What it does
In game theory a Nash equilibrium is a choice of strategies where no player can do better by changing only their own choice. This tool finds every equilibrium of a two-player game in which each player has 2 to 6 strategies (a “bimatrix” game), including the mixed ones, where players pick at random with set probabilities. It also:
- marks each player’s best responses in the payoff table, so pure equilibria stand out;
- removes dominated strategies step by step (strategies that are always worse than another one);
- for zero-sum games (one player’s gain is the other’s loss), gives the value of the game and each player’s safest (maximin) mix, by linear programming.
Everything runs in your browser.
It works well with the reliability and agreement, the uncertainty propagation calculator and the CSV profiler and SQL playground.
How to use it
- Pick a classic game to start from, or set the number of strategies for each player and start from a blank table.
- In each cell type the row player’s payoff, then the column player’s payoff, such as
3, 2. Use a dot for decimals. Tick Zero-sum to type only the row player’s payoff. - Rename the strategies in the header cells if you like.
- Read the list of equilibria: what each player does (probabilities shown as fractions where they are simple) and the expected payoff of each. Download them as CSV, or the whole report as an image, PDF or Word file.
How it works
For every possible support (the strategies a player uses with positive probability) the tool solves the equations that make the other player indifferent among their best replies, then keeps the solutions that are true probabilities with no better reply outside. Supports of every size are tried, which is the same as enumerating the vertices of the best-response polytopes, so the tool also finds the extreme equilibria of degenerate games (where some strategy has two equally good replies). For such games a note explains that mixtures of the listed equilibria can be equilibria too.
Worked examples
Battle of the sexes (Opera 3, 2 / Football 2, 3): two pure equilibria, both at Opera (payoffs 3 and 2) or both at Football (2 and 3), and one mixed one where the row player chooses Opera with probability 3/5 and the column player with probability 2/5, giving each an expected 1.2.
Prisoner’s dilemma: Quiet is strictly dominated by Confess for both players, so dominance alone solves the game: both confess.
Rock, paper, scissors: one equilibrium, each move with probability 1/3; the value is 0.
Made-up 3 × 3 game: three equilibria: Top / Right (2, 5), Middle / Centre (5, 4), and a mixed one with the row player on Top 1/3 and Middle 2/3 and the column player on Centre 1/3 and Right 2/3, paying 1.667 and 3.
FAQ
Why does the number of equilibria matter? A non-degenerate game always has an odd number of equilibria. An even count signals a degenerate game.
Is an equilibrium the best outcome? No. In the prisoner’s dilemma both players would prefer to stay quiet, but that is not an equilibrium.
How large can the game be? Up to 6 × 6; the tool checks every support pair, which takes a moment at most.
Where the numbers come from
- Methods: Nash (1950) equilibrium; support and vertex enumeration as in Nisan, Roughgarden, Tardos and Vazirani (eds.), Algorithmic Game Theory (2007), chapter 3; von Neumann's minimax theorem solved as a linear programme. Classic textbook games (prisoner's dilemma, battle of the sexes, chicken, stag hunt, matching pennies, rock-paper-scissors) are common knowledge; no tables copied.
- Written by fibub. Checked in the tests against nashpy 0.0.43 (vertex enumeration) on textbook and random games up to 6 × 6, including degenerate ones, and against SciPy linprog for zero-sum values.
- The 3 × 3 example game is made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.