Nash Equilibrium and Game Theory

Nash Equilibrium and Best Responses

  • Nash Equilibrium Definition: A strategy profile where no player has an incentive to unilaterally deviate.
  • Each strategy in a Nash equilibrium is rationalizable because they can be justified by the equilibrium strategies of the other players.
  • Example: Meeting in New York game has two unique Nash equilibria - (Empire State, Empire State) and (Grand Central, Grand Central).

Key Concepts in Game Theory

  • Rationalizable Strategies: All strategies that can be the best response to some beliefs about the other players’ strategies.
  • Nash equilibrium provides sharper predictions than just rationalizability, particularly when beliefs align.

Best-Response Correspondence

  • Player i's best-response correspondence, denoted ( bi : S{-i} \to S_i ), assigns to each strategy profile of others the best responses for player i.
  • A strategy profile (s1, …, sn) is a Nash equilibrium if ( si \in bi(s_{-i}) ) for all players i.

Arguments for Nash Equilibrium

  1. Rational Inference: Players can infer opponents' strategies correctly based on their rationality.
  2. Unique Outcome: If only one outcome is rationally predictable, it must align with Nash equilibrium.
  3. Focal Points: Certain outcomes are naturally appealing (e.g., meeting at a well-known location).
  4. Self-Enforcing Agreements: Agreements must be Nash equilibria to hold significance.
  5. Social Conventions: Stable strategies develop over time as common practices, like walking on a specific side of the street.

Mixed Strategy Nash Equilibria

  • Definition: A mixed strategy profile ( \sigma = (\sigma1, …, \sigman) ) is a Nash Equilibrium if player i's expected payoff from each strategy he plays aligns.
  • Example: In Matching Pennies, players randomizing with equal probability leads to mixed strategy equilibrium.
  • Indifference principle: Players randomizing must find each pure strategy played with positive probability equally beneficial.

Existence of Nash Equilibria

  1. Proposition 8.D.2: Every finite game has a Nash equilibrium in mixed strategies.
  2. Proposition 8.D.3: A finite game possessing certain conditions ensures existence of a pure strategy Nash equilibrium.

Incomplete Information Games

  • Players may not know each other's payoffs; the model adapts through beliefs of players which can lead to Bayesian Nash Equilibria.

Trembling-Hand Perfection

  • Refines the Nash equilibrium by excluding predictions that involve weakly dominated strategies (those that might be played with slight mistakes).
  • A trembling-hand perfect Nash equilibrium is robust under small mistakes by players.

Backward Induction and Subgame Perfect Equilibria

  1. Subgame Definition: A part of the game indicating a complete set of moves, which can be analyzed as a standalone game.
  2. Subgame Perfect Nash Equilibrium (SPNE): A profile of strategies that must induce a Nash equilibrium in every subgame.
  3. Backward Induction Procedure: In sequential games, one determines optimal actions at final decision nodes, then moves backward to set strategies at preceding nodes.
    • Example: If each player’s strategy leads to a Nash equilibrium at every subgame resulting from the backward induction, it indicates a SPNE.

Summary of Key Results

  • Every finite game of perfect information has a pure strategy SPNE through backward induction.
  • If no two terminal nodes provide the same payoff, the SPNE is unique.
  • The concept captures rational behavior in dynamically strategic interactions, emphasizing each player's strategy must uphold both optimality and credibility.