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Expected Payoff Matrix Calculator

Enter the four payoffs and the row/column probabilities to find the expected payoff.

How it is calculated

The matrix is read as a = row 1 / column 1, b = row 1 / column 2, c = row 2 / column 1 and d = row 2 / column 2, all payoffs for the row player. p is the probability of playing row 1 and q is the probability of column 1, so row 2 has 1 − p and column 2 has 1 − q. The expected payoff is EV = pq·a + p(1−q)·b + (1−p)q·c + (1−p)(1−q)·d, shown to four decimals. The tool also reports the largest and smallest of the four payoffs. Probabilities must be between 0 and 1 (not percent), otherwise there is no result. Payoffs can be negative or decimal. Only one player's payoffs are used: the tool does not find equilibrium probabilities for you.

Worked example

Payoffs: a = 3, b = 0, c = 5, d = 1, with p = 0.6 and q = 0.5. Cell weights: pq = 0.30, p(1−q) = 0.30, (1−p)q = 0.20, (1−p)(1−q) = 0.20. Weighted payoffs: 0.30 × 3 = 0.90, 0.30 × 0 = 0, 0.20 × 5 = 1.00, 0.20 × 1 = 0.20. Sum: 2.1000. The best payoff in the matrix is 5.00 and the worst is 0.00.

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Common mistakes

  • Entering probabilities as percentages. 60 is rejected and gives no result; use 0.6.
  • Mixing up which cell is a, b, c and d. The order is row 1 then row 2, with column 1 before column 2, so swapping b and c changes the answer.
  • Reading p as the column player's probability. p belongs to the rows and q to the columns, and swapping them gives a different expected value.
  • Entering the opponent's payoffs. The matrix holds the row player's payoff only, so a second matrix is needed for the other player.
  • Treating the best and worst values as results of the mixed strategy. They are the highest and lowest single cells, so the expected value can sit anywhere between them.

About this calculator

In game theory, a payoff matrix lays out the outcome for every combination of two players' strategies. When each player mixes their strategies with certain probabilities rather than committing to one choice, the expected payoff is the probability-weighted average across all four cells: it sums each payoff multiplied by the joint probability of the row and column strategies that produce it. This calculator takes a 2×2 matrix plus the row and column probabilities and returns that expected value, alongside the best and worst outcomes under pure strategies.

This calculation underlies mixed-strategy Nash equilibrium analysis, taught in economics, game theory and decision science courses to model situations like pricing competition, auctions or the classic matching-pennies and battle-of-the-sexes games. Analysts and strategists use the same expected-payoff logic outside the classroom too — in negotiation modeling, risk assessment and any decision where outcomes depend on an opponent's or counterparty's uncertain choice.

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Comments (1)

  • Editorial team

    With payoffs 3, 0, 5, 1 and q = 0.5, setting p = 1 gives 1.5000, p = 0.6 gives 2.1000, p = 0.5 gives 2.2500 and p = 0 gives 3.0000. The change is steady because, for a fixed q, the expected value moves in a straight line with p. So the best row mix against a known column mix is a pure row, and mixing only matters when both sides are uncertain. What decision are you modelling, and how did you pick q?

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