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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?

Nash equilibrium is a concept within game theory where the ideal result of a game is the place where there is no impetus to veer off from their underlying technique. Generally speaking, an individual can get no steady profit by evolving activities, assuming other players stay consistent in their techniques.

How do you find the Nash equilibrium?

To discover the Nash equilibria, we look at each activity profile. Neither one of the players can expand her result by picking an activity not the same as her present one. Accordingly, this activity profile is a Nash equilibrium. By pickingA rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action.

Why is Nash equilibrium important?

Nash equilibrium likewise takes into account the likelihood that leaders follow randomized methodologies. aking into account randomization is significant for the arithmetic of game hypothesis since it ensures that each game has a Nash equilibrium

Can there be no Nash equilibrium?

It additionally shows a case of games without an equilibrum. Nash's hypothesis shows that each game with a limited number of players and a limited number of unadulterated strategies has in any event one Nash harmony. Accordingly, a game with vastly numerous strategies may have no equilibria.

Does Nash equilibrium always exist?

There does not always exist a pure Nash equilibrium.

How do you calculate Nash equilibrium?

To discover the Nash equilibria, we look at each activity profile. Neither one of the players can expand her result by picking an activity not the same as her present one. Accordingly, this activity profile is a Nash equilibrium. By pickingA rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action.

Where can I find pure Nash equilibrium?

In this game, both (L, l) and (R, r) are Nash equilibria. If Player 1 chooses L then Player 2 gets 1 by playing l and 0 by playing r; if Player 1 chooses R then Player 2 gets 2 by playing r and 0 by playing l.

Further Reading

Nash equilibria in models of fiscal competitionNash 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 fundsA 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 contextOn 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 reliefA 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 trustEffects 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 theoryNash 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 gamesUniform 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 externalitiesForcing 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 controlFeedback 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?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 …


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