NE Nash equilibrium

Nash equilibrium is a concept in game theory that is named after its creator, John Nash. It is a fundamental solution concept that describes a state in which each player in a game, knowing the strategies of the other players, has no incentive to unilaterally change their strategy. In other words, a Nash equilibrium is a set of strategies in which no player can improve their outcome by changing their strategy, given that the other players' strategies remain unchanged.

To understand Nash equilibrium, it is important to have a basic understanding of game theory. Game theory is the study of strategic decision-making, where the outcome of an individual's choice depends on the choices made by other participants in the game. In a game, each player has a set of strategies available to them, and the payoff for each player depends on the combination of strategies chosen by all the players.

In a game with multiple players, a strategy profile is a combination of strategies, where each player chooses one strategy from their set of available strategies. A Nash equilibrium occurs when no player can improve their payoff by unilaterally changing their strategy, given the strategies chosen by the other players. In other words, a Nash equilibrium is a stable state in which no player has an incentive to deviate from their chosen strategy.

To illustrate Nash equilibrium, let's consider a classic example known as the Prisoner's Dilemma. In this game, two prisoners are held in separate cells and are being interrogated by the police. Each prisoner has two options: to cooperate with their accomplice by remaining silent or to betray their accomplice by confessing to the crime. The possible outcomes and payoffs for each player are as follows:

1. If both prisoners remain silent (cooperate), they each receive a moderate sentence, resulting in a payoff of -2 for each player.
2. If both prisoners confess (betray), they each receive a harsher sentence, resulting in a payoff of -5 for each player.
3. If one prisoner confesses while the other remains silent, the confessor receives a reduced sentence (payoff of -1) while the silent prisoner receives a severe sentence (payoff of -10).

To analyze this game, we need to consider the strategies available to each player and their preferences over the outcomes. In this case, the strategies are "remain silent" and "confess" for each player. Let's assume that both players simultaneously choose their strategies without communication.

If Player A chooses to remain silent, Player B's best response would be to confess, as it leads to a reduced sentence. Similarly, if Player A chooses to confess, Player B's best response would still be to confess, as it minimizes the harshness of the sentence. This implies that Player B's strategy of confessing is a best response to any strategy chosen by Player A.

Likewise, if Player B chooses to remain silent, Player A's best response would be to confess to avoid the severe sentence. If Player B chooses to confess, Player A's best response would still be to confess to minimize the sentence. This means that Player A's strategy of confessing is a best response to any strategy chosen by Player B.

In this case, both players choosing to confess is a Nash equilibrium because neither player can unilaterally change their strategy to improve their payoff. If Player A deviates and chooses to remain silent, Player B's best response would still be to confess, resulting in a higher payoff for Player B. The same logic applies if Player B deviates and chooses to remain silent.

It is worth noting that the outcome of both players confessing is suboptimal from a collective perspective. If both players remained silent, they would receive a higher collective payoff. However, each player has a dominant strategy of confessing, regardless of the other player's choice, which leads to the suboptimal outcome.

Nash equilibrium extends beyond simple two-player games to more complex games with multiple players and a variety of strategies. It provides a useful framework for analyzing strategic interactions and predicting the likely outcomes of such interactions.

In summary, Nash equilibrium is a solution concept in game theory that describes a stable state in which no player has an incentive to unilaterally change their strategy, given the strategies chosen by other players. It helps analyze the likely outcomes of strategic interactions and is applicable in a wide range of fields, including economics, political science, and evolutionary biology. By understanding Nash equilibrium, researchers and decision-makers can gain insights into the dynamics of strategic decision-making and make more informed choices.