Payoff Matrices & Dominance

Build payoff matrices, find dominant strategies, and explore classic 2x2 games

Strategic Form Games

Two players, two choices each, and one small table of payoffs. Change the numbers and watch what each player should do.

Payoff Matrix

Edit the payoffs or load a classic game. Nash equilibria light up green and dominated strategies are struck through in red; change the Prisoner's Dilemma payoffs and watch the equilibrium shift.

Column Player
CooperateDefect
Row PlayerCooperate
,
,
Defect
,
,
Green = Row payoffBlue = Col payoffHighlighted = Nash equilibrium

Analysis

Dominated strategies: Cooperate (row player, strictly dominated by Defect); Cooperate (col player, strictly dominated by Defect)
Nash equilibria: Pure (Defect, Defect) → payoffs (1, 1)
Go deeper:Strategic Form Games

Game theory studies strategic interactions — situations where your best choice depends on what others choose. The simplest model is the 2-player normal-form game, represented as a payoff matrix. Each cell shows the payoffs to both players for a given combination of strategies.

A strategy is dominant if it gives a higher payoff than any alternative regardless of what the opponent does. When both players have dominant strategies, the outcome is determined — but it may not be optimal for either player.

Key insight: In the Prisoner's Dilemma, both players have a dominant strategy (Defect), leading to an outcome worse for both than mutual cooperation. This tension between individual and collective rationality is the heart of game theory.

Go deeper:Classic Games Gallery

Try loading each preset game and notice the differences:

  • Prisoner's Dilemma — one Nash equilibrium (Defect, Defect), both would prefer (Cooperate, Cooperate)
  • Chicken — two pure Nash equilibria and one mixed; no dominant strategies
  • Stag Hunt — two pure Nash equilibria, one risk-dominant and one payoff-dominant
  • Battle of the Sexes — coordination problem with two asymmetric equilibria
  • Matching Pennies — zero-sum; only a mixed-strategy Nash equilibrium

Key Takeaways

  • Payoff matrix — encodes the strategic structure of a game in one table
  • Dominant strategy — best regardless of opponent's choice
  • The Dilemma — individual rationality can lead to collectively bad outcomes

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