# What is nash equilibrium?

## What is nash equilibrium?

Nash equilibrium is a fundamental concept in game theory that represents a stable state of a strategic interaction between multiple participants, where no player has an incentive to unilaterally change their strategy. It is named after mathematician John Nash, who introduced the concept in his seminal work in the 1950s.

To understand Nash equilibrium, it's important to first grasp the basics of game theory. Game theory is a mathematical framework that analyzes decision-making in strategic situations where the outcome of an individual's choice depends on the choices made by others. These situations are often referred to as games.

In a game, players have a set of available strategies that determine their actions or decisions. The outcome or payoff of each player depends not only on their own chosen strategy but also on the strategies chosen by the other players. The goal of game theory is to analyze the possible outcomes and strategies in these situations to identify stable states or equilibria.

A Nash equilibrium is reached when each player in a game selects their best response strategy, given the strategies chosen by the other players. In other words, it is a state where no player can benefit by unilaterally changing their strategy, assuming the other players' strategies remain unchanged. At a Nash equilibrium, players are effectively "locked in" to their strategies, and no player has an incentive to deviate from their current course of action.

Formally, in a game with multiple players and strategies, a Nash equilibrium is a set of strategies, one for each player, where no player can improve their payoff by unilaterally changing their strategy. It represents a stable point in the game where players have found an optimal response to the strategies of others.

It's important to note that a Nash equilibrium may not always result in the best possible outcome for all players involved. It simply represents a situation where no player can improve their own payoff by changing their strategy independently. This means that Nash equilibria can lead to suboptimal or inefficient outcomes from a global perspective.

To illustrate the concept, let's consider the classic example of the Prisoner's Dilemma. In this game, two suspects are arrested and held in separate cells. The district attorney offers each prisoner a deal: if one prisoner confesses and cooperates with the prosecution while the other remains silent, the one who cooperates will receive a reduced sentence, while the other will receive a severe sentence. If both prisoners confess, they will each receive a moderately reduced sentence. If both prisoners remain silent, they will each receive a reduced sentence.

In this scenario, the Nash equilibrium occurs when both prisoners confess, even though both would be better off if they both remained silent. Each prisoner fears that the other will confess, leading to a severe sentence if they remain silent. Therefore, they choose the strategy that minimizes their own potential losses, resulting in a Nash equilibrium where both confess.

Nash equilibrium has widespread applications in various fields, including economics, political science, biology, and evolutionary psychology. It provides a framework for analyzing strategic interactions and predicting the likely outcomes in situations where multiple players are involved.

One key aspect of Nash equilibrium is that it does not require perfect rationality or knowledge of the other players' strategies. Players only need to choose their best response strategy given their understanding of the game and the strategies of others. This concept allows for the analysis of situations where players have limited information or face uncertainty about the actions of others.

In more complex games with multiple equilibria, researchers often use additional concepts such as dominance, backward induction, or refinements like subgame perfect equilibrium to refine the set of possible equilibria and predict the most likely outcome. These refinements help address situations where multiple equilibria exist, but some equilibria are considered more plausible or compelling based on additional assumptions or reasoning.

In summary, Nash equilibrium is a fundamental concept in game theory that represents a stable state in a strategic interaction where no player has an incentive to unilaterally change their strategy. It provides insights into the likely outcomes of games and has applications in various fields. While Nash equilibrium does not guarantee the best outcome for all players, it helps analyze strategic decision-making and understand the dynamics of complex interactions.

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