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<title>Bayesian game</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Bayesian game</span></span>

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<p>In <a href="Game_theory" title="Game theory">game theory</a>, a <b>Bayesian game</b> is a strategic decision-making model which assumes players have incomplete information. Players may hold private information relevant to the game, meaning that the payoffs are not <a href="Common_knowledge_(logic)" title="Common knowledge (logic)">common knowledge</a>.<sup id="cite_ref-zamir_1-0" class="reference"><a href="#cite_note-zamir-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Bayesian games model the outcome of player interactions using aspects of <a href="Bayesian_probability" title="Bayesian probability">Bayesian probability</a>. They are notable because they allowed the specification of the solutions to games with <a href="Complete_information" title="Complete information">incomplete information</a> for the first time in game theory.

</p><p>Hungarian economist <a href="John_C._Harsanyi" class="mw-redirect" title="John C. Harsanyi">John C. Harsanyi</a> introduced the concept of Bayesian games in three papers from 1967 and 1968:<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><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> He was awarded the <a href="Nobel_Memorial_Prize_in_Economic_Sciences" title="Nobel Memorial Prize in Economic Sciences">Nobel Memorial Prize in Economic Sciences</a> for these and other contributions to game theory in 1994. Roughly speaking, Harsanyi defined Bayesian games in the following way: players are assigned a set of characteristics by nature at the start of the game. By mapping <a href="Probability_distribution" title="Probability distribution">probability distributions</a> to these characteristics and by calculating the outcome of the game using Bayesian probability, the result is a game whose solution is, for <a href="#Normal_form_games_with_incomplete_information">technical reasons</a>, far easier to calculate than a similar game in a non-Bayesian context.

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<div class="mw-heading mw-heading2"><h2 id="Normal_form_games_with_incomplete_information">Normal form games with incomplete information</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Elements">Elements</h3></div>

<p>A Bayesian game is defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (N\!,A,T,u,p)}">

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<annotation encoding="application/x-tex">{\displaystyle (N\!,A,T,u,p)}</annotation>

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</math></span><img src="./8f58bd6c77ae46c967a87d11c3009d65dc228a17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.5ex; height:2.843ex;" alt="{\displaystyle (N\!,A,T,u,p)}" loading="lazy"></span>, where it consists of the following elements:<sup id="cite_ref-kajii1997robustness_5-0" class="reference"><a href="#cite_note-kajii1997robustness-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<dl><dt>Set of players, <i>N</i></dt>

<dd>The set of players within the game</dd>

<dt>Action sets, <i>a<sub>i</sub></i></dt>

<dd>The set of actions available to Player <i>i</i>. An action profile <i>a</i> = (<i>a</i><sub>1</sub>, . . . , <i>a<sub>N</sub></i>) is a list of actions, one for each player</dd>

<dd>The set of types of players <i>i</i>. "Types" capture the private information a player can have. A type profile <i>t</i> = (<i>t</i><sub>1</sub>, . . . , <i>t<sub>N</sub></i>) is a list of types, one for each player</dd>

<dt>Payoff functions, <i>u</i></dt>

<dd>Assign a payoff to a player given their type and the action profile. A payoff function, <i>u</i> = (<i>u</i><sub>1</sub>, . . . , <i>u<sub>N</sub></i>) denotes the utilities of player <i>i</i></dd>

<dt>Prior, <i>p</i></dt>

<dd>A probability distribution over all possible type profiles, where <i>p</i>(<i>t</i>) = <i>p</i>(<i>t</i><sub>1</sub>, . . . , <i>t<sub>N</sub></i>) is the probability that Player 1 has type <i>t</i><sub>1</sub> and Player <i>N</i> has type <i>t<sub>N</sub></i>.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Pure_strategies">Pure strategies</h3></div>

<p>In a strategic game, a <i>pure strategy</i> is a player's choice of action at each point where the player must make a decision.<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>

<div class="mw-heading mw-heading3"><h3 id="Three_stages">Three stages</h3></div>

<p>There are three stages of Bayesian games, each describing the players' knowledge of types within the game.

</p>

<ol><li><i>Ex-ante stage game.</i> Players do not know their types or those of other players. A player recognizes payoffs as expected values based on a prior distribution of all possible types.</li>

<li><i>Interim stage game.</i> Players know their type but only a probability distribution of other players. When considering payoffs, a player studies the expected value of the other player's type.</li>

<li><i>Ex-post stage game.</i> Players know their types and those of other players. The payoffs are known to players.<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></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Improvements_over_non-Bayesian_games">Improvements over non-Bayesian games</h3></div>

<p>There are two important and novel aspects to Bayesian games that were themselves specified by Harsanyi.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The first is that Bayesian games should be considered and structured identically to complete information games. However, by attaching probability to the game, the final game functions as an incomplete information game. Therefore, players can be essentially modeled as having incomplete information, and the probability space of the game still follows the <a href="Law_of_total_probability" title="Law of total probability">law of total probability</a>. Bayesian games are also useful because they do not require infinite sequential calculations, which is typical of strategic thinking in <a href="Repeated_games" class="mw-redirect" title="Repeated games">repeated games</a>. Infinite sequential calculations would arise where players try to "get into each other's heads." For example, one may ask questions and decide, "If I expect some action from player B, then player B will anticipate that I expect that action, so then I should anticipate that anticipation" <i>ad infinitum</i>. Bayesian games allow for the calculation of these outcomes in one move by assigning different probability weights to different outcomes simultaneously. The effect of this is that Bayesian games allow for the modeling of a number of games that in a non-Bayesian setting would be <a href="Rationality_(economics)" class="mw-redirect" title="Rationality (economics)">irrational</a> to compute.

</p>

<div class="mw-heading mw-heading2"><h2 id="Bayesian_Nash_equilibrium">Bayesian Nash equilibrium</h2></div>

<p>A <b>Bayesian Nash Equilibrium</b> (BNE) is a Nash equilibrium for a Bayesian game, which is derived from the ex-ante normal form game associated with the Bayesian framework.

</p><p>In a traditional (non-Bayesian) game, a strategy profile is a <a href="Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a> if every player's strategy is a <a href="Best_response" title="Best response">best response</a> to the other players' strategies. In this situation, no player can unilaterally change their strategy to achieve a higher payoff, given the strategies chosen by the other players.

</p><p>For a Bayesian game, the concept of Nash equilibrium extends to include the uncertainty about the state of nature: Each player maximizes their expected payoff based on their beliefs about the state of nature, which are formed using <a href="Bayes'_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a>. A strategy profile <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =(\sigma _{1},\sigma _{2},\dots ,\sigma _{N})}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =(\sigma _{1},\sigma _{2},\dots ,\sigma _{N})}</annotation>

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</math></span><img src="./e4f301b9b22c975ada66304ba3ff8b6448d2d909.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.232ex; height:2.843ex;" alt="{\displaystyle \sigma =(\sigma _{1},\sigma _{2},\dots ,\sigma _{N})}" loading="lazy"></span> is a Bayesian Nash equilibrium if, for every player <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, the strategy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation>

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</math></span><img src="./6ab3208a7d0c634ef720e03ff5a9949e8310edc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}" loading="lazy"></span> maximizes player <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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<ul><li>Their beliefs about the state of nature (based on their type),</li>

<li>The strategies played by other players.<sup id="cite_ref-kajii1997robustness_5-1" class="reference"><a href="#cite_note-kajii1997robustness-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Mathematically:

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<mtext> maximizes </mtext>

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<mtext>type of player </mtext>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{i}{\text{ maximizes }}\mathbb {E} [u_{i}(\sigma _{i},\sigma _{-i})\mid {\text{type of player }}i].}</annotation>

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</p><p>For finite Bayesian games (where the action and type spaces are finite), the BNE can be represented in two equivalent ways:

</p>

<ol><li>Agent-Form Game: The number of players is expanded from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |N|}">

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<annotation encoding="application/x-tex">{\displaystyle |N|}</annotation>

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</math></span><img src="./f9becad1e12f3234bef53f3d619b7c9c8dfc2b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.357ex; height:2.843ex;" alt="{\displaystyle |N|}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum _{i=1}^{|N|}|\Theta _{i}|}">

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<annotation encoding="application/x-tex">{\textstyle \sum _{i=1}^{|N|}|\Theta _{i}|}</annotation>

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</math></span><img src="./fff589df6190b75ff0a5f516314a4b6f15f001f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.643ex; height:3.676ex;" alt="{\textstyle \sum _{i=1}^{|N|}|\Theta _{i}|}" loading="lazy"></span>, where each type of a player is treated as a separate "player." This is detailed in Theorem 9.51 of the book <i>Game Theory</i>.<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></li>

<li>Induced <a href="Normal-form_game" title="Normal-form game">Normal Form Game</a>: The number of players remains <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |N|}">

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<annotation encoding="application/x-tex">{\displaystyle |N|}</annotation>

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</math></span><img src="./f9becad1e12f3234bef53f3d619b7c9c8dfc2b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.357ex; height:2.843ex;" alt="{\displaystyle |N|}" loading="lazy"></span>, but the action space for each player <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> is expanded from <span class="mwe-math-element mwe-math-element-inline"><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}|}">

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle |A_{i}|}</annotation>

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</math></span><img src="./0f68b806478e831ad08926160aa2c69e0ec55386.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.836ex; height:2.843ex;" alt="{\displaystyle |A_{i}|}" loading="lazy"></span> to <span class="mwe-math-element mwe-math-element-inline"><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}|^{|\Theta _{i}|}}">

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<annotation encoding="application/x-tex">{\displaystyle |A_{i}|^{|\Theta _{i}|}}</annotation>

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</math></span><img src="./ad2ec1fb132362f1e101a6e7ce7dbdef0061fd68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.887ex; height:3.509ex;" alt="{\displaystyle |A_{i}|^{|\Theta _{i}|}}" loading="lazy"></span>. This means the strategy now specifies an action for every type of player. This representation is discussed in Section 6.3.3 of the book <i>Multiagent Systems</i>.<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></li></ol>

<p>In both cases, the Nash equilibrium for the game can be computed using these representations, and the BNE can be recovered from the results. A linear program can be formulated to compute the BNE efficiently for two-player Bayesian games with a zero-sum objective.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Extensive_form_games_with_incomplete_information">Extensive form games with incomplete information</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Elements_of_extensive_form_games">Elements of extensive form games</h3></div>

<p><a href="Extensive-form_game" title="Extensive-form game">Extensive form games</a> with perfect or imperfect information, have the following elements:<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p>

<ol><li>Set of players</li>

<li>Set of decision nodes</li>

<li>A player function assigning a player to each decision node</li>

<li>Set of actions for each player at each of her decision nodes</li>

<li>Set of terminal nodes</li>

<li>A payoff function for each player</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Nature_and_information_sets">Nature and information sets</h3></div>

<p>An unfilled circle usually denotes nature's node. Its strategy is always specified and completely mixed. Although Nature is generally at the tree's root, it can also move to other points.

</p><p>An information set of player <i>i</i> is a subset of player <i>i'</i>s decision nodes that she cannot distinguish between. If player <i>i</i> is at one of her decision nodes in an information set, she does not know which node within the information set she is at.

</p><p>For two decision nodes to be in the same <a href="Information_set_(game_theory)" title="Information set (game theory)">information set</a>, they must<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p>

<ol><li>Belong to the same player; and</li>

<li>Have the same set of actions</li></ol>

<p>Information sets are denoted by dotted lines, the most common notation today.

</p>

<div class="mw-heading mw-heading3"><h3 id="The_role_of_beliefs">The role of beliefs</h3></div>

<p>In Bayesian games, players' beliefs about the game are denoted by a probability distribution over various types.

</p><p>If players do not have private information, the probability distribution over types is known as a <i>common prior</i>.<sup id="cite_ref-zamir_1-1" class="reference"><a href="#cite_note-zamir-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Bayes'_rule">Bayes' rule</h3></div>

<p>An assessment of an extensive form game is a pair <span class="nowrap">⟨<i>b, μ</i>⟩</span>

</p>

<ol><li><a href="Strategy_(game_theory)#Behavior_strategy" title="Strategy (game theory)">Behavior Strategy</a> profile; and</li>

<li>Belief system</li></ol>

<p>An assessment <span class="nowrap">⟨<i>b, μ</i>⟩</span> satisfies <a href="Bayes'_theorem" title="Bayes' theorem">Bayes' rule</a> if<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <i>μ</i>(<i>x</i>|<i>h<sub>i</sub></i>) = <i>Pr</i>[<i>x</i> is reached given <i>b</i><sub>−<i>i</i></sub> ] / Σ <i>Pr</i>[<i>x′</i> is reached given <i>b</i><sub>−<i>i</i></sub> ] whenever <i>h<sub>i</sub></i> is reached with strictly positive probability according to <span class="texhtml"><i>b</i><sub>−<i>i</i></sub></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Perfect_Bayesian_equilibrium">Perfect Bayesian equilibrium</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Perfect_Bayesian_equilibrium" title="Perfect Bayesian equilibrium">Perfect Bayesian equilibrium</a></div>

<p>A <a href="Perfect_Bayesian_equilibrium" title="Perfect Bayesian equilibrium">perfect Bayesian equilibrium</a> in an extensive form game is a combination of strategies and a specification of beliefs such that the following two conditions are satisfied:<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<ol><li>Bayesian consistency: the beliefs are consistent with the strategies under consideration;</li>

<li>Sequential rationality: the players choose optimally given their beliefs.</li></ol>

<p>Bayesian Nash equilibrium can result in implausible equilibria in dynamic games, where players move sequentially rather than simultaneously. As in games of complete information, these can arise via <a href="Non-credible_threat" title="Non-credible threat">non-credible</a> strategies off the equilibrium path. In games of incomplete information, non-credible beliefs are also possible.

</p><p>To address these issues, Perfect Bayesian equilibrium, according to <a href="Subgame_perfect_equilibrium" title="Subgame perfect equilibrium">subgame perfect equilibrium</a>, requires that subsequent play be optimal starting from any information set. It also requires that beliefs be updated consistently with Bayes' rule on every path of play that occurs with a positive probability.

</p>

<div class="mw-heading mw-heading3"><h3 id="Stochastic_Bayesian_games">Stochastic Bayesian games</h3></div>

<p>Stochastic Bayesian games<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> combine the definitions of Bayesian games and <a href="Stochastic_game" title="Stochastic game">stochastic game</a> to represent environment states (e.g., physical world states) with stochastic transitions between states as well as uncertainty about the types of different players in each state. The resulting model is solved via a recursive combination of the Bayesian Nash equilibrium and the <a href="Bellman_equation" title="Bellman equation">Bellman optimality equation</a>. Stochastic Bayesian games have been used to address diverse problems, including defense and security planning,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> cybersecurity of power plants,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> autonomous driving,<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> mobile edge computing,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> self-stabilization in dynamic systems,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and misbehavior treating in crowdsourcing IoT.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Incomplete_information_over_collective_agency">Incomplete information over collective agency</h3></div>

<p>The definition of Bayesian games and Bayesian equilibrium has been extended to deal with collective <a href="Agency_(sociology)" title="Agency (sociology)">agency</a>. One approach is to continue to treat individual players as reasoning in isolation but to allow them, with some probability, to reason from the perspective of a collective.<sup id="cite_ref-bacharach1999interactive_23-0" class="reference"><a href="#cite_note-bacharach1999interactive-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Another approach is to assume that players within any collective agent know that the agent exists but that other players do not know this, although they suspect it with some probability.<sup id="cite_ref-Newton2019agency_24-0" class="reference"><a href="#cite_note-Newton2019agency-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> For example, Alice and Bob may sometimes optimize as individuals and sometimes collude as a team, depending on the state of nature, but other players may not know which of these is the case.

</p>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Sheriff's_dilemma">Sheriff's dilemma</h3></div>

<p>A sheriff faces an armed suspect. Both must simultaneously decide whether to shoot the other or not.

</p><p>The suspect can either be of type "criminal" or "civilian". The sheriff has only one type. The suspect knows its type and the Sheriff's type, but the Sheriff does not know the suspect's type. Thus, there is <a href="Complete_information" title="Complete information">incomplete information</a> (because the suspect has private information), making it a Bayesian game. There is a probability <i>p</i> that the suspect is a criminal and a probability <i>1-p</i> that the suspect is a civilian; both players are aware of this probability (common prior assumption, which can be converted into a complete-information game with <a href="Perfect_information" title="Perfect information">imperfect information</a>).

</p><p>The sheriff would rather defend himself and shoot if the suspect shoots or not shoot if the suspect does not (even if the suspect is a criminal). The suspect would rather shoot if he is a criminal, even if the sheriff does not shoot, but would rather not shoot if he is a civilian, even if the sheriff shoots. Thus, the payoff matrix of this <a href="Normal-form_game" title="Normal-form game">Normal-form game</a> for both players depends on the type of the suspect. This game is defined by <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (N,A,T,p,u)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (N,A,T,p,u)}</annotation>

</semantics>

</math></span><img src="./853b535c5911f43b32f8ebeae406fb90d0788ab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.887ex; height:2.843ex;" alt="{\displaystyle (N,A,T,p,u)}" loading="lazy"></span>⁠</span>, where:

</p>

<ul><li><i>N</i> = {Suspect, Sheriff}</li>

<li><i>A</i><sub>Suspect</sub> = {Shoot, Not} , <i>A</i><sub>Sheriff</sub> = {Shoot, Not}</li>

<li><i>T</i><sub>Suspect</sub> = {Criminal, Civilian} , <i>T</i><sub>Sheriff</sub> = {*}</li>

<li><i>p</i><sub>Criminal</sub> = <i>p</i> , <i>p</i><sub>Civilian</sub> = (1 − <i>p</i>)</li>

<li>It is assumed that the payoffs, <i>u</i>, are given as follows:</li></ul>

</caption>

<th colspan="2" rowspan="2">Type = "Criminal"

</th>

<th colspan="2">Sheriff's action

</th></tr>

<tr>

<th>Shoot

</th>

<th>Not

</th></tr>

<tr>

<th rowspan="2">Suspect's action

</th>

<th>Shoot

</th>

<td>0, 0

</td>

<td>2, −2

</td></tr>

<tr>

<th>Not

</th>

<td>−2, −1

</td>

<td>−1, −1

</td></tr></tbody></table>

</caption>

<th colspan="2" rowspan="2">Type = "Civilian"

</th>

<th colspan="2">Sheriff's action

</th></tr>

<tr>

<th>Shoot

</th>

<th>Not

</th></tr>

<tr>

<th rowspan="2">Suspect's action

</th>

<th>Shoot

</th>

<td>−3, −1

</td>

<td>−1, −2

</td></tr>

<tr>

<th>Not

</th>

<td>−2, −1

</td>

<td>0, 0

</td></tr></tbody></table>

<p>If both players are rational and both know that both players are rational and everything that any player knows is known to be known by every player (i.e., player 1 knows player 2 knows that player 1 is rational and player 2 knows this, etc. <i>ad infinitum</i> – <a href="Common_knowledge_(logic)" title="Common knowledge (logic)">common knowledge</a>), play in the game will be as follows according to perfect Bayesian equilibrium:<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup>

</p><p>When the type is "criminal", the <a href="Strategic_dominance" title="Strategic dominance">dominant strategy</a> for the suspect is to shoot, and when the type is "civilian", the dominant strategy for the suspect is not to shoot; alternative strictly dominated strategy can thus be removed. Given this, if the sheriff shoots, he will have a payoff of 0 with probability <i>p</i> and a payoff of −1 with probability <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation>

</semantics>

</math></span><img src="./9633a8692121eedfa99cace406205e5d1511ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}" loading="lazy"></span>⁠</span>, i.e., an expected payoff of <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p-1}</annotation>

</semantics>

</math></span><img src="./f356ae51988add41a7da343e6b6d48fa968da162.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p-1}" loading="lazy"></span>⁠</span>; if the sheriff does not shoot, he will have a payoff of −2 with probability <i>p</i> and a payoff of 0 with probability <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -2p}</annotation>

</semantics>

</math></span><img src="./38409eaca4d181f6b3d903678befbb2f8a36cd50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.14ex; height:2.509ex;" alt="{\displaystyle -2p}" loading="lazy"></span>⁠</span>. Thus, the Sheriff will always shoot if <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1>-2p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p-1>-2p}</annotation>

</semantics>

</math></span><img src="./49cc1b272652a5473245e4988693f836bc3dac64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.5ex; height:2.509ex;" alt="{\displaystyle p-1>-2p}" loading="lazy"></span>⁠</span>, i.e., when <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p>1/3}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p>1/3}</annotation>

</semantics>

</math></span><img src="./e246c7ccb4ca986cca81365863be04c2de14ac78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.845ex; height:2.843ex;" alt="{\displaystyle p>1/3}" loading="lazy"></span>⁠</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="The_market_for_lemons">The market for lemons</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="The_Market_for_Lemons" title="The Market for Lemons">The Market for Lemons</a></div>

<p>The Market for Lemons is related to a concept known as <a href="Adverse_selection" title="Adverse selection">adverse selection</a>.

</p><p><b>Set up</b>

</p><p>There is a used car. Player 1 is a potential buyer who is interested in the vehicle. Player 2 owns the car and knows its value (how good it is, etc.). Player 1 does not and believes that the car's value to the owner (Player 2) is distributed uniformly between 0 and 100 (i.e., each of two value sub-intervals of [0, 100] of equal length is equally likely).

</p><p>Player 1 can bid p between 0 and 100 (inclusive) I. Player 2 can then accept or reject the offer. The payoffs are as follows:

</p>

<ul><li>Player 1's payoff: Bid Accepted is <span class="nowrap">⁠<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3}{2}}v-p}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}v-p}</annotation>

</semantics>

</math></span><img src="./c98f8db24a422ef002b3f45af69e53c3f884fdf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.136ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{2}}v-p}" loading="lazy"></span>⁠</span>, Bid Rejected is 0</li>

<li>Player 2's payoff: Bid Accepted is <i>p</i>, Bid Rejected is <i>v</i></li></ul>

<p><b>Side point: cut-off strategy</b>

</p><p>Player 2's strategy: Accept all bids above a certain cut-off <i>P</i><sup>∗</sup>, and Reject and bid below <i>P</i><sup>∗</sup>, is known as a cut-off strategy, where <i>P</i><sup>∗</sup> is called the cut-off.

</p>

<ul><li>Only "lemons" (used cars in bad conditions, specifically with value at most equal to <i>p</i>) are traded</li>

<li>Player 1 can guarantee herself a payoff of zero by bidding zero; hence, in equilibrium, <i>p</i> = 0</li>

<li>Since only "lemons" (used cars in bad conditions) are traded, the market collapses</li>

<li>No trade is possible even when trade would be <a href="Economically_efficient" class="mw-redirect" title="Economically efficient">economically efficient</a><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Enter_the_monopolized_market">Enter the monopolized market</h3></div>

<p>A new company (player1) that wants to enter a market that a large company monopolizes will encounter two types of monopolist (player2): type1 is prevented, and type2 is allowed. Player1 will never have complete information about player2, but may be able to infer the probability of type1 and type2 appearing from whether the previous firm entering the market was blocked, it is a Bayesian game. The reason for these judgments is that there are blocking costs for player2, which may need to make significant price cuts to prevent player1 from entering the market, so it will block player1 when the profit it steals from entering the market is greater than the blocking costs.

</p>

<ul><li><a href="Bayesian-optimal_mechanism" title="Bayesian-optimal mechanism">Bayesian-optimal mechanism</a></li>

<li><a href="Bayesian-optimal_pricing" title="Bayesian-optimal pricing">Bayesian-optimal pricing</a></li>

<li><a href="Bayesian_programming" title="Bayesian programming">Bayesian programming</a></li>

<li><a href="Bayesian_inference" title="Bayesian inference">Bayesian inference</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><cite id="CITEREFBernhardPollokKnoll2019" class="citation book cs1">Bernhard, Julian; Pollok, Stefan; Knoll, Alois (2019). "Addressing Inherent Uncertainty: Risk-Sensitive Behavior Generation for Automated Driving using Distributional Reinforcement Learning". <i>2019 IEEE Intelligent Vehicles Symposium (IV)</i>. Paris, France: IEEE. pp. <span class="nowrap">2148–</span>2155. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2102.03119">2102.03119</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIVS.2019.8813791">10.1109/IVS.2019.8813791</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-7281-0560-4</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:201811314">201811314</a>.</cite></span>

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<li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><cite id="CITEREFAsheralievaNiyato2021" class="citation journal cs1">Asheralieva, Alia; Niyato, Dusit (2021). "Fast and Secure Computational Offloading With Lagrange Coded Mobile Edge Computing". <i>IEEE Transactions on Vehicular Technology</i>. <b>70</b> (5): <span class="nowrap">4924–</span>4942. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTVT.2021.3070723">10.1109/TVT.2021.3070723</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9545">0018-9545</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:234331661">234331661</a>.</cite></span>

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<li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><cite id="CITEREFRamtinTowsley2021" class="citation arxiv cs1">Ramtin, Amir Reza; Towsley, Don (2021). "A Game-Theoretic Approach to Self-Stabilization with Selfish Agents". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2108.07362">2108.07362</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.DC">cs.DC</a>].</cite></span>

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<li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><cite id="CITEREFSuSfarNatalizioMoyal2023" class="citation conference cs1">Su, Runbo; Sfar, Arbia Riahi; Natalizio, Enrico; Moyal, Pascal; Song, Ye-Qiong (2023-09-11). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/10287527">"A Game Theoretical Model addressing Misbehavior in Crowdsourcing IoT"</a>. <a rel="nofollow" class="external text" href="https://hal.science/hal-04205286/file/SECON_Runbo_Su.pdf"><i>2023 20th Annual IEEE International Conference on Sensing, Communication, and Networking (SECON)</i></a> <span class="cs1-format">(PDF)</span>. IEEE. pp. <span class="nowrap">195–</span>203. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FSECON58729.2023.10287527">10.1109/SECON58729.2023.10287527</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>979-8-3503-0052-9</bdi>.</cite></span>

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</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFGibbons1992" class="citation book cs1">Gibbons, Robert (1992). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8ygxf2WunAIC&pg=PA144"><i>Game Theory for Applied Economists</i></a>. Princeton University Press. pp. <span class="nowrap">144–</span>52. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1400835887</bdi>.</cite></li>

<li><cite id="CITEREFLevin2002" class="citation web cs1">Levin, Jonathan (2002). <a rel="nofollow" class="external text" href="https://web.stanford.edu/~jdlevin/Econ%20203/Bayesian.pdf">"Games with Incomplete Information"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">25 August</span> 2016</span>.</cite></li></ul>

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</style><div id="Game_theory698" style="font-size:114%;margin:0 4em"><a href="Game_theory" title="Game theory">Game theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Glossary_of_game_theory" title="Glossary of game theory">Glossary</a></li>

<li><a href="List_of_game_theorists" title="List of game theorists">Game theorists</a></li>

<li><a href="List_of_games_in_game_theory" title="List of games in game theory">Games</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Traditional_game_theory698" style="font-size:114%;margin:0 4em">Traditional <a href="Game_theory" title="Game theory">game theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Game_theory#Basic_concepts" title="Game theory">Definitions</a></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="Asynchrony_(game_theory)" title="Asynchrony (game theory)">Asynchrony</a></li>

<li><a href="Bayesian_regret" title="Bayesian regret">Bayesian regret</a></li>

<li><a href="Best_response" title="Best response">Best response</a></li>

<li><a href="Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li>

<li><a href="Cheap_talk" title="Cheap talk">Cheap talk</a></li>

<li><a href="Coalition" title="Coalition">Coalition</a></li>

<li><a href="Complete_contract" title="Complete contract">Complete contract</a></li>

<li><a href="Complete_information" title="Complete information">Complete information</a></li>

<li><a href="Complete_mixing" title="Complete mixing">Complete mixing</a></li>

<li><a href="Confrontation_analysis" title="Confrontation analysis">Confrontation analysis</a></li>

<li><a href="Conjectural_variation" title="Conjectural variation">Conjectural variation</a></li>

<li><a href="Contingent_cooperator" title="Contingent cooperator">Contingent cooperator</a></li>

<li><a href="Coopetition" title="Coopetition">Coopetition</a></li>

<li><a href="Cooperative_game_theory" title="Cooperative game theory">Cooperative game theory</a></li>

<li><a href="Dynamic_inconsistency" title="Dynamic inconsistency">Dynamic inconsistency</a></li>

<li><a href="Escalation_of_commitment" title="Escalation of commitment">Escalation of commitment</a></li>

<li><a href="Farsightedness_(game_theory)" title="Farsightedness (game theory)">Farsightedness</a></li>

<li><a href="Game_semantics" title="Game semantics">Game semantics</a></li>

<li><a href="Hierarchy_of_beliefs" title="Hierarchy of beliefs">Hierarchy of beliefs</a></li>

<li><a href="Imperfect_information" class="mw-redirect" title="Imperfect information">Imperfect information</a></li>

<li><a href="Incomplete_information" class="mw-redirect" title="Incomplete information">Incomplete information</a></li>

<li><a href="Information_set_(game_theory)" title="Information set (game theory)">Information set</a></li>

<li><a href="Move_by_nature" title="Move by nature">Move by nature</a></li>

<li><a href="Mutual_knowledge" title="Mutual knowledge">Mutual knowledge</a></li>

<li><a href="Non-cooperative_game_theory" title="Non-cooperative game theory">Non-cooperative game theory</a></li>

<li><a href="Non-credible_threat" title="Non-credible threat">Non-credible threat</a></li>

<li><a href="Outcome_(game_theory)" title="Outcome (game theory)">Outcome</a></li>

<li><a href="Perfect_information" title="Perfect information">Perfect information</a></li>

<li><a href="Perfect_recall_(game_theory)" title="Perfect recall (game theory)">Perfect recall</a></li>

<li><a href="Preference_(economics)" title="Preference (economics)">Preference</a></li>

<li><a href="Rationality" title="Rationality">Rationality</a></li>

<li><a href="Sequential_game" title="Sequential game">Sequential game</a></li>

<li><a href="Simultaneous_action_selection" title="Simultaneous action selection">Simultaneous action selection</a></li>

<li><a href="Spite_(game_theory)" title="Spite (game theory)">Spite</a></li>

<li><a href="Strategic_complements" title="Strategic complements">Strategic complements</a></li>

<li><a href="Strategic_dominance" title="Strategic dominance">Strategic dominance</a></li>

<li><a href="Strategic_form" class="mw-redirect" title="Strategic form">Strategic form</a></li>

<li><a href="Strategic_interaction" class="mw-redirect" title="Strategic interaction">Strategic interaction</a></li>

<li><a href="Strategic_move" title="Strategic move">Strategic move</a></li>

<li><a href="Strategy_(game_theory)" title="Strategy (game theory)">Strategy</a></li>

<li><a href="Subgame" title="Subgame">Subgame</a></li>

<li><a href="Succinct_game" title="Succinct game">Succinct game</a></li>

<li><a href="Topological_game" title="Topological game">Topological game</a></li>

<li><a href="Tragedy_of_the_commons" title="Tragedy of the commons">Tragedy of the commons</a></li>

<li><a href="Uncorrelated_asymmetry" title="Uncorrelated asymmetry">Uncorrelated asymmetry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Economic_equilibrium" title="Economic equilibrium">Equilibrium<br>concepts</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="Backward_induction" title="Backward induction">Backward induction</a></li>

<li><a href="Bayes_correlated_equilibrium" title="Bayes correlated equilibrium">Bayes correlated equilibrium</a></li>

<li><a href="Bayesian_efficiency" title="Bayesian efficiency">Bayesian efficiency</a></li>

<li><a href="Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a></li>

<li><a href="Berge_equilibrium" title="Berge equilibrium">Berge equilibrium</a></li>

<li><a href="Bertrand%E2%80%93Edgeworth_model" title="Bertrand–Edgeworth model">Bertrand–Edgeworth model</a></li>

<li><a href="Coalition-proof_Nash_equilibrium" title="Coalition-proof Nash equilibrium">Coalition-proof Nash equilibrium</a></li>

<li><a href="Correlated_equilibrium" title="Correlated equilibrium">Correlated equilibrium</a></li>

<li><a href="Cursed_equilibrium" title="Cursed equilibrium">Cursed equilibrium</a></li>

<li><a href="Edgeworth_price_cycle" title="Edgeworth price cycle">Edgeworth price cycle</a></li>

<li><a href="Epsilon-equilibrium" title="Epsilon-equilibrium">Epsilon-equilibrium</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs equilibrium</a></li>

<li><a href="Incomplete_contracts" title="Incomplete contracts">Incomplete contracts</a></li>

<li><a href="Inequity_aversion" title="Inequity aversion">Inequity aversion</a></li>

<li><a href="Individual_rationality" class="mw-redirect" title="Individual rationality">Individual rationality</a></li>

<li><a href="Iterated_elimination_of_dominated_strategies" class="mw-redirect" title="Iterated elimination of dominated strategies">Iterated elimination of dominated strategies</a></li>

<li><a href="Markov_perfect_equilibrium" title="Markov perfect equilibrium">Markov perfect equilibrium</a></li>

<li><a href="Mertens-stable_equilibrium" title="Mertens-stable equilibrium">Mertens-stable equilibrium</a></li>

<li><a href="Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a></li>

<li><a href="Open-loop_model" title="Open-loop model">Open-loop model</a></li>

<li><a href="Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li>

<li><a href="Payoff_dominance" class="mw-redirect" title="Payoff dominance">Payoff dominance</a></li>

<li><a href="Perfect_Bayesian_equilibrium" title="Perfect Bayesian equilibrium">Perfect Bayesian equilibrium</a></li>

<li><a href="Price_of_anarchy" title="Price of anarchy">Price of anarchy</a></li>

<li><a href="Program_equilibrium" title="Program equilibrium">Program equilibrium</a></li>

<li><a href="Proper_equilibrium" title="Proper equilibrium">Proper equilibrium</a></li>

<li><a href="Quantal_response_equilibrium" title="Quantal response equilibrium">Quantal response equilibrium</a></li>

<li><a href="Quasi-perfect_equilibrium" title="Quasi-perfect equilibrium">Quasi-perfect equilibrium</a></li>

<li><a href="Rational_agent" title="Rational agent">Rational agent</a></li>

<li><a href="Rationalizability" class="mw-redirect" title="Rationalizability">Rationalizability</a></li>

<li><a href="Rationalizable_strategy" title="Rationalizable strategy">Rationalizable strategy</a></li>

<li><a href="Satisfaction_equilibrium" title="Satisfaction equilibrium">Satisfaction equilibrium</a></li>

<li><a href="Self-confirming_equilibrium" title="Self-confirming equilibrium">Self-confirming equilibrium</a></li>

<li><a href="Sequential_equilibrium" title="Sequential equilibrium">Sequential equilibrium</a></li>

<li><a href="Shapley_value" title="Shapley value">Shapley value</a></li>

<li><a href="Strong_Nash_equilibrium" title="Strong Nash equilibrium">Strong Nash equilibrium</a></li>

<li><a href="Subgame_perfect_equilibrium" title="Subgame perfect equilibrium">Subgame perfect equilibrium</a></li>

<li><a href="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="Strategy_(game_theory)" title="Strategy (game theory)">Strategies</a></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="Appeasement" title="Appeasement">Appeasement</a></li>

<li><a href="Bid_shading" title="Bid shading">Bid shading</a></li>

<li><a href="Cheap_talk" title="Cheap talk">Cheap talk</a></li>

<li><a href="Collusion" title="Collusion">Collusion</a></li>

<li><a href="Commitment_device" title="Commitment device">Commitment device</a></li>

<li><a href="De-escalation" title="De-escalation">De-escalation</a></li>

<li><a href="Deterrence_theory" title="Deterrence theory">Deterrence</a></li>

<li><a href="Conflict_escalation" title="Conflict escalation">Escalation</a></li>

<li><a href="Fictitious_play" title="Fictitious play">Fictitious play</a></li>

<li><a href="Focal_point_(game_theory)" title="Focal point (game theory)">Focal point</a></li>

<li><a href="Grim_trigger" title="Grim trigger">Grim trigger</a></li>

<li><a href="Hobbesian_trap" title="Hobbesian trap">Hobbesian trap</a></li>

<li><a href="Markov_strategy" title="Markov strategy">Markov strategy</a></li>

<li><a href="Max-dominated_strategy" title="Max-dominated strategy">Max-dominated strategy</a></li>

<li><a href="Strategy_(game_theory)#Mixed_strategy" title="Strategy (game theory)">Mixed strategy</a></li>

<li><a href="Strategy_(game_theory)" title="Strategy (game theory)">Pure strategy</a></li>

<li><a href="Win%E2%80%93stay%2C_lose%E2%80%93switch" title="Win–stay, lose–switch">Win–stay, lose–switch</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="All-pay_auction" title="All-pay auction">All-pay auction</a></li>

<li><a href="Battle_of_the_sexes_(game_theory)" title="Battle of the sexes (game theory)">Battle of the sexes</a></li>

<li><a href="Bargaining_problem" class="mw-redirect" title="Bargaining problem">Nash bargaining game</a></li>

<li><a href="Bertrand_competition" title="Bertrand competition">Bertrand competition</a></li>

<li><a href="Blotto_game" title="Blotto game">Blotto game</a></li>

<li><a href="Centipede_game" title="Centipede game">Centipede game</a></li>

<li><a href="Coordination_game" title="Coordination game">Coordination game</a></li>

<li><a href="Cournot_competition" title="Cournot competition">Cournot competition</a></li>

<li><a href="Deadlock_(game_theory)" title="Deadlock (game theory)">Deadlock</a></li>

<li><a href="Dictator_game" title="Dictator game">Dictator game</a></li>

<li><a href="Dictator_game#Trust_game" title="Dictator game">Trust game</a></li>

<li><a href="Unscrupulous_diner's_dilemma" title="Unscrupulous diner's dilemma">Diner's dilemma</a></li>

<li><a href="Dollar_auction" title="Dollar auction">Dollar auction</a></li>

<li><a href="El_Farol_Bar_problem" title="El Farol Bar problem">El Farol Bar problem</a></li>

<li><a href="Electronic_mail_game" title="Electronic mail game">Electronic mail game</a></li>

<li><a href="Gift-exchange_game" title="Gift-exchange game">Gift-exchange game</a></li>

<li><a href="Guess_2/3_of_the_average" title="Guess 2/3 of the average">Guess 2/3 of the average</a></li>

<li><a href="Keynesian_beauty_contest" title="Keynesian beauty contest">Keynesian beauty contest</a></li>

<li><a href="Kuhn_poker" title="Kuhn poker">Kuhn poker</a></li>

<li><a href="Lewis_signaling_game" title="Lewis signaling game">Lewis signaling game</a></li>

<li><a href="Matching_pennies" title="Matching pennies">Matching pennies</a></li>

<li><a href="Obligationes" title="Obligationes">Obligationes</a></li>

<li><a href="Optional_prisoner's_dilemma" title="Optional prisoner's dilemma">Optional prisoner's dilemma</a></li>

<li><a href="Pirate_game" title="Pirate game">Pirate game</a></li>

<li><a href="Prisoner's_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li>

<li><a href="Public_goods_game" title="Public goods game">Public goods game</a></li>

<li><a href="Rendezvous_problem" title="Rendezvous problem">Rendezvous problem</a></li>

<li><a href="Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a></li>

<li><a href="Stackelberg_competition" title="Stackelberg competition">Stackelberg competition</a></li>

<li><a href="Traveler's_dilemma" title="Traveler's dilemma">Traveler's dilemma</a></li>

<li><a href="Ultimatum_game" title="Ultimatum game">Ultimatum game</a></li>

<li><a href="Volunteer's_dilemma" title="Volunteer's dilemma">Volunteer's dilemma</a></li>

<li><a href="War_of_attrition_(game)" title="War of attrition (game)">War of attrition</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Game_theory#Theorems" title="Game theory">Theorems</a></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="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a></li>

<li><a href="Aumann's_agreement_theorem" title="Aumann's agreement theorem">Aumann's agreement theorem</a></li>

<li><a href="Brouwer_fixed-point_theorem" title="Brouwer fixed-point theorem">Brouwer fixed-point theorem</a></li>

<li><a href="Competitive_altruism" title="Competitive altruism">Competitive altruism</a></li>

<li><a href="Folk_theorem_(game_theory)" title="Folk theorem (game theory)">Folk theorem</a></li>

<li><a href="Gibbard%E2%80%93Satterthwaite_theorem" title="Gibbard–Satterthwaite theorem">Gibbard–Satterthwaite theorem</a></li>

<li><a href="Gibbs_lemma" title="Gibbs lemma">Gibbs lemma</a></li>

<li><a href="Glicksberg's_theorem" title="Glicksberg's theorem">Glicksberg's theorem</a></li>

<li><a href="Kakutani_fixed-point_theorem" title="Kakutani fixed-point theorem">Kakutani fixed-point theorem</a></li>

<li><a href="Kuhn's_theorem" title="Kuhn's theorem">Kuhn's theorem</a></li>

<li><a href="One-shot_deviation_principle" title="One-shot deviation principle">One-shot deviation principle</a></li>

<li><a href="Prim%E2%80%93Read_theory" title="Prim–Read theory">Prim–Read theory</a></li>

<li><a href="Rational_ignorance" title="Rational ignorance">Rational ignorance</a></li>

<li><a href="Rational_irrationality" title="Rational irrationality">Rational irrationality</a></li>

<li><a href="Sperner's_lemma" title="Sperner's lemma">Sperner's lemma</a></li>

<li><a href="Zermelo's_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%">Subfields</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="Algorithmic_game_theory" title="Algorithmic game theory">Algorithmic game theory</a></li>

<li><a href="Behavioral_game_theory" title="Behavioral game theory">Behavioral game theory</a></li>

<li><a href="Behavioral_strategy" title="Behavioral strategy">Behavioral strategy</a></li>

<li><a href="Compositional_game_theory" title="Compositional game theory">Compositional game theory</a></li>

<li><a href="Contract_theory" title="Contract theory">Contract theory</a></li>

<li><a href="Drama_theory" title="Drama theory">Drama theory</a></li>

<li><a href="Graphical_game_theory" title="Graphical game theory">Graphical game theory</a></li>

<li><a href="Heresthetic" title="Heresthetic">Heresthetic</a></li>

<li><a href="Mean-field_game_theory" title="Mean-field game theory">Mean-field game theory</a></li>

<li><a href="Negotiation_theory" title="Negotiation theory">Negotiation theory</a></li>

<li><a href="Quantum_game_theory" title="Quantum game theory">Quantum game theory</a></li>

<li><a href="Social_software_(research_field)" title="Social software (research field)">Social software</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key people</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="Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a></li>

<li><a href="Alvin_E._Roth" title="Alvin E. Roth">Alvin E. Roth</a></li>

<li><a href="Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li>

<li><a href="Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Antoine Augustin Cournot</a></li>

<li><a href="Ariel_Rubinstein" title="Ariel Rubinstein">Ariel Rubinstein</a></li>

<li><a href="David_Gale" title="David Gale">David Gale</a></li>

<li><a href="David_K._Levine" title="David K. Levine">David K. Levine</a></li>

<li><a href="David_M._Kreps" title="David M. Kreps">David M. Kreps</a></li>

<li><a href="Donald_B._Gillies" title="Donald B. Gillies">Donald B. Gillies</a></li>

<li><a href="Drew_Fudenberg" title="Drew Fudenberg">Drew Fudenberg</a></li>

<li><a href="Eric_Maskin" title="Eric Maskin">Eric Maskin</a></li>

<li><a href="Harold_W._Kuhn" title="Harold W. Kuhn">Harold W. Kuhn</a></li>

<li><a href="Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a></li>

<li><a href="Herbert_Scarf" title="Herbert Scarf">Herbert Scarf</a></li>

<li><a href="Herv%C3%A9_Moulin" title="Hervé Moulin">Hervé Moulin</a></li>

<li><a href="Jean_Tirole" title="Jean Tirole">Jean Tirole</a></li>

<li><a href="Jean-Fran%C3%A7ois_Mertens" title="Jean-François Mertens">Jean-François Mertens</a></li>

<li><a href="Jennifer_Tour_Chayes" title="Jennifer Tour Chayes">Jennifer Tour Chayes</a></li>

<li><a href="Ken_Binmore" class="mw-redirect" title="Ken Binmore">Ken Binmore</a></li>

<li><a href="Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a></li>

<li><a href="Leonid_Hurwicz" title="Leonid Hurwicz">Leonid Hurwicz</a></li>

<li><a href="Lloyd_Shapley" title="Lloyd Shapley">Lloyd Shapley</a></li>

<li><a href="Martin_Shubik" title="Martin Shubik">Martin Shubik</a></li>

<li><a href="Melvin_Dresher" title="Melvin Dresher">Melvin Dresher</a></li>

<li><a href="Merrill_M._Flood" title="Merrill M. Flood">Merrill M. Flood</a></li>

<li><a href="Olga_Bondareva" title="Olga Bondareva">Olga Bondareva</a></li>

<li><a href="Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a></li>

<li><a href="Paul_Milgrom" title="Paul Milgrom">Paul Milgrom</a></li>

<li><a href="Peyton_Young" title="Peyton Young">Peyton Young</a></li>

<li><a href="Reinhard_Selten" title="Reinhard Selten">Reinhard Selten</a></li>

<li><a href="Robert_Aumann" title="Robert Aumann">Robert Aumann</a></li>

<li><a href="Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li>

<li><a href="Robert_B._Wilson" title="Robert B. Wilson">Robert B. Wilson</a></li>

<li><a href="Roger_Myerson" title="Roger Myerson">Roger Myerson</a></li>

<li><a href="Samuel_Bowles_(economist)" title="Samuel Bowles (economist)">Samuel Bowles</a></li>

<li><a href="Suzanne_Scotchmer" title="Suzanne Scotchmer">Suzanne Scotchmer</a></li>

<li><a href="Thomas_Schelling" title="Thomas Schelling">Thomas Schelling</a></li>

<li><a href="William_Vickrey" title="William Vickrey">William Vickrey</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_game_theory698" style="font-size:114%;margin:0 4em"><a href="Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Core<br>concepts</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="Combinatorial_explosion" title="Combinatorial explosion">Combinatorial explosion</a></li>

<li><a href="Determinacy" title="Determinacy">Determinacy</a></li>

<li><a href="Disjunctive_sum" title="Disjunctive sum">Disjunctive sum</a></li>

<li><a href="First-player_and_second-player_win" title="First-player and second-player win">First-player and second-player win</a></li>

<li><a href="Game_complexity" title="Game complexity">Game complexity</a></li>

<li><a href="Impartial_game" title="Impartial game">Impartial game</a></li>

<li><a href="Mis%C3%A8re" title="Misère">Misère</a></li>

<li><a href="Partisan_game" title="Partisan game">Partisan game</a></li>

<li><a href="Solved_game" title="Solved game">Solved game</a></li>

<li><a href="Sprague%E2%80%93Grundy_theorem" title="Sprague–Grundy theorem">Sprague–Grundy theorem</a></li>

<li><a href="Strategy-stealing_argument" title="Strategy-stealing argument">Strategy-stealing argument</a></li>

<li><a href="Zugzwang" title="Zugzwang">Zugzwang</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Games</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="Chess" title="Chess">Chess</a></li>

<li><a href="Chomp" title="Chomp">Chomp</a></li>

<li><a href="Clobber" title="Clobber">Clobber</a></li>

<li><a href="Domineering" title="Domineering">Domineering</a></li>

<li><a href="Hackenbush" title="Hackenbush">Hackenbush</a></li>

<li><a href="Notakto" title="Notakto">Notakto</a></li>

<li><a href="Subtract_a_square" title="Subtract a square">Subtract a square</a></li>

<li><a href="Sylver_coinage" title="Sylver coinage">Sylver coinage</a></li>

<li><a href="Toads_and_Frogs" title="Toads and Frogs">Toads and Frogs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematical<br>tools</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Nimber" title="Nimber">Nimber</a></li>

<li><a href="On_Numbers_and_Games" title="On Numbers and Games">On Numbers and Games</a></li>

<li><a href="Surreal_number" title="Surreal number">Surreal number</a></li>

<li><a href="Winning_Ways_for_Your_Mathematical_Plays" title="Winning Ways for Your Mathematical Plays">Winning Ways for Your Mathematical Plays</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Search<br>algorithms</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="Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li>

<li><a href="Expectiminimax" title="Expectiminimax">Expectiminimax</a></li>

<li><a href="Minimax" title="Minimax">Minimax</a></li>

<li><a href="Monte_Carlo_tree_search" title="Monte Carlo tree search">Monte Carlo tree search</a></li>

<li><a href="Negamax" title="Negamax">Negamax</a></li>

<li><a href="Paranoid_algorithm" title="Paranoid algorithm">Paranoid algorithm</a></li>

<li><a href="Principal_variation_search" title="Principal variation search">Principal variation search</a></li></ul>

<ul><li><a href="Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li>

<li><a href="John_Conway" class="mw-redirect" title="John Conway">John Conway</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Evolutionary_game_theory698" style="font-size:114%;margin:0 4em"><a href="Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Core<br>concepts</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="Bishop%E2%80%93Cannings_theorem" title="Bishop–Cannings theorem">Bishop–Cannings theorem</a></li>

<li><a href="Evolution_and_the_Theory_of_Games" title="Evolution and the Theory of Games">Evolution and the Theory of Games</a></li>

<li><a href="Evolutionarily_stable_set" title="Evolutionarily stable set">Evolutionarily stable set</a></li>

<li><a href="Evolutionarily_stable_state" title="Evolutionarily stable state">Evolutionarily stable state</a></li>

<li><a href="Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">Evolutionarily stable strategy</a></li>

<li><a href="Replicator_equation" title="Replicator equation">Replicator equation</a></li>

<li><a href="Risk_dominance" title="Risk dominance">Risk dominance</a></li>

<li><a href="Stochastically_stable_equilibrium" title="Stochastically stable equilibrium">Stochastically stable equilibrium</a></li>

<li><a href="Weak_evolutionarily_stable_strategy" title="Weak evolutionarily stable strategy">Weak evolutionarily stable strategy</a></li></ul>

<ul><li><a href="Chicken_(game)" title="Chicken (game)">Chicken</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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="Cultural_group_selection" title="Cultural group selection">Cultural group selection</a></li>

<li><a href="Fisher's_principle" title="Fisher's principle">Fisher's principle</a></li>

<li><a href="Mobbing_(animal_behavior)" title="Mobbing (animal behavior)">Mobbing</a></li>

<li><a href="Terminal_investment_hypothesis" title="Terminal investment hypothesis">Terminal investment hypothesis</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key people</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="John_Maynard_Smith" title="John Maynard Smith">John Maynard Smith</a></li>

<li><a href="Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Mechanism_design698" style="font-size:114%;margin:0 4em"><a href="Mechanism_design" title="Mechanism design">Mechanism design</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Core<br>concepts</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="Algorithmic_mechanism_design" title="Algorithmic mechanism design">Algorithmic mechanism design</a></li>

<li><a href="Bayesian-optimal_mechanism" title="Bayesian-optimal mechanism">Bayesian-optimal mechanism</a></li>

<li><a href="Incentive_compatibility" title="Incentive compatibility">Incentive compatibility</a></li>

<li><a href="Market_design" title="Market design">Market design</a></li>

<li><a href="Monotonicity_(mechanism_design)" title="Monotonicity (mechanism design)">Monotonicity</a></li>

<li><a href="Participation_constraint_(mechanism_design)" title="Participation constraint (mechanism design)">Participation constraint</a></li>

<li><a href="Revelation_principle" title="Revelation principle">Revelation principle</a></li>

<li><a href="Strategyproofness" title="Strategyproofness">Strategyproofness</a></li>

<li><a href="Vickrey%E2%80%93Clarke%E2%80%93Groves_mechanism" title="Vickrey–Clarke–Groves mechanism">Vickrey–Clarke–Groves mechanism</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" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Myerson%E2%80%93Satterthwaite_theorem" title="Myerson–Satterthwaite theorem">Myerson–Satterthwaite theorem</a></li>

<li><a href="Revenue_equivalence" title="Revenue equivalence">Revenue equivalence</a></li></ul>

<ul><li><a href="Digital_goods_auction" title="Digital goods auction">Digital goods auction</a></li>

<li><a href="Knapsack_auction" title="Knapsack auction">Knapsack auction</a></li>

<li><a href="Truthful_cake-cutting" title="Truthful cake-cutting">Truthful cake-cutting</a></li></ul>

<ul><li><a href="Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand paradox</a></li>

<li><a href="Chainstore_paradox" title="Chainstore paradox">Chainstore paradox</a></li>

<li><a href="Computational_complexity_of_games" class="mw-redirect" title="Computational complexity of games">Computational complexity of games</a></li>

<li><a href="Helly_metric" title="Helly metric">Helly metric</a></li>

<li><a href="Multi-agent_system" title="Multi-agent system">Multi-agent system</a></li>

<li><a href="PPAD_(complexity)" title="PPAD (complexity)">PPAD-complete</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Game_theory" class="extiw external" title="commons:Category:Game theory">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span><b>WikiProject</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-07-11" href="https://en.wikipedia.org/wiki/?title=Bayesian_game&oldid=1300018430">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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