Enter payoffs for a 2×2 game. The calculator finds the mixed strategy Nash equilibrium: the probability each player should assign to each strategy so that neither can improve their expected payoff by changing their mix.
| P2: Strategy A | P2: Strategy B | |
|---|---|---|
| P1: Strategy A | , | , |
| P1: Strategy B | , | , |
A pure strategy means always choosing the same action. A mixed strategy means randomizing between actions according to specific probabilities. In many games, no pure strategy equilibrium exists — the only stable outcome is one where both players mix.
The classic example is rock-paper-scissors: no single choice is always best. The Nash equilibrium is to play each option with equal probability (1/3 each). Any other mix can be exploited by an opponent who notices the pattern.
The key insight is the indifference condition. At a mixed strategy equilibrium, each player's mixing probabilities are chosen so that the opponent is indifferent between their own strategies. If the opponent weren't indifferent, they'd prefer one pure strategy, breaking the equilibrium.
For a 2×2 game, the mixed strategy equilibrium can be found algebraically. Player 1 chooses probability p (for Strategy A) to make Player 2 indifferent between their two strategies. Player 2 chooses probability q (for Strategy A) to make Player 1 indifferent between theirs. Solving these two linear equations gives the equilibrium mixing probabilities.
If the resulting probabilities fall outside [0, 1], no interior mixed equilibrium exists — one strategy dominates, and the equilibrium is pure.
In Tactiko, the keeper chooses 1 of several positions to cover. The shooter picks 1 of several targets. Both choose blindly and simultaneously — a textbook mixed strategy game. From distance, it's a coin flip (50% scoring chance). Close to goal from the sides, the attacker scores 66% of the time. Right in front of goal, the attacker scores 75%. The attacker's optimal strategy? Randomize across targets. The keeper's optimal strategy? Randomize across positions. Any pattern can be exploited.
This is why the best Tactiko players vary their shots and their positioning — predictability is a weakness. The math of mixed strategies explains exactly why.