Nash Equilibrium Calculator

What Is a Nash Equilibrium? A Nash Equilibrium (named after mathematician John Forbes Nash Jr., who published the concept in 1950 in the United States) is a set of strategies where no player can improve their outcome by changing their strategy alone. It is the cornerstone concept in game theory. Nash received the Nobel Prize in Economics in 1994 for this work.

The 2x2 Payoff Matrix In a two-player game, each player has two strategies. The payoff matrix shows the outcome for each combination. Each cell contains two numbers: the payoff for Player 1 (row player) and the payoff for Player 2 (column player).

Finding Pure Strategy Nash Equilibria Check each cell: Is Player 1’s payoff the best in that column? Is Player 2’s payoff the best in that row? If both conditions hold, that cell is a Nash Equilibrium. There can be 0, 1, or 2 pure strategy equilibria.

Finding Mixed Strategy Nash Equilibria When no pure strategy equilibrium exists (or in addition to pure ones), players can randomize. Player 1 chooses a probability p for their first strategy such that Player 2 is indifferent between their two strategies. Similarly, Player 2 chooses probability q to make Player 1 indifferent. The mixed strategy equilibrium gives each player’s optimal randomization probabilities.

Famous Examples The Prisoner’s Dilemma: Two suspects can cooperate (stay silent) or defect (betray). The Nash Equilibrium is (Defect, Defect) even though (Cooperate, Cooperate) would be better for both. Battle of the Sexes: A couple wants to go out together but prefers different activities. There are two pure equilibria plus one mixed equilibrium. Matching Pennies: No pure equilibrium exists. The mixed equilibrium is 50/50 for each player.