What Is Nash Equilibrium?
A Nash Equilibrium (NE) is the most fundamental solution concept in game theory. Named after mathematician John Forbes Nash Jr., who proved its existence in 1950, a Nash Equilibrium is a profile of strategies — one for each player — such that no player can improve their payoff by unilaterally deviating. In other words, each player's strategy is a best response to the strategies of all other players.
Mixed Strategies
Not every game has a Nash Equilibrium in pure strategies (where each player deterministically chooses one action). However, Nash's existence theorem guarantees that every finite game has at least one equilibrium in mixed strategies — probability distributions over pure strategies. A player mixes so that their opponent is exactly indifferent between their own options, removing any incentive to deviate.
The 2x2 Bimatrix Game
This simulator models a symmetric 2x2 game where Player 1 chooses between rows (Cooperate or Defect) and Player 2 chooses between columns. The four parameters a1, b1, c1, d1 define Player 1's payoffs in each cell. By adjusting these values, you can recreate classic games: the Prisoner's Dilemma (c1 > a1 > d1 > b1), the Stag Hunt (a1 > c1 > d1 > b1), the Chicken Game (c1 > a1 > b1 > d1), or pure coordination games.
Best Response Functions
The lower diagram shows the best response correspondences plotted on the unit square, where the x-axis represents Player 1's probability of cooperating (q) and the y-axis represents Player 2's probability (p). Each player's best response is a step function: they cooperate with probability 1 when the opponent's mixing probability exceeds a threshold, and 0 below it. The Nash Equilibrium lies at the intersection point of these two best-response functions — the white pulsing dot.
Interpreting the Results
Move the payoff sliders to see how the equilibrium shifts. When the temptation to defect is high (c1 >> a1), cooperation probability drops. When mutual cooperation is very rewarding (a1 >> d1), the equilibrium favors more cooperation. Notice that the expected payoff at a mixed NE is typically lower than the mutual cooperation payoff — this is the price of strategic uncertainty. The inefficiency of Nash Equilibrium in social dilemmas is one of the central insights of modern game theory.