A Nash equilibrium is a set of strategies, one for each player, where each player’s strategy is optimal given the strategies of the other players. In other words, it is a point where all players are satisfied with their chosen strategies, and there is no incentive for any player to change their strategy.

Key Concepts:

• Players: Individuals or groups making decisions in a game.
• Strategies: The choices available to each player.
• Nash Equilibrium: A set of strategies, one for each player, where each strategy is optimal given the strategies of the other players.
• Optimal Strategy: The strategy that maximizes a player’s payoff, given the strategies of the other players.
• Payoff: The benefit or cost associated with each possible combination of strategies.

Examples:

• Prisoner’s Dilemma: Two prisoners are interrogated separately and asked to testify against each other. If both prisoners testify, they each receive a penalty of 5 years. If one prisoner testifies and the other remains silent, the testifying prisoner receives a penalty of 2 years and the silent prisoner receives a penalty of 3 years. If both prisoners remain silent, they receive a penalty of 1 year. In this case, the Nash equilibrium is (T,T), where both prisoners testify.
• Zero-Sum Game: Two players have two strategies each, and the sum of their payoffs is always zero. In a zero-sum game, there is always a Nash equilibrium, as each player can guarantee themselves a non-negative payoff by playing their best move.

Characteristics:

• Non-cooperative: Players do not cooperate in choosing their strategies.
• Strategic: Strategies are based on the expectations of other players’ actions.
• Ex-ante: Strategies are chosen before the game begins.
• Unique: In some games, there can be multiple Nash equilibria.

Applications:

Nash equilibrium is used in various fields, including:

• Game theory
• Economics
• Computer science
• Artificial intelligence

Key Takeaways:

• Nash equilibrium is a concept in game theory that describes a point where all players are satisfied with their chosen strategies.
• The Nash equilibrium is a fundamental concept in game theory and has applications in various fields.
• The existence of a Nash equilibrium depends on the specific game and the players’ strategies.