simplest of any program entered, containing only four lines of <a href="/wiki/BASIC" title="BASIC">BASIC</a>,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> and won the contest. The strategy is simply to cooperate on the first iteration of the game; after that, the player does what his or her opponent did on the previous move.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Depending on the situation, a slightly better strategy can be "tit for tat with forgiveness": when the opponent defects, on the next move, the player sometimes cooperates anyway, with a small probability (around 1–5%, depending on the lineup of opponents). This allows for occasional recovery from getting trapped in a cycle of defections. </p><p>After analyzing the top-scoring strategies, Axelrod stated several conditions necessary for a strategy to succeed:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <ul><li><b>Nice</b>: The strategy will not be the first to defect (this is sometimes referred to as an "optimistic" algorithm<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag may use weasel words or too-vague attribution. (June 2024)">by whom?</span></a></i>]</sup>), i.e., it will not "cheat" on its opponent for purely self-interested reasons first. Almost all the top-scoring strategies were nice.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup></li> <li><b>Retaliating</b>: The strategy must sometimes retaliate. An example of a non-retaliating strategy is Always Cooperate, a very bad choice that will frequently be exploited by "nasty" strategies.</li> <li><b>Forgiving</b>: Successful strategies must be forgiving. Though players will retaliate, they will cooperate again if the opponent does not continue to defect. This can stop long runs of revenge and counter-revenge, maximizing points.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup></li> <li><b>Non-envious</b>: The strategy must not strive to score more than the opponent.</li></ul> <p>In contrast to the one-time prisoner's dilemma game, the optimal strategy in the iterated prisoner's dilemma depends upon the strategies of likely opponents, and how they will react to defections and cooperation. For example, if a population consists entirely of players who always defect, except for one who follows the tit-for-tat strategy, that person is at a slight disadvantage because of the loss on the first turn. In such a population, the optimal strategy is to defect every time. More generally, given a population with a certain percentage of always-defectors with the rest being tit-for-tat players, the optimal strategy depends on the percentage and number of iterations played.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Other_strategies">Other strategies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=7" title="Edit section: Other strategies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Deriving the optimal strategy is generally done in two ways: </p> <ul><li><a href="/wiki/Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a>: If the statistical distribution of opposing strategies can be determined an optimal counter-strategy can be derived analytically.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo</a> simulations of populations have been made, where individuals with low scores die off, and those with high scores reproduce (a <a href="/wiki/Genetic_algorithm" title="Genetic algorithm">genetic algorithm</a> for finding an optimal strategy). The mix of algorithms in the final population generally depends on the mix in the initial population. The introduction of mutation (random variation during reproduction) lessens the dependency on the initial population; empirical experiments with such systems tend to produce tit-for-tat players,<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag may need clarification or removal of jargon. (August 2016)">clarification needed</span></a></i>]</sup> but no analytic proof exists that this will always occur.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup></li></ul> <p>In the strategy called <a href="/wiki/Win-stay,_lose-switch" class="mw-redirect" title="Win-stay, lose-switch">win-stay, lose-switch</a>, faced with a failure to cooperate, the player switches strategy the next turn.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In certain circumstances,<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="Statement needs to be more specific about the content to which it refers. (November 2012)">specify</span></a></i>]</sup> Pavlov beats all other strategies by giving preferential treatment to co-players using a similar strategy. </p><p>Although tit-for-tat is considered the most <a href="/wiki/Robust" class="mw-redirect" title="Robust">robust</a> basic strategy, a team from <a href="/wiki/Southampton_University" class="mw-redirect" title="Southampton University">Southampton University</a> in England introduced a more successful strategy at the 20th-anniversary iterated prisoner's dilemma competition. It relied on collusion between programs to achieve the highest number of points for a single program. The university submitted 60 programs to the competition, which were designed to recognize each other through a series of five to ten moves at the start.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Once this recognition was made, one program would always cooperate and the other would always defect, assuring the maximum number of points for the defector. If the program realized that it was playing a non-Southampton player, it would continuously defect in an attempt to minimize the competing program's score. As a result, the 2004 Prisoners' Dilemma Tournament results show <a href="/wiki/University_of_Southampton" title="University of Southampton">University of Southampton</a>'s strategies in the first three places (and a number of positions towards the bottom), despite having fewer wins and many more losses than the GRIM strategy. The Southampton strategy takes advantage of the fact that multiple entries were allowed in this particular competition and that a team's performance was measured by that of the highest-scoring player (meaning that the use of self-sacrificing players was a form of <a href="/wiki/Minmaxing" class="mw-redirect" title="Minmaxing">minmaxing</a>). </p><p>Because of this new rule, this competition also has little theoretical significance when analyzing single-agent strategies as compared to Axelrod's seminal tournament. But it provided a basis for analyzing how to achieve cooperative strategies in multi-agent frameworks, especially in the presence of noise. </p><p>Long before this new-rules tournament was played, Dawkins, in his book <i><a href="/wiki/The_Selfish_Gene" title="The Selfish Gene">The Selfish Gene</a></i>, pointed out the possibility of such strategies winning if multiple entries were allowed, but remarked that Axelrod would most likely not have allowed them if they had been submitted. It also relies on circumventing the rule that no communication is allowed between players, which the Southampton programs arguably did with their preprogrammed "ten-move dance" to recognize one another, reinforcing how valuable communication can be in shifting the balance of the game. </p><p>Even without implicit collusion between <a href="/wiki/Computer_program" title="Computer program">software strategies</a>, tit-for-tat is not always the absolute winner of any given tournament; more precisely, its long-run results over a series of tournaments outperform its rivals, but this does not mean it is the most successful in the short term. The same applies to tit-for-tat with forgiveness and other optimal strategies. </p><p>This can also be illustrated using the Darwinian <a href="/wiki/Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">ESS</a> simulation. In such a simulation, tit-for-tat will almost always come to dominate, though nasty strategies will drift in and out of the population because a tit-for-tat population is penetrable by non-retaliating nice strategies, which in turn are easy prey for the nasty strategies. Dawkins showed that here, no static mix of strategies forms a stable equilibrium, and the system will always oscillate between bounds.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="Unsure if the original author meant to continue to cite The Selfish Gene here. (April 2023)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Stochastic_iterated_prisoner's_dilemma"><span id="Stochastic_iterated_prisoner.27s_dilemma"></span>Stochastic iterated prisoner's dilemma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=8" title="Edit section: Stochastic iterated prisoner's dilemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Undue_weight_section plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span class="skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Unbalanced_scales.svg/45px-Unbalanced_scales.svg.png" decoding="async" width="45" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Unbalanced_scales.svg/68px-Unbalanced_scales.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Unbalanced_scales.svg/90px-Unbalanced_scales.svg.png 2x" data-file-width="400" data-file-height="354" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>lend <a href="/wiki/Wikipedia:Neutral_point_of_view#Due_and_undue_weight" title="Wikipedia:Neutral point of view">undue weight</a> to certain ideas, incidents, or controversies</b>. Please help to <a href="/wiki/Wikipedia:STRUCTURE" class="mw-redirect" title="Wikipedia:STRUCTURE">create a more balanced presentation</a>. Discuss and <a href="/wiki/Wikipedia:Dispute_resolution" title="Wikipedia:Dispute resolution">resolve</a> this issue before removing this message. <span class="date-container"><i>(<span class="date">May 2023</span>)</i></span></div></td></tr></tbody></table> <p>In a stochastic iterated prisoner's dilemma game, strategies are specified in terms of "cooperation probabilities".<sup id="cite_ref-Press2012_22-0" class="reference"><a href="#cite_note-Press2012-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In an encounter between player <i>X</i> and player <i>Y</i>, <i>X</i><span class="nowrap" style="padding-left:0.1em;">'</span>s strategy is specified by a set of probabilities <i>P</i> of cooperating with <i>Y</i>. <i>P</i> is a function of the outcomes of their previous encounters or some subset thereof. If <i>P</i> is a function of only their most recent <i>n</i> encounters, it is called a "memory-n" strategy. A memory-1 strategy is then specified by four cooperation probabilities: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{P_{cc},P_{cd},P_{dc},P_{dd}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\{P_{cc},P_{cd},P_{dc},P_{dd}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6049cd9b759ca645ad577ed9677915c00c4fa29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.456ex; height:2.843ex;" alt="{\displaystyle P=\{P_{cc},P_{cd},P_{dc},P_{dd}\}}"></span>, where <i>P<sub>cd</sub></i> is the probability that <i>X</i> will cooperate in the present encounter given that the previous encounter was characterized by <i>X</i> cooperating and <i>Y</i> defecting. If each of the probabilities are either 1 or 0, the strategy is called deterministic. An example of a deterministic strategy is the tit-for-tat strategy written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{1,0,1,0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\{1,0,1,0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a10b30586da3c5bb10cbb7bdc0271862174defb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.92ex; height:2.843ex;" alt="{\displaystyle P=\{1,0,1,0\}}"></span>, in which <i>X</i> responds as <i>Y</i> did in the previous encounter. Another is the win-stay, lose switch strategy written as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\{1,0,0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\{1,0,0,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/baaf8ca51e63a71c872ebeab1647b02d8e73794e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.92ex; height:2.843ex;" alt="{\displaystyle P=\{1,0,0,1\}}"></span>. It has been shown that for any memory-n strategy there is a corresponding memory-1 strategy that gives the same statistical results, so that only memory-1 strategies need be considered.<sup id="cite_ref-Press2012_22-1" class="reference"><a href="#cite_note-Press2012-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is defined as the above 4-element strategy vector of <i>X</i> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{Q_{cc},Q_{cd},Q_{dc},Q_{dd}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>d</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\{Q_{cc},Q_{cd},Q_{dc},Q_{dd}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e102e855f62f2cc7e5f197f19a6275a31f1b434" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.933ex; height:2.843ex;" alt="{\displaystyle Q=\{Q_{cc},Q_{cd},Q_{dc},Q_{dd}\}}"></span> as the 4-element strategy vector of <i>Y</i> (where the indices are from <i>Y</i>'s point of view), a transition matrix <i>M</i> may be defined for <i>X</i> whose <i>ij</i>-th entry is the probability that the outcome of a particular encounter between <i>X</i> and <i>Y</i> will be <i>j</i> given that the previous encounter was <i>i</i>, where <i>i</i> and <i>j</i> are one of the four outcome indices: <i>cc</i>, <i>cd</i>, <i>dc</i>, or <i>dd</i>. For example, from <i>X</i><span class="nowrap" style="padding-left:0.1em;">'</span>s point of view, the probability that the outcome of the present encounter is <i>cd</i> given that the previous encounter was <i>cd</i> is equal to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{cd,cd}=P_{cd}(1-Q_{dc})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> <mo>,</mo> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{cd,cd}=P_{cd}(1-Q_{dc})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc930c8ddbf26a0ed1e868c21982154bbf63fc6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.936ex; height:3.009ex;" alt="{\displaystyle M_{cd,cd}=P_{cd}(1-Q_{dc})}"></span>. Under these definitions, the iterated prisoner's dilemma qualifies as a <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> and <i>M</i> is a <a href="/wiki/Stochastic_matrix" title="Stochastic matrix">stochastic matrix</a>, allowing all of the theory of stochastic processes to be applied.<sup id="cite_ref-Press2012_22-2" class="reference"><a href="#cite_note-Press2012-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>One result of stochastic theory is that there exists a stationary vector <i>v</i> for the matrix <i>v</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\cdot M=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>⋅</mo> <mi>M</mi> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\cdot M=v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4391ee63cdd73b80bea8b4cd7807305ffa9b28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.475ex; height:2.176ex;" alt="{\displaystyle v\cdot M=v}"></span>. Without loss of generality, it may be specified that <i>v</i> is normalized so that the sum of its four components is unity. The <i>ij</i>-th entry in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655dc6ee87dfc8866fb4adaa13ed2eb37f519a1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.717ex; height:2.343ex;" alt="{\displaystyle M^{n}}"></span> will give the probability that the outcome of an encounter between <i>X</i> and <i>Y</i> will be <i>j</i> given that the encounter <i>n</i> steps previous is <i>i</i>. In the limit as <i>n</i> approaches infinity, <i>M</i> will converge to a matrix with fixed values, giving the long-term probabilities of an encounter producing <i>j</i> independent of <i>i</i>. In other words, the rows of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28929b01548f9a3f737aed54493dd719dee1d49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.374ex; height:2.343ex;" alt="{\displaystyle M^{\infty }}"></span> will be identical, giving the long-term equilibrium result probabilities of the iterated prisoner's dilemma without the need to explicitly evaluate a large number of interactions. It can be seen that <i>v</i> is a stationary vector for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/655dc6ee87dfc8866fb4adaa13ed2eb37f519a1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.717ex; height:2.343ex;" alt="{\displaystyle M^{n}}"></span> and particularly <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28929b01548f9a3f737aed54493dd719dee1d49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.374ex; height:2.343ex;" alt="{\displaystyle M^{\infty }}"></span>, so that each row of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28929b01548f9a3f737aed54493dd719dee1d49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.374ex; height:2.343ex;" alt="{\displaystyle M^{\infty }}"></span> will be equal to <i>v</i>. Thus, the stationary vector specifies the equilibrium outcome probabilities for <i>X</i>. Defining <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{x}=\{R,S,T,P\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>R</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mi>P</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}=\{R,S,T,P\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b088554d9d2b52d1f6aa9de2a7673585c3fcdd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.768ex; height:2.843ex;" alt="{\displaystyle S_{x}=\{R,S,T,P\}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{y}=\{R,T,S,P\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>R</mi> <mo>,</mo> <mi>T</mi> <mo>,</mo> <mi>S</mi> <mo>,</mo> <mi>P</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{y}=\{R,T,S,P\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446cf67988646d5db122d4efec2f8657c8691c13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.644ex; height:3.009ex;" alt="{\displaystyle S_{y}=\{R,T,S,P\}}"></span> as the short-term payoff vectors for the {<i>cc,cd,dc,dd</i>} outcomes (from <i>X</i><span class="nowrap" style="padding-left:0.1em;">'</span>s point of view), the equilibrium payoffs for <i>X</i> and <i>Y</i> can now be specified as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x}=v\cdot S_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mo>⋅</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{x}=v\cdot S_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bf287447ee2a31d42bc00cf771dc6f5de7ea26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.766ex; height:2.509ex;" alt="{\displaystyle s_{x}=v\cdot S_{x}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{y}=v\cdot S_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mo>⋅</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{y}=v\cdot S_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e26c0518d10aaeee1ec53f2bc598a47240195c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.519ex; height:2.843ex;" alt="{\displaystyle s_{y}=v\cdot S_{y}}"></span>, allowing the two strategies <i>P</i> and <i>Q</i> to be compared for their long-term payoffs. </p> <div class="mw-heading mw-heading4"><h4 id="Zero-determinant_strategies">Zero-determinant strategies</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=9" title="Edit section: Zero-determinant strategies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Undue_weight_section plainlinks metadata ambox ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span class="skin-invert-image" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Unbalanced_scales.svg/45px-Unbalanced_scales.svg.png" decoding="async" width="45" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Unbalanced_scales.svg/68px-Unbalanced_scales.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Unbalanced_scales.svg/90px-Unbalanced_scales.svg.png 2x" data-file-width="400" data-file-height="354" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section may <b>lend <a href="/wiki/Wikipedia:Neutral_point_of_view#Due_and_undue_weight" title="Wikipedia:Neutral point of view">undue weight</a> to certain ideas, incidents, or controversies</b>. Please help to <a href="/wiki/Wikipedia:STRUCTURE" class="mw-redirect" title="Wikipedia:STRUCTURE">create a more balanced presentation</a>. Discuss and <a href="/wiki/Wikipedia:Dispute_resolution" title="Wikipedia:Dispute resolution">resolve</a> this issue before removing this message. <span class="date-container"><i>(<span class="date">May 2023</span>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Iterated_Prisoners_Dilemma_Venn-Diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Iterated_Prisoners_Dilemma_Venn-Diagram.svg/330px-Iterated_Prisoners_Dilemma_Venn-Diagram.svg.png" decoding="async" width="330" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Iterated_Prisoners_Dilemma_Venn-Diagram.svg/495px-Iterated_Prisoners_Dilemma_Venn-Diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Iterated_Prisoners_Dilemma_Venn-Diagram.svg/660px-Iterated_Prisoners_Dilemma_Venn-Diagram.svg.png 2x" data-file-width="600" data-file-height="500" /></a><figcaption>The relationship between zero-determinant (ZD), cooperating and defecting strategies in the iterated prisoner's dilemma (iterated prisoner's dilemma)</figcaption></figure> <p>In 2012, <a href="/wiki/William_H._Press" title="William H. Press">William H. Press</a> and <a href="/wiki/Freeman_Dyson" title="Freeman Dyson">Freeman Dyson</a> published a new class of strategies for the stochastic iterated prisoner's dilemma called "zero-determinant" (ZD) strategies.<sup id="cite_ref-Press2012_22-3" class="reference"><a href="#cite_note-Press2012-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The long term payoffs for encounters between <i>X</i> and <i>Y</i> can be expressed as the determinant of a matrix which is a function of the two strategies and the short term payoff vectors: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{x}=D(P,Q,S_{x})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{x}=D(P,Q,S_{x})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfdaf95696e005f8fd55ed4c1a56f998f1c81481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.344ex; height:2.843ex;" alt="{\displaystyle s_{x}=D(P,Q,S_{x})}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{y}=D(P,Q,S_{y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{y}=D(P,Q,S_{y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acac39c6528344d71063e838e90d50c270fe3b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.098ex; height:3.009ex;" alt="{\displaystyle s_{y}=D(P,Q,S_{y})}"></span>, which do not involve the stationary vector <i>v</i>. Since the determinant function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{y}=D(P,Q,f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{y}=D(P,Q,f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82b9d3488898743cf7e8fef292ecb2876eb2cdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.902ex; height:3.009ex;" alt="{\displaystyle s_{y}=D(P,Q,f)}"></span> is linear in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, it follows that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha s_{x}+\beta s_{y}+\gamma =D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mo>=</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <mi>α</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha s_{x}+\beta s_{y}+\gamma =D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8930e2a276f19aca0ed87691a91f1f8e15e40357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.266ex; height:3.009ex;" alt="{\displaystyle \alpha s_{x}+\beta s_{y}+\gamma =D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)}"></span> (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\{1,1,1,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\{1,1,1,1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b386d2190691f1b530defc0a118734286b7633c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.958ex; height:2.843ex;" alt="{\displaystyle U=\{1,1,1,1\}}"></span>). Any strategies for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <mi>α</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46745dc035ae94029dbc29a4b48bd746921822d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.263ex; height:3.009ex;" alt="{\displaystyle D(P,Q,\alpha S_{x}+\beta S_{y}+\gamma U)=0}"></span> are by definition a ZD strategy, and the long-term payoffs obey the relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha s_{x}+\beta s_{y}+\gamma =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>β</mi> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha s_{x}+\beta s_{y}+\gamma =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/240d75a0c5a515d7254713636ea476fd43d4b81d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.426ex; height:2.843ex;" alt="{\displaystyle \alpha s_{x}+\beta s_{y}+\gamma =0}"></span>. </p><p>Tit-for-tat is a ZD strategy which is "fair", in the sense of not gaining advantage over the other player. But the ZD space also contains strategies that, in the case of two players, can allow one player to unilaterally set the other player's score or alternatively force an evolutionary player to achieve a payoff some percentage lower than his own. The extorted player could defect, but would thereby hurt himself by getting a lower payoff. Thus, extortion solutions turn the iterated prisoner's dilemma into a sort of <a href="/wiki/Ultimatum_game" title="Ultimatum game">ultimatum game</a>. Specifically, <i>X</i> is able to choose a strategy for which <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(P,Q,\beta S_{y}+\gamma U)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo>,</mo> <mi>β</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>γ</mi> <mi>U</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(P,Q,\beta S_{y}+\gamma U)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03e4a74271ab0e66ad82ba2c69a55a4a6495aa55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.338ex; height:3.009ex;" alt="{\displaystyle D(P,Q,\beta S_{y}+\gamma U)=0}"></span>, unilaterally setting <i>s<sub>y</sub></i> to a specific value within a particular range of values, independent of <i>Y</i><span class="nowrap" style="padding-left:0.1em;">'</span>s strategy, offering an opportunity for <i>X</i> to "extort" player <i>Y</i> (and vice versa). But if <i>X</i> tries to set <i>s<sub>x</sub></i> to a particular value, the range of possibilities is much smaller, consisting only of complete cooperation or complete defection.<sup id="cite_ref-Press2012_22-4" class="reference"><a href="#cite_note-Press2012-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>An extension of the iterated prisoner's dilemma is an evolutionary stochastic iterated prisoner's dilemma, in which the relative abundance of particular strategies is allowed to change, with more successful strategies relatively increasing. This process may be accomplished by having less successful players imitate the more successful strategies, or by eliminating less successful players from the game, while multiplying the more successful ones. It has been shown that unfair ZD strategies are not <a href="/wiki/Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">evolutionarily stable</a>. The key intuition is that an evolutionarily stable strategy must not only be able to invade another population (which extortionary ZD strategies can do) but must also perform well against other players of the same type (which extortionary ZD players do poorly because they reduce each other's surplus).<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p>Theory and simulations confirm that beyond a critical population size, ZD extortion loses out in evolutionary competition against more cooperative strategies, and as a result, the average payoff in the population increases when the population is larger. In addition, there are some cases in which extortioners may even catalyze cooperation by helping to break out of a face-off between uniform defectors and <a href="/wiki/Win%E2%80%93stay,_lose%E2%80%93switch" title="Win–stay, lose–switch">win–stay, lose–switch</a> agents.<sup id="cite_ref-Hilbe2013_24-0" class="reference"><a href="#cite_note-Hilbe2013-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>While extortionary ZD strategies are not stable in large populations, another ZD class called "generous" strategies is both stable and robust. When the population is not too small, these strategies can supplant any other ZD strategy and even perform well against a broad array of generic strategies for iterated prisoner's dilemma, including win–stay, lose–switch. This was proven specifically for the <a href="#Special_case:_Donation_game">donation game</a> by Alexander Stewart and Joshua Plotkin in 2013.<sup id="cite_ref-Stewart2013_25-0" class="reference"><a href="#cite_note-Stewart2013-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Generous strategies will cooperate with other cooperative players, and in the face of defection, the generous player loses more utility than its rival. Generous strategies are the intersection of ZD strategies and so-called "good" strategies, which were defined by Ethan Akin to be those for which the player responds to past mutual cooperation with future cooperation and splits expected payoffs equally if he receives at least the cooperative expected payoff.<sup id="cite_ref-Akin2013_26-0" class="reference"><a href="#cite_note-Akin2013-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Among good strategies, the generous (ZD) subset performs well when the population is not too small. If the population is very small, defection strategies tend to dominate.<sup id="cite_ref-Stewart2013_25-1" class="reference"><a href="#cite_note-Stewart2013-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Continuous_iterated_prisoner's_dilemma"><span id="Continuous_iterated_prisoner.27s_dilemma"></span>Continuous iterated prisoner's dilemma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=10" title="Edit section: Continuous iterated prisoner's dilemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most work on the iterated prisoner's dilemma has focused on the discrete case, in which players either cooperate or defect, because this model is relatively simple to analyze. However, some researchers have looked at models of the continuous iterated prisoner's dilemma, in which players are able to make a variable contribution to the other player. Le and Boyd<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> found that in such situations, cooperation is much harder to evolve than in the discrete iterated prisoner's dilemma. In a continuous prisoner's dilemma, if a population starts off in a non-cooperative equilibrium, players who are only marginally more cooperative than non-cooperators get little benefit from <a href="/wiki/Assortative_mating" title="Assortative mating">assorting</a> with one another. By contrast, in a discrete prisoner's dilemma, tit-for-tat cooperators get a big payoff boost from assorting with one another in a non-cooperative equilibrium, relative to non-cooperators. Since nature arguably offers more opportunities for variable cooperation rather than a strict dichotomy of cooperation or defection, the continuous prisoner's dilemma may help explain why real-life examples of tit-for-tat-like cooperation are extremely rare<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> even though tit-for-tat seems robust in theoretical models. </p> <div class="mw-heading mw-heading2"><h2 id="Real-life_examples">Real-life examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=11" title="Edit section: Real-life examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many instances of human interaction and natural processes have payoff matrices like the prisoner's dilemma's. It is therefore of interest to the <a href="/wiki/Social_science" title="Social science">social sciences</a>, such as <a href="/wiki/Economics" title="Economics">economics</a>, <a href="/wiki/Politics" title="Politics">politics</a>, and <a href="/wiki/Sociology" title="Sociology">sociology</a>, as well as to the biological sciences, such as <a href="/wiki/Ethology" title="Ethology">ethology</a> and <a href="/wiki/Evolutionary_biology" title="Evolutionary biology">evolutionary biology</a>. Many natural processes have been abstracted into models in which living beings are engaged in endless games of prisoner's dilemma. </p> <div class="mw-heading mw-heading3"><h3 id="Environmental_studies">Environmental studies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=12" title="Edit section: Environmental studies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Environmental_studies" title="Environmental studies">environmental studies</a>, the dilemma is evident in crises such as global <a href="/wiki/Climate_change" title="Climate change">climate change</a>. It is argued all countries will benefit from a stable climate, but any single country is often hesitant to curb <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">CO<sub style="font-size: 80%;vertical-align: -0.35em">2</sub></a> emissions. The immediate benefit to any one country from maintaining current behavior is perceived to be greater than the purported eventual benefit to that country if all countries' behavior was changed, therefore explaining the impasse concerning climate-change in 2007.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>An important difference between climate-change politics and the prisoner's dilemma is uncertainty; the extent and pace at which pollution can change climate is not known. The dilemma faced by governments is therefore different from the prisoner's dilemma in that the payoffs of cooperation are unknown. This difference suggests that states will cooperate much less than in a real iterated prisoner's dilemma, so that the probability of avoiding a possible climate catastrophe is much smaller than that suggested by a game-theoretical analysis of the situation using a real iterated prisoner's dilemma.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>Thomas Osang and Arundhati Nandy provide a theoretical explanation with proofs for a regulation-driven win-win situation along the lines of <a href="/wiki/Michael_Porter" title="Michael Porter">Michael Porter</a>'s hypothesis, in which government regulation of competing firms is substantial.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Animals">Animals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=13" title="Edit section: Animals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cooperative behavior of many animals can be understood as an example of the iterated prisoner's dilemma. Often animals engage in long-term partnerships; for example, <a href="/wiki/Guppy" title="Guppy">guppies</a> inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Vampire_bat" title="Vampire bat">Vampire bats</a> are social animals that engage in reciprocal food exchange. Applying the payoffs from the prisoner's dilemma can help explain this behavior.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Psychology">Psychology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=14" title="Edit section: Psychology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Addiction" title="Addiction">addiction</a> research and <a href="/wiki/Behavioral_economics" title="Behavioral economics">behavioral economics</a>, <a href="/wiki/George_Ainslie_(psychologist)" title="George Ainslie (psychologist)">George Ainslie</a> points out that addiction can be cast as an intertemporal prisoner's dilemma problem between the present and future selves of the addict. In this case, "defecting" means relapsing, where not relapsing both today and in the future is by far the best outcome. The case where one abstains today but relapses in the future is the worst outcome: in some sense, the discipline and self-sacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where they started and will have to start over. Relapsing today and tomorrow is a slightly "better" outcome, because while the addict is still addicted, they haven't put the effort in to trying to stop. The final case, where one engages in the addictive behavior today while abstaining tomorrow, has the problem that (as in other prisoner's dilemmas) there is an obvious benefit to defecting "today", but tomorrow one will face the same prisoner's dilemma, and the same obvious benefit will be present then, ultimately leading to an endless string of defections.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>In <i>The Science of Trust</i>, <a href="/wiki/John_Gottman" title="John Gottman">John Gottman</a> defines good relationships as those where partners know not to enter into mutual defection behavior, or at least not to get dynamically stuck there in a loop. In <a href="/wiki/Cognitive_neuroscience" title="Cognitive neuroscience">cognitive neuroscience</a>, fast brain signaling associated with processing different rounds may indicate choices at the next round. Mutual cooperation outcomes entail brain activity changes predictive of how quickly a person will cooperate in kind at the next opportunity;<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> this activity may be linked to basic homeostatic and motivational processes, possibly increasing the likelihood of short-cutting into mutual cooperation. </p> <div class="mw-heading mw-heading3"><h3 id="Economics">Economics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=15" title="Edit section: Economics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The prisoner's dilemma has been called the <i><a href="/wiki/Escherichia_coli" title="Escherichia coli">E. coli</a></i> of social psychology, and it has been used widely to research various topics such as <a href="/wiki/Oligopoly" title="Oligopoly">oligopolistic</a> competition and collective action to produce a collective good.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p><p>Advertising is sometimes cited as a real example of the prisoner's dilemma. When <a href="/wiki/Cigarette_advertising" class="mw-redirect" title="Cigarette advertising">cigarette advertising</a> was legal in the United States, competing cigarette manufacturers had to decide how much money to spend on advertising. The effectiveness of Firm A's advertising was partially determined by the advertising conducted by Firm B. Likewise, the profit derived from advertising for Firm B is affected by the advertising conducted by Firm A. If both Firm A and Firm B chose to advertise during a given period, then the advertisement from each firm negates the other's, receipts remain constant, and expenses increase due to the cost of advertising. Both firms would benefit from a reduction in advertising. However, should Firm B choose not to advertise, Firm A could benefit greatly by advertising. Nevertheless, the optimal amount of advertising by one firm depends on how much advertising the other undertakes. As the best strategy is dependent on what the other firm chooses there is no dominant strategy, which makes it slightly different from a prisoner's dilemma. The outcome is similar, though, in that both firms would be better off were they to advertise less than in the equilibrium. </p><p>Sometimes cooperative behaviors do emerge in business situations. For instance, cigarette manufacturers endorsed the making of laws banning cigarette advertising, understanding that this would reduce costs and increase profits across the industry.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup> </p><p>Without enforceable agreements, members of a <a href="/wiki/Cartel" title="Cartel">cartel</a> are also involved in a (multi-player) prisoner's dilemma.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> "Cooperating" typically means agreeing to a <a href="/wiki/Price_floor" title="Price floor">price floor</a>, while "defecting" means selling under this minimum level, instantly taking business from other cartel members. <a href="/wiki/Anti-trust" class="mw-redirect" title="Anti-trust">Anti-trust</a> authorities want potential cartel members to mutually defect, ensuring the lowest possible prices for consumers. </p> <div class="mw-heading mw-heading3"><h3 id="Sport">Sport</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=16" title="Edit section: Sport"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Doping_in_sport" title="Doping in sport">Doping in sport</a> has been cited as an example of a prisoner's dilemma. Two competing athletes have the option to use an illegal and/or dangerous drug to boost their performance. If neither athlete takes the drug, then neither gains an advantage. If only one does, then that athlete gains a significant advantage over the competitor, reduced by the legal and/or medical dangers of having taken the drug. But if both athletes take the drug, the benefits cancel out and only the dangers remain, putting them both in a worse position than if neither had doped.<sup id="cite_ref-wired_40-0" class="reference"><a href="#cite_note-wired-40"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="International_politics">International politics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=17" title="Edit section: International politics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/International_politics" class="mw-redirect" title="International politics">international relations theory</a>, the prisoner's dilemma is often used to demonstrate why cooperation fails in situations when cooperation between states is collectively optimal but individually suboptimal.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> A classic example is the <a href="/wiki/Security_dilemma" title="Security dilemma">security dilemma</a>, whereby an increase in one state's security (such as increasing its military strength) leads other states to fear for their own security out of fear of offensive action.<sup id="cite_ref-:1_43-0" class="reference"><a href="#cite_note-:1-43"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Consequently, security-increasing measures can lead to tensions, escalation or conflict with one or more other parties, producing an outcome which no party truly desires.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_43-1" class="reference"><a href="#cite_note-:1-43"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_46-0" class="reference"><a href="#cite_note-:3-46"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:02_47-0" class="reference"><a href="#cite_note-:02-47"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> The security dilemma is particularly intense in situations when it is hard to distinguish offensive weapons from defensive weapons, and offense has the advantage in any conflict over defense.<sup id="cite_ref-:1_43-2" class="reference"><a href="#cite_note-:1-43"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>The prisoner's dilemma has frequently been used by <a href="/wiki/Realism_(international_relations)" title="Realism (international relations)">realist</a> international relations theorists to demonstrate the why all states (regardless of their internal policies or professed ideology) under <a href="/wiki/Anarchy_(international_relations)" title="Anarchy (international relations)">international anarchy</a> will struggle to cooperate with one another even when all benefit from such cooperation. </p><p>Critics of realism argue that iteration and extending the shadow of the future are solutions to the prisoner's dilemma. When actors play the prisoner's dilemma once, they have incentives to defect, but when they expect to play it repeatedly, they have greater incentives to cooperate.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Multiplayer_dilemmas">Multiplayer dilemmas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=18" title="Edit section: Multiplayer dilemmas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many real-life dilemmas involve multiple players.<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Although metaphorical, <a href="/wiki/Garrett_Hardin" title="Garrett Hardin">Garrett Hardin</a>'s <a href="/wiki/Tragedy_of_the_commons" title="Tragedy of the commons">tragedy of the commons</a> may be viewed as an example of a multi-player generalization of the prisoner's dilemma: each villager makes a choice for personal gain or restraint. The collective reward for unanimous or frequent defection is very low payoffs and the destruction of the commons. </p><p>The commons are not always exploited: <a href="/wiki/William_Poundstone" title="William Poundstone">William Poundstone</a>, in a book about the prisoner's dilemma, describes a situation in New Zealand where newspaper boxes are left unlocked. It is possible for people to <a href="/wiki/Excludability" title="Excludability">take a paper without paying</a> (defecting), but very few do, feeling that if they do not pay then neither will others, destroying the system.<sup id="cite_ref-FOOTNOTEPoundstone1993126–127_50-0" class="reference"><a href="#cite_note-FOOTNOTEPoundstone1993126–127-50"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Subsequent research by <a href="/wiki/Elinor_Ostrom" title="Elinor Ostrom">Elinor Ostrom</a>, winner of the 2009 <a href="/wiki/Nobel_Memorial_Prize_in_Economic_Sciences" title="Nobel Memorial Prize in Economic Sciences">Nobel Memorial Prize in Economic Sciences</a>, hypothesized that the tragedy of the commons is oversimplified, with the negative outcome influenced by outside influences. Without complicating pressures, groups communicate and manage the commons among themselves for their mutual benefit, enforcing social norms to preserve the resource and achieve the maximum good for the group, an example of effecting the best-case outcome for prisoner's dilemma.<sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Academic_settings">Academic settings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=19" title="Edit section: Academic settings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The prisoner's dilemma has been used in various academic settings to illustrate the complexities of cooperation and competition. One notable example is the classroom experiment conducted by sociology professor Dan Chambliss at <a href="/wiki/Hamilton_College" title="Hamilton College">Hamilton College</a> in the 1980s. Starting in 1981, Chambliss proposed that if no student took the final exam, everyone would receive an A, but if even one student took it, those who didn't would receive a zero. In 1988, <a href="/wiki/John_K._Werner" title="John K. Werner">John Werner</a>, a first-year student, successfully organized his classmates to boycott the exam, demonstrating a practical application of game theory and the prisoner's dilemma concept.<sup id="cite_ref-insidehighered_53-0" class="reference"><a href="#cite_note-insidehighered-53"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> </p><p>Nearly 25 years later, a similar incident occurred at <a href="/wiki/Johns_Hopkins_University" title="Johns Hopkins University">Johns Hopkins University</a> in 2013. Professor Peter Fröhlich's grading policy scaled final exams according to the highest score, meaning that if everyone received the same score, they would all get an A. Students in Fröhlich's classes organized a boycott of the final exam, ensuring that no one took it. As a result, every student received an A, successfully solving the prisoner's dilemma in a mutually optimal way without iteration.<sup id="cite_ref-nyt_54-0" class="reference"><a href="#cite_note-nyt-54"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-baltimorefishbowl_55-0" class="reference"><a href="#cite_note-baltimorefishbowl-55"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> These examples highlight how the prisoner's dilemma can be used to explore cooperative behavior and strategic decision-making in educational contexts. </p> <div class="mw-heading mw-heading2"><h2 id="Related_games">Related games</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=20" title="Edit section: Related games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Closed-bag_exchange">Closed-bag exchange</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=21" title="Edit section: Closed-bag exchange"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Prisoner%27s_Dilemma_briefcase_exchange_(colorized).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Prisoner%27s_Dilemma_briefcase_exchange_%28colorized%29.svg/220px-Prisoner%27s_Dilemma_briefcase_exchange_%28colorized%29.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Prisoner%27s_Dilemma_briefcase_exchange_%28colorized%29.svg/330px-Prisoner%27s_Dilemma_briefcase_exchange_%28colorized%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Prisoner%27s_Dilemma_briefcase_exchange_%28colorized%29.svg/440px-Prisoner%27s_Dilemma_briefcase_exchange_%28colorized%29.svg.png 2x" data-file-width="586" data-file-height="586" /></a><figcaption>The prisoner's dilemma as a briefcase exchange</figcaption></figure> <p><a href="/wiki/Douglas_Hofstadter" title="Douglas Hofstadter">Douglas Hofstadter</a><sup id="cite_ref-dh_56-0" class="reference"><a href="#cite_note-dh-56"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> suggested that people often find problems such as the prisoner's dilemma problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was "closed bag exchange": </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>Two people meet and exchange closed bags, with the understanding that one of them contains money, and the other contains a purchase. Either player can choose to honor the deal by putting into his or her bag what he or she agreed, or he or she can defect by handing over an empty bag.</p></blockquote> <div class="mw-heading mw-heading3"><h3 id="Friend_or_Foe?"><span id="Friend_or_Foe.3F"></span><i>Friend or Foe?</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=22" title="Edit section: Friend or Foe?"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Friend_or_Foe%3F_(TV_series)" class="mw-redirect" title="Friend or Foe? (TV series)">Friend or Foe?</a></i> is a game show that aired from 2002 to 2003 on the <a href="/wiki/Game_Show_Network" title="Game Show Network">Game Show Network</a> in the US. On the game show, three pairs of people compete. When a pair is eliminated, they play a game similar to the prisoner's dilemma to determine how the winnings are split. If they both cooperate (Friend), they share the winnings 50–50. If one cooperates and the other defects (Foe), the defector gets all the winnings, and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the reward matrix is slightly different from the standard one given above, as the rewards for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, <i>Friend or Foe</i> has a rewards model between prisoner's dilemma and the <a href="/wiki/Chicken_(game)" title="Chicken (game)">game of Chicken</a>. </p><p>This is the rewards matrix: </p> <table class="wikitable"> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">Pair 2</span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">Pair 1</span></div> </th> <th scope="col" style="width:6em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">"Friend"<br />(cooperate)</span> </th> <th scope="col" style="width:6em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">"Foe"<br />(defect)</span> </th></tr> <tr> <th scope="row" style="width:6em;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">"Friend"<br />(cooperate)</span> </th> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">1</span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">1</span></div> </td> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">2</span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">0</span></div> </td></tr> <tr> <th scope="row"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">"Foe"<br />(defect)</span> </th> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">0</span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">2</span></div> </td> <td style="background:transparent;background:linear-gradient(to top right,transparent 49%,#AAA 49.5%,#AAA 50.5%,transparent 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#900">0</span></div><div style="margin-right:2em;text-align:left"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:#009">0</span></div> </td></tr></tbody></table> <p>This payoff matrix has also been used on the <a href="/wiki/United_Kingdom" title="United Kingdom">British</a> <a href="/wiki/Television" title="Television">television</a> programs <i>Trust Me</i>, <i><a href="/wiki/Shafted" title="Shafted">Shafted</a></i>, <i><a href="/wiki/The_Bank_Job_(TV_series)" class="mw-redirect" title="The Bank Job (TV series)">The Bank Job</a></i> and <i><a href="/wiki/Golden_Balls" title="Golden Balls">Golden Balls</a></i>, and on the <a href="/wiki/United_States" title="United States">American</a> game show <i><a href="/wiki/Take_It_All_(game_show)" title="Take It All (game show)">Take It All</a></i>, as well as for the winning couple on the reality shows <i><a href="/wiki/Bachelor_Pad" title="Bachelor Pad">Bachelor Pad</a></i> and <i><a href="/wiki/Love_Island_(2015_TV_series)" title="Love Island (2015 TV series)">Love Island</a></i>. Game data from the <i><a href="/wiki/Golden_Balls" title="Golden Balls">Golden Balls</a></i> series has been analyzed by a team of economists, who found that cooperation was "surprisingly high" for amounts of money that would seem consequential in the real world but were comparatively low in the context of the game.<sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Iterated_snowdrift">Iterated snowdrift</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=23" title="Edit section: Iterated snowdrift"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Researchers from the <a href="/wiki/University_of_Lausanne" title="University of Lausanne">University of Lausanne</a> and the <a href="/wiki/University_of_Edinburgh" title="University of Edinburgh">University of Edinburgh</a> have suggested that the "Iterated Snowdrift Game" may more closely reflect real-world social situations, although this model is actually a <a href="/wiki/Chicken_game" class="mw-redirect" title="Chicken game">chicken game</a>. In this model, the risk of being exploited through defection is lower, and individuals always gain from taking the cooperative choice. The snowdrift game imagines two drivers who are stuck on opposite sides of a <a href="/wiki/Snowdrift" title="Snowdrift">snowdrift</a>, each of whom is given the option of shoveling snow to clear a path or remaining in their car. A player's highest payoff comes from leaving the opponent to clear all the snow by themselves, but the opponent is still nominally rewarded for their work. </p><p>This may better reflect real-world scenarios, the researchers giving the example of two scientists collaborating on a report, both of whom would benefit if the other worked harder. "But when your collaborator doesn't do any work, it's probably better for you to do all the work yourself. You'll still end up with a completed project."<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p> <table> <tbody><tr> <td> <table class="wikitable" style="text-align:center;"> <caption>Example snowdrift payouts (A, B) </caption> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">B </div><div style="margin-right:2em;text-align:left"> A</div></th> <th>Cooperates</th> <th>Defects </th></tr> <tr> <th>Cooperates </th> <td>500, 500</td> <td>200, 800 </td></tr> <tr> <th>Defects </th> <td>800, 200</td> <td>0, 0 </td></tr></tbody></table> </td> <td> <table class="wikitable" style="text-align:center; margin-left:2em;"> <caption>Example prisoner's dilemma payouts (A, B) </caption> <tbody><tr> <th style="background:#EAECF0;background:linear-gradient(to top right,#EAECF0 49%,#AAA 49.5%,#AAA 50.5%,#EAECF0 51%);line-height:1.2;padding:0.1em 0.4em;"><div style="margin-left:2em;text-align:right">B </div><div style="margin-right:2em;text-align:left"> A</div></th> <th>Cooperates</th> <th>Defects </th></tr> <tr> <th>Cooperates </th> <td>500, 500</td> <td>−200, 1200 </td></tr> <tr> <th>Defects </th> <td>1200, −200</td> <td>0, 0 </td></tr></tbody></table> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Coordination_games">Coordination games</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=24" title="Edit section: Coordination games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Coordination_game" title="Coordination game">Coordination game</a></div> <p>In coordination games, players must coordinate their strategies for a good outcome. An example is two cars that abruptly meet in a blizzard; each must choose whether to swerve left or right. If both swerve left, or both right, the cars do not collide. The local <a href="/wiki/Left-_and_right-hand_traffic" title="Left- and right-hand traffic">left- and right-hand traffic</a> convention helps to co-ordinate their actions. </p><p>Symmetrical co-ordination games include <a href="/wiki/Stag_hunt" title="Stag hunt">Stag hunt</a> and <a href="/wiki/Bach_or_Stravinsky" class="mw-redirect" title="Bach or Stravinsky">Bach or Stravinsky</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Asymmetric_prisoner's_dilemmas"><span id="Asymmetric_prisoner.27s_dilemmas"></span>Asymmetric prisoner's dilemmas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=25" title="Edit section: Asymmetric prisoner's dilemmas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A more general set of games is asymmetric. As in the prisoner's dilemma, the best outcome is cooperation, and there are motives for defection. Unlike the symmetric prisoner's dilemma, though, one player has more to lose and/or more to gain than the other. Some such games have been described as a prisoner's dilemma in which one prisoner has an <a href="/wiki/Alibi" title="Alibi">alibi</a>, hence the term "alibi game".<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> </p><p>In experiments, players getting unequal payoffs in repeated games may seek to maximize profits, but only under the condition that both players receive equal payoffs; this may lead to a stable equilibrium strategy in which the disadvantaged player defects every X game, while the other always co-operates. Such behavior may depend on the experiment's social norms around fairness.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Software">Software</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=26" title="Edit section: Software"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several software packages have been created to run simulations and tournaments of the prisoner's dilemma, some of which have their source code available: </p> <ul><li>The source code for the <a href="/wiki/The_Evolution_of_Cooperation" title="The Evolution of Cooperation">second tournament</a> run by Robert Axelrod (written by Axelrod and many contributors in <a href="/wiki/Fortran" title="Fortran">Fortran</a>)<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup></li> <li>Prison,<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> a library written in <a href="/wiki/Java_(programming_language)" title="Java (programming language)">Java</a>, last updated in 1998</li> <li>Axelrod-Python,<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> written in <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a></li> <li>Evoplex,<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> a fast agent-based modeling program released in 2018 by Marcos Cardinot</li></ul> <div class="mw-heading mw-heading2"><h2 id="In_fiction">In fiction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=27" title="Edit section: In fiction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hannu_Rajaniemi" title="Hannu Rajaniemi">Hannu Rajaniemi</a> set the opening scene of his <i><a href="/wiki/The_Quantum_Thief" title="The Quantum Thief">The Quantum Thief</a></i> trilogy in a "dilemma prison". The main theme of the series has been described as the "inadequacy of a binary universe" and the ultimate antagonist is a character called the All-Defector. The first book in the series was published in 2010, with the two sequels, <i><a href="/wiki/The_Fractal_Prince" title="The Fractal Prince">The Fractal Prince</a></i> and <i><a href="/wiki/The_Causal_Angel" title="The Causal Angel">The Causal Angel</a></i>, published in 2012 and 2014, respectively. </p><p>A game modeled after the iterated prisoner's dilemma is a central focus of the 2012 video game <i><a href="/wiki/Zero_Escape:_Virtue%27s_Last_Reward" title="Zero Escape: Virtue's Last Reward">Zero Escape: Virtue's Last Reward</a></i> and a minor part in its 2016 sequel <i><a href="/wiki/Zero_Escape:_Zero_Time_Dilemma" class="mw-redirect" title="Zero Escape: Zero Time Dilemma">Zero Escape: Zero Time Dilemma</a></i>. </p><p>In <i><a href="/wiki/The_Mysterious_Benedict_Society_and_the_Prisoner%27s_Dilemma" title="The Mysterious Benedict Society and the Prisoner's Dilemma">The Mysterious Benedict Society and the Prisoner's Dilemma</a></i> by <a href="/wiki/Trenton_Lee_Stewart" title="Trenton Lee Stewart">Trenton Lee Stewart</a>, the main characters start by playing a version of the game and escaping from the "prison" altogether. Later, they become actual prisoners and escape once again. </p><p>In <i><a href="/wiki/The_Adventure_Zone" title="The Adventure Zone">The Adventure Zone</a>: Balance</i> during <i>The Suffering Game</i> subarc, the player characters are twice presented with the prisoner's dilemma during their time in two liches' domain, once cooperating and once defecting. </p><p>In the eighth novel from the author James S. A. Corey, <i><a href="/wiki/Tiamat%27s_Wrath" title="Tiamat's Wrath">Tiamat's Wrath</a></i>, Winston Duarte explains the prisoner's dilemma to his 14-year-old daughter, Teresa, to train her in strategic thinking.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (April 2020)">citation needed</span></a></i>]</sup> </p><p>The 2008 film <i><a href="/wiki/The_Dark_Knight" title="The Dark Knight">The Dark Knight</a></i> includes a scene loosely based on the problem in which the <a href="/wiki/Joker_(The_Dark_Knight)" title="Joker (The Dark Knight)">Joker</a> rigs two ferries, one containing prisoners and the other containing civilians, arming both groups with the means to detonate the bomb on each other's ferries, threatening to detonate them both if they hesitate.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_moral_philosophy">In moral philosophy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=28" title="Edit section: In moral philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The prisoner's dilemma is commonly used as a thinking tool in <a href="/wiki/Moral_philosophy" class="mw-redirect" title="Moral philosophy">moral philosophy</a> as an illustration of the potential tension between the benefit of the individual and the benefit of the community. </p><p>Both the one-shot and the iterated prisoner's dilemma have applications in moral philosophy. Indeed, many of the moral situations, such as <a href="/wiki/Genocide" title="Genocide">genocide</a>, are not easily repeated more than once. Moreover, in many situations, the previous rounds' outcomes are unknown to the players, since they are not necessarily the same (e.g. interaction with a panhandler on the street).<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> </p><p>The philosopher <a href="/wiki/David_Gauthier" title="David Gauthier">David Gauthier</a> uses the prisoner's dilemma to show how morality and rationality can conflict.<sup id="cite_ref-sk_69-0" class="reference"><a href="#cite_note-sk-69"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p><p>Some game theorists have criticized the use of the prisoner's dilemma as a thinking tool in moral philosophy.<sup id="cite_ref-sk_69-1" class="reference"><a href="#cite_note-sk-69"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Kenneth_Binmore" title="Kenneth Binmore">Kenneth Binmore</a> argued that the prisoner's dilemma does not accurately describe the game played by humanity, which he argues is closer to a <a href="/wiki/Coordination_game" title="Coordination game">coordination game</a>. <a href="/wiki/Brian_Skyrms" title="Brian Skyrms">Brian Skyrms</a> shares this perspective. </p><p><a href="/wiki/Steven_Kuhn" title="Steven Kuhn">Steven Kuhn</a> suggests that these views may be reconciled by considering that moral behavior can modify the payoff matrix of a game, transforming it from a prisoner's dilemma into other games.<sup id="cite_ref-sk_69-2" class="reference"><a href="#cite_note-sk-69"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pure_and_impure_prisoner's_dilemma"><span id="Pure_and_impure_prisoner.27s_dilemma"></span>Pure and impure prisoner's dilemma</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=29" title="Edit section: Pure and impure prisoner's dilemma"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A prisoner's dilemma is considered "impure" if a mixed strategy may give better expected payoffs than a pure strategy. This creates the interesting possibility that the moral action from a utilitarian perspective (i.e., aiming at maximizing the good of an action) may require randomization of one's strategy, such as cooperating with 80% chance and defecting with 20% chance.<sup id="cite_ref-70" class="reference"><a href="#cite_note-70"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=30" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 18em;"> <ul><li><a href="/wiki/Abilene_paradox" title="Abilene paradox">Abilene paradox</a></li> <li><a href="/wiki/Centipede_game" title="Centipede game">Centipede game</a></li> <li><a href="/wiki/Collective_action_problem" title="Collective action problem">Collective action problem</a></li> <li><a href="/wiki/Externality" title="Externality">Externality</a></li> <li><a href="/wiki/Folk_theorem_(game_theory)" title="Folk theorem (game theory)">Folk theorem (game theory)</a></li> <li><a href="/wiki/Free-rider_problem" title="Free-rider problem">Free-rider problem</a></li> <li><a href="/wiki/Gift-exchange_game" title="Gift-exchange game">Gift-exchange game</a></li> <li><a href="/wiki/Hobbesian_trap" title="Hobbesian trap">Hobbesian trap</a></li> <li><a href="/wiki/Innocent_prisoner%27s_dilemma" title="Innocent prisoner's dilemma">Innocent prisoner's dilemma</a></li> <li><i><a href="/wiki/Liar_Game" title="Liar Game">Liar Game</a></i></li> <li><a href="/wiki/Metagame" title="Metagame">Metagame</a></li> <li><a href="/wiki/Optional_prisoner%27s_dilemma" title="Optional prisoner's dilemma">Optional prisoner's dilemma</a></li> <li><a href="/wiki/Robert_H._Frank#Prisoner's_dilemma_and_cooperation" title="Robert H. Frank">Prisoner's dilemma and cooperation</a></li> <li><a href="/wiki/Public_goods_game" title="Public goods game">Public goods game</a></li> <li><a href="/wiki/Reciprocal_altruism" title="Reciprocal altruism">Reciprocal altruism</a></li> <li><a href="/wiki/Rent-seeking" title="Rent-seeking">Rent-seeking</a></li> <li><a href="/wiki/Social_preferences" title="Social preferences">Social preferences</a></li> <li><a href="/wiki/Superrationality" title="Superrationality">Superrationality</a></li> <li><a href="/wiki/Swift_trust_theory" title="Swift trust theory">Swift trust theory</a></li> <li><a href="/wiki/Tragedy_of_the_commons" title="Tragedy of the commons">Tragedy of the commons</a></li> <li><a href="/wiki/Traveler%27s_dilemma" title="Traveler's dilemma">Traveler's dilemma</a></li> <li><a href="/wiki/Unscrupulous_diner%27s_dilemma" title="Unscrupulous diner's dilemma">Unscrupulous diner's dilemma</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=31" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">The tournament has two rounds. In the first round, each of the top eight strategies were nice, and not one of the bottom seven were nice. In the second round (strategy designers could take into account the results of the first round), all but one of the top fifteen strategies were nice (and that one ranked eighth). Of the bottom fifteen strategies, all but one were not nice.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">In contrast to strategies like <a href="/wiki/Grim_trigger" title="Grim trigger">grim trigger</a> (also called Friedman), which is never first to defect, but once the other defects even once, grim trigger defects from then on.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">For example see the 2003 study<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> for discussion of the concept and whether it can apply in real <a href="/wiki/Economic" class="mw-redirect" title="Economic">economic</a> or strategic situations.</span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text">This argument for the development of cooperation through trust is given in <i><a href="/wiki/The_Wisdom_of_Crowds" title="The Wisdom of Crowds">The Wisdom of Crowds</a></i>, where it is argued that long-distance <a href="/wiki/Capitalism" title="Capitalism">capitalism</a> was able to form around a nucleus of <a href="/wiki/Religious_Society_of_Friends" class="mw-redirect" title="Religious Society of Friends">Quakers</a>, who always dealt honourably with their business partners (rather than defecting and reneging on promises – a phenomenon that had discouraged earlier long-term unenforceable overseas contracts). It is argued that dealings with reliable merchants allowed the <a href="/wiki/Meme" title="Meme">meme</a> for cooperation to spread to other traders, who spread it further until a high degree of cooperation became a profitable strategy in general <a href="/wiki/Commerce" title="Commerce">commerce</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=32" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/prisoner-dilemma/">"Prisoner's Dilemma"</a>. <i><a href="/wiki/Stanford_Encyclopedia_of_Philosophy" title="Stanford Encyclopedia of Philosophy">Stanford Encyclopedia of Philosophy</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">10 March</span> 2024</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=Prisoner%27s+Dilemma&rft_id=http%3A%2F%2Fplato.stanford.edu%2Fentries%2Fprisoner-dilemma%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEPoundstone19938,_117-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPoundstone19938,_117_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPoundstone1993">Poundstone 1993</a>, pp. 8, 117.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFPoundstone1993">Poundstone 1993</a>, p. 118: "A typical contemporary version of the story goes like this: Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of speaking to or exchanging messages with the other. The police admit they don't have enough evidence to convict the pair on the principal charge. They plan to sentence both to a year in prison on a lesser charge. Simultaneously, the police offer each prisoner a Faustian bargain. If he testifies against his partner, he will go free while the partner will get three years in prison on the main charge. Oh, yes, there is a catch ... If both prisoners testify against each other, both will be sentenced to two years in jail. The prisoners are given a little time to think this over, but in no case may either learn what the other has decided until he has irrevocably made his decision. Each is informed that the other prisoner is being offered the very same deal. Each prisoner is concerned only with his own welfare—with minimizing his own prison sentence."</span> </li> <li id="cite_note-FOOTNOTEPoundstone1993118-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPoundstone1993118_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPoundstone1993">Poundstone 1993</a>, p. 118.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrofmanPool1977" class="citation journal cs1">Grofman, Bernard; Pool, Jonathan (January 1977). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1080/0022250x.1977.9989871">"How to make cooperation the optimizing strategy in a two-person game"</a>. <i>The Journal of Mathematical Sociology</i>. <b>5</b> (2): 173–186. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0022250x.1977.9989871">10.1080/0022250x.1977.9989871</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-250X">0022-250X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Mathematical+Sociology&rft.atitle=How+to+make+cooperation+the+optimizing+strategy+in+a+two-person+game&rft.volume=5&rft.issue=2&rft.pages=173-186&rft.date=1977-01&rft_id=info%3Adoi%2F10.1080%2F0022250x.1977.9989871&rft.issn=0022-250X&rft.aulast=Grofman&rft.aufirst=Bernard&rft.au=Pool%2C+Jonathan&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1080%2F0022250x.1977.9989871&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></span> </li> <li id="cite_note-Shy-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Shy_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShy1995" class="citation book cs1">Shy, Oz (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tr4CjJ5LlRcC&q=industrial+organization+theory+and+applications&pg=PR13"><i>Industrial Organization: Theory and Applications</i></a>. Massachusetts Institute of Technology Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0262193665" title="Special:BookSources/978-0262193665"><bdi>978-0262193665</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">February 27,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Industrial+Organization%3A+Theory+and+Applications&rft.pub=Massachusetts+Institute+of+Technology+Press&rft.date=1995&rft.isbn=978-0262193665&rft.aulast=Shy&rft.aufirst=Oz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dtr4CjJ5LlRcC%26q%3Dindustrial%2Borganization%2Btheory%2Band%2Bapplications%26pg%3DPR13&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAumann2016" class="citation cs2">Aumann, Robert J. (2016-03-02), <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9781400882168-018/html">"16. Acceptable Points in General Cooperative n-Person Games"</a>, <i>Contributions to the Theory of Games (AM-40), Volume IV</i>, Princeton University Press, pp. 287–324, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400882168-018">10.1515/9781400882168-018</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-8216-8" title="Special:BookSources/978-1-4008-8216-8"><bdi>978-1-4008-8216-8</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2024-05-14</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=16.+Acceptable+Points+in+General+Cooperative+n-Person+Games&rft.btitle=Contributions+to+the+Theory+of+Games+%28AM-40%29%2C+Volume+IV&rft.pages=287-324&rft.pub=Princeton+University+Press&rft.date=2016-03-02&rft_id=info%3Adoi%2F10.1515%2F9781400882168-018&rft.isbn=978-1-4008-8216-8&rft.aulast=Aumann&rft.aufirst=Robert+J.&rft_id=https%3A%2F%2Fwww.degruyter.com%2Fdocument%2Fdoi%2F10.1515%2F9781400882168-018%2Fhtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCooperDeJongForsytheRoss1996" class="citation journal cs1">Cooper, Russell; DeJong, Douglas V.; Forsythe, Robert; Ross, Thomas W. 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New York: Anchor. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-385-41580-X" title="Special:BookSources/0-385-41580-X"><bdi>0-385-41580-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Prisoner%27s+Dilemma&rft.place=New+York&rft.edition=1st+Anchor+Books&rft.pub=Anchor&rft.date=1993&rft.isbn=0-385-41580-X&rft.aulast=Poundstone&rft.aufirst=William&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprisonersdilemma00poun&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=34" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <ul><li>Amadae, S. (2016). <a rel="nofollow" class="external text" href="http://www.amadae.com/Amadae__Prisoners_of_Reason__Prisoners_Dilemma.pdf">"Prisoner's Dilemma"</a>, <i>Prisoners of Reason</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, NY, pp. 24–61.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAumann1959" class="citation book cs1"><a href="/wiki/Robert_Aumann" title="Robert Aumann">Aumann, Robert</a> (1959). "Acceptable points in general cooperative <i>n</i>-person games". In Luce, R. D.; Tucker, A. W. (eds.). <i>Contributions to the Theory 23 of Games IV</i>. Annals of Mathematics Study. Vol. 40. Princeton NJ: Princeton University Press. pp. 287–324. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0104521">0104521</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Acceptable+points+in+general+cooperative+n-person+games&rft.btitle=Contributions+to+the+Theory+23+of+Games+IV&rft.place=Princeton+NJ&rft.series=Annals+of+Mathematics+Study&rft.pages=287-324&rft.pub=Princeton+University+Press&rft.date=1959&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0104521%23id-name%3DMR&rft.aulast=Aumann&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxelrod2006" class="citation book cs1">Axelrod, Robert (2006). <i><a href="/wiki/The_Evolution_of_Cooperation" title="The Evolution of Cooperation">The Evolution of Cooperation</a></i> (Revised ed.). Basic Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-465-00564-0" title="Special:BookSources/0-465-00564-0"><bdi>0-465-00564-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Evolution+of+Cooperation&rft.edition=Revised&rft.pub=Basic+Books&rft.date=2006&rft.isbn=0-465-00564-0&rft.aulast=Axelrod&rft.aufirst=Robert&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li> <li><a href="/wiki/Cristina_Bicchieri" title="Cristina Bicchieri">Bicchieri, Cristina</a> (1993). Rationality and Coordination. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChess1988" class="citation journal cs1">Chess, David M. (December 1988). <a rel="nofollow" class="external text" href="http://www.complex-systems.com/pdf/02-6-4.pdf">"Simulating the evolution of behavior: the iterated prisoners' dilemma problem"</a> <span class="cs1-format">(PDF)</span>. <i>Complex Systems</i>. <b>2</b> (6): 663–70. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150530031939/http://www.complex-systems.com/pdf/02-6-4.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2015-05-30.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Complex+Systems&rft.atitle=Simulating+the+evolution+of+behavior%3A+the+iterated+prisoners%27+dilemma+problem&rft.volume=2&rft.issue=6&rft.pages=663-70&rft.date=1988-12&rft.aulast=Chess&rft.aufirst=David+M.&rft_id=http%3A%2F%2Fwww.complex-systems.com%2Fpdf%2F02-6-4.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCollins2022" class="citation journal cs1">Collins, Rory W. (March 2022). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1477175621000464">"The prisoner's dilemma paradox: Rationality, morality, and reciprocity"</a>. <i>Think: Philosophy for Everyone</i>. <b>21</b> (61): 45–55. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS1477175621000464">10.1017/S1477175621000464</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Think%3A+Philosophy+for+Everyone&rft.atitle=The+prisoner%27s+dilemma+paradox%3A+Rationality%2C+morality%2C+and+reciprocity&rft.volume=21&rft.issue=61&rft.pages=45-55&rft.date=2022-03&rft_id=info%3Adoi%2F10.1017%2FS1477175621000464&rft.aulast=Collins&rft.aufirst=Rory+W.&rft_id=https%3A%2F%2Fdoi.org%2F10.1017%252FS1477175621000464&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li> <li><a href="/wiki/Melvin_Dresher" title="Melvin Dresher">Dresher, M.</a> (1961). <i>The Mathematics of Games of Strategy: Theory and Applications</i> <a href="/wiki/Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>, Englewood Cliffs, NJ.</li> <li>Greif, A. (2006). <i>Institutions and the Path to the Modern Economy: Lessons from Medieval Trade.</i> Cambridge University Press, <a href="/wiki/Cambridge" title="Cambridge">Cambridge</a>, UK.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKopelman2020" class="citation journal cs1">Kopelman, Shirli (February 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fncmr.12172">"Tit for tat and beyond: the legendary work of Anatol Rapoport"</a>. <i>Negotiation and Conflict Management Research</i>. <b>13</b> (1): 60–84. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fncmr.12172">10.1111/ncmr.12172</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F153763">2027.42/153763</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Negotiation+and+Conflict+Management+Research&rft.atitle=Tit+for+tat+and+beyond%3A+the+legendary+work+of+Anatol+Rapoport&rft.volume=13&rft.issue=1&rft.pages=60-84&rft.date=2020-02&rft_id=info%3Ahdl%2F2027.42%2F153763&rft_id=info%3Adoi%2F10.1111%2Fncmr.12172&rft.aulast=Kopelman&rft.aufirst=Shirli&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fncmr.12172&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li> <li><a href="/wiki/Anatol_Rapoport" title="Anatol Rapoport">Rapoport, Anatol</a> and Albert M. Chammah (1965). <i>Prisoner's Dilemma</i>. <a href="/wiki/University_of_Michigan_Press" title="University of Michigan Press">University of Michigan Press</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchneiderShields2022" class="citation journal cs1">Schneider, Mark; Shields, Timothy W. (June 2022). <a rel="nofollow" class="external text" href="https://digitalcommons.chapman.edu/cgi/viewcontent.cgi?article=1368&context=esi_working_papers">"Motives for cooperation in the one-shot prisoner's dilemma"</a>. <i>Journal of Behavioral Finance</i>. <b>23</b> (4): 438–456. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F15427560.2022.2081974">10.1080/15427560.2022.2081974</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:249903572">249903572</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Behavioral+Finance&rft.atitle=Motives+for+cooperation+in+the+one-shot+prisoner%27s+dilemma&rft.volume=23&rft.issue=4&rft.pages=438-456&rft.date=2022-06&rft_id=info%3Adoi%2F10.1080%2F15427560.2022.2081974&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A249903572%23id-name%3DS2CID&rft.aulast=Schneider&rft.aufirst=Mark&rft.au=Shields%2C+Timothy+W.&rft_id=https%3A%2F%2Fdigitalcommons.chapman.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1368%26context%3Desi_working_papers&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Prisoner%27s_dilemma&action=edit&section=35" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Prisoner%27s_dilemma" class="extiw" title="commons:Category:Prisoner's dilemma">Prisoner's dilemma</a> at Wikimedia Commons</li> <li><a rel="nofollow" class="external text" href="http://www.msri.org/ext/larryg/pages/15.htm">The Bowerbird's Dilemma</a> The Prisoner's Dilemma in ornithology – mathematical cartoon by Larry Gonick.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDixitNalebuff2008" class="citation encyclopaedia cs1"><a href="/wiki/Avinash_Dixit" title="Avinash Dixit">Dixit, Avinash</a>; <a href="/wiki/Barry_Nalebuff" title="Barry Nalebuff">Nalebuff, Barry</a> (2008). <a rel="nofollow" class="external text" href="http://www.econlib.org/library/Enc/PrisonersDilemma.html">"Prisoner's Dilemma"</a>. In <a href="/wiki/David_R._Henderson" title="David R. Henderson">David R. Henderson</a> (ed.). <i><a href="/wiki/Concise_Encyclopedia_of_Economics" class="mw-redirect" title="Concise Encyclopedia of Economics">Concise Encyclopedia of Economics</a></i> (2nd ed.). Indianapolis: <a href="/wiki/Library_of_Economics_and_Liberty" class="mw-redirect" title="Library of Economics and Liberty">Library of Economics and Liberty</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0865976658" title="Special:BookSources/978-0865976658"><bdi>978-0865976658</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/237794267">237794267</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Prisoner%27s+Dilemma&rft.btitle=Concise+Encyclopedia+of+Economics&rft.place=Indianapolis&rft.edition=2nd&rft.pub=Library+of+Economics+and+Liberty&rft.date=2008&rft_id=info%3Aoclcnum%2F237794267&rft.isbn=978-0865976658&rft.aulast=Dixit&rft.aufirst=Avinash&rft.au=Nalebuff%2C+Barry&rft_id=http%3A%2F%2Fwww.econlib.org%2Flibrary%2FEnc%2FPrisonersDilemma.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3APrisoner%27s+dilemma" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=I71mjZefg8g">Dawkins: Nice Guys Finish First</a></li> <li><a rel="nofollow" class="external text" href="https://axelrod.readthedocs.io/en/stable/">Axelrod</a> Iterated Prisoner's Dilemma <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> library</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20141011014608/http://demo.otree.org/demo/Prisoner%27s+Dilemma/">Play Prisoner's Dilemma on <i>oTree</i></a> (N/A 11-5-17)</li> <li>Nicky Case's <a rel="nofollow" class="external text" href="https://web.archive.org/web/20181229222135/https://ncase.me/trust/">Evolution of Trust</a>, an example of the donation game</li> <li><a rel="nofollow" class="external text" href="http://iterated-prisoners-dilemma.info">Iterated Prisoner's Dilemma online game</a> by Wayne Davis</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=mScpHTIi-kM">What The Prisoner's Dilemma Reveals About Life, The Universe, and Everything</a> by <a href="/wiki/Veritasium" class="mw-redirect" title="Veritasium">Veritasium</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist 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href="/wiki/Template:Paradoxes" title="Template:Paradoxes"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Paradoxes" title="Template talk:Paradoxes"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Paradoxes" title="Special:EditPage/Template:Paradoxes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Notable_paradoxes" style="font-size:114%;margin:0 4em">Notable <a href="/wiki/Paradox" title="Paradox">paradoxes</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Philosophical</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Paradox_of_analysis" title="Paradox of analysis">Analysis</a></li> <li><a href="/wiki/Buridan%27s_bridge" title="Buridan's bridge">Buridan's bridge</a></li> <li><a href="/wiki/Dream_argument" title="Dream argument">Dream argument</a></li> <li><a href="/wiki/Epicurean_paradox" title="Epicurean paradox">Epicurean</a></li> <li><a href="/wiki/Paradox_of_fiction" title="Paradox of fiction">Fiction</a></li> <li><a href="/wiki/Fitch%27s_paradox_of_knowability" title="Fitch's paradox of knowability">Fitch's knowability</a></li> <li><a href="/wiki/Argument_from_free_will" title="Argument from free will">Free will</a></li> <li><a href="/wiki/New_riddle_of_induction" title="New riddle of induction">Goodman's</a></li> <li><a href="/wiki/Paradox_of_hedonism" title="Paradox of hedonism">Hedonism</a></li> <li><a href="/wiki/Liberal_paradox" title="Liberal paradox">Liberal</a></li> <li><a href="/wiki/Meno" title="Meno">Meno's</a></li> <li><a href="/wiki/Mere_addition_paradox" title="Mere addition paradox">Mere addition</a></li> <li><a href="/wiki/Moore%27s_paradox" title="Moore's paradox">Moore's</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Paradox_of_nihilism" title="Paradox of nihilism">Nihilism</a></li> <li><a href="/wiki/Omnipotence_paradox" title="Omnipotence paradox">Omnipotence</a></li> <li><a href="/wiki/Preface_paradox" title="Preface paradox">Preface</a></li> <li><a href="/wiki/Wittgenstein_on_Rules_and_Private_Language" title="Wittgenstein on Rules and Private Language">Rule-following</a></li> <li><a href="/wiki/Sorites_paradox" title="Sorites paradox">Sorites</a></li> <li><a href="/wiki/Ship_of_Theseus" title="Ship of Theseus">Theseus' ship</a></li> <li><a href="/wiki/White_Horse_Dialogue" title="White Horse Dialogue">White horse</a></li> <li><a href="/wiki/Zeno%27s_paradoxes" title="Zeno's paradoxes">Zeno's</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Logical</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Barber_paradox" title="Barber paradox">Barber</a></li> <li><a href="/wiki/Berry_paradox" title="Berry paradox">Berry</a></li> <li><a href="/wiki/Bhartrhari%27s_paradox" class="mw-redirect" title="Bhartrhari's paradox">Bhartrhari's</a></li> <li><a href="/wiki/Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti</a></li> <li><a href="/wiki/Paradox_of_the_Court" title="Paradox of the Court">Court</a></li> <li><a href="/wiki/Crocodile_dilemma" title="Crocodile dilemma">Crocodile</a></li> <li><a href="/wiki/Curry%27s_paradox" title="Curry's paradox">Curry's</a></li> <li><a href="/wiki/Epimenides_paradox" title="Epimenides paradox">Epimenides</a></li> <li><a href="/wiki/Free_choice_inference" title="Free choice inference">Free choice paradox</a></li> <li><a href="/wiki/Grelling%E2%80%93Nelson_paradox" title="Grelling–Nelson paradox">Grelling–Nelson</a></li> <li><a href="/wiki/Kleene%E2%80%93Rosser_paradox" title="Kleene–Rosser paradox">Kleene–Rosser</a></li> <li><a href="/wiki/Liar_paradox" title="Liar paradox">Liar</a> <ul><li><a href="/wiki/Card_paradox" title="Card paradox">Card</a></li> <li><a href="/wiki/No%E2%80%93no_paradox" title="No–no paradox">No-no</a></li> <li><a href="/wiki/Pinocchio_paradox" title="Pinocchio paradox">Pinocchio</a></li> <li><a href="/wiki/Quine%27s_paradox" title="Quine's paradox">Quine's</a></li> <li><a href="/wiki/Yablo%27s_paradox" class="mw-redirect" title="Yablo's paradox">Yablo's</a></li></ul></li> <li><a href="/wiki/Opposite_Day" title="Opposite Day">Opposite Day</a></li> <li><a href="/wiki/Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes of set theory</a></li> <li><a href="/wiki/Richard%27s_paradox" title="Richard's paradox">Richard's</a></li> <li><a href="/wiki/Russell%27s_paradox" title="Russell's paradox">Russell's</a></li> <li><a href="/wiki/I_know_that_I_know_nothing" title="I know that I know nothing">Socratic</a></li> <li><a href="/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel" title="Hilbert's paradox of the Grand Hotel">Hilbert's Hotel</a></li> <li><a href="/wiki/Temperature_paradox" title="Temperature paradox">Temperature paradox</a></li> <li><a href="/wiki/Barbershop_paradox" title="Barbershop paradox">Barbershop</a></li> <li><a href="/wiki/Catch-22_(logic)" title="Catch-22 (logic)">Catch-22</a></li> <li><a href="/wiki/Chicken_or_the_egg" title="Chicken or the egg">Chicken or the egg</a></li> <li><a href="/wiki/Drinker_paradox" title="Drinker paradox">Drinker</a></li> <li><a href="/wiki/Paradoxes_of_material_implication" title="Paradoxes of material implication">Entailment</a></li> <li><a href="/wiki/Lottery_paradox" title="Lottery paradox">Lottery</a></li> <li><a href="/wiki/Plato%27s_beard" title="Plato's beard">Plato's beard</a></li> <li><a href="/wiki/Raven_paradox" title="Raven paradox">Raven</a></li> <li><a href="/wiki/Imperative_logic#Ross's_paradox" title="Imperative logic">Ross's</a></li> <li><a href="/wiki/Unexpected_hanging_paradox" title="Unexpected hanging paradox">Unexpected hanging</a></li> <li>"<a href="/wiki/What_the_Tortoise_Said_to_Achilles" title="What the Tortoise Said to Achilles">What the Tortoise Said to Achilles</a>"</li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Olbers%27s_paradox" title="Olbers's paradox">Olbers's paradox</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Economic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Allais_paradox" title="Allais paradox">Allais</a></li> <li><a href="/wiki/The_Antitrust_Paradox" title="The Antitrust Paradox">Antitrust</a></li> <li><a href="/wiki/Arrow_information_paradox" title="Arrow information paradox">Arrow information</a></li> <li><a href="/wiki/Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand</a></li> <li><a href="/wiki/Braess%27s_paradox" title="Braess's paradox">Braess's</a></li> <li><a href="/wiki/Paradox_of_competition" title="Paradox of competition">Competition</a></li> <li><a href="/wiki/Income_and_fertility" title="Income and fertility">Income and fertility</a></li> <li><a href="/wiki/Downs%E2%80%93Thomson_paradox" title="Downs–Thomson paradox">Downs–Thomson</a></li> <li><a href="/wiki/Easterlin_paradox" title="Easterlin paradox">Easterlin</a></li> <li><a href="/wiki/Edgeworth_paradox" title="Edgeworth paradox">Edgeworth</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/European_paradox" title="European paradox">European</a></li> <li><a href="/wiki/Gibson%27s_paradox" title="Gibson's paradox">Gibson's</a></li> <li><a href="/wiki/Giffen_good" title="Giffen good">Giffen good</a></li> <li><a href="/wiki/Icarus_paradox" title="Icarus paradox">Icarus</a></li> <li><a href="/wiki/Jevons_paradox" title="Jevons paradox">Jevons</a></li> <li><a href="/wiki/Leontief_paradox" title="Leontief paradox">Leontief</a></li> <li><a href="/wiki/Lerner_paradox" title="Lerner paradox">Lerner</a></li> <li><a href="/wiki/Lucas_paradox" title="Lucas paradox">Lucas</a></li> <li><a href="/wiki/Mandeville%27s_paradox" title="Mandeville's paradox">Mandeville's</a></li> <li><a href="/wiki/Mayfield%27s_paradox" title="Mayfield's paradox">Mayfield's</a></li> <li><a href="/wiki/Metzler_paradox" title="Metzler paradox">Metzler</a></li> <li><a href="/wiki/Resource_curse" title="Resource curse">Plenty</a></li> <li><a href="/wiki/Productivity_paradox" title="Productivity paradox">Productivity</a></li> <li><a href="/wiki/Paradox_of_prosperity" title="Paradox of prosperity">Prosperity</a></li> <li><a href="/wiki/Scitovsky_paradox" title="Scitovsky paradox">Scitovsky</a></li> <li><a href="/wiki/Service_recovery_paradox" title="Service recovery paradox">Service recovery</a></li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg</a></li> <li><a href="/wiki/Paradox_of_thrift" title="Paradox of thrift">Thrift</a></li> <li><a href="/wiki/Paradox_of_toil" title="Paradox of toil">Toil</a></li> <li><a href="/wiki/Tullock_paradox" class="mw-redirect" title="Tullock paradox">Tullock</a></li> <li><a href="/wiki/Paradox_of_value" title="Paradox of value">Value</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abilene_paradox" title="Abilene paradox">Abilene</a></li> <li><a href="/wiki/Apportionment_paradox" title="Apportionment paradox">Apportionment</a> <ul><li><a href="/wiki/House_monotonicity" title="House monotonicity">Alabama</a></li> <li><a href="/wiki/Coherence_(fairness)" title="Coherence (fairness)">New states</a></li> <li><a href="/wiki/State-population_monotonicity" class="mw-redirect" title="State-population monotonicity">Population</a></li></ul></li> <li><a href="/wiki/Arrow%27s_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's</a></li> <li><a href="/wiki/Buridan%27s_ass" title="Buridan's ass">Buridan's ass</a></li> <li><a href="/wiki/Chainstore_paradox" title="Chainstore paradox">Chainstore</a></li> <li><a href="/wiki/Condorcet_paradox" title="Condorcet paradox">Condorcet's</a></li> <li><a href="/wiki/Decision-making_paradox" title="Decision-making paradox">Decision-making</a></li> <li><a href="/wiki/Paradox_of_voting" title="Paradox of voting">Downs</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg</a></li> <li><a href="/wiki/Fenno%27s_paradox" title="Fenno's paradox">Fenno's</a></li> <li><a href="/wiki/Fredkin%27s_paradox" title="Fredkin's paradox">Fredkin's</a></li> <li><a href="/wiki/The_Green_Paradox" title="The Green Paradox">Green</a></li> <li><a href="/wiki/Hedgehog%27s_dilemma" title="Hedgehog's dilemma">Hedgehog's</a></li> <li><a href="/wiki/Inventor%27s_paradox" title="Inventor's paradox">Inventor's</a></li> <li><a href="/wiki/Kavka%27s_toxin_puzzle" title="Kavka's toxin puzzle">Kavka's toxin puzzle</a></li> <li><a href="/wiki/Morton%27s_fork" title="Morton's fork">Morton's fork</a></li> <li><a href="/wiki/Navigation_paradox" title="Navigation paradox">Navigation</a></li> <li><a href="/wiki/Newcomb%27s_paradox" title="Newcomb's paradox">Newcomb's</a></li> <li><a href="/wiki/Parrondo%27s_paradox" title="Parrondo's paradox">Parrondo's</a></li> <li><a href="/wiki/Preparedness_paradox" title="Preparedness paradox">Preparedness</a></li> <li><a href="/wiki/Prevention_paradox" title="Prevention paradox">Prevention</a></li> <li><a class="mw-selflink selflink">Prisoner's dilemma</a></li> <li><a href="/wiki/Paradox_of_tolerance" title="Paradox of tolerance">Tolerance</a></li> <li><a href="/wiki/Willpower_paradox" title="Willpower paradox">Willpower</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_paradoxes" title="List of paradoxes">List</a></li> <li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Paradoxes" title="Category:Paradoxes">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Topics_of_game_theory" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Game_theory" title="Template:Game theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Game_theory" title="Template talk:Game theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Game_theory" title="Special:EditPage/Template:Game theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topics_of_game_theory" style="font-size:114%;margin:0 4em">Topics of <a href="/wiki/Game_theory" title="Game theory">game theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Definitions</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Congestion_game" title="Congestion game">Congestion game</a></li> <li><a href="/wiki/Cooperative_game_theory" title="Cooperative game theory">Cooperative game</a></li> <li><a href="/wiki/Determinacy" title="Determinacy">Determinacy</a></li> <li><a href="/wiki/Escalation_of_commitment" title="Escalation of commitment">Escalation of commitment</a></li> <li><a href="/wiki/Extensive-form_game" title="Extensive-form game">Extensive-form game</a></li> <li><a href="/wiki/First-player_and_second-player_win" title="First-player and second-player win">First-player and second-player win</a></li> <li><a href="/wiki/Game_complexity" title="Game complexity">Game complexity</a></li> <li><a href="/wiki/Graphical_game_theory" title="Graphical game theory">Graphical game</a></li> <li><a href="/wiki/Hierarchy_of_beliefs" title="Hierarchy of beliefs">Hierarchy of beliefs</a></li> <li><a href="/wiki/Information_set_(game_theory)" title="Information set (game theory)">Information set</a></li> <li><a href="/wiki/Normal-form_game" title="Normal-form game">Normal-form game</a></li> <li><a href="/wiki/Preference_(economics)" title="Preference (economics)">Preference</a></li> <li><a href="/wiki/Sequential_game" title="Sequential game">Sequential game</a></li> <li><a href="/wiki/Simultaneous_game" title="Simultaneous game">Simultaneous game</a></li> <li><a href="/wiki/Simultaneous_action_selection" title="Simultaneous action selection">Simultaneous action selection</a></li> <li><a href="/wiki/Solved_game" title="Solved game">Solved game</a></li> <li><a href="/wiki/Succinct_game" title="Succinct game">Succinct game</a></li> <li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Economic_equilibrium" title="Economic equilibrium">Equilibrium</a><br /><a href="/wiki/Solution_concept" title="Solution concept">concepts</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayes_correlated_equilibrium" title="Bayes correlated equilibrium">Bayes correlated equilibrium</a></li> <li><a href="/wiki/Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a></li> <li><a href="/wiki/Berge_equilibrium" title="Berge equilibrium">Berge equilibrium</a></li> <li><a href="/wiki/Core_(game_theory)" title="Core (game theory)"> Core</a></li> <li><a href="/wiki/Correlated_equilibrium" title="Correlated equilibrium">Correlated equilibrium</a></li> <li><a href="/wiki/Coalition-proof_Nash_equilibrium" title="Coalition-proof Nash equilibrium">Coalition-proof Nash equilibrium</a></li> <li><a href="/wiki/Epsilon-equilibrium" title="Epsilon-equilibrium">Epsilon-equilibrium</a></li> <li><a href="/wiki/Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">Evolutionarily stable strategy</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs equilibrium</a></li> <li><a href="/wiki/Mertens-stable_equilibrium" title="Mertens-stable equilibrium">Mertens-stable equilibrium</a></li> <li><a href="/wiki/Markov_perfect_equilibrium" title="Markov perfect equilibrium">Markov perfect equilibrium</a></li> <li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a></li> <li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li> <li><a href="/wiki/Perfect_Bayesian_equilibrium" title="Perfect Bayesian equilibrium">Perfect Bayesian equilibrium</a></li> <li><a href="/wiki/Proper_equilibrium" title="Proper equilibrium">Proper equilibrium</a></li> <li><a href="/wiki/Quantal_response_equilibrium" title="Quantal response equilibrium">Quantal response equilibrium</a></li> <li><a href="/wiki/Quasi-perfect_equilibrium" title="Quasi-perfect equilibrium">Quasi-perfect equilibrium</a></li> <li><a href="/wiki/Risk_dominance" title="Risk dominance">Risk dominance</a></li> <li><a href="/wiki/Satisfaction_equilibrium" title="Satisfaction equilibrium">Satisfaction equilibrium</a></li> <li><a href="/wiki/Self-confirming_equilibrium" title="Self-confirming equilibrium">Self-confirming equilibrium</a></li> <li><a href="/wiki/Sequential_equilibrium" title="Sequential equilibrium">Sequential equilibrium</a></li> <li><a href="/wiki/Shapley_value" title="Shapley value">Shapley value</a></li> <li><a href="/wiki/Strong_Nash_equilibrium" title="Strong Nash equilibrium">Strong Nash equilibrium</a></li> <li><a href="/wiki/Subgame_perfect_equilibrium" title="Subgame perfect equilibrium">Subgame perfection</a></li> <li><a href="/wiki/Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">Trembling hand equilibrium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Strategy_(game_theory)" title="Strategy (game theory)">Strategies</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Appeasement" title="Appeasement">Appeasement</a></li> <li><a href="/wiki/Backward_induction" title="Backward induction">Backward induction</a></li> <li><a href="/wiki/Bid_shading" title="Bid shading">Bid shading</a></li> <li><a href="/wiki/Collusion" title="Collusion">Collusion</a></li> <li><a href="/wiki/Cheap_talk" title="Cheap talk">Cheap talk</a></li> <li><a href="/wiki/De-escalation" title="De-escalation">De-escalation</a></li> <li><a href="/wiki/Deterrence_theory" title="Deterrence theory">Deterrence</a></li> <li><a href="/wiki/Conflict_escalation" title="Conflict escalation">Escalation</a></li> <li><a href="/wiki/Forward_induction" class="mw-redirect" title="Forward induction">Forward induction</a></li> <li><a href="/wiki/Grim_trigger" title="Grim trigger">Grim trigger</a></li> <li><a href="/wiki/Markov_strategy" title="Markov strategy">Markov strategy</a></li> <li><a href="/wiki/Pairing_strategy" title="Pairing strategy">Pairing strategy</a></li> <li><a href="/wiki/Strategic_dominance" title="Strategic dominance">Dominant strategies</a></li> <li><a href="/wiki/Strategy_(game_theory)" title="Strategy (game theory)">Pure strategy</a></li> <li><a href="/wiki/Strategy_(game_theory)#Mixed_strategy" title="Strategy (game theory)">Mixed strategy</a></li> <li><a href="/wiki/Strategy-stealing_argument" title="Strategy-stealing argument">Strategy-stealing argument</a></li> <li><a href="/wiki/Tit_for_tat" title="Tit for tat">Tit for tat</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Game_theory_game_classes" title="Category:Game theory game classes">Classes<br />of games</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Auction" title="Auction">Auction</a></li> <li><a href="/wiki/Bargaining_problem" class="mw-redirect" title="Bargaining problem">Bargaining problem</a></li> <li><a href="/wiki/Global_game" title="Global game">Global game</a></li> <li><a href="/wiki/Intransitive_game" title="Intransitive game">Intransitive game</a></li> <li><a href="/wiki/Mean-field_game_theory" title="Mean-field game theory">Mean-field game</a></li> <li><a href="/wiki/N-player_game" title="N-player game"><i>n</i>-player game</a></li> <li><a href="/wiki/Perfect_information" title="Perfect information">Perfect information</a></li> <li><a href="/wiki/Poisson_games" class="mw-redirect" title="Poisson games">Large Poisson game</a></li> <li><a href="/wiki/Potential_game" title="Potential game">Potential game</a></li> <li><a href="/wiki/Repeated_game" title="Repeated game">Repeated game</a></li> <li><a href="/wiki/Screening_game" title="Screening game">Screening game</a></li> <li><a href="/wiki/Signaling_game" title="Signaling game">Signaling game</a></li> <li><a href="/wiki/Strictly_determined_game" title="Strictly determined game">Strictly determined game</a></li> <li><a href="/wiki/Stochastic_game" title="Stochastic game">Stochastic game</a></li> <li><a href="/wiki/Symmetric_game" title="Symmetric game">Symmetric game</a></li> <li><a href="/wiki/Zero-sum_game" title="Zero-sum game">Zero-sum game</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_games_in_game_theory" title="List of games in game theory">Games</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Go_(game)" title="Go (game)">Go</a></li> <li><a href="/wiki/Chess" title="Chess">Chess</a></li> <li><a href="/wiki/Infinite_chess" title="Infinite chess">Infinite chess</a></li> <li><a href="/wiki/Draughts" class="mw-redirect" title="Draughts">Checkers</a></li> <li><a href="/wiki/All-pay_auction" title="All-pay auction">All-pay auction</a></li> <li><a class="mw-selflink selflink">Prisoner's dilemma</a></li> <li><a href="/wiki/Gift-exchange_game" title="Gift-exchange game">Gift-exchange game</a></li> <li><a href="/wiki/Optional_prisoner%27s_dilemma" title="Optional prisoner's dilemma">Optional prisoner's dilemma</a></li> <li><a href="/wiki/Traveler%27s_dilemma" title="Traveler's dilemma">Traveler's dilemma</a></li> <li><a href="/wiki/Coordination_game" title="Coordination game">Coordination game</a></li> <li><a href="/wiki/Chicken_(game)" title="Chicken (game)">Chicken</a></li> <li><a href="/wiki/Centipede_game" title="Centipede game">Centipede game</a></li> <li><a href="/wiki/Lewis_signaling_game" title="Lewis signaling game">Lewis signaling game</a></li> <li><a href="/wiki/Volunteer%27s_dilemma" title="Volunteer's dilemma">Volunteer's dilemma</a></li> <li><a href="/wiki/Dollar_auction" title="Dollar auction">Dollar auction</a></li> <li><a href="/wiki/Battle_of_the_sexes_(game_theory)" title="Battle of the sexes (game theory)">Battle of the sexes</a></li> <li><a href="/wiki/Stag_hunt" title="Stag hunt">Stag hunt</a></li> <li><a href="/wiki/Matching_pennies" title="Matching pennies">Matching pennies</a></li> <li><a href="/wiki/Ultimatum_game" title="Ultimatum game">Ultimatum game</a></li> <li><a href="/wiki/Electronic_mail_game" title="Electronic mail game">Electronic mail game</a></li> <li><a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a></li> <li><a href="/wiki/Pirate_game" title="Pirate game">Pirate game</a></li> <li><a href="/wiki/Dictator_game" title="Dictator game">Dictator game</a></li> <li><a href="/wiki/Public_goods_game" title="Public goods game">Public goods game</a></li> <li><a href="/wiki/Blotto_game" title="Blotto game">Blotto game</a></li> <li><a href="/wiki/War_of_attrition_(game)" title="War of attrition (game)">War of attrition</a></li> <li><a href="/wiki/El_Farol_Bar_problem" title="El Farol Bar problem">El Farol Bar problem</a></li> <li><a href="/wiki/Fair_division" title="Fair division">Fair division</a></li> <li><a href="/wiki/Fair_cake-cutting" title="Fair cake-cutting">Fair cake-cutting</a></li> <li><a href="/wiki/Bertrand_competition" title="Bertrand competition">Bertrand competition</a></li> <li><a href="/wiki/Cournot_competition" title="Cournot competition">Cournot competition</a></li> <li><a href="/wiki/Stackelberg_competition" title="Stackelberg competition">Stackelberg competition</a></li> <li><a href="/wiki/Deadlock_(game_theory)" title="Deadlock (game theory)">Deadlock</a></li> <li><a href="/wiki/Unscrupulous_diner%27s_dilemma" title="Unscrupulous diner's dilemma">Diner's dilemma</a></li> <li><a href="/wiki/Guess_2/3_of_the_average" title="Guess 2/3 of the average">Guess 2/3 of the average</a></li> <li><a href="/wiki/Kuhn_poker" title="Kuhn poker">Kuhn poker</a></li> <li><a href="/wiki/Bargaining_problem" class="mw-redirect" title="Bargaining problem">Nash bargaining game</a></li> <li><a href="/wiki/Induction_puzzles" title="Induction puzzles">Induction puzzles</a></li> <li><a href="/wiki/Dictator_game#Trust_game" title="Dictator game">Trust game</a></li> <li><a href="/wiki/Princess_and_monster_game" title="Princess and monster game">Princess and monster game</a></li> <li><a href="/wiki/Rendezvous_problem" title="Rendezvous problem">Rendezvous problem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aumann%27s_agreement_theorem" title="Aumann's agreement theorem">Aumann's agreement theorem</a></li> <li><a href="/wiki/Folk_theorem_(game_theory)" title="Folk theorem (game theory)">Folk theorem</a></li> <li><a href="/wiki/Minimax" title="Minimax">Minimax theorem</a></li> <li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash's theorem</a></li> <li><a href="/wiki/Negamax" title="Negamax">Negamax theorem</a></li> <li><a href="/wiki/Purification_theorem" title="Purification theorem">Purification theorem</a></li> <li><a href="/wiki/Revelation_principle" title="Revelation principle">Revelation principle</a></li> <li><a href="/wiki/Sprague%E2%80%93Grundy_theorem" title="Sprague–Grundy theorem">Sprague–Grundy theorem</a></li> <li><a href="/wiki/Zermelo%27s_theorem_(game_theory)" title="Zermelo's theorem (game theory)">Zermelo's theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key<br />figures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a></li> <li><a href="/wiki/Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li> <li><a href="/wiki/Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Antoine Augustin Cournot</a></li> <li><a href="/wiki/Ariel_Rubinstein" title="Ariel Rubinstein">Ariel Rubinstein</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a></li> <li><a href="/wiki/David_K._Levine" title="David K. Levine">David K. Levine</a></li> <li><a href="/wiki/David_M._Kreps" title="David M. Kreps">David M. Kreps</a></li> <li><a href="/wiki/Donald_B._Gillies" title="Donald B. Gillies">Donald B. Gillies</a></li> <li><a href="/wiki/Drew_Fudenberg" title="Drew Fudenberg">Drew Fudenberg</a></li> <li><a href="/wiki/Eric_Maskin" title="Eric Maskin">Eric Maskin</a></li> <li><a href="/wiki/Harold_W._Kuhn" title="Harold W. Kuhn">Harold W. Kuhn</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a></li> <li><a href="/wiki/Herv%C3%A9_Moulin" title="Hervé Moulin">Hervé Moulin</a></li> <li><a href="/wiki/John_Conway" class="mw-redirect" title="John Conway">John Conway</a></li> <li><a href="/wiki/Jean_Tirole" title="Jean Tirole">Jean Tirole</a></li> <li><a href="/wiki/Jean-Fran%C3%A7ois_Mertens" title="Jean-François Mertens">Jean-François Mertens</a></li> <li><a href="/wiki/Jennifer_Tour_Chayes" title="Jennifer Tour Chayes">Jennifer Tour Chayes</a></li> <li><a href="/wiki/John_Harsanyi" title="John Harsanyi">John Harsanyi</a></li> <li><a href="/wiki/John_Maynard_Smith" title="John Maynard Smith">John Maynard Smith</a></li> <li><a href="/wiki/John_Forbes_Nash_Jr." title="John Forbes Nash Jr.">John Nash</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a></li> <li><a href="/wiki/Kenneth_Binmore" title="Kenneth Binmore">Kenneth Binmore</a></li> <li><a href="/wiki/Leonid_Hurwicz" title="Leonid Hurwicz">Leonid Hurwicz</a></li> <li><a href="/wiki/Lloyd_Shapley" title="Lloyd Shapley">Lloyd Shapley</a></li> <li><a href="/wiki/Melvin_Dresher" title="Melvin Dresher">Melvin Dresher</a></li> <li><a href="/wiki/Merrill_M._Flood" title="Merrill M. Flood">Merrill M. Flood</a></li> <li><a href="/wiki/Olga_Bondareva" title="Olga Bondareva">Olga Bondareva</a></li> <li><a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a></li> <li><a href="/wiki/Paul_Milgrom" title="Paul Milgrom">Paul Milgrom</a></li> <li><a href="/wiki/Peyton_Young" title="Peyton Young">Peyton Young</a></li> <li><a href="/wiki/Reinhard_Selten" title="Reinhard Selten">Reinhard Selten</a></li> <li><a href="/wiki/Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li> <li><a href="/wiki/Robert_Aumann" title="Robert Aumann">Robert Aumann</a></li> <li><a href="/wiki/Robert_B._Wilson" title="Robert B. Wilson">Robert B. Wilson</a></li> <li><a href="/wiki/Roger_Myerson" title="Roger Myerson">Roger Myerson</a></li> <li><a href="/wiki/Samuel_Bowles_(economist)" title="Samuel Bowles (economist)"> Samuel Bowles</a></li> <li><a href="/wiki/Suzanne_Scotchmer" title="Suzanne Scotchmer">Suzanne Scotchmer</a></li> <li><a href="/wiki/Thomas_Schelling" title="Thomas Schelling">Thomas Schelling</a></li> <li><a href="/wiki/William_Vickrey" title="William Vickrey">William Vickrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Search optimizations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li> <li><a href="/wiki/Aspiration_window" title="Aspiration window">Aspiration window</a></li> <li><a href="/wiki/Principal_variation_search" title="Principal variation search">Principal variation search</a></li> <li><a href="/wiki/Max%5En_algorithm" title="Max^n algorithm">max^n algorithm</a></li> <li><a href="/wiki/Paranoid_algorithm" title="Paranoid algorithm">Paranoid algorithm</a></li> <li><a href="/wiki/Lazy_SMP" title="Lazy SMP">Lazy SMP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li> <li><a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a></li> <li><a href="/wiki/Confrontation_analysis" title="Confrontation analysis">Confrontation analysis</a></li> <li><a href="/wiki/Coopetition" title="Coopetition">Coopetition</a></li> <li><a href="/wiki/Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a></li> <li><a href="/wiki/Glossary_of_game_theory" title="Glossary of game theory">Glossary of game theory</a></li> <li><a href="/wiki/List_of_game_theorists" title="List of game theorists">List of game theorists</a></li> <li><a href="/wiki/List_of_games_in_game_theory" title="List of games in game theory">List of games in game theory</a></li> <li><a href="/wiki/No-win_situation" title="No-win situation">No-win situation</a></li> <li><a href="/wiki/Topological_game" title="Topological game">Topological game</a></li> <li><a href="/wiki/Tragedy_of_the_commons" title="Tragedy of the commons">Tragedy of the commons</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link 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