Grim Trigger

Game Theory · Axiom Academy

Understanding the Ultimate Punishment Strategy in Repeated Games The strategy is "grim" because it shows no forgiveness. One defection triggers eternal punishment through mutual defection. This creates a powerful threat to deter deviation from cooperation. When both players use grim trigger, is this a Nash equilibrium? We need to check if either player has an incentive to deviate. Suppose both players follow grim trigger. They cooperate forever, earning the cooperation payoff in each period. Consider a player deviating by defecting once: Let's denote the payoffs in a standard prisoner's dilemma: C = cooperation payoff (both cooperate) D = defection payoff (both defect) T = temptation payoff (I defect, opponent cooperates) S = sucker payoff (I cooperate, opponent defects) With discount factor δ, compare present values: For grim trigger to be a Nash equilibrium, cooperation must be at least as valuable as deviation: Rearranging this inequality step by step: 5. When Grim Trigger Sustains Cooperation The critical threshold δ* determines when players are patient enough for grim trigger to work: This formula shows that cooperation is sustainable when: Players are sufficiently patient (δ is large enough) The temptation gain (T - C) is small relative to cooperation value The punishment is severe (C is much larger than D)

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