Game Theory Payoff Matrix Solver

What Is Game Theory? Game theory is the mathematical study of strategic decision-making between rational agents. It was founded by John von Neumann and Oskar Morgenstern with their 1944 book “Theory of Games and Economic Behavior” in the United States. John Nash extended it to non-zero-sum games in 1950, winning the Nobel Prize in Economics in 1994. Applications: economics, evolutionary biology, political science, military strategy, network routing, and auction design.

The Payoff Matrix A normal-form game is described by a payoff matrix. Rows represent Player 1’s strategies, columns represent Player 2’s strategies. Each cell contains the payoffs: (Player 1 payoff, Player 2 payoff). In a zero-sum game, payoffs sum to zero — one player’s gain is the other’s loss.

Dominant Strategies A strategy is strictly dominant if it gives a higher payoff than any other strategy, regardless of what the opponent does. A strategy is weakly dominant if it is at least as good as any other strategy against all opponent moves. If a dominant strategy exists, rational players always choose it — no need for equilibrium analysis. Dominated strategies can be iteratively eliminated to simplify complex games.

Nash Equilibrium A Nash equilibrium is a strategy profile where no player can unilaterally improve their payoff. Named after John Nash, a mathematician from the United States. Every finite game has at least one Nash equilibrium (possibly in mixed strategies). A pure strategy NE: neither player wants to deviate given the other’s strategy. Multiple NE can exist — choosing among them is a coordination problem.

Mixed Strategy Equilibrium When no pure strategy NE exists, players randomize to make the opponent indifferent. Player 2 mixes so Player 1 gets the same expected payoff from all their strategies, and vice versa. For a 2×2 game: if Player 1 plays Row 1 with probability p, set Player 2’s expected payoffs equal and solve for p. Famous example: tennis serve — mixing between left and right prevents the receiver from always guessing correctly.

Minimax Theorem (Zero-Sum Games) In two-player zero-sum games, the minimax strategy minimizes the maximum possible loss. Von Neumann proved (1928 in Germany): max_p min_q E[payoff] = min_q max_p E[payoff] — the minimax equals the maximin. This guarantees a well-defined optimal value for every finite zero-sum game. Rock-Paper-Scissors: the optimal strategy is to play each with probability 1/3.

Famous Games Prisoner’s Dilemma: both players defecting is a Nash equilibrium, even though mutual cooperation is better — a social dilemma. Battle of the Sexes: two Nash equilibria in pure strategies (coordination game). Matching Pennies: a zero-sum game with only a mixed strategy equilibrium. Chicken (Hawk-Dove): models nuclear deterrence and animal conflict.