Mixed Strategy
What is Mixed Strategy?
A mixed strategy is a plan to randomize over your moves with fixed probabilities, used when always making the same predictable choice would be exploited.
A pure strategy commits you to one move every time. A mixed strategy assigns a probability to each move, so the opponent cannot predict you and cannot tailor a response. Games where any predictable pattern gets punished, such as a penalty kick or a pitcher choosing pitches, often have no equilibrium in pure strategies at all, and the only stable outcome is both sides randomizing. The equilibrium probabilities have a surprising property: each player mixes in the way that leaves the opponent indifferent between their own options. John Nash proved that every finite game has at least one equilibrium once mixed strategies are allowed, which is why a game that looks unsolvable in pure strategies still has an answer.
Mixed Strategy: a worked example
A kicker aims Left or Right while a keeper dives Left or Right at the same instant, with payoffs as scoring chances in percent written (kicker, keeper): Left against a Left dive gives (30, 70), Left against Right gives (80, 20), Right against Left gives (90, 10), and Right against Right gives (20, 80). Suppose the keeper dives Left half the time. Kicking Left then scores 30 half the time and 80 half the time, an average of 55. Kicking Right scores 90 half the time and 20 half the time, also 55. Because both aims are worth the same, the kicker has no pattern worth exploiting, and a fifty-fifty dive is the keeper's equilibrium mix.
The mistake students make with mixed strategy
Many students assume a mixed strategy just means guessing, or that you pick the probabilities that make your own payoff largest. Neither is right. At the equilibrium mix your expected payoff is identical across the moves you are mixing over, so your own probabilities cannot be chosen to raise it. Those probabilities are pinned down by the other player's payoffs, because their job is to leave the other player with no better reply.
Mixed Strategy questions
When does a game need a mixed strategy?
A game needs a mixed strategy when it has no Nash equilibrium in pure strategies, so no pair of fixed choices is stable. You spot this by checking every cell and finding that one player always wants to switch. Games of pure conflict, like matching pennies or a penalty kick, are the standard examples.
Why does the equilibrium mix make the opponent indifferent?
The mix makes the opponent indifferent because if it did not, the opponent would have a strictly better move and would play that move every time. You would then be facing a predictable rival and would change your own mix in response, so nothing would be stable. Indifference is the only condition under which neither side wants to adjust.
Is a pure strategy also a mixed strategy?
Yes, a pure strategy is the special case of a mixed strategy that puts a probability of 1 on one move and 0 on the rest. Treating pure strategies this way is what lets a single definition of Nash equilibrium cover both cases. It is also why every finite game is guaranteed at least one equilibrium.
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