Mixed Strategies & Indifference

A mixed strategy assigns a probability distribution over a player's pure strategies. In a mixed strategy Nash equilibrium, each player randomizes in a way that makes the opponent indifferent among all strategies in their support. The indifference principle: If player 1 mixes over strategies A and B, then in equilibrium player 2's mixed strategy must make player 1's expected payoff from A equal to their expected payoff from B. Otherwise, player 1 would prefer one pure strategy over the other and would not mix. Concrete example — Matching Pennies: Player 1 chooses H or T, Player 2 chooses H or T. Player 1 wins if they match, Player 2 wins if they differ. Payoff matrix: (H,H)=(1,-1), (H,T)=(-1,1), (T,H)=(-1,1), (T,T)=(1,-1). Let Player 2 play H with probability $q$. Player 1's payoff from H: $q(1) + (1-q)(-1) = 2q-1$. From T: $q(-1) + (1-q)(1) = 1-2q$. Setting equal: $2q-1 = 1-2q \Rightarrow q =$$\frac{1}{2}$. By symmetry, Player 1 also mixes 50-50. Computing mixed equilibria (2$\times$2 games): 1. Assign probability $p$ to Player 1's first strategy 2. Write Player 2's expected payoffs from each pure strategy as functions of $p$ 3. Set them equal and solve for $p$ 4. Repeat for Player 2's mixing probability $q$ When to use: Zero-sum games (poker bluffing, penalty kicks), security games, and any setting where predictability can be exploited. In market making, randomizing quote placement prevents adversarial pattern detection.