How many nash equilibria does a game have

Imagine that a heinous crime has been committed by two people. The authorities know they are guilty, but they do not have enough evidence to convict them. Therefore, they develop a plan to try to trick the suspects. Each suspect or player is placed in an isolated prison cell and given two choices.

Each player can cooperate with the authorities, or , or refuse, or. By cooperating, the player admits guilt and turns over evidence to the authorities. By refusing, the player claims innocence and refuses to help the authorities. The cost represents the number of years that the player will be sentenced to prison. The cost matrices are assigned as. What should the players do? What would you do? If they could make a coordinated decision, then it seems that a good choice would be for both to refuse, which results in costs.

In this case, however, there would be regret because each player would think that he had a chance to go free receiving cost 0 by refusing. If they were to play the game a second time, they might be inclined to change their decisions. The Nash equilibrium for this problem is for both of them to cooperate, which results in.

More specifically, 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 their 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. 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. To quickly find the Nash equilibrium or see if it even exists, reveal each player's strategy to the other players.

If no one changes their strategy, then the Nash equilibrium is proven. Nash equilibrium is often compared alongside dominant strategy, both being strategies of game theory. The Nash equilibrium states that the optimal strategy for an actor is to stay the course of their initial strategy while knowing the opponent's strategy and that all players maintain the same strategy, as long as all other players do not change their strategy.

Dominant strategy asserts that the chosen strategy of an actor will lead to better results out of all the possible strategies that can be used, regardless of the strategy that the opponent uses.

All models of game theory only work if the players involved are "rational agents," meaning that they desire specific outcomes, operate in attempting to choose the most optimal outcome, incorporate uncertainty in their decisions, and are realistic in their options.

Both the terms are similar but slightly different. Nash equilibrium states that nothing is gained if any of the players change their strategy if all other players maintain their strategy. Dominant strategy asserts that a player will choose a strategy that will lead to the best outcome regardless of the strategies that other plays have chosen. Dominant strategy can be included in Nash equilibrium whereas a Nash equilibrium may not be the best strategy in a game. Imagine a game between Tom and Sam.

If you revealed Sam's strategy to Tom and vice versa, you see that no player deviates from the original choice. Knowing the other player's move means little and doesn't change either player's behavior. Outcome A represents a Nash equilibrium. 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 in game theory is a situation in which a player will continue with their chosen strategy, having no incentive to deviate from it, after taking into consideration the opponent's strategy. To find the Nash equilibrium in a game, one would have to model out each of the possible scenarios to determine the results and then choose what the optimal strategy would be. In a two-person game, this would take into consideration the possible strategies that both players could choose.

If neither player changes their strategy knowing all of the information, a Nash equilibrium has occurred. By comparison, the number of seconds that have elapsed since the Big Bang is only about 2 This communication bottleneck means that every possible method for adapting strategies from round to round is going to fail to guide players efficiently to a Nash equilibrium for at least some complex games such as a player restaurant game with complicated preferences.

After all, in each round, the players learn only a bit of new information about each other: how happy they are with the single dinner arrangement that got played. Many of the games economists use to model the real world have additional structure that greatly reduces the amount of information each player must communicate.

That means your collection of preferences will have a high degree of symmetry, and you can potentially convey its entirety in a couple of well-chosen sentences instead of 2 of them. Economists could use such arguments to justify why Nash equilibrium might be attainable for particular games. But the two fields have very different mindsets, which can hamper interdisciplinary communication: Economists tend to look for simple models that capture the essence of a complex interaction, while theoretical computer scientists are often more interested in understanding what happens as the models grow increasingly complex.

And even though a mediator can give many different kinds of advice, the set of correlated equilibria of a game, which is represented by a collection of linear equations and inequalities, is more mathematically tractable than the set of Nash equilibria. When it comes to repeated rounds of play, many of the most natural ways that players could choose to adapt their strategies converge, in a particular sense, to correlated equilibria.

For many regret-minimizing approaches, researchers have shown that play will rapidly converge to a correlated equilibrium in the following surprising sense: after maybe rounds have been played, the game history will look essentially the same as if a mediator had been advising the players all along. The process is reminiscent of what happens in real life, Roughgarden said, as societal norms about which equilibrium should be played gradually evolve.

The fact that humanity came up with the idea of Nash equilibrium before correlated equilibrium may be just an accident of history, Myerson said. This means, Rubinstein pointed out, that regret minimization approaches are not always an ideal choice for rational players in any given round.