Game Theory Payoff Matrix Explorer

A normal-form game lists the players, the choices each can make, and the payoff each receives for every combination of choices. In a 2x2 game two players each pick one of two strategies.

The payoff matrix shows both payoffs in each cell. The first number goes to player 1 who picks the row, the second to player 2 who picks the column.

A best response is the choice that maximizes a player's payoff given what the other player does. The tool underlines each player's best-response payoff so you can read the structure off the matrix.

A strictly dominant strategy is a best response no matter what the opponent picks. When a player has one, rational play is settled before the game even begins.

A Nash equilibrium is a choice for every player where no one can do better by switching alone. In a pure equilibrium each player commits to a single strategy.

A cell is a pure Nash equilibrium exactly when both payoffs are best responses, so both values are underlined. A game can have zero, one, or several pure equilibria.

When no pure equilibrium exists, players randomize. In a mixed equilibrium each player chooses probabilities that keep the opponent indifferent between their two options.

The tool solves the 2x2 closed form. Player 1 plays the first row with probability p, player 2 the first column with probability q, and it reports the expected payoffs.

In the prisoner's dilemma defection strictly dominates cooperation for both players, so the only Nash equilibrium is mutual defection.

That outcome pays both players less than mutual cooperation would. The dilemma is that individual rationality leads to a jointly worse result.

• Stag hunt. Two equilibria, one safe and one rewarding, model the tension between trust and caution.
• Chicken. Each player wants the other to yield, with the worst outcome when neither does.
• Matching pennies. A zero-sum game with no pure equilibrium, only a 50-50 mix.
• Battle of the sexes. Two players prefer to coordinate but disagree on which option.

When the prisoner's dilemma repeats, cooperation can pay because today's choice shapes the opponent's future moves. The tool runs the chosen strategies for a set number of rounds and shows the move history and cumulative scores.

Tit for tat cooperates first, then copies the opponent's last move. It is simple, never defects first, and punishes defection once before forgiving. Grim trigger is harsher and defects forever after any betrayal. The round-robin tournament pits all five strategies against each other so you can compare totals.