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<title>Equilibrium selection</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Equilibrium selection</span></span>

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<p><b>Equilibrium selection</b> is a concept from <a href="Game_theory" title="Game theory">game theory</a> which seeks to address reasons for players of a game to select a certain equilibrium over another. The concept is especially relevant in <a href="Evolutionary_game_theory" title="Evolutionary game theory">evolutionary game theory</a>, where the different methods of equilibrium selection respond to different ideas of what equilibria will be stable and persistent for one player to play even in the face of deviations (and mutations) of the other players. This is important because there are various <a href="Equilibrium_concept" class="mw-redirect" title="Equilibrium concept">equilibrium concepts</a>, and for many particular concepts, such as the <a href="Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a>, many games have multiple equilibria.

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<div class="mw-heading mw-heading2"><h2 id="Equilibrium_Selection_with_Repeated_Games">Equilibrium Selection with Repeated Games</h2></div>

<p>A stage game is an <a href="N-player_game" title="N-player game">n-player game</a> where players choose from a finite set of actions, and there is a payoff profile for their choices. A <a href="Repeated_game" title="Repeated game">repeated game</a> is playing a number of repetitions of a stage game in discrete periods of time (Watson, 2013). A player's reputation affects the actions and behavior of the other players. In other words, how a player behaves in preceding rounds determines the actions of their opponents in subsequent rounds. An example is the interaction between an employee and an employer where the employee shirks their responsibility for a short-term gain then loses on the bonus that the employer discontinues after observing the employee's behavior (Watson, 2013). The dynamics of equilibrium selection for repeated games can be illustrated with a two-period game. With every action from the players in one period, a new subgame is initiated based on that action profile. For the Nash Equilibrium of the entire game, a subgame perfect equilibrium of every game is required. Hence, in the last period of a repeated game, the players will choose a stage game Nash Equilibrium. Equilibria stipulation actions that are not Nash Equilibrium strategies in the stage game are still supported. This can be achieved by establishing a reputation of "cooperation" in the preceding periods that leads to the opponent selecting a more favorable Nash Equilibrium strategy in the final period. If a player builds a reputation of deviating instead of co-operating, then the opponent can "punish" the player by choosing a less favorable Equilibrium in the final period of the repeated game.

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<div class="mw-heading mw-heading2"><h2 id="Focal_point">Focal point</h2></div>

<p>Another concept that can help to select an equilibrium is <a href="Focal_point_(game_theory)" title="Focal point (game theory)">focal point</a>. This concept was first introduced by Thomas Schelling, a Nobel-winning game theorist, in his book The Strategy of Conflict in 1960 (Schelling, 1960). When the participants are in a coordinate game where the players do not have a chance to discuss their strategies beforehand, focal point is a solution that somehow stands out as the natural answer.  

</p><p>For example, in an experiment conducted in 1990 by Mehta et al. (1994), the researchers let the participants answer a questionnaire, which contained the question "name a year" or "name a city in England". The participants were asked to provide the first answer that came to their minds, and many provided their birth year or hometown city.

</p><p>However, when they had the incentive to coordinate - the participants were told they would be paid if they managed to answer the question the same way as an anonymous partner - most of them chose 1990 (the year at the time) and London (the largest city in the UK). These are not the first answers that came to their minds, but they are the best bets after deliberation while trying to find a partner without prior discussion. In this case, the year 1990, or the city of London, is the focal point of this game, to help the players to select the best equilibrium in this coordinate game.

</p><p>Besides, even in the situation where the game players are allowed to communicate with each other, such as negotiation, focal point can still be useful for them selecting an appropriate equilibrium: When the negotiation is about to the end, each player must make the last-minute decision about how aggressive they should be and to what extent they should trust their opponents (Hyde, 2017).

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

<p>Various authors have studied games with <a href="Symmetric_game" title="Symmetric game">symmetries</a> (over players or actions), especially in the case of fully cooperative games. For example, the team game <a href="Hanabi_(card_game)" title="Hanabi (card game)">Hanabi</a> has been considered, in which the different players and suits are symmetric. Relatedly, some authors have considered how equilibrium selection relates between isomorphic games. Generally, it has been argued that a group of players shouldn't choose strategies that arbitrarily break these symmetries, i.e., should play symmetric strategies to aim for symmetric equilibria.<sup id="cite_ref-Hu2020_1-0" class="reference"><a href="#cite_note-Hu2020-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Similarly they should play isomorphic games isomorphically.<sup id="cite_ref-Harsanyi1988_2-0" class="reference"><a href="#cite_note-Harsanyi1988-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> For example, in Hanabi, hints corresponding to the different suits should have analogous (symmetric) meanings. In team games, the optimal symmetric strategy is also a (symmetric) Nash equilibrium.<sup id="cite_ref-Emmons2022_3-0" class="reference"><a href="#cite_note-Emmons2022-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> While breaking symmetries may allow for higher utilities, it results in unnatural, inhuman strategies.<sup id="cite_ref-Hu2020_1-1" class="reference"><a href="#cite_note-Hu2020-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Every finite symmetric game (including non-team games) has a symmetric Nash equilibrium.<sup id="cite_ref-Nash1951_4-0" class="reference"><a href="#cite_note-Nash1951-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Examples_of_equilibrium_selection_concepts">Examples of equilibrium selection concepts</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Risk_&_Payoff_dominance">Risk & Payoff dominance</h3></div>

<p><b>Definition</b>: Consider a situation where a game has multiple Nash equilibria (NE), the equilibrium can be classified into two categories:

</p>

<ul><li>A Nash equilibrium is considered <a href="Risk_dominance" title="Risk dominance">risk dominant</a> if it has the largest basin of attraction (i.e. is less risky).</li>

<li>A Nash equilibrium is considered <a href="Risk_dominance" title="Risk dominance">payoff dominant</a> if it is Pareto superior to all other Nash equilibria in the game.</li></ul>

<p><b>Explanation:</b> A risk dominant NE is chosen when the player wants to avoid big losses while a payoff dominant NE is considered for an optimal payoff solution. Note that either the risk dominant or payoff dominant is a typical type of NE.

</p><p><b>Example:</b> Take the normal form payoff matrix of a game as an example:

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<caption><b>Table1: Normal form of the example game</b>

</caption>

<th>

</th>

<th>L

</th>

<th>R

</th></tr>

<tr>

<td>U

</td>

<td>10, 10

</td>

<td>0, 9

</td></tr>

<tr>

<td>D

</td>

<td>9, 0

</td>

<td>5, 5

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

<p>There are two NEs in this game, i.e. (U, L) and (D,R). Here (U, L) is a payoff dominant NE since such a strategy can return an optimal overall payoff. However, given the uncertainty of an opponent's action, one of the players may consider a more conservative strategy (D for player 1 and R for player two), which can avoid the situation of "a great loss", i.e. getting 0 payoff once the opponent deviates. Hence the (D, R) is a risk dominant NE.

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<div class="mw-heading mw-heading3"><h3 id="1/2_dominance">1/2 dominance</h3></div>

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

<li><a href="Equilibrium_refinement" class="mw-redirect" title="Equilibrium refinement">Equilibrium refinement</a></li></ul>

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

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</style><cite id="CITEREFHuLererPeysakhovichFoerster2020" class="citation conference cs1">Hu, Hengyuan; Lerer, Adam; Peysakhovich, Alex; Foerster, Jakob (2020). "'Other-Play' for Zero-Shot Coordination". <i>Proceedings of the 37th <a href="International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">International Conference on Machine Learning</a></i>. <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/2003.02979">2003.02979</a></span>.</cite></span>

</li>

<li id="cite_note-Harsanyi1988-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Harsanyi1988_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFHarsanyiSelten1988" class="citation book cs1">Harsanyi, John C.; Selten, Reinhard (1988). <i>A General Theory of Equilibrium Selection</i>. MIT Press.</cite></span>

</li>

<li id="cite_note-Emmons2022-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Emmons2022_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFEmmonsOesterheldCritchConitzer2022" class="citation conference cs1">Emmons, Scott; Oesterheld, Caspar; Critch, Andrew; Conitzer, Vincent; Russell, Stuart (2022). "For Learning in Symmetric Teams, Local Optima are Global Nash Equilibria". <i>Proceedings of the 39th <a href="International_Conference_on_Machine_Learning" title="International Conference on Machine Learning">International Conference on Machine Learning</a></i>. pp. <span class="nowrap">5924–</span>5943.</cite></span>

</li>

<li id="cite_note-Nash1951-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nash1951_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFNash1951" class="citation journal cs1">Nash, John (1951). "Non-cooperative games". <i>Annals of Mathematics</i>. <b>54</b> (2): <span class="nowrap">286–</span>295. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969529">10.2307/1969529</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969529">1969529</a>.</cite></span>

</li>

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

<ul><li>Harsanyi, John C. and Selten, Reinhard, <i>A General Theory of Equilibrium Selection in Games</i>, MIT Press (1988)</li>

<li>Watson, J. (2013). Repeated Games and Reputation. In Strategy: An introduction to game theory (3rd ed., pp. 291–305). essay, Norton & Company.</li>

<li>Hyde, T., 2017. Can Schelling's focal points help us understand high-stakes negotiations?. [online] Aeaweb.org. Available at: <https://www.aeaweb.org/research/can-schellings-focal-points-help-us-understand-high-stakes-negotiations> [Accessed 9 December 2021].</li>

<li>Mehta, J., Starmer, C., & Sugden, R. (1994). The Nature of Salience: An Experimental Investigation of Pure Coordination Games. The American Economic Review, 84(3), 658–673. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2118074">2118074</a></li>

<li>Schelling, T. C. (1960). The strategy of conflict. Cambridge, Mass.</li></ul>

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<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_game" title="Bayesian game">Bayesian game</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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