Payoff Matrix Expected Value Calculator for Mixed Strategies

Enter the row player's payoff matrix and a probability for each opponent action. The calculator multiplies each payoff by the matching column probability, sums the products by row, and identifies the pure strategy or tied strategies with the highest expected payoff. It is useful for game theory exercises, strategic decision analysis, economics courses, and transparent checks of mixed-strategy reasoning.

Arrange the payoff matrix from the row player's perspective

Begin with a matrix whose rows represent the pure strategies you want to evaluate and whose columns represent the opponent's possible pure strategies. Every cell must contain the payoff received by the row player for that particular pair of choices. If a textbook displays ordered payoff pairs such as (3, 1), enter only the first number when the row player is the first player; use only the second number if you are instead evaluating the second player's choices after transposing the strategic viewpoint. Keep every row the same length because each row must correspond to the same ordered set of opponent actions. Negative payoffs, zero, fractions, and decimal values are all valid. Optional strategy names make the result easier to interpret, but they must follow the matrix row order. Without names, the calculator assigns Strategy 1, Strategy 2, and so forth. This explicit structure prevents a common error: combining one player's payoff entries with probabilities that refer to a differently ordered set of opponent actions.

Enter and interpret the opponent's mixed strategy

Supply one probability for each matrix column, in exactly the same order as the opponent strategies represented by those columns. Probabilities must be between zero and one and must add to one. A probability of zero is allowed and means that the corresponding opponent action contributes nothing to any expected payoff. For each row, the calculator forms a probability-weighted average: it multiplies every payoff in that row by its column probability and then adds those products. The returned expected payoff is therefore the long-run average payoff associated with committing to that pure row strategy while the opponent repeatedly randomizes according to the supplied distribution. It is not a guaranteed payoff in a single play, nor does it predict which action the opponent will actually choose. The probability sum is returned as a check, and inputs with a sum outside a small floating-point tolerance are rejected instead of silently normalized. That behavior keeps the stated mixed strategy distinct from accidental or incomplete weights.

Compare pure strategies and identify a best response

Read the expected_payoffs list to compare all row strategies under the same opponent distribution. The maximum_expected_payoff field gives the largest computed value, while best_strategies lists every strategy attaining that value. Returning all maximizers matters because a player can be indifferent between two or more pure strategies. Any mixture supported entirely on tied best responses also has the same expected payoff against this fixed opponent strategy, although this calculator does not solve for equilibrium probabilities. A best response here is conditional on the probabilities you supplied: changing the opponent's mixture can change the ranking, create a tie, or remove one. Use the result to verify a candidate mixed-strategy equilibrium by checking whether strategies claimed to be in a player's support are tied and no excluded pure strategy pays more. For automated use, the API price is $0.002 per request. The deterministic calculation also runs without network calls or random sampling, so repeated identical inputs produce identical numeric results.