Expected Payoff vs Mixed Opponent

This formula computes your average payoff when the opponent randomizes among several actions. You first find the payoff you would get against each possible opponent move, then weight each payoff by the chance of that move and add them up; this is just the probability rule for an average outcome. It works because expected value is the long-run mean of all possible results under the opponent’s mixed strategy.

Use this whenever the opponent’s strategy is not a single fixed action, but a mixture with known probabilities, such as in mixed-strategy game theory or randomized decision models. It is especially useful for comparing your own possible actions against an uncertain opponent.

• Forgetting to multiply each payoff by the opponent’s probability before adding
• Using your own strategy probabilities instead of the opponent’s mixed strategy probabilities
• Treating expected payoff as a guaranteed outcome rather than an average over possible opponent moves