What is a Nash equilibrium?

In game theory, a Nash equilibrium occurs in a noncooperative game involving multiple players when none of the players can improve their expected outcome by changing their own strategy. It is a key concept in game theory that defines the solution to N-player noncooperative games. American mathematician John Nash established the mathematical principles of game theory and demonstrated that in any finite game, all players can achieve an optimal outcome, known as the Nash equilibrium, by considering the possible actions of other players. He was awarded the 1994 Nobel Prize for Economics for his contributions to game theory.

Who came up with the idea of Nash equilibrium?

The concept of Nash equilibrium is attributed to American mathematician John Nash , who introduced it in his doctoral thesis titled Non-Cooperative Games . This work established the mathematical principles of game theory, illustrating that in any finite game, all players can achieve an optimal outcome, known as the Nash equilibrium, by considering the possible actions of others. Nash was awarded the 1994 Nobel Prize in Economics for his contributions to game theory.

In what type of games does Nash equilibrium occur?

Nash equilibrium occurs in noncooperative games where no player can improve their outcome by changing their strategy. These games involve interdependent decisions, with players unable to make binding agreements. A classic example is the prisoner's dilemma, where the Nash equilibrium results in both prisoners confessing, even though they would be better off remaining silent. Nash equilibria can also be found in situations with multiple players or asymmetrical situations, such as contract negotiations. In some games, the Nash equilibrium is a "mixed" equilibrium where players employ a specific mix of strategies. An example of this is rock-paper-scissors, where each player should choose each option one-third of the time.

Why is Nash equilibrium important in game theory?

In game theory, Nash equilibrium is a solution concept for noncooperative games involving two or more players, where no player can improve their expected outcome by unilaterally changing their strategy. American mathematician John Nash , who won the 1994 Nobel Prize in Economics, established the mathematical principles of game theory and demonstrated that in any finite game, all players can reach an optimal outcome or Nash equilibrium by considering the possible actions of other players. Nash equilibrium is important because it identifies stable outcomes in strategic interactions, where no player has an incentive to deviate, offering a way to predict behavior in various real-world scenarios, including economics and military strategy.

How can you identify a Nash equilibrium in a simple game?

In game theory, a Nash equilibrium occurs when no player can improve their outcome by changing their strategy, assuming the other players' strategies remain constant. To identify a Nash equilibrium in a simple game: 1. List each player's possible strategies. 2. Create a payoff matrix showing the outcomes for all strategy combinations. 3. Examine each cell in the matrix to see if any player can benefit by unilaterally changing their strategy. If no player can benefit, that combination is a Nash equilibrium. A game may have multiple Nash equilibria or none. Some games have a "mixed" Nash equilibrium where players use a mix of strategies.

Can a game have more than one Nash equilibrium or none at all?

In game theory, a Nash equilibrium occurs when no player can improve their expected outcome by changing their strategy. A game can have multiple Nash equilibria. For instance, if two friends want to see a movie together but disagree on which one, each friend seeing either movie constitutes a Nash equilibrium if they would rather see a movie together than alone. It is also possible for a game to have no Nash equilibrium. An example of this is a game that was discovered by William F. Lucas, an American mathematician.

Nash equilibrium

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A diagram illustrating the prisoners' dilemma in game theory, showing two prisoners, A and B, with four possible outcomes based on whether each confesses or remains silent. Years of prison time are indicated for each scenario, with visual icons showing happy or sad prisoners in cells or free.

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Also called:: Nash solution

Key People:: John Nash

Related Topics:: game theory

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Nash equilibrium

mathematics

Also known as: Nash solution

Written by

Stephen Eldridge

Fact-checked by

Britannica Editors

Last updated

Oct. 1, 2026 •History

Top Questions

What is a Nash equilibrium?
In game theory, a Nash equilibrium occurs in a noncooperative game involving multiple players when none of the players can improve their expected outcome by changing their own strategy. It is a key concept in game theory that defines the solution to N-player noncooperative games.

American mathematician John Nash established the mathematical principles of game theory and demonstrated that in any finite game, all players can achieve an optimal outcome, known as the Nash equilibrium, by considering the possible actions of other players. He was awarded the 1994 Nobel Prize for Economics for his contributions to game theory.
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1. John Nash
2. Nash equilibrium
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Who came up with the idea of Nash equilibrium?
The concept of Nash equilibrium is attributed to American mathematician John Nash, who introduced it in his doctoral thesis titled Non-Cooperative Games. This work established the mathematical principles of game theory, illustrating that in any finite game, all players can achieve an optimal outcome, known as the Nash equilibrium, by considering the possible actions of others. Nash was awarded the 1994 Nobel Prize in Economics for his contributions to game theory.
2 Britannica Sources
1. Nash equilibrium
2. John Nash
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In what type of games does Nash equilibrium occur?
Nash equilibrium occurs in noncooperative games where no player can improve their outcome by changing their strategy. These games involve interdependent decisions, with players unable to make binding agreements.

A classic example is the prisoner's dilemma, where the Nash equilibrium results in both prisoners confessing, even though they would be better off remaining silent. Nash equilibria can also be found in situations with multiple players or asymmetrical situations, such as contract negotiations. In some games, the Nash equilibrium is a "mixed" equilibrium where players employ a specific mix of strategies. An example of this is rock-paper-scissors, where each player should choose each option one-third of the time.
2 Britannica Sources
1. Nash equilibrium
2. game theory: Sequential and simultaneous truels
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Why is Nash equilibrium important in game theory?
In game theory, Nash equilibrium is a solution concept for noncooperative games involving two or more players, where no player can improve their expected outcome by unilaterally changing their strategy. American mathematician John Nash, who won the 1994 Nobel Prize in Economics, established the mathematical principles of game theory and demonstrated that in any finite game, all players can reach an optimal outcome or Nash equilibrium by considering the possible actions of other players.

Nash equilibrium is important because it identifies stable outcomes in strategic interactions, where no player has an incentive to deviate, offering a way to predict behavior in various real-world scenarios, including economics and military strategy.
3 Britannica Sources
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
How can you identify a Nash equilibrium in a simple game?
In game theory, a Nash equilibrium occurs when no player can improve their outcome by changing their strategy, assuming the other players' strategies remain constant.

To identify a Nash equilibrium in a simple game:
1. List each player's possible strategies.
2. Create a payoff matrix showing the outcomes for all strategy combinations.
3. Examine each cell in the matrix to see if any player can benefit by unilaterally changing their strategy. If no player can benefit, that combination is a Nash equilibrium.

A game may have multiple Nash equilibria or none. Some games have a "mixed" Nash equilibrium where players use a mix of strategies.
2 Britannica Sources
1. Nash equilibrium
2. game theory: Power in voting: the paradox of the chair’s position
Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.
Can a game have more than one Nash equilibrium or none at all?
In game theory, a Nash equilibrium occurs when no player can improve their expected outcome by changing their strategy.

A game can have multiple Nash equilibria. For instance, if two friends want to see a movie together but disagree on which one, each friend seeing either movie constitutes a Nash equilibrium if they would rather see a movie together than alone.

It is also possible for a game to have no Nash equilibrium. An example of this is a game that was discovered by William F. Lucas, an American mathematician.
2 Britannica Sources
1. Nash equilibrium
2. game theory: The von Neumann–Morgenstern theory
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Nash equilibrium, in game theory, an outcome in a noncooperative game for two or more players in which no player’s expected outcome can be improved by changing one’s own strategy. The Nash equilibrium is a key concept in game theory, in which it defines the solution of N-player noncooperative games. It is named for American mathematician John Nash, who was awarded the 1994 Nobel Prize for Economics for his contributions to game theory.

Game theory uses mathematics to model and analyze situations in which decisions are interdependent. While it can be used to model recreational games such as Monopoly or poker, it is often used to analyze topics of real-world interest, including economics and military strategy. In game theory, a game may be any situation in which there are interdependent decisions, and the players are all the decision-making entities.

A game is noncooperative as long as no mechanism exists for the players to make binding agreements with one another. For example, in the famous prisoner’s dilemma, two prisoners have been accused of a crime and are asked to confess. If one confesses and the other does not, the one who confesses will be released, and the one who does not will receive a harsh sentence. If both confess, both will receive a serious, but not harsh, sentence. If neither confesses, both will receive a very light sentence. Because there is no outside authority enforcing any agreement between the prisoners, the game is noncooperative; neither prisoner suffers a penalty for betraying the other.

A colorful payoff matrix for a normal form game in game theory, showing strategy options-support, oppose, evade-for both party A and party B. Each cell contains cartoon figures representing each party's action and lists percentage payoffs for A and B. The bottom-right cell is labeled saddlepoint, highlighting the equilibrium.

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game theory: The Nash solution

A payoff matrix is often used to help determine the optimal strategy for the players in the game. In the payoff matrix, each row represents one possible strategy for one player, and each column represents one possible strategy for the other. In the example above, the matrix would look like the figure below.

Each player (prisoner A or prisoner B) will attempt to adopt the strategy (confess or remain silent) that results in the least amount of jail time (0, 1, 5, or 20 years). The best outcome for the prisoners is for both to remain silent, as this results in a total sentence of only 2 years (as opposed to 20, if only one chooses to remain silent, or 10, if both choose to confess). This collection of strategies results in the best payoff for the players collectively. However, it is not the Nash equilibrium, because either prisoner’s payoff can be improved by choosing a different strategy.

If prisoner A remains silent, then prisoner B can either remain silent and receive a 1-year sentence or confess and go free. Prisoner B’s own payoff therefore can be improved by confessing. However, one prisoner confessing and the other remaining silent is also not a Nash equilibrium, because the payoff of the prisoner who remains silent can be improved by changing strategies. If prisoner A confesses, then prisoner B can either remain silent and face a 20-year sentence or confess and face a 5-year sentence. Thus, prisoner B’s payoff can be improved by switching from remaining silent to confessing.

The only collection of strategies in which no player’s payoff can be improved by switching strategies is if both prisoners confess. In this scenario, either prisoner choosing to switch strategies will result in a lower payoff. Despite this being worse for both players (resulting in a total 10-year sentence) than if both were to remain silent, it is the Nash equilibrium.

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It is possible for there to be multiple Nash equilibria to a given problem. For example, suppose two friends wish to see a movie together but disagree on which movie. If both would rather see either movie together than see a movie alone, then both friends seeing either movie constitutes a Nash equilibrium, as neither can opt to see the other film without suffering a worse outcome.

It is also possible that a Nash equilibrium is a “mixed” equilibrium, meaning that at least one player should employ a specific mix of strategies rather than employing the same strategy consistently (a “pure” Nash equilibrium). For example, in the game rock-paper-scissors, the Nash equilibrium is that each player should choose each option exactly one-third of the time, because if a player chooses one option more than the others, the other player can exploit that tendency to win a greater percentage of the matches.

Nash equilibria may be found for situations involving many players (such as individual use of common resources) or for asymmetrical situations (such as contract negotiations between an individual and a business). Nash proved that if mixed strategies are allowed, then there is at least one Nash equilibrium for every noncooperative game with a finite number of players choosing from a finite number of strategies.

Stephen Eldridge

Nash equilibrium

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