Grim trigger

In game theory, grim trigger (also called the grim strategy or just grim) is a trigger strategy for a repeated game. Initially, a player using grim trigger will cooperate, but as soon as the opponent defects (thus satisfying the trigger condition), the player using grim trigger will defect for the remainder of the iterated game. Since a single defect by the opponent triggers defection forever, grim trigger is the most strictly unforgiving of strategies in an iterated game. In Robert Axelrod's book The Evolution of Cooperation, grim trigger is called "Friedman", for a 1971 paper by James Friedman, which uses the concept. The infinitely repeated prisoners’ dilemma is a well-known example for the grim trigger strategy. The normal game for two prisoners is as follows: In the prisoners' dilemma, each player has two choices in each stage: Cooperate Defect for an immediate gain If a player defects, he will be punished for the remainder of the game. In fact, both players are better off to stay silent (cooperate) than to betray the other, so playing (C, C) is the cooperative profile while playing (D, D), also the unique Nash equilibrium in this game, is the punishment profile. In the grim trigger strategy, a player cooperates in the first round and in the subsequent rounds as long as his opponent does not defect from the agreement. Once the player finds that the opponent has betrayed in the previous game, he will then defect forever. In order to evaluate the subgame perfect equilibrium (SPE) for the following grim trigger strategy of the game, strategy S* for players i and j is as follows: Play C in every period unless someone has ever played D in the past Play D forever if someone has played D in the past Then, the strategy is an SPE only if the discount factor is . In other words, neither Player 1 or Player 2 is incentivized to defect from the cooperation profile if the discount factor is greater than one half. To prove that the strategy is a SPE, cooperation should be the best response to the other player's cooperation, and the defection should be the best response to the other player's defection.