Enter the payoffs for a 2×2 game below. Each cell has two values: the payoff for Player 1 and Player 2. Each player has two strategies (A and B). Click "Solve" to find all pure and mixed strategy Nash equilibria.
| P2: Strategy A | P2: Strategy B | |
|---|---|---|
| P1: Strategy A | , | , |
| P1: Strategy B | , | , |
A Nash equilibrium is a set of strategies — one for each player — where no player can improve their payoff by unilaterally changing their strategy. In a pure strategy equilibrium, each player picks a single action. In a mixed strategy equilibrium, players randomize between actions with specific probabilities that make the opponent indifferent between their own options.
Named after mathematician John Nash (made famous by the film "A Beautiful Mind"), this concept is foundational to economics, political science, biology, and computer science. Nash proved that every finite game has at least one equilibrium — though it may be in mixed strategies.
For a 2×2 game, the calculator checks all four cells for pure strategy equilibria by verifying that neither player can profitably deviate. For the mixed equilibrium, it solves the indifference conditions: Player 1 chooses their mixing probability to make Player 2 indifferent, and vice versa. The expected payoffs are then computed from these optimal mixing probabilities.
Every turn in Tactiko is a simultaneous-move game. Both teams submit their moves at the same time — just like a 2-player game theory problem. Shooting is a direct analogy: the attacker picks a target, the keeper picks a position, neither knows the other's choice. The optimal strategy? Mix randomly across your options, weighted by the payoffs.
Try it yourself at playtactiko.com — it's free, with no ads.