Zero-sum, Game

In <a title="Game theory" href="https://en.wikipedia.org/wiki/Game_theory">game theory</a> and <a class="mw-redirect" title="Economic theory" href="https://en.wikipedia.org/wiki/Economic_theory">economic theory</a>, a zero-sum game is a <a title="Mathematical model" href="https://en.wikipedia.org/wiki/Mathematical_model">mathematical representation</a> of a situation in which each participant's gain or loss of <a title="Utility" href="https://en.wikipedia.org/wiki/Utility">utility</a> is exactly balanced by the losses or gains of the utility of the other participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, <a title="Fair cake-cutting" href="https://en.wikipedia.org/wiki/Fair_cake-cutting">cutting a cake</a>, where taking a larger piece reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of cake equally (see <a title="Marginal utility" href="https://en.wikipedia.org/wiki/Marginal_utility">marginal utility</a>).

In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. A zero-sum game is also called a strictly competitive game while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the <a title="Minimax theorem" href="https://en.wikipedia.org/wiki/Minimax_theorem">minimax theorem</a> which is closely related to <a class="mw-redirect" title="LP duality" href="https://en.wikipedia.org/wiki/LP_duality">linear programming duality</a>,<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-Binmore2007-1">[1]</a> or with <a title="Nash equilibrium" href="https://en.wikipedia.org/wiki/Nash_equilibrium">Nash equilibrium</a>.

<h2>Contents</h2>

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<ul>

<li class="toclevel-1 tocsection-1"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Definition">1Definition</a></li>

<li class="toclevel-1 tocsection-2"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Solution">2Solution</a>

<ul>

<li class="toclevel-2 tocsection-3"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Example">2.1Example</a></li>

<li class="toclevel-2 tocsection-4"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Solving">2.2Solving</a></li>

<li class="toclevel-2 tocsection-5"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Universal_solution">2.3Universal solution</a></li>

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</li>

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<li class="toclevel-2 tocsection-7"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Psychology">3.1Psychology</a></li>

<li class="toclevel-2 tocsection-8"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Complexity">3.2Complexity</a></li>

</ul>

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<li class="toclevel-1 tocsection-9"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Extensions">4Extensions</a></li>

<li class="toclevel-1 tocsection-10"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Misunderstandings">5Misunderstandings</a></li>

<li class="toclevel-1 tocsection-11"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Zero-sum_thinking">6Zero-sum thinking</a></li>

<li class="toclevel-1 tocsection-13"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#References">8References</a></li>

<li class="toclevel-1 tocsection-14"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#Further_reading">9Further reading</a></li>

<li class="toclevel-1 tocsection-15"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#External_links">10External links</a></li>

</ul>

</div>

<h2>Definition[<a title="Edit section: Definition" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=1">edit</a>]</h2>

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<td>Choice 1Choice</td>

<td>2Choice</td>

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<td>B, -BChoice</td>

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<td>, DGeneric</td>

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The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is <a class="mw-redirect" title="Pareto optimal" href="https://en.wikipedia.org/wiki/Pareto_optimal">Pareto optimal</a>. Generally, any game where all strategies are Pareto optimal is called a conflict game.<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-2">[2]</a>

Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation.

Situations where participants can all gain or suffer together are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with.

The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent’s payoff at a favorable cost to himself rather to prefer more than less. The punishing-the-opponent standard can be used in both zero-sum games (e.g. warfare game, Chess) and non-zero-sum games (e.g. pooling selection games).<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-3">[3]</a>

<h2>Solution[<a title="Edit section: Solution" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=2">edit</a>]</h2>

For two-player finite zero-sum games, the different <a title="Game theory" href="https://en.wikipedia.org/wiki/Game_theory">game theoretic</a> <a title="Solution concept" href="https://en.wikipedia.org/wiki/Solution_concept">solution concepts</a> of <a title="Nash equilibrium" href="https://en.wikipedia.org/wiki/Nash_equilibrium">Nash equilibrium</a>, <a title="Minimax" href="https://en.wikipedia.org/wiki/Minimax">minimax</a>, and <a class="mw-redirect" title="Maximin (decision theory)" href="https://en.wikipedia.org/wiki/Maximin_(decision_theory)">maximin</a> all give the same solution. If the players are allowed to play a <a class="mw-redirect" title="Mixed strategy" href="https://en.wikipedia.org/wiki/Mixed_strategy">mixed strategy</a>, the game always has an equilibrium.

<h3>Example[<a title="Edit section: Example" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=3">edit</a>]</h3>

<table class="wikitable"><caption align="bottom">A zero-sum gameBlueRedABC1-303010-10-2020210-10-202020-20</caption>

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A game's <a class="mw-redirect" title="Payoff matrix" href="https://en.wikipedia.org/wiki/Payoff_matrix">payoff matrix</a> is a convenient representation. Consider for example the two-player zero-sum game pictured at right or above.

The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.

Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.

Now, in this example game both players know the payoff matrix and attempt to maximize the number of their points. What should they do?

Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, while with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. But what happens if Blue anticipates Red's reasoning and choice of action 1, and goes for action B, so as to win 10 points? Or if Red, in turn, anticipates this devious trick and goes for action 2, so as to win 20 points after all?

<a title="Émile Borel" href="https://en.wikipedia.org/wiki/%C3%89mile_Borel">Émile Borel</a> and <a title="John von Neumann" href="https://en.wikipedia.org/wiki/John_von_Neumann">John von Neumann</a> had the fundamental and surprising insight that <a title="Probability" href="https://en.wikipedia.org/wiki/Probability">probability</a> provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum <a title="Expected value" href="https://en.wikipedia.org/wiki/Expected_value">expected</a> point-loss independent of the opponent's strategy. This leads to a <a title="Linear programming" href="https://en.wikipedia.org/wiki/Linear_programming">linear programming</a> problem with the optimal strategies for each player. This <a title="Minimax" href="https://en.wikipedia.org/wiki/Minimax">minimax</a> method can compute provably optimal strategies for all two-player zero-sum games.

For the example given above, it turns out that Red should choose action 1 with probability 4/7 and action 2 with probability 3/7, while Blue should assign the probabilities 0, 4/7, and 3/7 to the three actions A, B, and C. Red will then win 20/7 points on average per game.

<h3>Solving[<a title="Edit section: Solving" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=4">edit</a>]</h3>

The <a title="Nash equilibrium" href="https://en.wikipedia.org/wiki/Nash_equilibrium">Nash equilibrium</a> for a two-player, zero-sum game can be found by solving a <a title="Linear programming" href="https://en.wikipedia.org/wiki/Linear_programming">linear programming</a> problem. Suppose a zero-sum game has a payoff matrix M where element {\displaystyle M{i,j}}<img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0471da86bad57963868302a83a9decf6804d4197" alt="M{i,j}" /> is the payoff obtained when the minimizing player chooses pure strategy i and the maximizing player chooses pure strategy j (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of M is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (see ref. [2], page 740) by solving the following linear program to find a vector u:

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<dd>Minimize:

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<dd>{\displaystyle \sum {i}u{i}}<img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92bbf18fcc71c8a142918c1584f6de3f1bd5f223" alt="\sum {i}u{i}" /></dd>

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<dd>Subject to the constraints:

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The first constraint says each element of the u vector must be nonnegative, and the second constraint says each element of the M u vector must be at least 1. For the resulting u vector, the inverse of the sum of its elements is the value of the game. Multiplying u by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.

If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.

The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of M (adding a constant so it's positive), then solving the resulting game.

If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general.[<a title="Wikipedia:Citation needed" href="https://en.wikipedia.org/wiki/Wikipedia:Citation_needed">citation needed</a>]

<h3>Universal solution[<a title="Edit section: Universal solution" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=5">edit</a>]</h3>

If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or incredible after the play is started, such as Poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it’s interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-4">[4]</a>

<h3>Psychology[<a title="Edit section: Psychology" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=7">edit</a>]</h3>

The most common or simple example from the subfield of <a title="Social psychology" href="https://en.wikipedia.org/wiki/Social_psychology">social psychology</a> is the concept of "<a title="Social trap" href="https://en.wikipedia.org/wiki/Social_trap">social traps</a>". In some cases pursuing our personal interests can enhance our collective well-being, but in others personal interest results in mutually destructive behavior.

<h3>Complexity[<a title="Edit section: Complexity" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=8">edit</a>]</h3>

It has been theorized by <a title="Robert Wright (journalist)" href="https://en.wikipedia.org/wiki/Robert_Wright(journalist)">Robert Wright</a> in his book <a title="Nonzero: The Logic of Human Destiny" href="https://en.wikipedia.org/wiki/Nonzero:The_Logic_of_Human_Destiny">Nonzero: The Logic of Human Destiny</a>, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.

<h2>Extensions[<a title="Edit section: Extensions" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=9">edit</a>]</h2>

In 1944, <a title="John von Neumann" href="https://en.wikipedia.org/wiki/John_von_Neumann">John von Neumann</a> and <a title="Oskar Morgenstern" href="https://en.wikipedia.org/wiki/Oskar_Morgenstern">Oskar Morgenstern</a> proved that any non-zero-sum game for n players is equivalent to a zero-sum game with n + 1 players; the (n + 1)th player representing the global profit or loss.<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-5">[5]</a>

<h2>Misunderstandings[<a title="Edit section: Misunderstandings" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=10">edit</a>]</h2>

Zero-sum games and particularly their solutions are commonly misunderstood by critics of <a title="Game theory" href="https://en.wikipedia.org/wiki/Game_theory">game theory</a>, usually with respect to the independence and <a title="Rational choice theory" href="https://en.wikipedia.org/wiki/Rational_choice_theory">rationality</a> of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational <a title="Game" href="https://en.wikipedia.org/wiki/Game">games</a>.<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-Binmore2007-1">[1]</a>

Politics is sometimes called zero sum.<a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-6">[6]</a><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-7">[7]</a><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_note-8">[8]</a>

<h2>Zero-sum thinking[<a title="Edit section: Zero-sum thinking" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=11">edit</a>]</h2>

In psychology, <a title="Zero-sum thinking" href="https://en.wikipedia.org/wiki/Zero-sum_thinking">zero-sum thinking</a> refers to the perception that a situation is like a zero-sum game, where one person's gain is another's loss.

<ul>

<li><a title="Bimatrix game" href="https://en.wikipedia.org/wiki/Bimatrix_game">Bimatrix game</a></li>

<li><a title="Comparative advantage" href="https://en.wikipedia.org/wiki/Comparative_advantage">Comparative advantage</a></li>

<li><a title="Dutch disease" href="https://en.wikipedia.org/wiki/Dutch_disease">Dutch disease</a></li>

<li><a title="Gains from trade" href="https://en.wikipedia.org/wiki/Gains_from_trade">Gains from trade</a></li>

<li><a title="Lump of labour fallacy" href="https://en.wikipedia.org/wiki/Lump_of_labour_fallacy">Lump of labour fallacy</a></li>

<li><a class="mw-redirect" title="Positive-sum game" href="https://en.wikipedia.org/wiki/Positive-sum_game">Positive-sum game</a></li>

<li><a title="Zero-sum thinking" href="https://en.wikipedia.org/wiki/Zero-sum_thinking">Zero-sum thinking</a></li>

</ul>

<h2>References[<a title="Edit section: References" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=13">edit</a>]</h2>

<li id="cite_note-Binmore2007-1">^ <a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-Binmore2007_1-0">Jump up to:a</a> <a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-Binmore2007_1-1">b</a> <cite class="citation book"><a class="mw-redirect" title="Ken Binmore" href="https://en.wikipedia.org/wiki/Ken_Binmore">Ken Binmore</a> (2007). <a class="external text" href="https://books.google.com/books?id=eY0YhSk9ujsC" rel="nofollow">Playing for real: a text on game theory</a>. Oxford University Press US. <a title="International Standard Book Number" href="https://en.wikipedia.org/wiki/International_Standard_Book_Number">ISBN</a> <a title="Special:BookSources/978-0-19-530057-4" href="https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-530057-4">978-0-19-530057-4</a>.</cite>, chapters 1 & 7</li>

<li id="cite_note-2"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-2">Jump up^{</a> }<cite class="citation book">Bowles, Samuel (2004). Microeconomics: Behavior, Institutions, and Evolution. <a title="Princeton University Press" href="https://en.wikipedia.org/wiki/Princeton_University_Press">Princeton University Press</a>. pp. 33–36. <a title="International Standard Book Number" href="https://en.wikipedia.org/wiki/International_Standard_Book_Number">ISBN</a> <a title="Special:BookSources/0-691-09163-3" href="https://en.wikipedia.org/wiki/Special:BookSources/0-691-09163-3">0-691-09163-3</a>.</cite></li>

<li id="cite_note-3"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-3">Jump up^{</a> }Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. <a title="International Standard Book Number" href="https://en.wikipedia.org/wiki/International_Standard_Book_Number">ISBN</a> <a title="Special:BookSources/978-1507658246" href="https://en.wikipedia.org/wiki/Special:BookSources/978-1507658246">978-1507658246</a>. Chapter 1 and Chapter 4.</li>

<li id="cite_note-4"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-4">Jump up^{</a> }Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. <a title="International Standard Book Number" href="https://en.wikipedia.org/wiki/International_Standard_Book_Number">ISBN</a> <a title="Special:BookSources/978-1507658246" href="https://en.wikipedia.org/wiki/Special:BookSources/978-1507658246">978-1507658246</a>. Chapter 4.</li>

<li id="cite_note-5"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-5">Jump up^{</a> }<cite class="citation web"><a class="external text" href="https://press.princeton.edu/titles/7802.html" rel="nofollow">"Theory of Games and Economic Behavior"</a>. Princeton University Press (1953). June 25, 2005. Retrieved 2018-02-25.</cite></li>

<li id="cite_note-6"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-6">Jump up^{</a> }<cite class="citation web">Rubin, Jennifer (2013-10-04). <a class="external text" href="https://www.washingtonpost.com/blogs/right-turn/wp/2013/10/04/the-flaw-in-zero-sum-politics/" rel="nofollow">"The flaw in zero sum politics"</a>. The Washington Post. Retrieved 2017-03-08.</cite></li>

<li id="cite_note-7"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-7">Jump up^{</a> }<cite class="citation web"><a class="external text" href="https://www.economist.com/news/united-states/21595973-voters-think-both-parties-are-telling-truth-about-how-awful-other-lot-are-zero-sum" rel="nofollow">"Lexington: Zero-sum politics"</a>. The Economist. 2014-02-08. Retrieved 2017-03-08.</cite></li>

<li id="cite_note-8"><a href="https://en.wikipedia.org/wiki/Zero-sum_game#cite_ref-8">Jump up^{</a> }<cite class="citation web"><a class="external text" href="http://www.dictionary.com/browse/zero-sum-game" rel="nofollow">"Zero-sum game | Define Zero-sum game at"</a>. Dictionary.com. Retrieved 2017-03-08.</cite></li>

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<h2>Further reading[<a title="Edit section: Further reading" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=14">edit</a>]</h2>

<ul>

<li>Misstating the Concept of Zero-Sum Games within the Context of Professional Sports Trading Strategies, series Pardon the Interruption (2010-09-23) <a title="ESPN" href="https://en.wikipedia.org/wiki/ESPN">ESPN</a>, created by <a title="Tony Kornheiser" href="https://en.wikipedia.org/wiki/Tony_Kornheiser">Tony Kornheiser</a> and <a title="Michael Wilbon" href="https://en.wikipedia.org/wiki/Michael_Wilbon">Michael Wilbon</a>, performance by <a title="Bill Simmons" href="https://en.wikipedia.org/wiki/Bill_Simmons">Bill Simmons</a></li>

<li>Handbook of Game Theory - volume 2, chapter Zero-sum two-person games, (1994) <a title="Elsevier" href="https://en.wikipedia.org/wiki/Elsevier">Elsevier</a> Amsterdam, by Raghavan, T. E. S., Edited by Aumann and Hart, pp. 735–759, <a title="International Standard Book Number" href="https://en.wikipedia.org/wiki/International_Standard_Book_Number">ISBN</a> <a title="Special:BookSources/0-444-89427-6" href="https://en.wikipedia.org/wiki/Special:BookSources/0-444-89427-6">0-444-89427-6</a></li>

<li>Power:Its Forms, Bases and Uses (1997) Transaction Publishers, by <a title="Dennis Wrong" href="https://en.wikipedia.org/wiki/Dennis_Wrong">Dennis Wrong</a></li>

</ul>

<h2>External links[<a title="Edit section: External links" href="https://en.wikipedia.org/w/index.php?title=Zero-sum_game&action=edit&section=15">edit</a>]</h2>

<ul>

<li><a class="external text" href="http://www.egwald.ca/operationsresearch/twoperson.php" rel="nofollow">Play zero-sum games online</a> by Elmer G. Wiens.</li>

<li><a class="external text" href="https://web.archive.org/web/20070518035304/http://www.le.ac.uk/psychology/amc/gtaia.html" rel="nofollow">Game Theory & its Applications</a> - comprehensive text on psychology and game theory. (Contents and Preface to Second Edition.)</li>

<li><a class="external text" href="http://economics-games.com/mixed-nash" rel="nofollow">A playable zero-sum game</a> and its mixed strategy Nash equilibrium.</li>

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<div id="Topics_in_game_theory">Topics in <a title="Game theory" href="https://en.wikipedia.org/wiki/Game_theory">game theory</a></div>

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<th class="navbox-group" scope="row">Definitions</th>

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<ul>

<li><a class="mw-redirect" title="Cooperative game" href="https://en.wikipedia.org/wiki/Cooperative_game">Cooperative game</a></li>

<li><a title="Determinacy" href="https://en.wikipedia.org/wiki/Determinacy">Determinacy</a></li>

<li><a title="Escalation of commitment" href="https://en.wikipedia.org/wiki/Escalation_of_commitment">Escalation of commitment</a></li>

<li><a title="Extensive-form game" href="https://en.wikipedia.org/wiki/Extensive-form_game">Extensive-form game</a></li>

<li><a title="First-player and second-player win" href="https://en.wikipedia.org/wiki/First-player_and_second-player_win">First-player and second-player win</a></li>

<li><a title="Game complexity" href="https://en.wikipedia.org/wiki/Game_complexity">Game complexity</a></li>

<li><a title="Graphical game theory" href="https://en.wikipedia.org/wiki/Graphical_game_theory">Graphical game</a></li>

<li><a title="Hierarchy of beliefs" href="https://en.wikipedia.org/wiki/Hierarchy_of_beliefs">Hierarchy of beliefs</a></li>

<li><a title="Information set (game theory)" href="https://en.wikipedia.org/wiki/Information_set_(game_theory)">Information set</a></li>

<li><a title="Normal-form game" href="https://en.wikipedia.org/wiki/Normal-form_game">Normal-form game</a></li>

<li><a title="Preference (economics)" href="https://en.wikipedia.org/wiki/Preference_(economics)">Preference</a></li>

<li><a title="Sequential game" href="https://en.wikipedia.org/wiki/Sequential_game">Sequential game</a></li>

<li><a title="Simultaneous game" href="https://en.wikipedia.org/wiki/Simultaneous_game">Simultaneous game</a></li>

<li><a title="Simultaneous action selection" href="https://en.wikipedia.org/wiki/Simultaneous_action_selection">Simultaneous action selection</a></li>

<li><a title="Solved game" href="https://en.wikipedia.org/wiki/Solved_game">Solved game</a></li>

<li><a title="Succinct game" href="https://en.wikipedia.org/wiki/Succinct_game">Succinct game</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<th class="navbox-group" scope="row"><a title="Economic equilibrium" href="https://en.wikipedia.org/wiki/Economic_equilibrium">Equilibrium</a>
<a title="Solution concept" href="https://en.wikipedia.org/wiki/Solution_concept">concepts</a></th>

<div>

<ul>

<li><a title="Nash equilibrium" href="https://en.wikipedia.org/wiki/Nash_equilibrium">Nash equilibrium</a></li>

<li><a title="Subgame perfect equilibrium" href="https://en.wikipedia.org/wiki/Subgame_perfect_equilibrium">Subgame perfection</a></li>

<li><a title="Mertens-stable equilibrium" href="https://en.wikipedia.org/wiki/Mertens-stable_equilibrium">Mertens-stable equilibrium</a></li>

<li><a class="mw-redirect" title="Bayesian Nash equilibrium" href="https://en.wikipedia.org/wiki/Bayesian_Nash_equilibrium">Bayesian Nash equilibrium</a></li>

<li><a title="Perfect Bayesian equilibrium" href="https://en.wikipedia.org/wiki/Perfect_Bayesian_equilibrium">Perfect Bayesian equilibrium</a></li>

<li><a title="Trembling hand perfect equilibrium" href="https://en.wikipedia.org/wiki/Trembling_hand_perfect_equilibrium">Trembling hand</a></li>

<li><a title="Proper equilibrium" href="https://en.wikipedia.org/wiki/Proper_equilibrium">Proper equilibrium</a></li>

<li><a title="Epsilon-equilibrium" href="https://en.wikipedia.org/wiki/Epsilon-equilibrium">Epsilon-equilibrium</a></li>

<li><a title="Correlated equilibrium" href="https://en.wikipedia.org/wiki/Correlated_equilibrium">Correlated equilibrium</a></li>

<li><a title="Sequential equilibrium" href="https://en.wikipedia.org/wiki/Sequential_equilibrium">Sequential equilibrium</a></li>

<li><a title="Quasi-perfect equilibrium" href="https://en.wikipedia.org/wiki/Quasi-perfect_equilibrium">Quasi-perfect equilibrium</a></li>

<li><a title="Evolutionarily stable strategy" href="https://en.wikipedia.org/wiki/Evolutionarily_stable_strategy">Evolutionarily stable strategy</a></li>

<li><a title="Risk dominance" href="https://en.wikipedia.org/wiki/Risk_dominance">Risk dominance</a></li>

<li><a title="Shapley value" href="https://en.wikipedia.org/wiki/Shapley_value">Shapley value</a></li>

<li><a title="Pareto efficiency" href="https://en.wikipedia.org/wiki/Pareto_efficiency">Pareto efficiency</a></li>

<li><a title="Potential game" href="https://en.wikipedia.org/wiki/Potential_game#Bounded_Rational_Models">Gibbs equilibrium</a></li>

<li><a title="Quantal response equilibrium" href="https://en.wikipedia.org/wiki/Quantal_response_equilibrium">Quantal response equilibrium</a></li>

<li><a title="Self-confirming equilibrium" href="https://en.wikipedia.org/wiki/Self-confirming_equilibrium">Self-confirming equilibrium</a></li>

<li><a title="Strong Nash equilibrium" href="https://en.wikipedia.org/wiki/Strong_Nash_equilibrium">Strong Nash equilibrium</a></li>

<li><a title="Markov perfect equilibrium" href="https://en.wikipedia.org/wiki/Markov_perfect_equilibrium">Markov perfect equilibrium</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<th class="navbox-group" scope="row"><a title="Strategy (game theory)" href="https://en.wikipedia.org/wiki/Strategy_(game_theory)">Strategies</a></th>

<div>

<ul>

<li><a class="mw-redirect" title="Dominance (game theory)" href="https://en.wikipedia.org/wiki/Dominance_(game_theory)">Dominant strategies</a></li>

<li><a title="Strategy (game theory)" href="https://en.wikipedia.org/wiki/Strategy_(game_theory)">Pure strategy</a></li>

<li><a title="Strategy (game theory)" href="https://en.wikipedia.org/wiki/Strategy_(game_theory)#Mixed_strategy">Mixed strategy</a></li>

<li><a title="Strategy-stealing argument" href="https://en.wikipedia.org/wiki/Strategy-stealing_argument">Strategy-stealing argument</a></li>

<li><a title="Grim trigger" href="https://en.wikipedia.org/wiki/Grim_trigger">Grim trigger</a></li>

<li><a title="Collusion" href="https://en.wikipedia.org/wiki/Collusion">Collusion</a></li>

<li><a title="Backward induction" href="https://en.wikipedia.org/wiki/Backward_induction">Backward induction</a></li>

<li><a class="mw-redirect" title="Forward induction" href="https://en.wikipedia.org/wiki/Forward_induction">Forward induction</a></li>

<li><a title="Markov strategy" href="https://en.wikipedia.org/wiki/Markov_strategy">Markov strategy</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<th class="navbox-group" scope="row">Classes
of games</th>

<div>

<ul>

<li><a title="Symmetric game" href="https://en.wikipedia.org/wiki/Symmetric_game">Symmetric game</a></li>

<li><a title="Perfect information" href="https://en.wikipedia.org/wiki/Perfect_information">Perfect information</a></li>

<li><a title="Repeated game" href="https://en.wikipedia.org/wiki/Repeated_game">Repeated game</a></li>

<li><a title="Signaling game" href="https://en.wikipedia.org/wiki/Signaling_game">Signaling game</a></li>

<li><a title="Screening game" href="https://en.wikipedia.org/wiki/Screening_game">Screening game</a></li>

<li><a title="Cheap talk" href="https://en.wikipedia.org/wiki/Cheap_talk">Cheap talk</a></li>

<li><a title="Mechanism design" href="https://en.wikipedia.org/wiki/Mechanism_design">Mechanism design</a></li>

<li><a title="Bargaining problem" href="https://en.wikipedia.org/wiki/Bargaining_problem">Bargaining problem</a></li>

<li><a title="Stochastic game" href="https://en.wikipedia.org/wiki/Stochastic_game">Stochastic game</a></li>

<li><a title="N-player game" href="https://en.wikipedia.org/wiki/N-player_game">n-player game</a></li>

<li><a class="mw-redirect" title="Large Poisson game" href="https://en.wikipedia.org/wiki/Large_Poisson_game">Large Poisson game</a></li>

<li><a title="Nontransitive game" href="https://en.wikipedia.org/wiki/Nontransitive_game">Nontransitive game</a></li>

<li><a title="Global game" href="https://en.wikipedia.org/wiki/Global_game">Global game</a></li>

<li><a title="Strictly determined game" href="https://en.wikipedia.org/wiki/Strictly_determined_game">Strictly determined game</a></li>

<li><a title="Potential game" href="https://en.wikipedia.org/wiki/Potential_game">Potential game</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<th class="navbox-group" scope="row"><a title="List of games in game theory" href="https://en.wikipedia.org/wiki/List_of_games_in_game_theory">Games</a></th>

<div>

<ul>

<li><a title="Chess" href="https://en.wikipedia.org/wiki/Chess">Chess</a></li>

<li><a title="Infinite chess" href="https://en.wikipedia.org/wiki/Infinite_chess">Infinite chess</a></li>

<li><a title="Draughts" href="https://en.wikipedia.org/wiki/Draughts">Checkers</a></li>

<li><a title="Prisoner's dilemma" href="https://en.wikipedia.org/wiki/Prisoner%27s_dilemma">Prisoner's dilemma</a></li>

<li><a title="Optional prisoner's dilemma" href="https://en.wikipedia.org/wiki/Optional_prisoner%27s_dilemma">Optional prisoner's dilemma</a></li>

<li><a title="Traveler's dilemma" href="https://en.wikipedia.org/wiki/Traveler%27s_dilemma">Traveler's dilemma</a></li>

<li><a title="Coordination game" href="https://en.wikipedia.org/wiki/Coordination_game">Coordination game</a></li>

<li><a title="Chicken (game)" href="https://en.wikipedia.org/wiki/Chicken_(game)">Chicken</a></li>

<li><a title="Centipede game" href="https://en.wikipedia.org/wiki/Centipede_game">Centipede game</a></li>

<li><a title="Volunteer's dilemma" href="https://en.wikipedia.org/wiki/Volunteer%27s_dilemma">Volunteer's dilemma</a></li>

<li><a title="Dollar auction" href="https://en.wikipedia.org/wiki/Dollar_auction">Dollar auction</a></li>

<li><a title="Battle of the sexes (game theory)" href="https://en.wikipedia.org/wiki/Battle_of_the_sexes_(game_theory)">Battle of the sexes</a></li>

<li><a title="Matching pennies" href="https://en.wikipedia.org/wiki/Matching_pennies">Matching pennies</a></li>

<li><a title="Ultimatum game" href="https://en.wikipedia.org/wiki/Ultimatum_game">Ultimatum game</a></li>

<li><a title="Rock–paper–scissors" href="https://en.wikipedia.org/wiki/Rock%E2%80%93paper%E2%80%93scissors">Rock–paper–scissors</a></li>

<li><a title="Pirate game" href="https://en.wikipedia.org/wiki/Pirate_game">Pirate game</a></li>

<li><a title="Dictator game" href="https://en.wikipedia.org/wiki/Dictator_game">Dictator game</a></li>

<li><a title="Public goods game" href="https://en.wikipedia.org/wiki/Public_goods_game">Public goods game</a></li>

<li><a title="Blotto game" href="https://en.wikipedia.org/wiki/Blotto_game">Blotto game</a></li>

<li><a title="War of attrition (game)" href="https://en.wikipedia.org/wiki/War_of_attrition_(game)">War of attrition</a></li>

<li><a title="El Farol Bar problem" href="https://en.wikipedia.org/wiki/El_Farol_Bar_problem">El Farol Bar problem</a></li>

<li><a title="Fair division" href="https://en.wikipedia.org/wiki/Fair_division">Fair division</a></li>

<li><a title="Fair cake-cutting" href="https://en.wikipedia.org/wiki/Fair_cake-cutting">Fair cake-cutting</a></li>

<li><a title="Cournot competition" href="https://en.wikipedia.org/wiki/Cournot_competition">Cournot game</a></li>

<li><a title="Deadlock (game theory)" href="https://en.wikipedia.org/wiki/Deadlock_(game_theory)">Deadlock</a></li>

<li><a title="Unscrupulous diner's dilemma" href="https://en.wikipedia.org/wiki/Unscrupulous_diner%27s_dilemma">Diner's dilemma</a></li>

<li><a title="Guess 2/3 of the average" href="https://en.wikipedia.org/wiki/Guess_2/3_of_the_average">Guess 2/3 of the average</a></li>

<li><a title="Kuhn poker" href="https://en.wikipedia.org/wiki/Kuhn_poker">Kuhn poker</a></li>

<li><a class="mw-redirect" title="Nash bargaining game" href="https://en.wikipedia.org/wiki/Nash_bargaining_game">Nash bargaining game</a></li>

<li><a class="mw-redirect" title="Prisoners and hats puzzle" href="https://en.wikipedia.org/wiki/Prisoners_and_hats_puzzle">Prisoners and hats puzzle</a></li>

<li><a title="Dictator game" href="https://en.wikipedia.org/wiki/Dictator_game#Trust_game">Trust game</a></li>

<li><a title="Princess and Monster game" href="https://en.wikipedia.org/wiki/Princess_and_Monster_game">Princess and Monster game</a></li>

<li><a title="Rendezvous problem" href="https://en.wikipedia.org/wiki/Rendezvous_problem">Rendezvous problem</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<th class="navbox-group" scope="row">Theorems</th>

<div>

<ul>

<li><a title="Minimax" href="https://en.wikipedia.org/wiki/Minimax">Minimax theorem</a></li>

<li><a title="Nash equilibrium" href="https://en.wikipedia.org/wiki/Nash_equilibrium">Nash's theorem</a></li>

<li><a title="Purification theorem" href="https://en.wikipedia.org/wiki/Purification_theorem">Purification theorem</a></li>

<li><a title="Zermelo's theorem (game theory)" href="https://en.wikipedia.org/wiki/Zermelo%27s_theorem_(game_theory)">Zermelo's theorem</a></li>

<li><a title="Folk theorem (game theory)" href="https://en.wikipedia.org/wiki/Folk_theorem_(game_theory)">Folk theorem</a></li>

<li><a title="Revelation principle" href="https://en.wikipedia.org/wiki/Revelation_principle">Revelation principle</a></li>

<li><a title="Arrow's impossibility theorem" href="https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem">Arrow's impossibility theorem</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<th class="navbox-group" scope="row">Key
figures</th>

<div>

<ul>

<li><a title="Albert W. Tucker" href="https://en.wikipedia.org/wiki/Albert_W._Tucker">Albert W. Tucker</a></li>

<li><a title="Amos Tversky" href="https://en.wikipedia.org/wiki/Amos_Tversky">Amos Tversky</a></li>

<li><a title="Ariel Rubinstein" href="https://en.wikipedia.org/wiki/Ariel_Rubinstein">Ariel Rubinstein</a></li>

<li><a title="Claude Shannon" href="https://en.wikipedia.org/wiki/Claude_Shannon">Claude Shannon</a></li>

<li><a title="Daniel Kahneman" href="https://en.wikipedia.org/wiki/Daniel_Kahneman">Daniel Kahneman</a></li>

<li><a title="David K. Levine" href="https://en.wikipedia.org/wiki/David_K._Levine">David K. Levine</a></li>

<li><a title="David M. Kreps" href="https://en.wikipedia.org/wiki/David_M._Kreps">David M. Kreps</a></li>

<li><a title="Donald B. Gillies" href="https://en.wikipedia.org/wiki/Donald_B._Gillies">Donald B. Gillies</a></li>

<li><a title="Drew Fudenberg" href="https://en.wikipedia.org/wiki/Drew_Fudenberg">Drew Fudenberg</a></li>

<li><a title="Eric Maskin" href="https://en.wikipedia.org/wiki/Eric_Maskin">Eric Maskin</a></li>

<li><a title="Harold W. Kuhn" href="https://en.wikipedia.org/wiki/Harold_W._Kuhn">Harold W. Kuhn</a></li>

<li><a title="Herbert A. Simon" href="https://en.wikipedia.org/wiki/Herbert_A._Simon">Herbert Simon</a></li>

<li><a title="Hervé Moulin" href="https://en.wikipedia.org/wiki/Herv%C3%A9_Moulin">Hervé Moulin</a></li>

<li><a title="Jean Tirole" href="https://en.wikipedia.org/wiki/Jean_Tirole">Jean Tirole</a></li>

<li><a title="Jean-François Mertens" href="https://en.wikipedia.org/wiki/Jean-Fran%C3%A7ois_Mertens">Jean-François Mertens</a></li>

<li><a title="John Harsanyi" href="https://en.wikipedia.org/wiki/John_Harsanyi">John Harsanyi</a></li>

<li><a title="John Maynard Smith" href="https://en.wikipedia.org/wiki/John_Maynard_Smith">John Maynard Smith</a></li>

<li><a title="Antoine Augustin Cournot" href="https://en.wikipedia.org/wiki/Antoine_Augustin_Cournot">Antoine Augustin Cournot</a></li>

<li><a title="John von Neumann" href="https://en.wikipedia.org/wiki/John_von_Neumann">John von Neumann</a></li>

<li><a title="Kenneth Arrow" href="https://en.wikipedia.org/wiki/Kenneth_Arrow">Kenneth Arrow</a></li>

<li><a title="Kenneth Binmore" href="https://en.wikipedia.org/wiki/Kenneth_Binmore">Kenneth Binmore</a></li>

<li><a title="Leonid Hurwicz" href="https://en.wikipedia.org/wiki/Leonid_Hurwicz">Leonid Hurwicz</a></li>

<li><a title="Lloyd Shapley" href="https://en.wikipedia.org/wiki/Lloyd_Shapley">Lloyd Shapley</a></li>

<li><a title="Melvin Dresher" href="https://en.wikipedia.org/wiki/Melvin_Dresher">Melvin Dresher</a></li>

<li><a title="Merrill M. Flood" href="https://en.wikipedia.org/wiki/Merrill_M._Flood">Merrill M. Flood</a></li>

<li><a title="Oskar Morgenstern" href="https://en.wikipedia.org/wiki/Oskar_Morgenstern">Oskar Morgenstern</a></li>

<li><a title="Paul Milgrom" href="https://en.wikipedia.org/wiki/Paul_Milgrom">Paul Milgrom</a></li>

<li><a title="Peyton Young" href="https://en.wikipedia.org/wiki/Peyton_Young">Peyton Young</a></li>

<li><a title="Reinhard Selten" href="https://en.wikipedia.org/wiki/Reinhard_Selten">Reinhard Selten</a></li>

<li><a title="Robert Axelrod" href="https://en.wikipedia.org/wiki/Robert_Axelrod">Robert Axelrod</a></li>

<li><a title="Robert Aumann" href="https://en.wikipedia.org/wiki/Robert_Aumann">Robert Aumann</a></li>

<li><a title="Robert B. Wilson" href="https://en.wikipedia.org/wiki/Robert_B._Wilson">Robert B. Wilson</a></li>

<li><a title="Roger Myerson" href="https://en.wikipedia.org/wiki/Roger_Myerson">Roger Myerson</a></li>

<li><a title="Samuel Bowles (economist)" href="https://en.wikipedia.org/wiki/Samuel_Bowles_(economist)">Samuel Bowles</a></li>

<li><a title="Thomas Schelling" href="https://en.wikipedia.org/wiki/Thomas_Schelling">Thomas Schelling</a></li>

<li><a title="William Vickrey" href="https://en.wikipedia.org/wiki/William_Vickrey">William Vickrey</a></li>

</ul>

</div>

</td>

</tr>

<tr>

<div>

<ul>

<li><a title="All-pay auction" href="https://en.wikipedia.org/wiki/All-pay_auction">All-pay auction</a></li>

<li><a title="Alpha–beta pruning" href="https://en.wikipedia.org/wiki/Alpha%E2%80%93beta_pruning">Alpha–beta pruning</a></li>

<li><a title="Bertrand paradox (economics)" href="https://en.wikipedia.org/wiki/Bertrand_paradox_(economics)">Bertrand paradox</a></li>

<li><a title="Bounded rationality" href="https://en.wikipedia.org/wiki/Bounded_rationality">Bounded rationality</a></li>

<li><a title="Combinatorial game theory" href="https://en.wikipedia.org/wiki/Combinatorial_game_theory">Combinatorial game theory</a></li>

<li><a title="Confrontation analysis" href="https://en.wikipedia.org/wiki/Confrontation_analysis">Confrontation analysis</a></li>

<li><a title="Coopetition" href="https://en.wikipedia.org/wiki/Coopetition">Coopetition</a></li>

<li><a title="First-move advantage in chess" href="https://en.wikipedia.org/wiki/First-move_advantage_in_chess">First-move advantage in chess</a></li>

<li><a title="Game mechanics" href="https://en.wikipedia.org/wiki/Game_mechanics">Game mechanics</a></li>

<li><a title="Glossary of game theory" href="https://en.wikipedia.org/wiki/Glossary_of_game_theory">Glossary of game theory</a></li>

<li><a title="List of game theorists" href="https://en.wikipedia.org/wiki/List_of_game_theorists">List of game theorists</a></li>

<li><a title="List of games in game theory" href="https://en.wikipedia.org/wiki/List_of_games_in_game_theory">List of games in game theory</a></li>

<li><a title="No-win situation" href="https://en.wikipedia.org/wiki/No-win_situation">No-win situation</a></li>

<li><a title="Solving chess" href="https://en.wikipedia.org/wiki/Solving_chess">Solving chess</a></li>

<li><a title="Topological game" href="https://en.wikipedia.org/wiki/Topological_game">Topological game</a></li>

<li><a title="Tragedy of the commons" href="https://en.wikipedia.org/wiki/Tragedy_of_the_commons">Tragedy of the commons</a></li>

<li><a title="Tyranny of small decisions" href="https://en.wikipedia.org/wiki/Tyranny_of_small_decisions">Tyranny of small decisions</a></li>

</ul>

</div>

</td>

</tr>

</tbody>

</table>

</div>

<div id="mw-normal-catlinks" class="mw-normal-catlinks"><a title="Help:Category" href="https://en.wikipedia.org/wiki/Help:Category">Categories</a>:

<ul>

<li><a title="Category:Non-cooperative games" href="https://en.wikipedia.org/wiki/Category:Non-cooperative_games">Non-cooperative games</a></li>

<li><a title="Category:International relations theory" href="https://en.wikipedia.org/wiki/Category:International_relations_theory">International relations theory</a></li>

<li><a title="Category:Game theory game classes" href="https://en.wikipedia.org/wiki/Category:Game_theory_game_classes">Game theory game classes</a></li>

</ul>

</div>

Zero-sum, Game

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