Define 2-player games with payoff matrices. Find Nash equilibria, dominant strategies, and mixed strategy solutions visually.
A payoff pair for every combination of row and column strategies defines the game this tool analyzes. It compares unilateral alternatives to identify dominant strategies and pure-strategy Nash equilibria, then determines mixed-strategy probabilities when applicable. Payoff order matters: in a pair written as (row, column), the first value belongs to the row player and the second to the column player. Reversing those values changes the best-response comparisons and can change every reported equilibrium.
A concrete input and expected output from the current implementation.
Input
Payoff order: (row player, column player) Columns: Heads, Tails Rows: Heads: (1,-1), (-1,1) Tails: (-1,1), (1,-1)
Expected output
Pure-strategy Nash equilibria: none. Dominant strategies: none for either player. Mixed-strategy Nash equilibrium: the row player chooses Heads with 50% probability and Tails with 50%; the column player does the same.
At every pure outcome, the player receiving -1 can switch strategies and receive 1. Against a 50-50 mixture, either pure strategy has expected payoff 0, so both players are indifferent and neither benefits from changing the mixture.
For the row player, compare the first payoffs vertically while holding the column fixed. For the column player, compare the second payoffs horizontally while holding the row fixed. A cell is a pure Nash equilibrium when both selected strategies are best responses. Dominance is stronger: the same strategy must remain preferable against every strategy available to the opponent.
In a 2 by 2 game, the row player's mixing probability is chosen to make the column player indifferent between the two columns, and the column player's probability makes the row player indifferent between the two rows. An interior mixed equilibrium uses probabilities strictly between 0 and 1. Boundary probabilities of 0 or 1 represent pure play.
Yes. Several cells can simultaneously satisfy the best-response condition, and a game can also have mixed equilibria. The tool's equilibrium results should therefore be read as a set of stable strategy profiles, not as a guarantee that players will select one particular outcome. Additional assumptions may be needed to predict which equilibrium occurs.
A player can randomize with positive probability between two strategies only when those strategies give the same expected payoff against the opponent's mixture. Otherwise, the player would place all probability on the better strategy. The reported probability for one player is therefore calculated from the other player's indifference condition, not from that player's own payoff equation.
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