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Encyclopedia > Nash equilibrium
Nash Equilibrium
A solution concept in game theory
Relationships
Subset of: Rationalizability, Correlated equilibrium
Superset of: Evolutionary stable strategy, Subgame perfect equilibrium, Perfect Bayesian equilibrium, Trembling hand perfect equilibrium
Significance
Proposed by: John Forbes Nash
Used for: All non-cooperative games
Example: Prisoner's dilemma
This box: view  talk  edit

In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium. In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ... In game theory, rationalizability or rationalizable equilibria is a solution concept which generalizes Nash equilibrium. ... In game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. ... The evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a central concept in game theory introduced by John Maynard Smith and George R. Price in 1973 (a full account is given by Maynard Smith, 1982). ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... In game theory, a Bayesian game is one in which information is incomplete. ... The trembling hand perfection is a notion that eliminates actions of players that are unsafe because they were chosen through a slip of the hand. ... John Forbes Nash, Jr. ... In game theory, a non-cooperative game is a one in which players can cooperate, but any cooperation must be self-enforcing. ... Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with... Game theory is often described as a branch of applied mathematics and economics that studies situations where multiple players make decisions in an attempt to maximize their returns. ... John Forbes Nash, Jr. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ...

The concept of the Nash equilibrium (NE) is not original to Nash (e.g., Antoine Augustin Cournot showed how to find what we now call the Nash equilibrium of the Cournot duopoly game). Consequently, some authors refer to it as a “Cournot-Nash equilibrium” (or as a “Nash-Cournot equilibrium”). However, Nash showed for the first time in his dissertation, Non-cooperative games (1950), that Nash equilibria (in mixed strategies) must exist for all finite games with any number of players. Until Nash, this had only been proven for 2-player zero-sum games by John von Neumann and Oskar Morgenstern (1947). Antoine Augustin Cournot Antoine Augustin Cournot (28 August 1801â€‘ 31 March 1877) was a French philosopher and mathematician. ... Cournot competition is an economics model used to describe industry structure. ... Zero-sum describes a situation in which a participants gain (or loss) is exactly balanced by the losses (or gains) of the other participant(s). ... John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics... Oskar Morgenstern (January 24, 1902 - July 26, 1977) was an German- American economist who, working with John von Neumann, helped found the mathematical field of game theory. ...

### Informal definition

Informally, a set of strategies is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy. As a heuristic, one can imagine that each player is told the strategies of the other players. If any player would want to do something different after being informed about the others' strategies, then that set of strategies is not a Nash equilibrium. If, however, the player does not want to switch (or is indifferent between switching and not) the set of strategies is a Nash equilibrium.

This can have a sometimes counter-intuitive result. Since the Nash equilibrium focuses on individual's preferences given that the others stay the same, there can be Nash equilibria where, if players could coordinate, they would all want to switch. The stag hunt presents an example of this phenomenon. In game theory, the Stag Hunt is a game first discussed by Jean-Jacques Rousseau. ...

### Formal definition

Let (S, f) be a game, where S is the set of strategy profiles and f is the set of payoff profiles. Let σ i be a strategy profile of all players except for player i. When each player $i in {1, ..., n}$ chooses strategy xi resulting in strategy profile x = (x1,...,xn) then player i obtains payoff fi(x). Note that the payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile $x^* in S$ is a Nash equilibrium (NE) if no unilateral deviation in strategy by any single player is profitable, that is In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players bahaviour. ...

A game can have a pure strategy NE or an NE in its mixed extension (that of choosing a pure strategy stochastically with a fixed frequency). Nash proved that, if we allow mixed strategies (players choose strategies randomly according to pre-assigned probabilities), then every n-player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium. A pure strategy is a term used to refer to strategies in Game theory. ... In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ... Stochastic, from the Greek stochos or goal, means of, relating to, or characterized by conjecture; conjectural; random. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ... In game theory, an n-player game is a game which is well defined for any number of players. ...

## Examples

### Competition game

Player 2 chooses '0' Player 2 chooses '1' Player 2 chooses '2' Player 2 chooses '3' 0, 0 2, -2 2, -2 2, -2 -2, 2 1, 1 3, -1 3, -1 -2, 2 -1, 3 2, 2 4, 0 -2, 2 -1, 3 0, 4 3, 3

Consider the following two-player game: both players simultaneously choose a whole number from 0 to 3. Both players then win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then he has to give up two points to the other. This game has a unique Nash equilibrium: both players choosing 0 (highlighted in light red). Any other choice of strategies can be improved if one of the players lowers his number to one less than the other player's number. In the table to the left, for example, when starting at the green square it is in player 1's interest to move to the purple square by choosing a smaller number, and it is in player 2's interest to move to the blue square by choosing a smaller number. If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 4 Nash equilibria.

### Coordination game

Main article: Coordination game
Player 2 adopts strategy 1 Player 2 adopts strategy 2 A, A B, C C, B D, D

The coordination game is a classic (symmetric) two player, two strategy game, with the payoff matrix shown to the right, where the payoffs are according to A>C and D>B. The players should thus cooperate on either of the two strategies to receive a high payoff. Players in the game have to agree on one of the two strategies in order to receive a high payoff. If the players do not agree, a lower payoff is rewarded. An example of a coordination game is the setting where two technologies are available to two firms with compatible products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game. In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, whereby the players reach an outcome to mutual advantage. ... In game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. ... In game theory, a players strategy, in a game or a business situation, is a complete plan of action for whatever situation might arise; this fully determines the players behaviour. ... It has been suggested that this article or section be merged with normal form game. ...

Driving on a road, and having to choose either to drive on the left or to drive on the right of the road, is also a coordination game. For example, with payoffs 100 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:

Drive on the Left Drive on the Right 100, 100 0, 0 0, 0 100, 100

In this case there are two pure strategy Nash equilibria,; when both choose to either drive on the left or on the right. If we admit mixed strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%,100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player is (50%, 50%). In game theory a mixed strategy is a strategy which chooses randomly between possible moves. ...

### Prisoner's dilemma

Main article: Prisoner's dilemma
(but watch out for differences in the orientation of the payoff matrix)

The Prisoner's Dilemma has the same payoff matrix as depicted for the Coordination Game, but now C > A > D > B. Because C > A and D > B, each player improves his situation by switching from strategy #1 to strategy #2, no matter what the other player decides. The Prisoner's Dilemma thus has a single Nash Equilibrium: both players choosing strategy #2 ("betraying"). What has long made this an interesting case to study is the fact that D < A ("both betray") is globally inferior to "both remain loyal". The globally optimal strategy is unstable; it is not an equilibrium. Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with...

As Ian Stewart put it, "sometimes rational decisions aren't sensible!" Ian Stewart, FRS (b. ...

### Nash equilibria in a payoff matrix

There is an easy numerical way to identify Nash Equilibria on a Payoff Matrix. It is especially helpful in two person games where players have more than two strategies. In this case formal analysis may become too long. This rule does not apply to the case where mixed (stochastic) strategies are of interest. The rule goes as follows: if the first payoff number, in the duplet of the cell, is the maximum of the column of the cell and if the second number is the maximum of the row of the cell - then the cell represents a Nash equilibrium.

We can apply this rule to a 3x3 matrix:

Option A Option B Option C 0, 0 25, 40 5, 10 40, 25 0, 0 5, 15 10, 5 15, 5 10, 10

Using the rule, we can very quickly (much faster than with formal analysis) see that the Nash Equlibria cells are (B,A), (A,B), and (C,C). Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row. For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C). For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns.

This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the tuple has maximum of the row. If these conditions are met, the cell represents a Nash Equilibrium. Check all columns this way to find all NE cells. An NxN matrix may have between 0 and NxN pure strategy Nash equilibria. A pure strategy is a term used to refer to strategies in Game theory. ...

## Stability

The concept of stability, useful in the analysis of many kinds of equilibrium, can also be applied to Nash equilibria. In mathematics, stability theory deals with the stability of the solutions of differential equations and dynamical systems. ... Look up equilibrium in Wiktionary, the free dictionary. ...

A Nash equilibrium for a mixed strategy game is stable if a small change (specifically, an infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:

1. the player who did not change has no better strategy in the new circumstance
2. the player who did change is now playing with a strictly worse strategy

If these cases are both met, then a player with the small change in his mixed-strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. John Nash showed that the latter situation could not arise in a range of well-defined games. John Forbes Nash, Jr. ...

In the "driving game" example above there are both stable and unstable equilibria. The equilibria involving mixed-strategies with 100% probabilities are stable. If either player changes his probabilities slightly, they will be both at a disadvantage, and his opponent will have no reason to change his strategy in turn. The (50%,50%) equilibrium is unstable. If either player changes his probabilities, then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).

Stability is crucial in practical applications of Nash equilibria, since the mixed-strategy of each player is not perfectly known, but has to be inferred from statistical distribution of his actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.

Note that stability of the equilibrium is related to, but distinct from, stability of a strategy.

A Coalition-Proof Nash Equilibrium (CPNE)(similar to a Strong Nash Equilibrium) occurs when players cannot do better even if they are allowed to communicate and collaborate before the game. Every correlated strategy supported by iterated strict dominance and on the pareto frontier is a CPNE[1]. Further, it is possible for a game to have a Nash equilibrium that is resilient against coalitions less than a specified size, k. CPNE is related to the theory of the core. In game theory, dominance occurs when one strategy is better or worse than another regardless of the strategies of a players opponents. ... Pareto efficiency, or Pareto optimality, is an important notion in neoclassical economics with broad applications in game theory, engineering and the social sciences. ... A core is the set of feasible allocations in an economy that cannot be improved upon by subset of the set of the economys consumers (a coalition). ...

## Occurrence

If a game has a unique Nash equilibrium and is played among players with certain characteristics, then (by definition of these characteristics) the NE strategy set will be adopted. Sufficient conditions to be met by the players are: In predicate logic and technical fields that depend on it, uniqueness quantification, or unique existential quantification, is an attempt to formalise the notion of something being true for exactly one thing, or exactly one thing of a certain type. ...

1. The players all will do their utmost to maximize their expected payoff as described by the game.
2. The players are flawless in execution.
3. The players have sufficient intelligence to deduce the solution.
4. There is common knowledge that all players meet these conditions, including this one. So, not only must each player know the other players meet the conditions, but also they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on.

### Where the conditions are not met

Examples of game theory problems in which these conditions are not met:

1. The first condition is not met if the game does not correctly describe the quantities a player wishes to maximize. In this case there is no particular reason for that player to adopt an equilibrium strategy. For instance, the prisoner’s dilemma is not a dilemma if either player is happy to be jailed indefinitely.
2. Pong has an equilibrium which can be played perfectly by a computer, but to make human vs. computer games interesting the programmers add small errors in execution, violating the second condition.
3. In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess[2]. Or, if known, it may not be known to all players, as when playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria).
4. The fourth criterion of common knowledge may not be met even if all players do, in fact, meet all the other criteria. Players wrongly distrusting each other's rationality may adopt counter-strategies to expected irrational play on their opponents’ behalf. This is a major consideration in “Chicken” or an arms race, for example.

PONG helped bring computerized video games into everyday life. ... Xiangqi (Chinese: &#35937;&#26827;; pinyin: xi , Wade-Giles: hsiang-chi; roughly pronounced shyang-chee; literally translated as elephant chess) is one of a family of strategic board games of which chess and shogi are also members. ... Tic-tac-toe, also called noughts and crosses and many other names, is a paper and pencil game between two players, O and X, who alternate in marking the spaces in a 3×3 board. ... The game of chicken (also referred to as playing chicken) is a game in which two players engage in an activity that will result in serious harm unless one of them backs down. ... The term arms race in its original usage describes a competition between two or more parties for military supremacy. ...

### Where the conditions are met

Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics, and evolutionary biology the NE has explanatory power. The payoff in economics is money, and in evolutionary biology gene transmission, both are the fundamental bottom line of survival. Researchers who apply games theory in these fields claim that agents failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.[verification needed] Face-to-face trading interactions on the New York Stock Exchange trading floor. ... This article or section does not cite any references or sources. ... In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. ...

## Proof of existence

As above, let σ i be a mixed strategy profile of all players except for player i. We can define a best response correspondence for player i, bi. bi is a relation from the set of all probability distributions over opponent player profiles to a set of player i's strategies, such that each element of In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ... In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings. ...

bii)

is a best response to σ i. Define

$b(sigma) = b_1(sigma_{-1}) times b_2(sigma_{-2}) times cdots times b_n(sigma_{-n})$.

One can use the Kakutani fixed point theorem to prove that b has a fixed point. That is, there is a σ * such that $sigma^* in b(sigma^*)$. Since b* ) represents the best response for all players to σ * , the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium. The Kakutani fixed point theorem is a fixed-point theorem that was famously used by John Nash in his description of Nash equilibrium. ...

In problems of fair division, the adjusted winner procedure is used to partition a bundle of goods between two players in such a way as to minimize envy and maximize efficiency and equitability. ... In game theory, the best response, is the strategy (or strategies) which produces the most favorable immediate outcome for the current player, taking other players strategies as given. ... Conflict resolution is any reduction in the severity of a conflict. ... In game theory, an evolutionarily stable strategy (or ESS; also evolutionary stable strategy) is a strategy which if adopted by a population cannot be invaded by any competing alternative strategy. ... Game theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. ... Hotellings law is an observation in economics that in many markets it is rational for producers to make their products as similar as possible. ... Minimax is a method in decision theory for minimizing the expected maximum loss. ... These are two closely related (and sometimes confused) terms: In contract bridge the optimum contract is that contract which offers the best chances, in unopposed bidding, of gaining bonus points for part-score, game or slam whilst minimising the risk of failure. ... Will the two prisoners cooperate to minimize total loss of liberty or will one of them, trusting the other to cooperate, betray him so as to go free? In game theory, the prisoners dilemma is a type of non-zero-sum game in which two players can cooperate with... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ... In game theory and economic modelling, a solution concept is a process via which equilibria of a game are identified. ... The Stackelberg leadership model is a model of duopoly in economics. ... Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. ... John Glen Wardrop was an English transport analyst who developed Wardrops first and second principles of equilibrium. ...

## References

• Fudenberg, Drew and Jean Tirole (1991) Game Theory MIT Press.
• Mehlmann, A. The Game's Afoot! Game Theory in Myth and Paradox, American Mathematical Society (2000).
• Morgenstern, Oskar and John von Neumann (1947) The Theory of Games and Economic Behavior Princeton University Press
• Nash, John (1950) "Equilibrium points in n-person games" Proceedings of the National Academy of Sciences 36(1):48-49.
• Nash, John (1951) "Non-Cooperative Games" The Annals of Mathematics 54(2):286-295.

Jean Tirole (born 9 August 1953) is a notable contemporary french economist, author of many works in economics, scientific director of the Industrial Economics Institute in Toulouse. ... Oskar Morgenstern (January 24, 1902 - July 26, 1977) was an German- American economist who, working with John von Neumann, helped found the mathematical field of game theory. ... John von Neumann (Hungarian Margittai Neumann JÃ¡nos Lajos) (born December 28, 1903 in Budapest, Austria-Hungary; died February 8, 1957 in Washington D.C., United States) was a Hungarian-born American mathematician who made contributions to quantum physics, functional analysis, set theory, topology, economics, computer science, numerical analysis, hydrodynamics... John Forbes Nash, Jr. ... John Forbes Nash, Jr. ...

## Notes

1. ^ Coalition-Proof Equilibrium. D. Moreno, J. Wooders, Games and Economic Behavior, Vol. 17 (1996), pp. 80-112
2. ^ Nash has proven that a perfect NE exists for this type of finite extensive form game – it can be represented as a strategy complying with his original conditions for a game with a NE. Such games may not have unique NE, but at least one of the many equilibrium strategies would be played by hypothetical players having perfect knowledge of all 10150 game trees.

It has been suggested that Game tree be merged into this article or section. ... In game theory, game complexity is a measure of the complexity of a game. ...

Results from FactBites:

 NationMaster - Encyclopedia: Nash equilibrium (5782 words) In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. Nash equilibrium · Subgame perfection · Bayes-Nash ;· Trembling hand · Proper equilibrium · Epsilon-equilibrium ;· Correlated equilibrium · Sequential equilibrium · Quasi-perfect equilibrium · Evolutionarily stable strategy ;· Risk dominance Subgame perfect equilibrium is an economics term used in game theory to describe an equilibrium such that players strategies constitute a Nash equilibrium in every subgame of the original game. A Nash Equilibrium is a set of mixed strategies for finite, non-cooperative games between two or more players whereby no player can improve his or her payoff by changing their strategy.
 Nash Equilibrium (1153 words) Other academic theorists used the concept of 'equilibrium' in the 19th century (Maxwell, Walrus, Gibbs), for chemical and economic equilibrium in the early stages of the 20th century (van der Waals, Onnes, Keynes) before Nash used it in the middle of the 20th century. As in a contemporary neo-Darwinian scientific notion of Nash Equilibrium, the gaming concepts of 'competition' and 'survival,' may be psychologically associated with the ideas of (non)'cooperation' and of 'aid' (Kroptokin), or with other conflict paradigms theorized in the human and/or social sciences. Equilibrium theory struggles to satisfy academic standards in contemporary social sciences (and economics), which require a double hermeneutical approach (Radder, 2003) in addition to the explanatory method given by the mathematical sciences, by neo-classical economics, and even in the new technological sciences.
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