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 W. Friedman, which uses the concept.

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 W. Friedman, which uses the concept. Then, the strategy is an SPE only if the discount factor is \delta \geq {\frac{1}{2}} . 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. Step 1: Suppose that D is never played so far. Player i's payoff from C : (1-\delta)[1+\delta+\delta^2+ ... ] = (1-\delta) \times \frac{1}{1-\delta} = 1 Player i's payoff from D : (1-\delta)[2+0+0+ ... ] = 2(1-\delta) Then, C is better than D if 1 \geq 2(1-\delta) . Step 2: Suppose that someone has played D previously, then Player j will play D no matter what. Player i's payoff from C : (1-\delta)[-1+\delta \times 0+\delta^2 \times 0 + ... ] = (1-\delta) \times -1 = \delta - 1 Player i's payoff from D : (1-\delta)[0+\delta \times 0+\delta^2 \times 0 + ... ] = 0 Since 0 \leq \delta \leq 1 , playing D is optimal. The preceding argument emphasizes that there is no incentive to deviate (no profitable deviation) from the cooperation profile if \delta \geq \frac{1}{2} , and this is true for every subgame. Therefore, the strategy for the infinitely repeated prisoners’ dilemma game is a Subgame Perfect Nash equilibrium. In iterated prisoner's dilemma strategy competitions, grim trigger performs poorly even without noise, and adding signal errors makes it even worse. Its ability to threaten permanent defection gives it a theoretically effective way to sustain trust, but because of its unforgiving nature and the inability to communicate this threat in advance, it performs poorly. ==See also==

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