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class="vector-toc-numb">3</span> <span>Mixed strategy</span> </div> </a> <button aria-controls="toc-Mixed_strategy-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 Mixed strategy subsection</span> </button> <ul id="toc-Mixed_strategy-sublist" class="vector-toc-list"> <li id="toc-Illustration" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Illustration"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Illustration</span> </div> </a> <ul id="toc-Illustration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Significance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Significance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Significance</span> </div> </a> <ul id="toc-Significance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Interpretations_of_mixed_strategies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Interpretations_of_mixed_strategies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Interpretations of mixed strategies</span> </div> </a> <ul id="toc-Interpretations_of_mixed_strategies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Behavior_strategy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Behavior_strategy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Behavior strategy</span> </div> </a> <button aria-controls="toc-Behavior_strategy-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 Behavior strategy subsection</span> </button> <ul id="toc-Behavior_strategy-sublist" class="vector-toc-list"> <li id="toc-Outcome_equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Outcome_equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Outcome equivalence</span> </div> </a> <ul id="toc-Outcome_equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strategy_with_perfect_recall" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strategy_with_perfect_recall"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Strategy with perfect recall</span> </div> </a> <ul id="toc-Strategy_with_perfect_recall-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul 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href="https://de.wikipedia.org/wiki/Strategie_(Spieltheorie)" title="Strategie (Spieltheorie) – German" lang="de" hreflang="de" data-title="Strategie (Spieltheorie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Estrategia_(teor%C3%ADa_de_juegos)" title="Estrategia (teoría de juegos) – Spanish" lang="es" hreflang="es" data-title="Estrategia (teoría de juegos)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%B3%D8%AA%D8%B1%D8%A7%D8%AA%DA%98%DB%8C_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%A8%D8%A7%D8%B2%DB%8C)" title="استراتژی (نظریه بازی) – Persian" lang="fa" hreflang="fa" data-title="استراتژی (نظریه بازی)" 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/Strat%C3%A9gie_(th%C3%A9orie_des_jeux)" title="Stratégie (théorie des jeux) – French" lang="fr" hreflang="fr" data-title="Stratégie (théorie des jeux)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EB%9E%B5_(%EA%B2%8C%EC%9E%84_%EC%9D%B4%EB%A1%A0)" title="전략 (게임 이론) – Korean" lang="ko" hreflang="ko" data-title="전략 (게임 이론)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Strategia_(teoria_dei_giochi)" title="Strategia (teoria dei giochi) – Italian" lang="it" hreflang="it" data-title="Strategia (teoria dei giochi)" 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%AA%D7%9B%D7%A1%D7%99%D7%A1_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A9%D7%97%D7%A7%D7%99%D7%9D)" title="תכסיס (תורת המשחקים) – Hebrew" lang="he" hreflang="he" data-title="תכסיס (תורת המשחקים)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Strategia_(teoria_gier)" title="Strategia (teoria gier) – Polish" lang="pl" hreflang="pl" data-title="Strategia (teoria gier)" 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/Estrat%C3%A9gia_(teoria_dos_jogos)" title="Estratégia (teoria dos jogos) – Portuguese" lang="pt" hreflang="pt" data-title="Estratégia (teoria dos jogos)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D0%B0%D1%82%D0%B5%D0%B3%D0%B8%D1%8F_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B8%D0%B3%D1%80)" title="Стратегия (теория игр) – Russian" lang="ru" hreflang="ru" data-title="Стратегия (теория игр)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D0%B0%D1%82%D0%B5%D0%B3%D0%B8%D1%98%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B8%D0%B3%D0%B0%D1%80%D0%B0)" title="Стратегија (теорија игара) – Serbian" lang="sr" hreflang="sr" data-title="Стратегија (теорија игара)" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D1%82%D1%80%D0%B0%D1%82%D0%B5%D0%B3%D1%96%D1%8F_(%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D1%96%D0%B3%D0%BE%D1%80)" title="Стратегія (теорія ігор) – Ukrainian" lang="uk" hreflang="uk" data-title="Стратегія (теорія ігор)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Complete plan on how a game player will behave in every possible game situation</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .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 of "Strategy", see <a href="/wiki/Strategy_(disambiguation)" class="mw-disambig" title="Strategy (disambiguation)">Strategy (disambiguation)</a>.</div> <p>In <a href="/wiki/Game_theory" title="Game theory">game theory</a>, a <b>move</b>, <b>action</b>, or <b>play</b> is any one of the options which a player can choose in a setting where the optimal outcome depends <i>not only</i> on their own actions <i>but</i> on the actions of others.<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> The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The term <b>strategy</b> is typically used to mean a complete <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for playing a game, telling a player what to do for every possible situation. A player's strategy determines the action the player will take at any stage of the game. However, the idea of a strategy is often confused or <a href="/wiki/Conflate" class="mw-redirect" title="Conflate">conflated</a> with that of a move or action, because of the correspondence between moves and <a href="/wiki/Pure_strategies" class="mw-redirect" title="Pure strategies">pure strategies</a> in <a href="/wiki/Normal-form_game" title="Normal-form game">most games</a>: for any move <i>X</i>, "always play move <i>X</i>" is an example of a valid strategy, and as a result every move can also be considered to be a strategy. Other authors treat strategies as being a different type of thing from actions, and therefore distinct. </p><p>It is helpful to think about a "strategy" as a list of directions, and a "move" as a single turn on the list of directions itself. This strategy is based on the payoff or outcome of each action. The goal of each agent is to consider their payoff based on a competitors action. For example, competitor A can assume competitor B enters the market. From there, Competitor A compares the payoffs they receive by entering and not entering. The next step is to assume Competitor B does not enter and then consider which payoff is better based on if Competitor A chooses to enter or not enter. This technique can identify dominant strategies where a player can identify an action that they can take no matter what the competitor does to try to maximize the payoff. </p><p>A <b>strategy profile</b> (sometimes called a <b>strategy combination</b>) is a set of strategies for all players which fully specifies all actions in a game. A strategy profile must include one and only one strategy for every player. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Strategy_set">Strategy set</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=1" title="Edit section: Strategy set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A player's <b>strategy set</b> defines what strategies are available for them to play. </p><p>A player has a <b>finite</b> strategy set if they have a number of discrete strategies available to them. For instance, a game of <a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">rock paper scissors</a> comprises a single move by each player—and each player's move is made without knowledge of the other's, not as a response—so each player has the finite strategy set {rock paper scissors}. </p><p>A strategy set is infinite otherwise. For instance the <a href="/wiki/Fair_division" title="Fair division">cake cutting game</a> has a bounded continuum of strategies in the strategy set {Cut anywhere between zero percent and 100 percent of the cake}. </p><p>In a <a href="/wiki/Dynamic_game" class="mw-redirect" title="Dynamic game">dynamic game</a>, games that are played over a series of time, the strategy set consists of the possible rules a player could give to a <a href="/wiki/Robot" title="Robot">robot</a> or <a href="/wiki/Software_agent" title="Software agent">agent</a> on how to play the game. For instance, in the <a href="/wiki/Ultimatum_game" title="Ultimatum game">ultimatum game</a>, the strategy set for the second player would consist of every possible rule for which offers to accept and which to reject. </p><p>In a <a href="/wiki/Bayesian_game" title="Bayesian game">Bayesian game</a>, or games in which players have incomplete information about one another, the strategy set is similar to that in a dynamic game. It consists of rules for what action to take for any possible private information. </p> <div class="mw-heading mw-heading3"><h3 id="Choosing_a_strategy_set">Choosing a strategy set</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=2" title="Edit section: Choosing a strategy set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In applied game theory, the definition of the strategy sets is an important part of the art of making a game simultaneously solvable and meaningful. The game theorist can use knowledge of the overall problem, that is the friction between two or more players, to limit the strategy spaces, and ease the solution. </p><p>For instance, strictly speaking in the Ultimatum game a player can have strategies such as: <i>Reject offers of ($1, $3, $5, ..., $19), accept offers of ($0, $2, $4, ..., $20)</i>. Including all such strategies makes for a very large strategy space and a somewhat difficult problem. A game theorist might instead believe they can limit the strategy set to: {Reject any offer ≤ <i>x</i>, accept any offer > <i>x</i>; for <i>x</i> in ($0, $1, $2, ..., $20)}. </p> <div class="mw-heading mw-heading2"><h2 id="Pure_and_mixed_strategies">Pure and mixed strategies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=3" title="Edit section: Pure and mixed strategies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>pure strategy</b> provides a complete definition of how a player will play a game. Pure strategy can be thought about as a singular concrete plan subject to the observations they make during the course of the game of play. In particular, it determines the move a player will make for any situation they could face. A player's <b>strategy set</b> is the set of pure strategies available to that player. </p><p>A <b>mixed strategy</b> is an assignment of a <a href="/wiki/Probability" title="Probability">probability</a> to each pure strategy. When enlisting mixed strategy, it is often because the game does not allow for a rational description in specifying a pure strategy for the game. This allows for a player to randomly select a pure strategy. (See the following section for an illustration.) Since probabilities are continuous, there are infinitely many mixed strategies available to a player. Since probabilities are being assigned to strategies for a specific player when discussing the payoffs of certain scenarios the payoff must be referred to as "expected payoff". </p><p>Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is selected with probability <i>1</i> and every other strategy with probability <i>0</i>. </p><p>A <b>totally mixed strategy</b> is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy. (Totally mixed strategies are important for <a href="/wiki/Equilibrium_refinement" class="mw-redirect" title="Equilibrium refinement">equilibrium refinement</a> such as <a href="/wiki/Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">trembling hand perfect equilibrium</a>.) </p> <div class="mw-heading mw-heading2"><h2 id="Mixed_strategy">Mixed strategy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=4" title="Edit section: Mixed strategy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Illustration">Illustration</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=5" title="Edit section: Illustration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a soccer penalty kick, the kicker must choose whether to kick to the right or left side of the goal, and simultaneously the goalie must decide which way to block it. Also, the kicker has a direction they are best at shooting, which is left if they are right-footed. The matrix for the soccer game illustrates this situation, a simplified form of the game studied by Chiappori, Levitt, and Groseclose (2002).<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> It assumes that if the goalie guesses correctly, the kick is blocked, which is set to the base payoff of 0 for both players. If the goalie guesses wrong, the kick is more likely to go in if it is to the left (payoffs of +2 for the kicker and -2 for the goalie) than if it is to the right (the lower payoff of +1 to kicker and -1 to goalie). </p> <table class="wikitable" style="background:white;color:maroon;text-align:center; float:right;"> <tbody><tr> <td style="background:white; border:1px solid white;" colspan="2" rowspan="2"></td> <td style="background:White; color:black; border:1px solid white; font-weight:bold;" align="center" colspan="2">Goalie </td></tr> <tr> <td style="color:cadetblue;">Lean Left</td> <td style="color:cadetblue;">Lean Right </td></tr> <tr> <td style="background:white; color:black; border:1px solid white; font-weight:bold;" valign="center" rowspan="2">Kicker</td> <td style="color:cadetblue;" align="left">Kick Left</td> <td> 0,  0</td> <td>+2, -2 </td></tr> <tr> <td style="color:cadetblue;" align="left">Kick Right</td> <td>+1, -1</td> <td> 0,  0 </td></tr> <tr> <td style="background:white; border:1px solid white;" colspan="4"> </td></tr> <tr> <td style="background:white; border:1px solid white;" colspan="4"> </td></tr> <tr> <td style="background:white; border:1px solid white;" align="left" colspan="4"> Payoff for the Soccer Game (Kicker, Goalie) </td></tr> </tbody></table> <p>This game has no pure-strategy equilibrium, because one player or the other would deviate from any profile of strategies—for example, (Left, Left) is not an equilibrium because the Kicker would deviate to Right and increase his payoff from 0 to 1. </p><p>The kicker's mixed-strategy equilibrium is found from the fact that they will deviate from randomizing unless their payoffs from Left Kick and Right Kick are exactly equal. If the goalie leans left with probability g, the kicker's expected payoff from Kick Left is g(0) + (1-g)(2), and from Kick Right is g(1) + (1-g)(0). Equating these yields g= 2/3. Similarly, the goalie is willing to randomize only if the kicker chooses mixed strategy probability k such that Lean Left's payoff of k(0) + (1-k)(-1) equals Lean Right's payoff of k(-2) + (1-k)(0), so k = 1/3. Thus, the mixed-strategy equilibrium is (Prob(Kick Left) = 1/3, Prob(Lean Left) = 2/3). </p><p>In equilibrium, the kicker kicks to their best side only 1/3 of the time. That is because the goalie is guarding that side more. Also, in equilibrium, the kicker is indifferent which way they kick, but for it to be an equilibrium they must choose exactly 1/3 probability. </p><p>Chiappori, Levitt, and Groseclose try to measure how important it is for the kicker to kick to their favored side, add center kicks, etc., and look at how professional players actually behave. They find that they do randomize, and that kickers kick to their favored side 45% of the time and goalies lean to that side 57% of the time. Their article is well-known as an example of how people in real life use mixed strategies. </p> <div class="mw-heading mw-heading3"><h3 id="Significance">Significance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=6" title="Edit section: Significance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In his famous paper, <a href="/wiki/John_Forbes_Nash" class="mw-redirect" title="John Forbes Nash">John Forbes Nash</a> proved that there is an <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">equilibrium</a> for every finite game. One can divide Nash equilibria into two types. <i>Pure strategy Nash equilibria</i> are Nash equilibria where all players are playing pure strategies. <i>Mixed strategy Nash equilibria</i> are equilibria where at least one player is playing a mixed strategy. While Nash proved that every finite game has a Nash equilibrium, not all have pure strategy Nash equilibria. For an example of a game that does not have a Nash equilibrium in pure strategies, see <a href="/wiki/Matching_pennies" title="Matching pennies">Matching pennies</a>. However, many games do have pure strategy Nash equilibria (e.g. the <a href="/wiki/Coordination_game" title="Coordination game">Coordination game</a>, the <a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a>, the <a href="/wiki/Stag_hunt" title="Stag hunt">Stag hunt</a>). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both players play either strategy with probability 1/2. </p> <div class="mw-heading mw-heading3"><h3 id="Interpretations_of_mixed_strategies">Interpretations of mixed strategies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=7" title="Edit section: Interpretations of mixed strategies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic", since they are weak Nash equilibria, and a player is indifferent about whether to follow their equilibrium strategy probability or deviate to some other probability.<sup id="cite_ref-Aumann1985_4-0" class="reference"><a href="#cite_note-Aumann1985-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-Rubinstein1991_5-0" class="reference"><a href="#cite_note-Rubinstein1991-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Game theorist <a href="/wiki/Ariel_Rubinstein" title="Ariel Rubinstein">Ariel Rubinstein</a> describes alternative ways of understanding the concept. The first, due to Harsanyi (1973),<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> is called <i><a href="/wiki/Purification_theorem" title="Purification theorem">purification</a></i>, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogenous factors.<sup id="cite_ref-Rubinstein1991_5-1" class="reference"><a href="#cite_note-Rubinstein1991-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents. </p><p>Later, Aumann and Brandenburger (1995),<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> re-interpreted Nash equilibrium as an equilibrium in <i>beliefs</i>, rather than actions. For instance, in <a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">rock paper scissors</a> an equilibrium in beliefs would have each player <i>believing</i> the other was equally likely to play each strategy. This interpretation weakens the descriptive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to <i>actually</i> play a pure strategy of Rock in each play of the game, even though over time the probabilities are those of the mixed strategy. </p> <div class="mw-heading mw-heading2"><h2 id="Behavior_strategy">Behavior strategy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=8" title="Edit section: Behavior strategy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While a mixed strategy assigns a probability distribution over pure strategies, a <b>behavior strategy</b> assigns at each information set a probability distribution over the set of possible actions. While the two concepts are very closely related in the context of normal form games, they have very different implications for extensive form games. Roughly, a mixed strategy randomly chooses a deterministic path through the <a href="/wiki/Game_tree" title="Game tree">game tree</a>, while a behavior strategy can be seen as a stochastic path. The relationship between mixed and behavior strategies is the subject of <a href="/wiki/Kuhn%27s_theorem" title="Kuhn's theorem">Kuhn's theorem</a>, a behavioral outlook on traditional game-theoretic hypotheses. The result establishes that in any finite extensive-form game with perfect recall, for any player and any mixed strategy, there exists a behavior strategy that, against all profiles of strategies (of other players), induces the same distribution over terminal nodes as the mixed strategy does. The converse is also true. </p><p>A famous example of why perfect recall is required for the equivalence is given by Piccione and Rubinstein (1997)<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources#What_information_to_include" title="Wikipedia:Citing sources"><span title="A complete citation is needed. (September 2018)">full citation needed</span></a></i>]</sup> with their <i>Absent-Minded Driver</i> game. </p> <div class="mw-heading mw-heading3"><h3 id="Outcome_equivalence">Outcome equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=9" title="Edit section: Outcome equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Outcome equivalence combines the mixed and behavioral strategy of Player i in relation to the pure strategy of Player i’s opponent. Outcome equivalence is defined as the situation in which, for any mixed and behavioral strategy that Player i takes, in response to any pure strategy that Player I’s opponent plays, the outcome distribution of the mixed and behavioral strategy must be equal. This equivalence can be described by the following formula: (Q^(U(i), S(-i)))(z) = (Q^(β(i), S(-i)))(z), where U(i) describes Player i's mixed strategy, β(i) describes Player i's behavioral strategy, and S(-i) is the opponent's strategy.<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> <div class="mw-heading mw-heading3"><h3 id="Strategy_with_perfect_recall">Strategy with perfect recall</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=10" title="Edit section: Strategy with perfect recall"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Perfect recall is defined as the ability of every player in game to remember and recall all past actions within the game. Perfect recall is required for equivalence as, in finite games with imperfect recall, there will be existing mixed strategies of Player I in which there is no equivalent behavior strategy. This is fully described in the <i>Absent-Minded Driver</i> game formulated by Piccione and Rubinstein. In short, this game is based on the decision-making of a driver with imperfect recall, who needs to take the second exit off the highway to reach home but does not remember which intersection they are at when they reach it. Figure [2] describes this game. </p><p>Without perfect information (i.e. imperfect information), players make a choice at each decision node without knowledge of the decisions that have preceded it. Therefore, a player’s mixed strategy can produce outcomes that their behavioral strategy cannot, and vice versa. This is demonstrated in the <i>Absent-minded Driver</i> game. With perfect recall and information, the driver has a single pure strategy, which is [continue, exit], as the driver is aware of what intersection (or decision node) they are at when they arrive to it. On the other hand, looking at the planning-optimal stage only, the maximum payoff is achieved by continuing at both intersections, maximized at p=2/3 (reference). This simple one player game demonstrates the importance of perfect recall for outcome equivalence, and its impact on normal and extended form games.<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-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strategy_(game_theory)&action=edit&section=11" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a></li> <li><a href="/wiki/Haven_(graph_theory)" title="Haven (graph theory)">Haven (graph theory)</a></li> <li><a href="/wiki/Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">Evolutionarily stable strategy</a></li></ul> <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=Strategy_(game_theory)&action=edit&section=12" title="Edit section: References"><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"> <div class="mw-references-wrap"><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"><a href="/wiki/Ben_Polak" title="Ben Polak">Ben Polak</a> <i>Game Theory: Lecture 1 Transcript</i> ECON 159, 5 September 2007, <a href="/wiki/Open_Yale_Courses" title="Open Yale Courses">Open Yale Courses</a>.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</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 id="CITEREFAumann2017" class="citation book cs1"><a href="/wiki/Robert_Aumann" title="Robert Aumann">Aumann, R.</a> (22 March 2017). <i>Game Theory. In: Palgrave Macmillan</i>. London: Palgrave Macmillan. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-349-95121-5" title="Special:BookSources/978-1-349-95121-5"><bdi>978-1-349-95121-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Game+Theory.+In%3A+Palgrave+Macmillan&rft.place=London&rft.pub=Palgrave+Macmillan&rft.date=2017-03-22&rft.isbn=978-1-349-95121-5&rft.aulast=Aumann&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrategy+%28game+theory%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChiapporiLevittGroseclose2002" class="citation journal cs1">Chiappori, P. -A.; Levitt, S.; Groseclose, T. 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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 class="mw-selflink selflink">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 class="mw-selflink selflink">Pure strategy</a></li> <li><a class="mw-selflink-fragment" href="#Mixed_strategy">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 href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">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></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q1546627#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4182285-7">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="strategie her"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph135592&CON_LNG=ENG">Czech Republic</a></span></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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