What is the formula for expected payoff when one player uses a mixed strategy?

Conditions

• The opponent has a set of pure strategies.
• The opponent assigns a specific probability to each of their pure strategies, and these probabilities sum to 1.
• The payoffs for the target player are known for each combination of pure strategies.

Reasoning, step by step

1. Identify the pure strategy of the target player for which you want to calculate the expected payoff.
2. List the opponent's pure strategies and their corresponding probabilities.
3. For each opponent's pure strategy, find the payoff to the target player when the target player plays their chosen pure strategy.
4. Multiply each payoff by the probability of the opponent playing that specific pure strategy.
5. Sum all the resulting products to get the expected payoff.

Example

If Player 1 plays R with probability $p$ and L with probability $1-p$, Player 2's expected payoff from choosing R is $\pi_2(R) = p(1) + (1-p)(0) = p$. The payoffs 1 and 0 are multiplied by the probabilities $p$ and $1-p$ respectively, and then summed.

Common misconceptions

• Forgetting to multiply the payoffs by the probabilities of the opponent's strategies.
• Assuming the expected payoff is simply the average of the payoffs without weighting them by probability.

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