Game Theory & Decision

Intermediate

Expected Payoff vs Mixed Opponent

Your payoff when the opponent mixes strategies.

Formula

E = \sum_j q_j u(a, b_j)

Variables

EExpected payoff

q_jOpponent's probability of b_j

uPayoff

Example

Weight each of your outcomes by how often the opponent plays each move.

Did You Know?

Best-response analysis — maximising this against the opponent's mix — is how equilibria are found.

game theory payoff mixed

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More in Game Theory & Decision

Expected Value

Probability-weighted average outcome of a decision.

Kelly Criterion

Optimal fraction of bankroll to bet for long-run growth.

Expected Utility

Weighted average of the utility of outcomes, not just their cash value.

Payoff Comparison

One strategy dominates another if its payoff is always higher.

Minimax Value

Best worst-case payoff a player can guarantee.

Mixed Strategy (Indifference)

Probability that makes an opponent indifferent between their options.