Multiple Equilibria

Game Theory · Axiom Academy

Unit 2 - Dominance and Nash Equilibrium 1. Games with Multiple Nash Equilibria A game can have zero, one, or multiple Nash equilibria. When multiple equilibria exist, each represents a stable outcome where no player wants to deviate unilaterally. The animation below shows a simple 2×2 game with two pure strategy Nash equilibria . Both (Top, Left) and (Bottom, Right) are Nash equilibria. 2. Coordination Game: Battle of the Sexes The classic Battle of the Sexes game illustrates the coordination problem. Two players want to spend time together, but they have different preferences about the activity. Nash Equilibria: (Opera, Opera) and (Football, Football). Both players prefer being together over being apart, but they disagree on which activity is better. 3. The Equilibrium Selection Problem When a game has multiple Nash equilibria, game theory alone cannot predict which equilibrium will be played. This is the equilibrium selection problem . Players must somehow coordinate their expectations about which equilibrium will occur. Several mechanisms can help: Communication: Pre-play discussion or agreements Focal points: Equilibria that "stand out" psychologically Precedent: Historical patterns or conventions Payoff dominance: Pareto superior equilibria 4. Focal Points (Schelling Points) Thomas Schelling observed that when players cannot communicate, they often converge on equilibria that are psychologically salient or prominent—these are called focal points .

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