In-Depth Notes on Nash Equilibrium and Game Theory Concepts

Nash Equilibrium

  • Definition: A Nash Equilibrium occurs when each player’s strategy is optimal given the strategies of all other players.
  • Unique Prediction: Example with Mr. Schelling and Mr. Thomas illustrates how Nash equilibria yield unique predictions in certain situations.

Rationalizable Strategies

  • Every strategy in a Nash equilibrium is considered rationalizable, meaning it can be justified based on the other players' strategies.
  • Often, Nash equilibrium leads to sharper predictions than rationalizability.

Meeting in New York Game Example

  • Game Scenario: Players can choose to meet at two locations: 1) Empire State, or 2) Grand Central Station.
  • Nash Equilibria: There are two equilibria: (Empire State, Empire State) and (Grand Central, Grand Central).
  • Mutual Expectations: Players must have correct expectations about the strategy choices of one another.

Best-Response Correspondence

  • Definition: Player's best-response correspondence assigns each strategy to a set of best responses based on others' strategies.
  • Restatement of Nash Equilibrium: A strategy profile is a Nash equilibrium if each player's strategy is in their own best response to the others.

Discussion of Nash Equilibrium

  • Rational players should ideally predict opponents' plays correctly.
Arguments for Nash Equilibrium
  1. Rational Inference: Rational players should correctly forecast rivals' actions. However, this isn't guaranteed always.
  2. Unique Outcomes: If there’s only one predicted outcome, rational players will understand it must be a Nash equilibrium.
  3. Focal Points: Certain outcomes can be preferred due to cultural familiarity or simplicity (e.g., where to meet).
  4. Self-Enforcing Agreements: Non-binding agreements must lead to Nash equilibria because players cannot secure strategies.
  5. Stable Social Conventions: Conventions that arise over time in repeated conditions can lead to Nash equilibria.

Mixed Strategies

  • Definition: A mixed strategy Nash equilibrium involves players randomizing their strategies.
  • Example of Matching Pennies: Each player randomizes choices to ensure that the other is indifferent between their options.
  • Indifference: For a mixed strategy to be Nash, players must earn the same payoffs regardless of their pure strategy choices.

Existence of Nash Equilibria

  • Proposition: A game with a finite number of strategies for each player will have a mixed strategy Nash equilibrium.
  • Pure Strategy Existence: For certain conditions involving compact and continuous payoffs, a pure strategy Nash equilibrium can be guaranteed.

Games of Incomplete Information

  • Incomplete information introduces uncertainty about other players’ preferences.
  • Bayesian Games: Each player's actions depend on the realized values of a random variable that indicates their type.

Bayesian Nash Equilibrium

  • A profile of decision rules is a Bayesian Nash equilibrium if players' strategies are optimal given their beliefs about other players' types.

Trembling-Hand Perfection

  • Concept: A refinement of Nash equilibrium that considers the possibility of unintended mistakes in players’ strategies.
  • Normal Form Trembling-Hand Perfect Nash Equilibrium: Such equilibria withstand small probability mistakes and avoid weakly dominated strategies.

Conclusion:

  • Understanding Nash Equilibria is crucial in predicting outcomes in strategic interactions over both complete and incomplete information under dynamic conditions.
  • Key concepts include backward induction, subgame perfection, and rational behavior underpinning rationalizable strategies.
  • The existence of equilibria is foundational both for pure strategies in finite games and mixed strategies across broader scenarios.