Mixed Strategy Nash Equilibrium

Game Theory · Axiom Academy

LESSON Mixed Strategy Nash Equilibrium Unit 2: Dominance and Nash Equilibrium 1. Mixed Strategies and the Indifference Condition When a player uses a mixed strategy (randomizing between actions with positive probability), they must be indifferent between those actions. This is the key insight: Why? If one action gave a strictly higher payoff than another, the player would only play that action. Mixing only makes sense when all actions in the mix yield equal expected payoffs. 2. Setting Up the Indifference Equations Consider the game Matching Pennies . Two players simultaneously choose Heads (H) or Tails (T). Player 1 wins if they match; Player 2 wins if they don't match. This game has no pure strategy Nash equilibrium. Let's find the mixed strategy equilibrium. Let p = probability Player 1 plays H, and q = probability Player 2 plays H. For Player 2 to be indifferent: The expected payoff from playing H must equal the expected payoff from playing T. This gives us the condition to solve for p . 3. Solving Player 2's Indifference Equation Player 2's expected payoff from H: When Player 2 plays H, they get -1 if Player 1 plays H (probability p ) and +1 if Player 1 plays T (probability 1-p ). Player 2's expected payoff from T: When Player 2 plays T, they get +1 if Player 1 plays H (probability p ) and -1 if Player 1 plays T (probability 1-p ). 4. Solving Player 1's Indifference Equation By symmetry, we can apply the same logic to find q (Player 2's mixing probability).

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