Nash equilibrium is a key concept in game theory. It's a situation where no player can gain by changing their strategy if others stick to theirs. This stable state reflects the best responses of all players to each other's strategies.

Understanding Nash equilibrium helps predict outcomes in strategic interactions. It shows how rational players might behave when their decisions affect each other. This concept is crucial for analyzing various real-world scenarios, from economics to politics.

Fundamentals of Nash Equilibrium

Definition and Components

• Nash equilibrium represents a stable state in a game where no player has an incentive to unilaterally change their strategy, given the strategies of the other players
• Best response refers to the optimal strategy a player can choose, considering the strategies chosen by the other players
• Strategy profile is a combination of strategies, one for each player in the game, that specifies the actions taken by all players
• Payoff matrix is a tabular representation of a game, showing the payoffs each player receives for every possible combination of strategies

Identifying Nash Equilibrium

• To find Nash equilibrium, analyze each player's best response to the strategies of the other players
• In a two-player game, a Nash equilibrium occurs when the strategies of both players intersect at a point where neither player can improve their payoff by unilaterally changing their strategy
• In games with more than two players, a Nash equilibrium is a strategy profile where each player's strategy is a best response to the strategies of the other players
• Multiple Nash equilibria can exist in a game, and finding all of them requires systematically examining each player's best responses

Characteristics of Nash Equilibrium

Strategic Stability

• Nash equilibrium is strategically stable because no player has an incentive to deviate from their chosen strategy, assuming the other players also stick to their equilibrium strategies
• This stability arises from the fact that each player's strategy is a best response to the strategies of the other players
• The absence of incentives to unilaterally change strategies makes Nash equilibrium a stable outcome in a game

Self-Enforcing Agreement

• Nash equilibrium can be seen as a self-enforcing agreement among players, as it is in each player's best interest to adhere to their equilibrium strategy
• Players do not need external enforcement mechanisms to maintain their strategies, as deviating from the equilibrium would lead to a worse outcome for the deviating player
• The self-enforcing nature of Nash equilibrium makes it a plausible outcome in real-world strategic interactions where binding contracts or external enforcement may not be feasible

Mutual Best Response

• In a Nash equilibrium, each player's strategy is a best response to the strategies of the other players
• This mutual best response property ensures that no player can improve their payoff by unilaterally changing their strategy
• The concept of mutual best response highlights the interdependence of players' strategies in reaching a Nash equilibrium
• When all players are simultaneously playing their best responses to each other's strategies, the game reaches a Nash equilibrium

Advanced Concepts

Equilibrium Selection

• Equilibrium selection refers to the process of choosing among multiple Nash equilibria in a game
• Games with multiple Nash equilibria raise the question of which equilibrium players are likely to coordinate on
• Focal points, such as salient or symmetric outcomes, can serve as equilibrium selection devices, helping players coordinate on a particular Nash equilibrium
• Payoff dominance, where one equilibrium offers higher payoffs for all players compared to another, can also guide equilibrium selection
• Risk dominance, which considers the riskiness of each equilibrium in terms of the potential losses from deviating, is another criterion for equilibrium selection
• Evolutionary stability, which examines the robustness of an equilibrium against small perturbations in the population of players, can also inform equilibrium selection in repeated games

Refinements of Nash Equilibrium

• Refinements of Nash equilibrium aim to narrow down the set of plausible equilibria by imposing additional criteria or restrictions
• Subgame perfect equilibrium requires that the strategies chosen by players constitute a Nash equilibrium in every subgame of the original game, eliminating non-credible threats
• Perfect Bayesian equilibrium combines the concepts of subgame perfection and Bayesian updating of beliefs in games with incomplete information
• Trembling hand perfect equilibrium assumes that players may make small mistakes (or "tremble") when choosing their strategies, selecting equilibria that are robust to such mistakes
• Proper equilibrium further refines trembling hand perfection by requiring that more costly mistakes are made with lower probability than less costly ones