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<title>Proper equilibrium</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Proper equilibrium</span></span>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="border-bottom: solid sienna 1px; font-size: 125%;">Proper equilibrium</th></tr><tr><td colspan="2" class="infobox-subheader"><a href="Solution_concept" title="Solution concept">Solution concept</a> in <a href="Game_theory" title="Game theory">game theory</a></td></tr><tr><th colspan="2" class="infobox-header" style="background-color: #fffdec; border-top: solid silver 1px; border-bottom: solid silver 1px;">Relationship</th></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Subset of</th><td class="infobox-data"><a href="Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">Trembling hand perfect equilibrium</a></td></tr><tr><th colspan="2" class="infobox-header" style="background-color: #fffdec; border-top: solid silver 1px; border-bottom: solid silver 1px;">Significance</th></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Proposed by</th><td class="infobox-data"><a href="Roger_B._Myerson" class="mw-redirect" title="Roger B. Myerson">Roger B. Myerson</a></td></tr></tbody></table>

<p><b>Proper equilibrium</b> in game theory is a refinement of <a href="Nash_Equilibrium" class="mw-redirect" title="Nash Equilibrium">Nash Equilibrium</a> by <a href="Roger_B._Myerson" class="mw-redirect" title="Roger B. Myerson">Roger B. Myerson</a>.

Proper equilibrium further refines <a href="Reinhard_Selten" title="Reinhard Selten">Reinhard Selten</a>'s notion of a

<a href="Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">trembling hand perfect equilibrium</a> by assuming that more costly trembles are made with

significantly smaller probability than less

costly ones.

</p>

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

<p>Given a <a href="Normal_form_(abstract_rewriting)" title="Normal form (abstract rewriting)">normal form</a> game and a parameter <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 \epsilon >0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon >0}</annotation>

</semantics>

</math></span><img src="./568095ad3924314374a5ab68fae17343661f2a71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.205ex; height:2.176ex;" alt="{\displaystyle \epsilon >0}" loading="lazy"></span>, a <a href="Mixed_strategy" class="mw-redirect" title="Mixed strategy">totally mixed</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 }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"></span> is defined to be <b><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 \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></span>-proper</b> if, whenever a player has two pure strategies s and s' such that the expected payoff of playing s is smaller than the expected payoff of

playing s' (that is <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 u(s,\sigma _{-i})<u(s',\sigma _{-i})}">

<msub>

</mrow>

</msub>

<msup>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(s,\sigma _{-i})<u(s',\sigma _{-i})}</annotation>

</semantics>

</math></span><img src="./4061fb5f3cb3e34dfe249bb92202f89d60896b23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.121ex; height:3.009ex;" alt="{\displaystyle u(s,\sigma _{-i})<u(s',\sigma _{-i})}" loading="lazy"></span>), then the probability assigned to s

is at most <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 \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</p><p>The strategy profile of the game is said to be a proper equilibrium

if it is a limit point, as <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 \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></span> approaches 0, of a sequence of <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 \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</p>

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

<p>The game to the right is a variant of <a href="Matching_Pennies" class="mw-redirect" title="Matching Pennies">Matching Pennies</a>.

</p>

<caption align="bottom"><i>Matching Pennies with a twist</i>

</caption>

<td>

</td>

<th><i>Guess heads up</i>

</th>

<th><i>Guess tails up</i>

</th>

<th><i>Grab penny</i>

</th></tr>

<tr>

<th><i>Hide heads up</i>

</th>

<td align="center">−1, 1

</td>

<td align="center">0, 0

</td>

<td align="center">−1, 1

</td></tr>

<tr>

<th><i>Hide tails up</i>

</th>

<td align="center">0, 0

</td>

<td align="center">−1, 1

</td>

<td align="center">−1, 1

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

<p>Player 1 (row player) hides a

penny and if Player 2 (column player) guesses correctly whether it is heads up or tails up, he gets the penny. In

this variant, Player 2 has a third option: grabbing the penny without guessing.

The <a href="Nash_equilibrium" title="Nash equilibrium">Nash equilibria</a> of the game are the strategy profiles where Player 2 grabs the penny

with probability 1. Any mixed strategy of Player 1 is in (Nash) equilibrium with this pure strategy

of Player 2. Any such pair is even <a href="Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">trembling hand perfect</a>.

Intuitively, since Player 1 expects Player 2 to grab the penny, he is not concerned about

leaving Player 2 uncertain about whether it is heads up or tails up. However, it can be seen

that the unique proper equilibrium of this game is the one where Player 1 hides the penny heads up with probability 1/2 and tails up with probability 1/2 (and Player 2 grabs the penny).

This unique proper equilibrium can be motivated

intuitively as follows: Player 1 fully expects Player 2 to grab the penny.

However, Player 1 still prepares for the unlikely event that Player 2 does not grab the

penny and instead for some reason decides to make a guess. Player 1 prepares for this event by

making sure that Player 2 has no information about whether the penny is heads up or tails up,

exactly as in the original <a href="Matching_Pennies" class="mw-redirect" title="Matching Pennies">Matching Pennies</a> game.

</p>

<div class="mw-heading mw-heading2"><h2 id="Proper_equilibria_of_extensive_games">Proper equilibria of extensive games</h2></div>

<p>One may apply the properness notion to <a href="Extensive_form_game" class="mw-redirect" title="Extensive form game">extensive form games</a> in two different ways, completely analogous to the

two different ways <a href="Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">trembling hand perfection</a>

is applied to extensive games. This leads to the notions of <b>normal form proper equilibrium</b>

and <b>extensive form proper equilibrium</b> of an extensive form game. It was shown by van

Damme that a normal form proper equilibrium of an extensive form game is behaviorally equivalent to

a <a href="Quasi-perfect_equilibrium" title="Quasi-perfect equilibrium">quasi-perfect equilibrium</a> of that game.

</p>

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

<ul><li>Roger B. Myerson. <a rel="nofollow" class="external text" href="https://www.kellogg.northwestern.edu/research/math/papers/295.pdf">Refinements of the Nash equilibrium concept</a>. <i>International Journal of Game Theory</i>, 15:133-154, 1978.</li>

<li><a href="Eric_van_Damme" title="Eric van Damme">Eric van Damme</a>. "<a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF01769861">A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games</a>." <i>International Journal of Game Theory</i> 13:1--13, 1984.</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="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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