What is a Nash equilibrium?

A Nash equilibrium is a combination of strategies where no player can gain by changing their own strategy while the others keep theirs. In a payoff matrix a cell is a pure equilibrium exactly when the row player’s payoff is the largest in its column and the column player’s payoff is the largest in its row — that is, both numbers are underlined. In the prisoner’s dilemma, mutual defection is the only equilibrium even though mutual cooperation would leave both players better off.

Why do matching pennies and rock–paper–scissors have no pure equilibrium?

Whatever your opponent plays, you always have a better reply, so someone wants to deviate from every cell. Nash proved that every finite game has an equilibrium in mixed strategies, where each player randomises so that the other is indifferent between the strategies they use. In matching pennies both players pick each side with probability 1/2, in rock–paper–scissors each move with probability 1/3, and the value of both games is 0.

How are mixed equilibria calculated?

By support enumeration: assume the row player uses a certain set of strategies (the support) and the column player a set of the same size, write equations making the opponent’s expected payoff equal across those strategies with probabilities adding up to 1, then check that the probabilities are non-negative and no strategy outside the support does better. Every solution that passes is an equilibrium. The arithmetic uses exact fractions, so you get 9/10 rather than a rounded decimal.

What does the value of a zero-sum game mean?

In a zero-sum game one player’s gain is the other’s loss. Von Neumann’s minimax theorem says the largest expected payoff the row player can guarantee (maximin) equals the smallest the column player can hold them to (minimax); that common number is the value, and the strategies achieving it are optimal. If the matrix has a saddle point — an entry that is the minimum of its row and the maximum of its column — pure strategies suffice; otherwise the tool finds the optimal mixed strategies with linear programming.

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♟️Game Theory Calculator - Nash Equilibrium & Zero-Sum Solver

Whether you’re studying game theory, checking homework or thinking through a strategic situation, type in the payoff matrix of a two-player normal-form game — 2 to 6 strategies each, with integers, decimals or fractions — and get a full analysis at once. Pure Nash equilibria: each player’s best responses are underlined in every cell, and cells where both payoffs are underlined are the equilibria, highlighted in green. Mixed Nash equilibria come from support enumeration, solving the indifference equations for every pair of equal-size supports in exact BigInt fractions, so you get precise probabilities such as 3/5 and 2/5 plus each player’s expected payoff. For nondegenerate games every equilibrium is found (there is always an odd number of them), and degenerate games are flagged. Iterated elimination of strictly dominated strategies lists who is dominated by whom, step by step; click a step to see the removed rows and columns greyed out, and optionally check for domination by mixed strategies. Zero-sum (and constant-sum) games are also solved with an exact-fraction simplex for the value of the game and both optimal mixed strategies, along with the pure maximin, minimax and any saddle points. Pareto optimal outcomes are marked, and 2×2 games get a best-response diagram whose intersections are the equilibria. Classic presets include the prisoner’s dilemma, chicken, battle of the sexes, matching pennies, rock–paper–scissors and the stag hunt, or roll a random game.

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How to use Game Theory Calculator

1Pick a classic game or set the number of strategies, then fill in the payoffs (row player left, column player right)
2Read the highlighted best responses and the pure and mixed Nash equilibria with expected payoffs
3Explore the elimination of dominated strategies, the minimax solution of zero-sum games and the 2×2 best-response diagram

FAQ

What is a Nash equilibrium?: A Nash equilibrium is a combination of strategies where no player can gain by changing their own strategy while the others keep theirs. In a payoff matrix a cell is a pure equilibrium exactly when the row player’s payoff is the largest in its column and the column player’s payoff is the largest in its row — that is, both numbers are underlined. In the prisoner’s dilemma, mutual defection is the only equilibrium even though mutual cooperation would leave both players better off.
Why do matching pennies and rock–paper–scissors have no pure equilibrium?: Whatever your opponent plays, you always have a better reply, so someone wants to deviate from every cell. Nash proved that every finite game has an equilibrium in mixed strategies, where each player randomises so that the other is indifferent between the strategies they use. In matching pennies both players pick each side with probability 1/2, in rock–paper–scissors each move with probability 1/3, and the value of both games is 0.
How are mixed equilibria calculated?: By support enumeration: assume the row player uses a certain set of strategies (the support) and the column player a set of the same size, write equations making the opponent’s expected payoff equal across those strategies with probabilities adding up to 1, then check that the probabilities are non-negative and no strategy outside the support does better. Every solution that passes is an equilibrium. The arithmetic uses exact fractions, so you get 9/10 rather than a rounded decimal.
What does the value of a zero-sum game mean?: In a zero-sum game one player’s gain is the other’s loss. Von Neumann’s minimax theorem says the largest expected payoff the row player can guarantee (maximin) equals the smallest the column player can hold them to (minimax); that common number is the value, and the strategies achieving it are optimal. If the matrix has a saddle point — an entry that is the minimum of its row and the maximum of its column — pure strategies suffice; otherwise the tool finds the optimal mixed strategies with linear programming.