CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Minimax - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"754a1166-5d9e-4d93-9c32-c597afe9b878","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Minimax","wgTitle":"Minimax","wgCurRevisionId":1250833808,"wgRevisionId":1250833808,"wgArticleId":19589,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description matches Wikidata","All articles with failed verification","Articles with failed verification from February 2015","Articles with example pseudocode","Detection theory","Game artificial intelligence","Graph algorithms","Optimization algorithms and methods","Search algorithms","Game theory","Theorems in discrete mathematics","Decision theory","Fixed points (mathematics)"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel": "wikitext","wgRelevantPageName":"Minimax","wgRelevantArticleId":19589,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q751319","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model", "platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips", "ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.6"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Minimax - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Minimax"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Minimax&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Minimax"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Minimax rootpage-Minimax skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Minimax" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Minimax" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Minimax" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Minimax" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Game_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Game_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Game theory</span> </div> </a> <button aria-controls="toc-Game_theory-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 Game theory subsection</span> </button> <ul id="toc-Game_theory-sublist" class="vector-toc-list"> <li id="toc-In_general_games" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_general_games"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>In general games</span> </div> </a> <ul id="toc-In_general_games-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_zero-sum_games" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_zero-sum_games"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>In zero-sum games</span> </div> </a> <ul id="toc-In_zero-sum_games-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Example</span> </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximin"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Maximin</span> </div> </a> <ul id="toc-Maximin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_repeated_games" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_repeated_games"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>In repeated games</span> </div> </a> <ul id="toc-In_repeated_games-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Combinatorial_game_theory" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Combinatorial_game_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Combinatorial game theory</span> </div> </a> <button aria-controls="toc-Combinatorial_game_theory-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 Combinatorial game theory subsection</span> </button> <ul id="toc-Combinatorial_game_theory-sublist" class="vector-toc-list"> <li id="toc-Minimax_algorithm_with_alternate_moves" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimax_algorithm_with_alternate_moves"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Minimax algorithm with alternate moves</span> </div> </a> <ul id="toc-Minimax_algorithm_with_alternate_moves-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudocode" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudocode"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Pseudocode</span> </div> </a> <ul id="toc-Pseudocode-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Example</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Minimax_for_individual_decisions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Minimax_for_individual_decisions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Minimax for individual decisions</span> </div> </a> <button aria-controls="toc-Minimax_for_individual_decisions-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 Minimax for individual decisions subsection</span> </button> <ul id="toc-Minimax_for_individual_decisions-sublist" class="vector-toc-list"> <li id="toc-Minimax_in_the_face_of_uncertainty" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimax_in_the_face_of_uncertainty"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Minimax in the face of uncertainty</span> </div> </a> <ul id="toc-Minimax_in_the_face_of_uncertainty-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimax_criterion_in_statistical_decision_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimax_criterion_in_statistical_decision_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Minimax criterion in statistical decision theory</span> </div> </a> <ul id="toc-Minimax_criterion_in_statistical_decision_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-probabilistic_decision_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-probabilistic_decision_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Non-probabilistic decision theory</span> </div> </a> <ul id="toc-Non-probabilistic_decision_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Minimax_in_politics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Minimax_in_politics"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Minimax in politics</span> </div> </a> <ul id="toc-Minimax_in_politics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximin_in_philosophy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Maximin_in_philosophy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Maximin in philosophy</span> </div> </a> <ul id="toc-Maximin_in_philosophy-sublist" class="vector-toc-list"> </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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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">Minimax</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 28 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-28" 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">28 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%D9%8A%D9%86%D9%8A_%D9%85%D8%A7%D9%83%D8%B3" title="ميني ماكس – Arabic" lang="ar" hreflang="ar" data-title="ميني ماكس" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Minimax" title="Minimax – Catalan" lang="ca" hreflang="ca" data-title="Minimax" 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/Minimax_(algoritmus)" title="Minimax (algoritmus) – Czech" lang="cs" hreflang="cs" data-title="Minimax (algoritmus)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Minimax-Algorithmus" title="Minimax-Algorithmus – German" lang="de" hreflang="de" data-title="Minimax-Algorithmus" 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/Minimax" title="Minimax – Spanish" lang="es" hreflang="es" data-title="Minimax" 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/%D9%85%DB%8C%D9%86%DB%8C%D9%85%D8%A7%DA%A9%D8%B3" title="مینیماکس – Persian" lang="fa" hreflang="fa" data-title="مینیماکس" 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/Algorithme_minimax" title="Algorithme minimax – French" lang="fr" hreflang="fr" data-title="Algorithme minimax" 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%B5%9C%EC%86%8C%EA%B7%B9%EB%8C%80%ED%99%94" title="최소극대화 – Korean" lang="ko" hreflang="ko" data-title="최소극대화" 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/%D5%84%D5%AB%D5%B6%D5%AB%D5%B4%D5%A1%D6%84%D5%BD" title="Մինիմաքս – Armenian" lang="hy" hreflang="hy" data-title="Մինիմաքս" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Minimax" title="Minimax – Indonesian" lang="id" hreflang="id" data-title="Minimax" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Minimax" title="Minimax – Italian" lang="it" hreflang="it" data-title="Minimax" 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%A2%D7%A5_%D7%9E%D7%99%D7%A0%D7%99%D7%9E%D7%A7%D7%A1" title="עץ מינימקס – Hebrew" lang="he" hreflang="he" data-title="עץ מינימקס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Minimax" title="Minimax – Javanese" lang="jv" hreflang="jv" data-title="Minimax" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Minimax_elv" title="Minimax elv – Hungarian" lang="hu" hreflang="hu" data-title="Minimax elv" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Minimax" title="Minimax – Dutch" lang="nl" hreflang="nl" data-title="Minimax" 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/%E3%83%9F%E3%83%8B%E3%83%9E%E3%83%83%E3%82%AF%E3%82%B9%E6%B3%95" title="ミニマックス法 – Japanese" lang="ja" hreflang="ja" data-title="ミニマックス法" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algorytm_min-max" title="Algorytm min-max – Polish" lang="pl" hreflang="pl" data-title="Algorytm min-max" 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/Minimax" title="Minimax – Portuguese" lang="pt" hreflang="pt" data-title="Minimax" 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/Minimax" title="Minimax – Romanian" lang="ro" hreflang="ro" data-title="Minimax" 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%9C%D0%B8%D0%BD%D0%B8%D0%BC%D0%B0%D0%BA%D1%81" title="Минимакс – Russian" lang="ru" hreflang="ru" data-title="Минимакс" 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/Minimax" title="Minimax – Simple English" lang="en-simple" hreflang="en-simple" data-title="Minimax" 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/Minimax_(algoritmus)" title="Minimax (algoritmus) – Slovak" lang="sk" hreflang="sk" data-title="Minimax (algoritmus)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B8%D0%BD%D0%B8%D0%BC%D0%B0%D0%BA%D1%81_(%D0%B0%D0%BB%D0%B3%D0%BE%D1%80%D0%B8%D1%82%D0%B0%D0%BC)" title="Минимакс (алгоритам) – Serbian" lang="sr" hreflang="sr" data-title="Минимакс (алгоритам)" 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-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%82%E0%B8%B1%E0%B9%89%E0%B8%99%E0%B8%95%E0%B8%AD%E0%B8%99%E0%B8%A7%E0%B8%B4%E0%B8%98%E0%B8%B5%E0%B8%A1%E0%B8%B4%E0%B8%99%E0%B8%B4%E0%B9%81%E0%B8%A1%E0%B8%81%E0%B8%8B%E0%B9%8C" title="ขั้นตอนวิธีมินิแมกซ์ – Thai" lang="th" hreflang="th" data-title="ขั้นตอนวิธีมินิแมกซ์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D1%96%D0%BD%D1%96%D0%BC%D0%B0%D0%BA%D1%81" title="Мінімакс – Ukrainian" lang="uk" hreflang="uk" data-title="Мінімакс" 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 href="https://vi.wikipedia.org/wiki/Minimax" title="Minimax – Vietnamese" lang="vi" hreflang="vi" data-title="Minimax" 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/%E6%9C%80%E5%A4%A7%E6%9C%80%E5%B0%8F%E5%8C%96" title="最大最小化 – Cantonese" lang="yue" hreflang="yue" data-title="最大最小化" 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/%E6%9E%81%E5%B0%8F%E5%8C%96%E6%9E%81%E5%A4%A7%E7%AE%97%E6%B3%95" title="极小化极大算法 – Chinese" lang="zh" hreflang="zh" data-title="极小化极大算法" 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/Q751319#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/Minimax" 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 href="/wiki/Talk:Minimax" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Minimax"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Minimax&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Minimax&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Minimax"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Minimax&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Minimax&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Minimax" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Minimax" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Minimax&oldid=1250833808" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Minimax&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Minimax&id=1250833808&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMinimax"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrKodu&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FMinimax"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Minimax&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Minimax&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://en.wikiquote.org/wiki/Minimax" hreflang="en"><span>Wikiquote</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q751319" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div 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">Decision rule used for minimizing the possible loss for a worst case scenario</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">This article is about the decision theory concept. For other uses, see <a href="/wiki/Minimax_(disambiguation)" class="mw-disambig" title="Minimax (disambiguation)">Minimax (disambiguation)</a>.</div> <p><b>Minimax</b> (sometimes <b>Minmax</b>, <b>MM</b><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> or <b>saddle point</b><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>) is a decision rule used in <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a>, <a href="/wiki/Game_theory" title="Game theory">game theory</a>, <a href="/wiki/Statistics" title="Statistics">statistics</a>, and <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> for <i>minimizing</i> the possible <a href="/wiki/Loss_function" title="Loss function">loss</a> for a <a href="/wiki/Worst-case_scenario" title="Worst-case scenario">worst case (<i>max</i>imum loss) scenario</a>. When dealing with gains, it is referred to as "maximin" – to maximize the minimum gain. Originally formulated for several-player <a href="/wiki/Zero-sum" class="mw-redirect" title="Zero-sum">zero-sum</a> <a href="/wiki/Game_theory" title="Game theory">game theory</a>, covering both the cases where players take alternate moves and those where they make simultaneous moves, it has also been extended to more complex games and to general decision-making in the presence of uncertainty. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Game_theory">Game theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=1" title="Edit section: Game theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="In_general_games">In general games</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=2" title="Edit section: In general games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>maximin value</b> is the highest value that the player can be sure to get without knowing the actions of the other players; equivalently, it is the lowest value the other players can force the player to receive when they know the player's action. Its formal definition is:<sup id="cite_ref-ZMS2013_3-0" class="reference"><a href="#cite_note-ZMS2013-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\underline {v_{i}}}=\max _{a_{i}}\min _{a_{-i}}{v_{i}(a_{i},a_{-i})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>_</mo> </munder> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {v_{i}}}=\max _{a_{i}}\min _{a_{-i}}{v_{i}(a_{i},a_{-i})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d2fe8fe2fc328093c7b0c19e83a0197004a5d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.111ex; height:4.343ex;" alt="{\displaystyle {\underline {v_{i}}}=\max _{a_{i}}\min _{a_{-i}}{v_{i}(a_{i},a_{-i})}}"></span></dd></dl> <p>Where: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">i</span> is the index of the player of interest.</li> <li><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 -i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91fddb9f89a520937db3a8821575068cdcc76f60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.611ex; height:2.343ex;" alt="{\displaystyle -i}"></span> denotes all other players except player <span class="texhtml mvar" style="font-style:italic;">i</span>.</li> <li><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 a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> is the action taken by player <span class="texhtml mvar" style="font-style:italic;">i</span>.</li> <li><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 a_{-i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{-i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ad62ea9270fed5d53c1ec1d7f41177e50d0abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.009ex;" alt="{\displaystyle a_{-i}}"></span> denotes the actions taken by all other players.</li> <li><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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}"></span> is the value function of player <span class="texhtml mvar" style="font-style:italic;">i</span>.</li></ul> <p>Calculating the maximin value of a player is done in a worst-case approach: for each possible action of the player, we check all possible actions of the other players and determine the worst possible combination of actions – the one that gives player <span class="texhtml mvar" style="font-style:italic;">i</span> the smallest value. Then, we determine which action player <span class="texhtml mvar" style="font-style:italic;">i</span> can take in order to make sure that this smallest value is the highest possible. </p><p>For example, consider the following game for two players, where the first player ("row player") may choose any of three moves, labelled <span class="texhtml mvar" style="font-style:italic;">T</span>, <span class="texhtml mvar" style="font-style:italic;">M</span>, or <span class="texhtml mvar" style="font-style:italic;">B</span>, and the second player ("column player") may choose either of two moves, <span class="texhtml mvar" style="font-style:italic;">L</span> or <span class="texhtml mvar" style="font-style:italic;">R</span>. The result of the combination of both moves is expressed in a payoff table: </p> <dl><dd><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 {\begin{array}{c|cc}\hline &L&R\\\hline T&3,1&2,-20\\M&5,0&-10,1\\B&-100,2&4,4\\\hline \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <menclose notation="top bottom"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none none" columnlines="solid none"> <mtr> <mtd /> <mtd> <mi>L</mi> </mtd> <mtd> <mi>R</mi> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> </mtd> <mtd> <mn>3</mn> <mo>,</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> </mtd> <mtd> <mn>5</mn> <mo>,</mo> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> </mtd> <mtd> <mo>−</mo> <mn>100</mn> <mo>,</mo> <mn>2</mn> </mtd> <mtd> <mn>4</mn> <mo>,</mo> <mn>4</mn> </mtd> </mtr> </mtable> </menclose> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{c|cc}\hline &L&R\\\hline T&3,1&2,-20\\M&5,0&-10,1\\B&-100,2&4,4\\\hline \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94218d6211047370d9243ce80081b8860b90c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:23.091ex; height:13.843ex;" alt="{\displaystyle {\begin{array}{c|cc}\hline &L&R\\\hline T&3,1&2,-20\\M&5,0&-10,1\\B&-100,2&4,4\\\hline \end{array}}}"></span></dd></dl> <p>(where the first number in each of the cell is the pay-out of the row player and the second number is the pay-out of the column player). </p><p>For the sake of example, we consider only <a href="/wiki/Strategy_(game_theory)#Pure_and_mixed_strategies" title="Strategy (game theory)">pure strategies</a>. Check each player in turn: </p> <ul><li>The row player can play <span class="texhtml mvar" style="font-style:italic;">T</span>, which guarantees them a payoff of at least <span class="nowrap"><span data-sort-value="7000200000000000000♠"></span>2</span> (playing <span class="texhtml mvar" style="font-style:italic;">B</span> is risky since it can lead to payoff <span class="nowrap"><span data-sort-value="2997900000000000000♠"></span>−100</span>, and playing <span class="texhtml mvar" style="font-style:italic;">M</span> can result in a payoff of <span class="nowrap"><span data-sort-value="2998900000000000000♠"></span>−10</span>). Hence: <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 {\underline {v_{row}}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>w</mi> </mrow> </msub> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {v_{row}}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2506f62ac53a3d64ade893dca7134bd4142a037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.877ex; margin-bottom: -0.795ex; width:8.339ex; height:3.509ex;" alt="{\displaystyle {\underline {v_{row}}}=2}"></span>.</li> <li>The column player can play <span class="texhtml mvar" style="font-style:italic;">L</span> and secure a payoff of at least <span class="nowrap"><span data-sort-value="5000000000000000000♠"></span>0</span> (playing <span class="texhtml mvar" style="font-style:italic;">R</span> puts them in the risk of getting <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 -20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498de4e7b3ddc127b4be006bd2efafed19fa120d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.133ex; height:2.343ex;" alt="{\displaystyle -20}"></span>). Hence: <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 {\underline {v_{col}}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {v_{col}}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ec335c2a2babeb8424432c8f1552b0c8dded90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.878ex; margin-bottom: -0.793ex; width:7.623ex; height:3.509ex;" alt="{\displaystyle {\underline {v_{col}}}=0}"></span>.</li></ul> <p>If both players play their respective maximin strategies <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,L)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (T,L)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20493aa06e4ba84a0057e0fc6a261dc87667ca8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.062ex; height:2.843ex;" alt="{\displaystyle (T,L)}"></span>, the payoff vector is <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 (3,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8933db1c87b5fefc8d54c6e2d157e4b343bb8b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (3,1)}"></span>. </p><p>The <b>minimax value</b> of a player is the smallest value that the other players can force the player to receive, without knowing the player's actions; equivalently, it is the largest value the player can be sure to get when they <i>know</i> the actions of the other players. Its formal definition is:<sup id="cite_ref-ZMS2013_3-1" class="reference"><a href="#cite_note-ZMS2013-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074c806d741e20cc0e770027e6efcb9796b72871" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.224ex; height:4.343ex;" alt="{\displaystyle {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}}"></span></dd></dl> <p>The definition is very similar to that of the maximin value – only the order of the maximum and minimum operators is inverse. In the above example: </p> <ul><li>The row player can get a maximum value of <span class="texhtml mvar" style="font-style:italic;">4</span> (if the other player plays <span class="texhtml mvar" style="font-style:italic;">R</span>) or <span class="nowrap"><span data-sort-value="7000500000000000000♠"></span>5</span> (if the other player plays <span class="texhtml mvar" style="font-style:italic;">L</span>), so: <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 {\overline {v_{row}}}=4\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>w</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>4</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {v_{row}}}=4\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bb013feb5d042115f8d5cc01f598af02e78ed1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.679ex; height:2.676ex;" alt="{\displaystyle {\overline {v_{row}}}=4\ .}"></span></li> <li>The column player can get a maximum value of <span class="texhtml mvar" style="font-style:italic;">1</span> (if the other player plays <span class="texhtml mvar" style="font-style:italic;">T</span>), <span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1</span> (if <span class="texhtml mvar" style="font-style:italic;">M</span>) or <span class="nowrap"><span data-sort-value="7000400000000000000♠"></span>4</span> (if <span class="texhtml mvar" style="font-style:italic;">B</span>). Hence: <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 {\overline {v_{col}}}=1\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mn>1</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {v_{col}}}=1\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/532e4596876de0ce798f69db3dbf268369e6c3ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.963ex; height:2.676ex;" alt="{\displaystyle {\overline {v_{col}}}=1\ .}"></span></li></ul> <p>For every player <span class="texhtml mvar" style="font-style:italic;">i</span>, the maximin is at most the minimax: </p> <dl><dd><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 {\underline {v_{i}}}\leq {\overline {v_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>_</mo> </munder> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {v_{i}}}\leq {\overline {v_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36dd33ead0af78f7bfbc890a8b6561a5203e7289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.877ex; margin-bottom: -0.795ex; width:7.07ex; height:3.676ex;" alt="{\displaystyle {\underline {v_{i}}}\leq {\overline {v_{i}}}}"></span></dd></dl> <p>Intuitively, in maximin the maximization comes after the minimization, so player <span class="texhtml mvar" style="font-style:italic;">i</span> tries to maximize their value before knowing what the others will do; in minimax the maximization comes before the minimization, so player <span class="texhtml mvar" style="font-style:italic;">i</span> is in a much better position – they maximize their value knowing what the others did. </p><p>Another way to understand the <i>notation</i> is by reading from right to left: When we write </p> <dl><dd><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 {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}=\min _{a_{-i}}{\Big (}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}{\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </munder> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">min</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}=\min _{a_{-i}}{\Big (}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}{\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77e920b4e68020c5447fb03276dfc107aa8c3a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.568ex; height:5.343ex;" alt="{\displaystyle {\overline {v_{i}}}=\min _{a_{-i}}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}=\min _{a_{-i}}{\Big (}\max _{a_{i}}{v_{i}(a_{i},a_{-i})}{\Big )}}"></span></dd></dl> <p>the initial set of outcomes <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_{i}(a_{i},a_{-i})\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ v_{i}(a_{i},a_{-i})\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81bb566eb8a83bcae4d5b2a53b4eefcdd71d733b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.269ex; height:2.843ex;" alt="{\displaystyle \ v_{i}(a_{i},a_{-i})\ }"></span> depends on both <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 \ {a_{i}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {a_{i}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cb382086935b075ac6de7671af9d0e8d3e7139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.191ex; height:2.009ex;" alt="{\displaystyle \ {a_{i}}\ }"></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 \ {a_{-i}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {a_{-i}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bbd96afafec00598355429d921db457429f8c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.116ex; height:2.009ex;" alt="{\displaystyle \ {a_{-i}}\ .}"></span> We first <i>marginalize away</i> <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 {a_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b451383d62588c35b768dd4595ba364ab417139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle {a_{i}}}"></span> from <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_{i}(a_{i},a_{-i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}(a_{i},a_{-i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d9fc159469cf6df16c86e019b298557ee7c751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.108ex; height:2.843ex;" alt="{\displaystyle v_{i}(a_{i},a_{-i})}"></span>, by maximizing over <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 \ {a_{i}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {a_{i}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cb382086935b075ac6de7671af9d0e8d3e7139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.191ex; height:2.009ex;" alt="{\displaystyle \ {a_{i}}\ }"></span> (for every possible value 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 {a_{-i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {a_{-i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9194a406a3213d04cc60dc4be73e3f5160bbe15d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.009ex;" alt="{\displaystyle {a_{-i}}}"></span>) to yield a set of marginal outcomes <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'_{i}(a_{-i})\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ v'_{i}(a_{-i})\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a093923d26d7efb76ba8cc299e28280fae84b4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.659ex; height:3.009ex;" alt="{\displaystyle \ v'_{i}(a_{-i})\,,}"></span> which depends only on <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 \ {a_{-i}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {a_{-i}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bbd96afafec00598355429d921db457429f8c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.116ex; height:2.009ex;" alt="{\displaystyle \ {a_{-i}}\ .}"></span> We then minimize over <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 \ {a_{-i}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> </mrow> </msub> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {a_{-i}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ff62123276cf371d055c30d87f143b69a5183f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.469ex; height:2.009ex;" alt="{\displaystyle \ {a_{-i}}\ }"></span> over these outcomes. (Conversely for maximin.) </p><p>Although it is always the case 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 \ {\underline {v_{row}}}\leq {\overline {v_{row}}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>w</mi> </mrow> </msub> <mo>_</mo> </munder> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mi>o</mi> <mi>w</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\underline {v_{row}}}\leq {\overline {v_{row}}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e848cffb8643e2f0942b8642955807c86dac4b25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.877ex; margin-bottom: -0.795ex; width:12.528ex; height:3.676ex;" alt="{\displaystyle \ {\underline {v_{row}}}\leq {\overline {v_{row}}}\ }"></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 \ {\underline {v_{col}}}\leq {\overline {v_{col}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <munder> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mo>_</mo> </munder> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>o</mi> <mi>l</mi> </mrow> </msub> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\underline {v_{col}}}\leq {\overline {v_{col}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b41fd0145bfb716a9496dea1dd4e94a3a9b14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.878ex; margin-bottom: -0.793ex; width:11.549ex; height:3.676ex;" alt="{\displaystyle \ {\underline {v_{col}}}\leq {\overline {v_{col}}}\,,}"></span> the payoff vector resulting from both players playing their minimax strategies, <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 \ (2,-20)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (2,-20)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8844bb63fc604748f97c0b22edfd9b561c3f70d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.3ex; height:2.843ex;" alt="{\displaystyle \ (2,-20)\ }"></span> in the case 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 \ (T,R)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </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/fd5d3332dfd6375116f8cb91d04ad41855c1f879" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.405ex; height:2.843ex;" alt="{\displaystyle \ (T,R)\ }"></span> or <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 (-10,1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>10</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-10,1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d864d6c2532b278cfcefb38f1b33d339f3b5d5f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.139ex; height:2.843ex;" alt="{\displaystyle (-10,1)}"></span> in the case 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,R)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (M,R)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918d3e7264b7e3bf37ae54d64c1369bf1f69a533" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.664ex; height:2.843ex;" alt="{\displaystyle \ (M,R)\,,}"></span> cannot similarly be ranked against the payoff vector <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 \ (3,1)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (3,1)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a5d36f237d70e3b479f2426a70b68128638da23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle \ (3,1)\ }"></span> resulting from both players playing their maximin strategy. </p> <div class="mw-heading mw-heading3"><h3 id="In_zero-sum_games">In zero-sum games</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=3" title="Edit section: In zero-sum games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span id="Minimax_theorem"></span> In two-player <a href="/wiki/Zero-sum_game" title="Zero-sum game">zero-sum games</a>, the minimax solution is the same as the <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a>. </p><p>In the context of zero-sum games, the <a href="/wiki/Minimax_theorem" title="Minimax theorem">minimax theorem</a> is equivalent to:<sup id="cite_ref-Osborne_4-0" class="reference"><a href="#cite_note-Osborne-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability"><span title="The material near this tag failed verification of its source citation(s). (February 2015)">failed verification</span></a></i>]</sup> </p> <blockquote><p>For every two-person <a href="/wiki/Zero-sum" class="mw-redirect" title="Zero-sum">zero-sum</a> game with finitely many strategies, there exists a value <span class="texhtml mvar" style="font-style:italic;">V</span> and a mixed strategy for each player, such that </p><dl><dd>(a) Given Player 2's strategy, the best payoff possible for Player 1 is <span class="texhtml mvar" style="font-style:italic;">V</span>, and</dd> <dd>(b) Given Player 1's strategy, the best payoff possible for Player 2 is −<span class="texhtml mvar" style="font-style:italic;">V</span>.</dd></dl> </blockquote> <p>Equivalently, Player 1's strategy guarantees them a payoff of <span class="texhtml mvar" style="font-style:italic;">V</span> regardless of Player 2's strategy, and similarly Player 2 can guarantee themselves a payoff of −<span class="texhtml mvar" style="font-style:italic;">V</span>. The name <i>minimax</i> arises because each player minimizes the maximum payoff possible for the other – since the game is zero-sum, they also minimize their own maximum loss (i.e., maximize their minimum payoff). See also <a href="/wiki/Example_of_a_game_without_a_value" class="mw-redirect" title="Example of a game without a value">example of a game without a value</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=4" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="text-align:center; float:right; margin-left:1em"> <caption align="bottom" style="caption-side: bottom">Payoff matrix for player A </caption> <tbody><tr> <th> </th> <th>B chooses B1 </th> <th>B chooses B2 </th> <th>B chooses B3 </th></tr> <tr> <th>A chooses A1 </th> <td>+3 </td> <td>−2 </td> <td>+2 </td></tr> <tr> <th>A chooses A2 </th> <td>−1 </td> <td><span style="visibility:hidden;color:transparent;">+</span>0 </td> <td>+4 </td></tr> <tr> <th>A chooses A3 </th> <td>−4 </td> <td>−3 </td> <td>+1 </td></tr></tbody></table> <p>The following example of a zero-sum game, where <b>A</b> and <b>B</b> make simultaneous moves, illustrates <i>maximin</i> solutions. Suppose each player has three choices and consider the <a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">payoff matrix</a> for <b>A</b> displayed on the table ("Payoff matrix for player A"). Assume the payoff matrix for <b>B</b> is the same matrix with the signs reversed (i.e., if the choices are A1 and B1 then <b>B</b> pays 3 to <b>A</b>). Then, the maximin choice for <b>A</b> is A2 since the worst possible result is then having to pay 1, while the simple maximin choice for <b>B</b> is B2 since the worst possible result is then no payment. However, this solution is not stable, since if <b>B</b> believes <b>A</b> will choose A2 then <b>B</b> will choose B1 to gain 1; then if <b>A</b> believes <b>B</b> will choose B1 then <b>A</b> will choose A1 to gain 3; and then <b>B</b> will choose B2; and eventually both players will realize the difficulty of making a choice. So a more stable strategy is needed. </p><p>Some choices are <i>dominated</i> by others and can be eliminated: <b>A</b> will not choose A3 since either A1 or A2 will produce a better result, no matter what <b>B</b> chooses; <b>B</b> will not choose B3 since some mixtures of B1 and B2 will produce a better result, no matter what <b>A</b> chooses. </p><p>Player <b>A</b> can avoid having to make an expected payment of more than <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span> by choosing A1 with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 6 </span></span>⁠</span> and A2 with probability <span class="nowrap"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den"> 6 </span></span>⁠</span>:</span> The expected payoff for <b>A</b> would be <span class="nowrap">  3 × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 6 </span></span>⁠</span> − 1 × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den"> 6 </span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span>  </span> in case <b>B</b> chose B1 and <span class="nowrap">  −2 × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">6 </span></span>⁠</span> + 0 × <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den"> 6 </span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠−<span class="sr-only">+</span><span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span>  </span> in case <b>B</b> chose B2. Similarly, <b>B</b> can ensure an expected gain of at least <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span>, no matter what <b>A</b> chooses, by using a randomized strategy of choosing B1 with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span> and B2 with probability <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">2</span><span class="sr-only">/</span><span class="den"> 3 </span></span>⁠</span>. These <a href="/wiki/Mixed_strategy" class="mw-redirect" title="Mixed strategy">mixed</a> minimax strategies cannot be improved and are now stable. </p> <div class="mw-heading mw-heading3"><h3 id="Maximin">Maximin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=5" title="Edit section: Maximin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Frequently, in game theory, <b>maximin</b> is distinct from minimax. Minimax is used in zero-sum games to denote minimizing the opponent's maximum payoff. In a <a href="/wiki/Zero-sum_game" title="Zero-sum game">zero-sum game</a>, this is identical to minimizing one's own maximum loss, and to maximizing one's own minimum gain. </p><p>"Maximin" is a term commonly used for non-zero-sum games to describe the strategy which maximizes one's own minimum payoff. In non-zero-sum games, this is not generally the same as minimizing the opponent's maximum gain, nor the same as the <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a> strategy. </p> <div class="mw-heading mw-heading3"><h3 id="In_repeated_games">In repeated games</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=6" title="Edit section: In repeated games"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The minimax values are very important in the theory of <a href="/wiki/Repeated_games" class="mw-redirect" title="Repeated games">repeated games</a>. One of the central theorems in this theory, the <a href="/wiki/Folk_theorem_(game_theory)" title="Folk theorem (game theory)">folk theorem</a>, relies on the minimax values. </p> <div class="mw-heading mw-heading2"><h2 id="Combinatorial_game_theory">Combinatorial game theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=7" title="Edit section: Combinatorial game theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">combinatorial game theory</a>, there is a minimax algorithm for game solutions. </p><p>A <b>simple</b> version of the minimax <i>algorithm</i>, stated below, deals with games such as <a href="/wiki/Tic-tac-toe" title="Tic-tac-toe">tic-tac-toe</a>, where each player can win, lose, or draw. If player A <i>can</i> win in one move, their best move is that winning move. If player B knows that one move will lead to the situation where player A <i>can</i> win in one move, while another move will lead to the situation where player A can, at best, draw, then player B's best move is the one leading to a draw. Late in the game, it's easy to see what the "best" move is. The minimax algorithm helps find the best move, by working backwards from the end of the game. At each step it assumes that player A is trying to <b>maximize</b> the chances of A winning, while on the next turn player B is trying to <b>minimize</b> the chances of A winning (i.e., to maximize B's own chances of winning). </p> <div class="mw-heading mw-heading3"><h3 id="Minimax_algorithm_with_alternate_moves">Minimax algorithm with alternate moves</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=8" title="Edit section: Minimax algorithm with alternate moves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>minimax algorithm</b><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> is a recursive <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for choosing the next move in an n-player <a href="/wiki/Game_theory" title="Game theory">game</a>, usually a two-player game. A value is associated with each position or state of the game. This value is computed by means of a <a href="/wiki/Evaluation_function" title="Evaluation function">position evaluation function</a> and it indicates how good it would be for a player to reach that position. The player then makes the move that maximizes the minimum value of the position resulting from the opponent's possible following moves. If it is <b>A</b>'s turn to move, <b>A</b> gives a value to each of their legal moves. </p><p>A possible allocation method consists in assigning a certain win for <b>A</b> as +1 and for <b>B</b> as −1. This leads to <a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">combinatorial game theory</a> as developed by <a href="/wiki/John_Horton_Conway" title="John Horton Conway">John H. Conway</a>. An alternative is using a rule that if the result of a move is an immediate win for <b>A</b>, it is assigned positive infinity and if it is an immediate win for <b>B</b>, negative infinity. The value to <b>A</b> of any other move is the maximum of the values resulting from each of <b>B</b>'s possible replies. For this reason, <b>A</b> is called the <i>maximizing player</i> and <b>B</b> is called the <i>minimizing player</i>, hence the name <i>minimax algorithm</i>. The above algorithm will assign a value of positive or negative infinity to any position since the value of every position will be the value of some final winning or losing position. Often this is generally only possible at the very end of complicated games such as <a href="/wiki/Chess" title="Chess">chess</a> or <a href="/wiki/Go_(board_game)" class="mw-redirect" title="Go (board game)">go</a>, since it is not computationally feasible to look ahead as far as the completion of the game, except towards the end, and instead, positions are given finite values as estimates of the degree of belief that they will lead to a win for one player or another. </p><p>This can be extended if we can supply a <a href="/wiki/Heuristic" title="Heuristic">heuristic</a> evaluation function which gives values to non-final game states without considering all possible following complete sequences. We can then limit the minimax algorithm to look only at a certain number of moves ahead. This number is called the "look-ahead", measured in "<a href="/wiki/Ply_(chess)" class="mw-redirect" title="Ply (chess)">plies</a>". For example, the chess computer <a href="/wiki/IBM_Deep_Blue" class="mw-redirect" title="IBM Deep Blue">Deep Blue</a> (the first one to beat a reigning world champion, <a href="/wiki/Garry_Kasparov" title="Garry Kasparov">Garry Kasparov</a> at that time) looked ahead at least 12 plies, then applied a heuristic evaluation function.<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> </p><p>The algorithm can be thought of as exploring the <a href="/wiki/Node_(computer_science)" title="Node (computer science)">nodes</a> of a <i><a href="/wiki/Game_tree" title="Game tree">game tree</a></i>. The <i>effective <a href="/wiki/Branching_factor" title="Branching factor">branching factor</a></i> of the tree is the average number of <a href="/wiki/Child_node" class="mw-redirect" title="Child node">children</a> of each node (i.e., the average number of legal moves in a position). The number of nodes to be explored usually <a href="/wiki/Exponential_growth" title="Exponential growth">increases exponentially</a> with the number of plies (it is less than exponential if evaluating <a href="/wiki/Forced_move" class="mw-redirect" title="Forced move">forced moves</a> or repeated positions). The number of nodes to be explored for the analysis of a game is therefore approximately the branching factor raised to the power of the number of plies. It is therefore <a href="/wiki/Computational_complexity_theory#Intractability" title="Computational complexity theory">impractical</a> to completely analyze games such as chess using the minimax algorithm. </p><p>The performance of the naïve minimax algorithm may be improved dramatically, without affecting the result, by the use of <a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">alpha–beta pruning</a>. Other heuristic pruning methods can also be used, but not all of them are guaranteed to give the same result as the unpruned search. </p><p>A naïve minimax algorithm may be trivially modified to additionally return an entire <a href="/wiki/Variation_(game_tree)#Principal_variation" title="Variation (game tree)">Principal Variation</a> along with a minimax score. </p> <div class="mw-heading mw-heading3"><h3 id="Pseudocode">Pseudocode</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=9" title="Edit section: Pseudocode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Pseudocode" title="Pseudocode">pseudocode</a> for the depth-limited minimax algorithm is given below. </p> <pre><b>function</b> minimax(node, depth, maximizingPlayer) <b>is</b> <b>if</b> depth = 0 <b>or</b> node is a terminal node <b>then</b> <b>return</b> the heuristic value of node <b>if</b> maximizingPlayer <b>then</b> value := −∞ <b>for each</b> child of node <b>do</b> value := max(value, minimax(child, depth − 1, FALSE)) <b>return</b> value <b>else</b> <i>(* minimizing player *)</i> value := +∞ <b>for each</b> child of node <b>do</b> value := min(value, minimax(child, depth − 1, TRUE)) <b>return</b> value </pre> <pre><i>(* Initial call *)</i> minimax(origin, depth, TRUE) </pre> <p>The minimax function returns a heuristic value for <a href="/wiki/Leaf_nodes" class="mw-redirect" title="Leaf nodes">leaf nodes</a> (terminal nodes and nodes at the maximum search depth). Non-leaf nodes inherit their value from a descendant leaf node. The heuristic value is a score measuring the favorability of the node for the maximizing player. Hence nodes resulting in a favorable outcome, such as a win, for the maximizing player have higher scores than nodes more favorable for the minimizing player. The heuristic value for terminal (game ending) leaf nodes are scores corresponding to win, loss, or draw, for the maximizing player. For non terminal leaf nodes at the maximum search depth, an evaluation function estimates a heuristic value for the node. The quality of this estimate and the search depth determine the quality and accuracy of the final minimax result. </p><p>Minimax treats the two players (the maximizing player and the minimizing player) separately in its code. Based on the observation 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 \ \max(a,b)=-\min(-a,-b)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \max(a,b)=-\min(-a,-b)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/783d0a48a7d334535ee3117c0795d5d51f65710e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.447ex; height:2.843ex;" alt="{\displaystyle \ \max(a,b)=-\min(-a,-b)\ ,}"></span> minimax may often be simplified into the <a href="/wiki/Negamax" title="Negamax">negamax</a> algorithm. </p> <div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=10" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Minimax.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Minimax.svg/400px-Minimax.svg.png" decoding="async" width="400" height="182" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Minimax.svg/600px-Minimax.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Minimax.svg/800px-Minimax.svg.png 2x" data-file-width="900" data-file-height="410" /></a><figcaption>A minimax tree example</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Plminmax.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Plminmax.gif/400px-Plminmax.gif" decoding="async" width="400" height="290" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e1/Plminmax.gif/600px-Plminmax.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/e/e1/Plminmax.gif 2x" data-file-width="800" data-file-height="580" /></a><figcaption>An animated pedagogical example that attempts to be human-friendly by substituting initial infinite (or arbitrarily large) values for emptiness and by avoiding using the <a href="/wiki/Negamax" title="Negamax">negamax</a> coding simplifications.</figcaption></figure> <p>Suppose the game being played only has a maximum of two possible moves per player each turn. The algorithm generates the <a href="/wiki/Game_tree" title="Game tree">tree</a> on the right, where the circles represent the moves of the player running the algorithm (<i>maximizing player</i>), and squares represent the moves of the opponent (<i>minimizing player</i>). Because of the limitation of computation resources, as explained above, the tree is limited to a <i>look-ahead</i> of 4 moves. </p><p>The algorithm evaluates each <i><a href="/wiki/Leaf_node" class="mw-redirect" title="Leaf node">leaf node</a></i> using a heuristic evaluation function, obtaining the values shown. The moves where the <i>maximizing player</i> wins are assigned with positive infinity, while the moves that lead to a win of the <i>minimizing player</i> are assigned with negative infinity. At level 3, the algorithm will choose, for each node, the <b>smallest</b> of the <i><a href="/wiki/Child_node" class="mw-redirect" title="Child node">child node</a></i> values, and assign it to that same node (e.g. the node on the left will choose the minimum between "10" and "+∞", therefore assigning the value "10" to itself). The next step, in level 2, consists of choosing for each node the <b>largest</b> of the <i>child node</i> values. Once again, the values are assigned to each <i><a href="/wiki/Parent_node" class="mw-redirect" title="Parent node">parent node</a></i>. The algorithm continues evaluating the maximum and minimum values of the child nodes alternately until it reaches the <i><a href="/wiki/Root_node" class="mw-redirect" title="Root node">root node</a></i>, where it chooses the move with the largest value (represented in the figure with a blue arrow). This is the move that the player should make in order to <i>minimize</i> the <i>maximum</i> possible <a href="/wiki/Loss_function" title="Loss function">loss</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Minimax_for_individual_decisions">Minimax for individual decisions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=11" title="Edit section: Minimax for individual decisions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Minimax_in_the_face_of_uncertainty">Minimax in the face of uncertainty</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=12" title="Edit section: Minimax in the face of uncertainty"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts. For example, deciding to prospect for minerals entails a cost, which will be wasted if the minerals are not present, but will bring major rewards if they are. One approach is to treat this as a game against <i>nature</i> (see <a href="/wiki/Move_by_nature" title="Move by nature">move by nature</a>), and using a similar mindset as <a href="/wiki/Murphy%27s_law" title="Murphy's law">Murphy's law</a> or <a href="/wiki/Resistentialism" title="Resistentialism">resistentialism</a>, take an approach which minimizes the maximum expected loss, using the same techniques as in the two-person zero-sum games. </p><p>In addition, <a href="/wiki/Expectiminimax_tree" class="mw-redirect" title="Expectiminimax tree">expectiminimax trees</a> have been developed, for two-player games in which chance (for example, dice) is a factor. </p> <div class="mw-heading mw-heading3"><h3 id="Minimax_criterion_in_statistical_decision_theory">Minimax criterion in statistical decision theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=13" title="Edit section: Minimax criterion in statistical decision theory"><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/Minimax_estimator" title="Minimax estimator">Minimax estimator</a></div> <p>In classical statistical <a href="/wiki/Decision_theory" title="Decision theory">decision theory</a>, we have an <a href="/wiki/Estimator" title="Estimator">estimator</a> <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 \ \delta \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f849a31e497be33fca8db9b71119138c5a9bb41b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.21ex; height:2.343ex;" alt="{\displaystyle \ \delta \ }"></span> that is used to estimate a <a href="/wiki/Parameter" title="Parameter">parameter</a> <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 \ \theta \in \Theta \ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>θ</mi> <mo>∈</mo> <mi mathvariant="normal">Θ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \theta \in \Theta \ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1ee28f61e8f32d56a28a269c670d5008afc2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.547ex; height:2.176ex;" alt="{\displaystyle \ \theta \in \Theta \ .}"></span> We also assume a <a href="/wiki/Risk_function" class="mw-redirect" title="Risk function">risk function</a> <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(\theta ,\delta )\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>R</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ R(\theta ,\delta )\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/122583f7f2a94391d791a95f8008dde15ee3560a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.554ex; height:2.843ex;" alt="{\displaystyle \ R(\theta ,\delta )\ .}"></span> usually specified as the integral of a <a href="/wiki/Loss_function" title="Loss function">loss function</a>. In this framework, <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 \ {\tilde {\delta }}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\tilde {\delta }}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5085e0b92ae861c2ae2b0141e75cf296392b4d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.464ex; height:2.676ex;" alt="{\displaystyle \ {\tilde {\delta }}\ }"></span> is called <b>minimax</b> if it satisfies </p> <dl><dd><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 \sup _{\theta }R(\theta ,{\tilde {\delta }})=\inf _{\delta }\ \sup _{\theta }\ R(\theta ,\delta )\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </munder> <mi>R</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>δ</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">inf</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> </mrow> </munder> <mtext> </mtext> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </munder> <mtext> </mtext> <mi>R</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sup _{\theta }R(\theta ,{\tilde {\delta }})=\inf _{\delta }\ \sup _{\theta }\ R(\theta ,\delta )\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/853be7f4311be94863637b4c9dc3534d0926e52f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.436ex; height:4.843ex;" alt="{\displaystyle \sup _{\theta }R(\theta ,{\tilde {\delta }})=\inf _{\delta }\ \sup _{\theta }\ R(\theta ,\delta )\ .}"></span></dd></dl> <p>An alternative criterion in the decision theoretic framework is the <a href="/wiki/Bayes_estimator" title="Bayes estimator">Bayes estimator</a> in the presence of a <a href="/wiki/Prior_distribution" class="mw-redirect" title="Prior distribution">prior distribution</a> <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 \Pi \ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Π</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pi \ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb0718e98dc8b2318fdbb26053e69168b3dbe44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \Pi \ .}"></span> An estimator is Bayes if it minimizes the <i><a href="/wiki/Average" title="Average">average</a></i> risk </p> <dl><dd><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 \int _{\Theta }R(\theta ,\delta )\ \operatorname {d} \Pi (\theta )\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Θ</mi> </mrow> </msub> <mi>R</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi mathvariant="normal">d</mi> <mo>⁡</mo> <mi mathvariant="normal">Π</mi> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\Theta }R(\theta ,\delta )\ \operatorname {d} \Pi (\theta )\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfd3dc17a8577945b20013f95605e7dd18cc247" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.454ex; height:5.676ex;" alt="{\displaystyle \int _{\Theta }R(\theta ,\delta )\ \operatorname {d} \Pi (\theta )\ .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Non-probabilistic_decision_theory">Non-probabilistic decision theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=14" title="Edit section: Non-probabilistic decision theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A key feature of minimax decision making is being non-probabilistic: in contrast to decisions using <a href="/wiki/Expected_value" title="Expected value">expected value</a> or <a href="/wiki/Expected_utility" class="mw-redirect" title="Expected utility">expected utility</a>, it makes no assumptions about the probabilities of various outcomes, just <a href="/wiki/Scenario_analysis" class="mw-redirect" title="Scenario analysis">scenario analysis</a> of what the possible outcomes are. It is thus <a href="https://en.wiktionary.org/wiki/robust" class="extiw" title="wikt:robust">robust</a> to changes in the assumptions, in contrast to these other decision techniques. Various extensions of this non-probabilistic approach exist, notably <a href="/wiki/Minimax_regret" class="mw-redirect" title="Minimax regret">minimax regret</a> and <a href="/wiki/Info-gap_decision_theory" title="Info-gap decision theory">Info-gap decision theory</a>. </p><p>Further, minimax only requires <a href="/wiki/Ordinal_measurement" class="mw-redirect" title="Ordinal measurement">ordinal measurement</a> (that outcomes be compared and ranked), not <i>interval</i> measurements (that outcomes include "how much better or worse"), and returns ordinal data, using only the modeled outcomes: the conclusion of a minimax analysis is: "this strategy is minimax, as the worst case is (outcome), which is less bad than any other strategy". Compare to expected value analysis, whose conclusion is of the form: "This strategy yields <span class="nowrap"> <span class="texhtml">ℰ</span>(<span class="texhtml mvar" style="font-style:italic;">X</span>) = <span class="texhtml mvar" style="font-style:italic;">n</span> ."</span> Minimax thus can be used on ordinal data, and can be more transparent. </p> <div class="mw-heading mw-heading2"><h2 id="Minimax_in_politics">Minimax in politics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=15" title="Edit section: Minimax in politics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The concept of "<a href="/wiki/Lesser_evil" class="mw-redirect" title="Lesser evil">lesser evil</a>" voting (LEV) can be seen as a form of the <a class="mw-selflink selflink">minimax</a> strategy where voters, when faced with two or more candidates, choose the one they perceive as the least harmful or the "lesser evil." To do so, "voting should not be viewed as a form of personal self-expression or moral judgement directed in retaliation towards major party candidates who fail to reflect our values, or of a corrupt system designed to limit choices to those acceptable to corporate elites," but rather as an opportunity to reduce harm or loss.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Maximin_in_philosophy">Maximin in philosophy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimax&action=edit&section=16" title="Edit section: Maximin in philosophy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In philosophy, the term "maximin" is often used in the context of <a href="/wiki/John_Rawls" title="John Rawls">John Rawls</a>'s <i><a href="/wiki/A_Theory_of_Justice" title="A Theory of Justice">A Theory of Justice</a>,</i> where he refers to it in the context of The <a href="/wiki/Difference_Principle" class="mw-redirect" title="Difference Principle">Difference Principle</a>.<sup id="cite_ref-Rawls-1971_8-0" class="reference"><a href="#cite_note-Rawls-1971-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Rawls defined this principle as the rule which states that social and economic inequalities should be arranged so that "they are to be of the greatest benefit to the least-advantaged members of society".<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><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> </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=Minimax&action=edit&section=17" 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: 15em;"> <ul><li><a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li> <li><a href="/wiki/Expectiminimax" title="Expectiminimax">Expectiminimax</a></li> <li><a href="/wiki/Maxn_algorithm" class="mw-redirect" title="Maxn algorithm">Maxn algorithm</a></li> <li><a href="/wiki/Computer_chess" title="Computer chess">Computer chess</a></li> <li><a href="/wiki/Horizon_effect" title="Horizon effect">Horizon effect</a></li> <li><a href="/wiki/Lesser_of_two_evils_principle" title="Lesser of two evils principle">Lesser of two evils principle</a></li> <li><a href="/wiki/Minimax_Condorcet" class="mw-redirect" title="Minimax Condorcet">Minimax Condorcet</a></li> <li><a href="/wiki/Regret_(decision_theory)#Minimax_regret" title="Regret (decision theory)">Minimax regret</a></li> <li><a href="/wiki/Monte_Carlo_tree_search" title="Monte Carlo tree search">Monte Carlo tree search</a></li> <li><a href="/wiki/Negamax" title="Negamax">Negamax</a></li> <li><a href="/wiki/Negascout" class="mw-redirect" title="Negascout">Negascout</a></li> <li><a href="/wiki/Sion%27s_minimax_theorem" class="mw-redirect" title="Sion's minimax theorem">Sion's minimax theorem</a></li> <li><a href="/wiki/Tit_for_Tat" class="mw-redirect" title="Tit for Tat">Tit for Tat</a></li> <li><a href="/wiki/Transposition_table" title="Transposition table">Transposition table</a></li> <li><a href="/wiki/Wald%27s_maximin_model" title="Wald's maximin model">Wald's maximin model</a></li> <li><a href="/wiki/Gamma-minimax_inference" title="Gamma-minimax inference">Gamma-minimax inference</a></li></ul> </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=Minimax&action=edit&section=18" 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 reflist-columns references-column-width" style="column-width: 25em;"> <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 id="CITEREFBacchus,_Barua2013" class="citation report cs1">Bacchus, Barua (January 2013). <a rel="nofollow" class="external text" href="http://www.fraserinstitute.org/uploadedFiles/fraser-ca/Content/research-news/research/publications/provincial-healthcare-index-2013.pdf">Provincial Healthcare Index 2013</a> <span class="cs1-format">(PDF)</span> (Report). Fraser Institute. p. 25.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=report&rft.btitle=Provincial+Healthcare+Index+2013&rft.pages=25&rft.pub=Fraser+Institute&rft.date=2013-01&rft.au=Bacchus%2C+Barua&rft_id=http%3A%2F%2Fwww.fraserinstitute.org%2FuploadedFiles%2Ffraser-ca%2FContent%2Fresearch-news%2Fresearch%2Fpublications%2Fprovincial-healthcare-index-2013.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation audio-visual cs1">Professor Raymond Flood. <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=fJltiCjPeMA&t=12m0s"><i>Turing and von Neumann</i></a> (video). <a href="/wiki/Gresham_College" title="Gresham College">Gresham College</a> – via <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Turing+and+von+Neumann&rft.place=Gresham+College&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DfJltiCjPeMA%26t%3D12m0s&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></span> </li> <li id="cite_note-ZMS2013-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-ZMS2013_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ZMS2013_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaschler,_MichaelSolan,_EilonZamir,_Shmuel2013" class="citation book cs1">Maschler, Michael; <a href="/wiki/Eilon_Solan" title="Eilon Solan">Solan, Eilon</a>; Zamir, Shmuel (2013). <i>Game Theory</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. 176–180. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107005488" title="Special:BookSources/9781107005488"><bdi>9781107005488</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&rft.pages=176-180&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=9781107005488&rft.au=Maschler%2C+Michael&rft.au=Solan%2C+Eilon&rft.au=Zamir%2C+Shmuel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></span> </li> <li id="cite_note-Osborne-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Osborne_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOsborne,_Martin_J.Rubinstein,_A.1994" class="citation book cs1">Osborne, Martin J.; <a href="/wiki/Ariel_Rubinstein" title="Ariel Rubinstein">Rubinstein, A.</a> (1994). <i>A Course in Game Theory</i> (print ed.). Cambridge, MA: MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262150415" title="Special:BookSources/9780262150415"><bdi>9780262150415</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Game+Theory&rft.place=Cambridge%2C+MA&rft.edition=print&rft.pub=MIT+Press&rft.date=1994&rft.isbn=9780262150415&rft.au=Osborne%2C+Martin+J.&rft.au=Rubinstein%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></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="CITEREFRussellNorvig2021" class="citation book cs1"><a href="/wiki/Stuart_J._Russell" title="Stuart J. Russell">Russell, Stuart J.</a>; <a href="/wiki/Peter_Norvig" title="Peter Norvig">Norvig, Peter.</a> (2021). <i><a href="/wiki/Artificial_Intelligence:_A_Modern_Approach" title="Artificial Intelligence: A Modern Approach">Artificial Intelligence: A Modern Approach</a></i> (4th ed.). Hoboken: Pearson. pp. 149–150. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780134610993" title="Special:BookSources/9780134610993"><bdi>9780134610993</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/20190474">20190474</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Artificial+Intelligence%3A+A+Modern+Approach&rft.place=Hoboken&rft.pages=149-150&rft.edition=4th&rft.pub=Pearson&rft.date=2021&rft_id=info%3Alccn%2F20190474&rft.isbn=9780134610993&rft.aulast=Russell&rft.aufirst=Stuart+J.&rft.au=Norvig%2C+Peter.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHsu1999" class="citation journal cs1">Hsu, Feng-Hsiung (1999). "IBM's Deep Blue chess grandmaster chips". <i>IEEE Micro</i>. <b>19</b> (2). Los Alamitos, CA, USA: IEEE Computer Society: 70–81. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F40.755469">10.1109/40.755469</a>. <q>During the 1997 match, the software search extended the search to about 40 plies along the forcing lines, even though the non-extended search reached only about 12 plies.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Micro&rft.atitle=IBM%27s+Deep+Blue+chess+grandmaster+chips&rft.volume=19&rft.issue=2&rft.pages=70-81&rft.date=1999&rft_id=info%3Adoi%2F10.1109%2F40.755469&rft.aulast=Hsu&rft.aufirst=Feng-Hsiung&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" 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"><a href="/wiki/Noam_Chomsky" title="Noam Chomsky">Noam Chomsky</a> and John Halle, "<a rel="nofollow" class="external text" href="https://newpol.org/eight-point-brief-lev-lesser-evil-voting/">An Eight Point Brief for LEV (Lesser Evil Voting)</a>," <i><a rel="nofollow" class="external text" href="https://newpol.org/about/">New Politics</a>,</i> June 15, 2016.</span> </li> <li id="cite_note-Rawls-1971-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Rawls-1971_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRawls,_J.1971" class="citation book cs1"><a href="/wiki/John_Rawls" title="John Rawls">Rawls, J.</a> (1971). <a href="/wiki/A_Theory_of_Justice" title="A Theory of Justice"><i>A Theory of Justice</i></a>. p. 152.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Theory+of+Justice&rft.pages=152&rft.date=1971&rft.au=Rawls%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArrow,_K.1973" class="citation journal cs1"><a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Arrow, K.</a> (May 1973). <a rel="nofollow" class="external text" href="https://www.pdcnet.org/jphil/content/jphil_1973_0070_0009_0245_0263">"Some ordinalist-utilitarian notes on Rawls's <i>Theory of Justice</i>"</a>. <i>Journal of Philosophy</i>. <b>70</b> (9): 245–263. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2025006">10.2307/2025006</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2025006">2025006</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+Philosophy&rft.atitle=Some+ordinalist-utilitarian+notes+on+Rawls%27s+Theory+of+Justice&rft.volume=70&rft.issue=9&rft.pages=245-263&rft.date=1973-05&rft_id=info%3Adoi%2F10.2307%2F2025006&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2025006%23id-name%3DJSTOR&rft.au=Arrow%2C+K.&rft_id=https%3A%2F%2Fwww.pdcnet.org%2Fjphil%2Fcontent%2Fjphil_1973_0070_0009_0245_0263&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarsanyi,_J.1975" class="citation journal cs1"><a href="/wiki/John_Harsanyi" title="John Harsanyi">Harsanyi, J.</a> (June 1975). <a rel="nofollow" class="external text" href="http://piketty.pse.ens.fr/files/Harsanyi1975.pdf">"Can the maximin principle serve as a basis for morality? a critique of John Rawls's theory"</a> <span class="cs1-format">(PDF)</span>. <i>American Political Science Review</i>. <b>69</b> (2): 594–606. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1959090">10.2307/1959090</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1959090">1959090</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:118261543">118261543</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Political+Science+Review&rft.atitle=Can+the+maximin+principle+serve+as+a+basis+for+morality%3F+a+critique+of+John+Rawls%27s+theory&rft.volume=69&rft.issue=2&rft.pages=594-606&rft.date=1975-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118261543%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1959090%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1959090&rft.au=Harsanyi%2C+J.&rft_id=http%3A%2F%2Fpiketty.pse.ens.fr%2Ffiles%2FHarsanyi1975.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></span> </li> </ol></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=Minimax&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/minimax" class="extiw" title="wiktionary:Special:Search/minimax">minimax</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiquote-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/34px-Wikiquote-logo.svg.png" decoding="async" width="34" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/51px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/68px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></a></span></div> <div class="side-box-text plainlist">Wikiquote has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Minimax" class="extiw" title="q:Special:Search/Minimax">Minimax</a></b></i>.</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Minimax_principle">"Minimax principle"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Minimax+principle&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMinimax_principle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/Games/MixedStrategies.shtml">"Mixed strategies"</a>. <i>cut-the-knot.org</i>. Curriculum: Games.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=cut-the-knot.org&rft.atitle=Mixed+strategies&rft_id=http%3A%2F%2Fwww.cut-the-knot.org%2FCurriculum%2FGames%2FMixedStrategies.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span> — A visualization applet</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060307183023/http://www.swif.uniba.it/lei/foldop/foldoc.cgi?maximin+principle">"Maximin principle"</a>. <i>Dictionary of Philosophical Terms and Names</i>. Archived from <a rel="nofollow" class="external text" href="http://www.swif.uniba.it/lei/foldop/foldoc.cgi?maximin+principle">the original</a> on 2006-03-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Maximin+principle&rft.btitle=Dictionary+of+Philosophical+Terms+and+Names&rft_id=http%3A%2F%2Fwww.swif.uniba.it%2Flei%2Ffoldop%2Ffoldoc.cgi%3Fmaximin%2Bprinciple&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/minimax.html">"Minimax"</a>. <i><a href="/wiki/Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a></i>. US <a href="/wiki/NIST" class="mw-redirect" title="NIST">NIST</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Minimax&rft.btitle=Dictionary+of+Algorithms+and+Data+Structures&rft.pub=US+NIST&rft_id=https%3A%2F%2Fxlinux.nist.gov%2Fdads%2FHTML%2Fminimax.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimax" class="Z3988"></span></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 dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Data_structures_and_algorithms" style="padding:3px"><table class="nowraplinks hlist 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"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Data_structures_and_algorithms" title="Template:Data structures and algorithms"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Data_structures_and_algorithms" title="Template talk:Data structures and algorithms"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Data_structures_and_algorithms" title="Special:EditPage/Template:Data structures and algorithms"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Data_structures_and_algorithms" style="font-size:114%;margin:0 4em"><a href="/wiki/Data_structure" title="Data structure">Data structures</a> and <a href="/wiki/Algorithm" title="Algorithm">algorithms</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Data structures</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/Array_(data_structure)" title="Array (data structure)">Array</a></li> <li><a href="/wiki/Associative_array" title="Associative array">Associative array</a></li> <li><a href="/wiki/Binary_search_tree" title="Binary search tree">Binary search tree</a></li> <li><a href="/wiki/Fenwick_tree" title="Fenwick tree">Fenwick tree</a></li> <li><a href="/wiki/Graph_(abstract_data_type)" title="Graph (abstract data type)">Graph</a></li> <li><a href="/wiki/Hash_table" title="Hash table">Hash table</a></li> <li><a href="/wiki/Heap_(data_structure)" title="Heap (data structure)">Heap</a></li> <li><a href="/wiki/Linked_list" title="Linked list">Linked list</a></li> <li><a href="/wiki/Queue_(abstract_data_type)" title="Queue (abstract data type)">Queue</a></li> <li><a href="/wiki/Segment_tree" title="Segment tree">Segment tree</a></li> <li><a href="/wiki/Stack_(abstract_data_type)" title="Stack (abstract data type)">Stack</a></li> <li><a href="/wiki/String_(computer_science)" title="String (computer science)">String</a></li> <li><a href="/wiki/Tree_(data_structure)" class="mw-redirect" title="Tree (data structure)">Tree</a></li> <li><a href="/wiki/Trie" title="Trie">Trie</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algorithms and <a href="/wiki/Algorithmic_paradigm" title="Algorithmic paradigm">algorithmic paradigms</a></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/Backtracking" title="Backtracking">Backtracking</a></li> <li><a href="/wiki/Binary_search" title="Binary search">Binary search</a></li> <li><a href="/wiki/Breadth-first_search" title="Breadth-first search">Breadth-first search</a></li> <li><a href="/wiki/Brute-force_search" title="Brute-force search">Brute-force search</a></li> <li><a href="/wiki/Depth-first_search" title="Depth-first search">Depth-first search</a></li> <li><a href="/wiki/Divide-and-conquer_algorithm" title="Divide-and-conquer algorithm">Divide and conquer</a></li> <li><a href="/wiki/Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li> <li><a href="/wiki/Graph_traversal" title="Graph traversal">Graph traversal</a></li> <li><a href="/wiki/Fold_(higher-order_function)" title="Fold (higher-order function)">Fold</a></li> <li><a href="/wiki/Greedy_algorithm" title="Greedy algorithm">Greedy</a></li> <li><a href="/wiki/Hash_function" title="Hash function">Hash function</a></li> <li><a class="mw-selflink selflink">Minimax</a></li> <li><a href="/wiki/Online_algorithm" title="Online algorithm">Online</a></li> <li><a href="/wiki/Randomized_algorithm" title="Randomized algorithm">Randomized</a></li> <li><a href="/wiki/Recursion_(computer_science)" title="Recursion (computer science)">Recursion</a></li> <li><a href="/wiki/Root-finding_algorithms" class="mw-redirect" title="Root-finding algorithms">Root-finding</a></li> <li><a href="/wiki/Sorting_algorithm" title="Sorting algorithm">Sorting</a></li> <li><a href="/wiki/Streaming_algorithm" title="Streaming algorithm">Streaming</a></li> <li><a href="/wiki/Sweep_line_algorithm" title="Sweep line algorithm">Sweep line</a></li> <li><a href="/wiki/String-searching_algorithm" title="String-searching algorithm">String-searching</a></li> <li><a href="/wiki/Topological_sorting" title="Topological sorting">Topological sorting</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_data_structures" title="List of data structures">List of data structures</a></li> <li><a href="/wiki/List_of_algorithms" title="List of algorithms">List of algorithms</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 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 class="mw-selflink selflink">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"></div><div role="navigation" class="navbox" aria-labelledby="Decision_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:Decision_theory" title="Template:Decision theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Decision_theory" title="Template talk:Decision theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Decision_theory" title="Special:EditPage/Template:Decision theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Decision_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/Decision_theory" title="Decision theory">Decision theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decisions</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/Expected_utility_hypothesis" title="Expected utility hypothesis">Expected utility hypothesis</a></li> <li><a href="/wiki/Intertemporal_choice" title="Intertemporal choice">Intertemporal choice</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</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/Decision-matrix_method" title="Decision-matrix method">Decision-matrix method</a></li> <li><a href="/wiki/Decision_matrix" title="Decision matrix">Decision matrix</a></li> <li><a href="/wiki/Expected_utility" class="mw-redirect" title="Expected utility">Expected utility</a></li> <li><a href="/wiki/Strategic_dominance" title="Strategic dominance">Strategic dominance</a></li> <li><a class="mw-selflink selflink">Minimax</a></li> <li><a href="/wiki/Leximin" class="mw-redirect" title="Leximin">Leximin</a></li> <li><a href="/wiki/Principle_of_indifference" title="Principle of indifference">Principle of indifference</a></li> <li><a href="/wiki/Risk" title="Risk">Risk</a></li> <li><a href="/wiki/Allais_paradox" title="Allais paradox">Allais paradox</a></li> <li><a href="/wiki/Ellsberg_paradox" title="Ellsberg paradox">Ellsberg paradox</a></li> <li><a href="/wiki/St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg paradox</a></li> <li><a href="/wiki/Heuristics_in_judgment_and_decision-making" class="mw-redirect" title="Heuristics in judgment and decision-making">Heuristics in judgment and decision-making</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability theory</a></li> <li><a href="/wiki/Bayesian_epistemology" title="Bayesian epistemology">Bayesian epistemology</a></li> <li><a href="/wiki/Risk_aversion_(psychology)" title="Risk aversion (psychology)">Risk aversion</a></li> <li><a href="/wiki/Game_theory" title="Game theory">Game theory</a></li> <li><a href="/wiki/Social_choice_theory" title="Social choice theory">Social choice theory</a></li> <li><a href="/wiki/Causal_decision_theory" title="Causal decision theory">Causal decision theory</a></li> <li><a href="/wiki/Emotional_choice_theory" title="Emotional choice theory">Emotional choice theory</a></li> <li><a href="/wiki/Evidential_decision_theory" title="Evidential decision theory">Evidential decision theory</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Minimax&oldid=1250833808">https://en.wikipedia.org/w/index.php?title=Minimax&oldid=1250833808</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Detection_theory" title="Category:Detection theory">Detection theory</a></li><li><a href="/wiki/Category:Game_artificial_intelligence" title="Category:Game artificial intelligence">Game artificial intelligence</a></li><li><a href="/wiki/Category:Graph_algorithms" title="Category:Graph algorithms">Graph algorithms</a></li><li><a href="/wiki/Category:Optimization_algorithms_and_methods" title="Category:Optimization algorithms and methods">Optimization algorithms and methods</a></li><li><a href="/wiki/Category:Search_algorithms" title="Category:Search algorithms">Search algorithms</a></li><li><a href="/wiki/Category:Game_theory" title="Category:Game theory">Game theory</a></li><li><a href="/wiki/Category:Theorems_in_discrete_mathematics" title="Category:Theorems in discrete mathematics">Theorems in discrete mathematics</a></li><li><a href="/wiki/Category:Decision_theory" title="Category:Decision theory">Decision theory</a></li><li><a href="/wiki/Category:Fixed_points_(mathematics)" title="Category:Fixed points (mathematics)">Fixed points (mathematics)</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_failed_verification" title="Category:All articles with failed verification">All articles with failed verification</a></li><li><a href="/wiki/Category:Articles_with_failed_verification_from_February_2015" title="Category:Articles with failed verification from February 2015">Articles with failed verification from February 2015</a></li><li><a href="/wiki/Category:Articles_with_example_pseudocode" title="Category:Articles with example pseudocode">Articles with example pseudocode</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 October 2024, at 19:57<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Minimax&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-55db797859-tnfgk","wgBackendResponseTime":143,"wgPageParseReport":{"limitreport":{"cputime":"0.530","walltime":"0.715","ppvisitednodes":{"value":2535,"limit":1000000},"postexpandincludesize":{"value":77338,"limit":2097152},"templateargumentsize":{"value":3700,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":4,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":80148,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 503.969 1 -total"," 29.34% 147.881 1 Template:Reflist"," 16.45% 82.894 3 Template:Navbox"," 16.05% 80.880 1 Template:Cite_report"," 15.54% 78.306 1 Template:Algorithmic_paradigms"," 12.59% 63.449 1 Template:Short_description"," 8.09% 40.758 2 Template:Pagetype"," 6.78% 34.194 1 Template:Failed_verification"," 6.74% 33.985 7 Template:Val"," 5.85% 29.458 1 Template:Fix"]},"scribunto":{"limitreport-timeusage":{"value":"0.285","limit":"10.000"},"limitreport-memusage":{"value":7273089,"limit":52428800}},"cachereport":{"origin":"mw-api-int.codfw.main-7d596b5cf-kltmn","timestamp":"20241218031352","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Minimax","url":"https:\/\/en.wikipedia.org\/wiki\/Minimax","sameAs":"http:\/\/www.wikidata.org\/entity\/Q751319","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q751319","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-02-25T15:43:11Z","dateModified":"2024-10-12T19:57:13Z","headline":"decision rule used for minimizing the possible loss for a worst case scenario"}</script> </body> </html>