Mixed Strategies

Game Theory · Axiom Academy

Unit 2 - Dominance and Nash Equilibrium A pure strategy is a deterministic choice: you always play the same action. A mixed strategy is a probability distribution over your pure strategies, introducing controlled randomness. For example, in Rock-Paper-Scissors, playing "Rock" every time is a pure strategy. Playing Rock 1/3 of the time, Paper 1/3, and Scissors 1/3 is a mixed strategy. 2. Probability Distributions and Notation We denote a mixed strategy for player i as σ i (sigma). This represents the probability distribution over player i 's pure strategies. The constraint that probabilities sum to 1 ensures we have a valid probability distribution: 3. Why Randomize? The Value of Unpredictability Randomization prevents opponents from exploiting predictable patterns. In competitive settings, being predictable allows your opponent to best-respond perfectly, often putting you at a disadvantage. Consider the classic game of Matching Pennies : you win if coins match, your opponent wins if they don't. If you always play Heads, your opponent will always play Tails (assuming they want to mismatch). But if you randomize 50-50, they can't gain an advantage! 4. Expected Payoff with Mixed Strategies When players use mixed strategies, payoffs become expected values calculated by weighting each outcome by its probability. If player 1 plays mixed strategy σ 1 and player 2 plays σ 2 , the expected payoff for player 1 is:

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