Nash Equilibrium

A Nash equilibrium is a set of strategies — one per player — such that no player benefits from changing their own strategy while the others hold fixed. Formally, for each player $i$: $$u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \quad \text{for all } s_i$$ Concrete example — Prisoner's Dilemma: Each player can Cooperate (C) or Defect (D). Payoffs: (C,C) = (3,3), (C,D) = (0,5), (D,C) = (5,0), (D,D) = (1,1). Check (D,D): if Player 1 deviates to C, they get 0 < 1. If Player 2 deviates to C, they get 0 < 1. Neither benefits from deviating, so (D,D) is a Nash equilibrium — even though (C,C) is better for both. Finding Nash equilibria: - Pure strategy: Check each strategy profile. For 2$\times$2 games, check all 4 cells: is each player's strategy a best response to the other? - Mixed strategy: Use the indifference principle — each player mixes so that the other player is indifferent among their strategies in the support. Key results: - Every finite game has at least one Nash equilibrium (possibly mixed) — Nash's theorem - A strict dominant strategy equilibrium is always unique - Multiple equilibria are common; refinements (subgame-perfect, trembling-hand) help select among them When to use: Any strategic interaction — pricing games, auction strategy, market making, trading decisions. The concept appears constantly in quant interviews, typically in 2$\times$2 matrix games where you must find all equilibria (pure and mixed).