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Zero-Sum Game Value Calculator

Enter the four row-player payoffs to find the game value.

How it is calculated

The payoff matrix is [[a, b], [c, d]]: rows are the row player's two strategies, columns the column player's, and each number is what the row player wins (negative means loses). The tool first looks for a saddle point. It takes the smaller number in each row and then the larger of those (maximin), and the larger number in each column and then the smaller of those (minimax). If the two are equal, that number is the game value, the answer is "Yes" for a saddle point, and both players can play a pure strategy.

If they differ, the value comes from mixed play: v = (a·d − b·c) ÷ (a + d − b − c), and the answer is "No". The tool reports only the value, not the probabilities of the mixed strategy. The value is from the row player's point of view; the column player's payoff is its negative. Payoffs may be negative or decimal, and results have four decimals.

Worked example

Matrix a = 3, b = −1, c = −2, d = 4. Row minimums: −1 and −2, so maximin = −1. Column maximums: 3 and 4, so minimax = 3. They differ, so there is no saddle point and the value is mixed. Numerator: 3×4 − (−1)×(−2) = 12 − 2 = 10. Denominator: 3 + 4 − (−1) − (−2) = 10. Value = 10 ÷ 10 = 1.0000. The tool answers "No" for a saddle point: with best play by both sides, the row player wins 1 on average.

Try this example →

Common mistakes

  • Entering the payoffs from the column player's point of view. All four numbers must be the row player's winnings, otherwise the value comes out with the wrong sign.
  • Assuming a positive value means the row player wins every round. It is an average over many rounds of random mixing; in a single round the payoff is one of the four entries.
  • Putting the numbers in the wrong cells. The first row is a, b and the second is c, d, so swapping b and c turns the game into its transpose and changes the answer.
  • Using the mixed formula by hand when a saddle point exists. For a = 3, b = 5, c = 2, d = 1 the value is 3 (a saddle point), while the formula would give 2.3333, which is wrong.
  • Applying it to a larger matrix or to a game that is not zero-sum. This tool handles only 2×2 games where one player's gain is exactly the other's loss.

About this calculator

In game theory, a 2×2 zero-sum game is described by a payoff matrix showing what the row player wins (and the column player equally loses) for each combination of their two possible strategies. If a saddle point exists — a cell that is simultaneously the minimum of its row and the maximum of its column, meaning the row player’s maximin equals the column player’s minimax — that cell’s value is the game’s value, and both players have a stable pure-strategy best response.

When no saddle point exists, neither player should play a single strategy predictably, and the game’s value instead comes from optimal mixed strategies (randomizing between the two options with specific probabilities); for a payoff matrix [[a, b], [c, d]], that mixed-strategy value works out to v = (ad − bc) / (a + d − b − c). A saddle point is checked first, and only when none exists does the mixed-strategy value formula apply, the same approach economists use for any contest where one side's gain is exactly the other's loss.

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Comments (1)

  • Editorial team

    The tool reports only the value, but when no saddle point exists you can get the row player's mixing probability by hand. The chance of playing the first row is (d - c) divided by (a + d - b - c). For the matrix 3, -1, -2, 4 that is 0.6, and both columns then give exactly 1. Is this for coursework or a real contest such as penalty kicks? Share your payoff matrix.

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