Game Theory — Definition, Formula & Examples
Game theory is the mathematical study of strategic interactions where the outcome for each participant depends on the choices made by all participants. It provides frameworks for analyzing situations involving cooperation, competition, and conflict.
Game theory is a branch of mathematics that models decision-making scenarios as formal games defined by a set of players, a set of strategies available to each player, and a payoff function that assigns a numerical outcome to each player for every possible combination of strategies. A central solution concept is the Nash equilibrium, a strategy profile in which no player can improve their payoff by unilaterally changing their strategy.
How It Works
A game is specified by listing the players, the strategies each can choose, and the payoffs that result from every combination of choices. You represent a two-player game as a payoff matrix, where rows correspond to one player's strategies and columns to the other's. Each cell contains the payoffs for both players. To find a Nash equilibrium, you check whether any player has an incentive to deviate from their current strategy, given what the other player is doing. A strategy profile where no one benefits from switching is a Nash equilibrium.
Worked Example
Problem: Two suspects are arrested and interrogated separately. Each can either Cooperate (stay silent) or Defect (betray the other). Payoffs represent years of reduced sentence: if both cooperate, each gets 3; if both defect, each gets 1; if one defects while the other cooperates, the defector gets 5 and the cooperator gets 0. Find the Nash equilibrium.
Step 1: Write the payoff matrix with Player A's strategies as rows and Player B's as columns. Each cell shows (A's payoff, B's payoff).
Step 2: Check Player A's best response. If B cooperates, A gets 3 by cooperating or 5 by defecting — so A prefers to defect. If B defects, A gets 0 by cooperating or 1 by defecting — A again prefers to defect. Defect is a dominant strategy for A.
Step 3: By symmetry, Defect is also a dominant strategy for Player B. Since neither player can improve their payoff by switching away from Defect, the strategy profile (Defect, Defect) is the Nash equilibrium.
Answer: The Nash equilibrium is (Defect, Defect) with payoffs (1, 1), even though mutual cooperation (3, 3) would make both players better off.
Why It Matters
Game theory is foundational in economics, political science, and computer science. Auction design, network routing algorithms, and evolutionary biology all rely on game-theoretic models. In graph theory specifically, games played on graphs — such as pursuit-evasion games and network formation games — connect these two fields directly.
Common Mistakes
Mistake: Assuming the Nash equilibrium always produces the best total outcome for all players.
Correction: A Nash equilibrium is defined by stability (no player wants to deviate), not by optimality. The Prisoner's Dilemma shows that the equilibrium outcome can be worse for everyone than a cooperative alternative.