Mixed Strategy Equilibrium Calculator
For a two-player game with two choices each — penalty kicks, serve direction, bluff or fold — the mix of choices each side should play so that the other cannot exploit it, from the payoffs, with the value of the game and whether a pure strategy dominates instead.
When neither player has a choice that is always best, the answer is to randomise — but in the exact proportion that makes the opponent indifferent between their options.
How the mixed strategy equilibrium calculator works
When neither player has a choice that is always best, the answer is to randomise — but in the exact proportion that makes the opponent indifferent between their options. For the row player that proportion comes from the column player's payoffs, and vice versa. The result is the Nash equilibrium of the game: a kicker who goes left 58% of the time and a keeper who dives left 42% of the time, say, and neither can do better by changing. If one choice is always better, the game has a pure equilibrium and no mixing is needed.
Formula: row plays A with p = (d − c) / ((a − b) − (c − d)) using the column player's payoffs; column plays X with q likewise from the row player's payoffs
Worked examples
| Inputs | Row player: share of A | Note |
|---|---|---|
| A penalty kick | 57.14% | both sides mix |
| A dominant choice | 100% | a pure equilibrium |
| Matching pennies | 50% | 50/50 each |
FAQFrequently asked questions
What is a mixed strategy?
Choosing between options at random in fixed proportions. In a game where any predictable choice can be exploited, the right randomisation is the only choice that cannot be.
Why does the row player's mix depend on the column player's payoffs?
Because its job is to make the column player indifferent — to remove any reason for them to favour one option. Your own payoffs decide the value of the game, not your mix.
When is there no mixing?
When one option is better for a player whatever the other does — a dominant strategy — or when the best responses meet at a cell. Then the equilibrium is pure and the page says so.
Does this apply to real sport?
Penalty kicks, serve direction and pitch selection have all been studied this way, and professionals turn out to mix close to the equilibrium. Amateurs usually do not — which is the edge.
What about bigger games?
Three or more options each need linear programming. The two-by-two case is the one with a closed formula and most of the intuition.
Where these figures come from
- World Handicap System — Rules of Handicapping — the score differential and handicap index formula
- International Cricket Council — Playing Conditions — over structure and run-rate definitions
- Cricket Australia — the national body for the sport these statistics most often describe here
Last checked: September 2026. Formulas follow the governing bodies where one publishes them — the World Handicap System for golf, the ICC playing conditions for cricket run rates.