## Nash Equilibrium

## Definition

In game theory, the Nash equilibrium, named after American mathematician John Forbes Nash Jr., is a solution concept of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy. If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium. The Nash equilibrium is one of the foundational concepts in game theory. The reality of the Nash equilibrium of a game can be tested using experimental economics methods.

### Nash Equilibrium

### What is the 'Nash Equilibrium'

The Nash Equilibrium is a concept of game theory where the optimal outcome of a game is one where no player has an incentive to deviate from his chosen strategy after considering an opponent's choice. Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. A game may have multiple Nash Equilibria or none at all.

### Explaining 'Nash Equilibrium'

Nash Equilibrium is named after its inventor, John Nash, an American mathematician. It is considered one of the most important concepts of Game Theory, which attempts to determine mathematically and logically the actions that participants of a game should take to secure the best outcomes for themselves. The reason why Nash Equilibrium is considered such an important concept of Game Theory relates to its applicability. The Nash Equilibrium can be incorporated into a wide range of disciplines, from economics to the social sciences.### Nash Equilibrium

The Nash Equilibrium is the solution to a game in which two or more players have a strategy, and with each participant considering an opponent’s choice, he has no incentive, nothing to gain, by switching his strategy. In the Nash Equilibrium, each player's strategy is optimal when considering the decisions of other players. Every player wins because everyone gets the outcome they desire. To quickly test if the Nash equilibrium exists, reveal each player's strategy to the other players. If no one changes his strategy, then the Nash Equilibrium is proven.

### Prisoner's Dilemma

The Prisoner's Dilemma is a common situation analyzed in Game Theory that can employ the Nash Equilibrium. In this game, two criminals are arrested and each is held in solitary confinement with no means of communicating with the other. The prosecutors do not have the evidence to convict the pair, so they offer each prisoner the opportunity to either betray the other by testifying that the other committed the crime or cooperate by remaining silent. If both prisoners betray each other, each serves five years in prison. If A betrays B but B remains silent, prisoner A is set free and prisoner B serves 10 years in prison, or vice versa. If each remains silent, then each serves just one year in prison. The Nash equilibrium in this example is for both players to betray each other. Even though mutual cooperation leads to a better outcome, if one prisoner chooses mutual cooperation and the other does not, one prisoner's outcome is worse.

### Nash Equilibrium FAQ

#### What is a Nash equilibrium in game theory?

#### How do you find the Nash equilibrium?

#### Why is Nash equilibrium important?

#### Can there be no Nash equilibrium?

#### Does Nash equilibrium always exist?

#### How do you calculate Nash equilibrium?

#### Where can I find pure Nash equilibrium?

### Further Reading

**Nash equilibria in models of fiscal competition**

www.sciencedirect.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**A stochastic Nash equilibrium portfolio game between two DC pension funds**

www.sciencedirect.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**On simulation of optimal strategies and Nash equilibrium in the financial market context**

link.springer.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**A Generalized Nash Equilibrium network model for post-disaster humanitarian relief**

www.sciencedirect.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**Effects of financial incentives on the breakdown of mutual trust**

journals.sagepub.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**Nash equilibrium and the history of economic theory**

www.aeaweb.org [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**Uniform payoff security and Nash equilibrium in compact games**

www.sciencedirect.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**Forcing firms to talk: Financial disclosure regulation and externalities**

academic.oup.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**Feedback Nash equilibria for non-linear differential games in pollution control**

www.sciencedirect.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …

**Do soccer players play the mixed-strategy Nash equilibrium?**

www.tandfonline.com [PDF]

… Beck, JH, 1983, Tax competition, uniform assessment, and the benefit principle, Journal of Urban Economics 13, 127-146. Bucovetsky, S., 1986, Nash equilibrium with tax competition, University of Western Ontario … 240 DE Wildasin, Nash equilibria in models of fiscal competition …