Grim trigger

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:

1. Cooperate
2. 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[2]

Then, the strategy is an SPE only if the discount factor is ${\textstyle \delta \geq {\frac {1}{2}}}$. 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.[3]

To prove that the strategy is an 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.[2]

Step 1: Suppose that D is never played so far.

• Player i's payoff from C : ${\displaystyle (1-\delta )[1+\delta +\delta ^{2}+...]=(1-\delta )\times {\frac {1}{1-\delta }}=1}$
• Player i's payoff from D : ${\displaystyle (1-\delta )[2+0+0+...]=2(1-\delta )}$

Then, C is better than D if ${\displaystyle 1\geq 2(1-\delta )}$. This shows that if ${\displaystyle \delta \geq {\frac {1}{2}}}$ , playing C is pareto optimal.

Step 2: Suppose that someone has played D previously, then Player j will play D no matter what.

• Player i's payoff from C : ${\displaystyle (1-\delta )[-1+\delta \times 0+\delta ^{2}\times 0+...]=(1-\delta )\times -1=\delta -1}$
• Player i's payoff from D : ${\displaystyle (1-\delta )[0+\delta \times 0+\delta ^{2}\times 0+...]=0}$

Since ${\displaystyle 0\leq \delta \leq 1}$, playing D is optimal.

The preceding argument emphasizes that there is no incentive to deviate(no profitable deviation) from the cooperation profile if ${\displaystyle \delta \geq {\frac {1}{2}}}$, 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.[4]

Under the grim trigger in international relations perspective, a nation cooperates only if its partner has never exploited it in the past. Because a nation will refuse to cooperate in all future periods once its partner defects once, the indefinite removal of cooperation becomes the threat that makes such strategy a limiting case.[5] While grim trigger is a limiting case, Folk theorem states that a perfect equilibrium can be made if both nations are patient.[6]

Game Theory has recently been used in developing future communications systems, and the user in the user-network interaction game employing the grim trigger strategy is one of such examples.[7] If the grim trigger is decided to be used in the user-network interaction game, the user stays in the network (cooperates) if the network maintains a certain quality, but punishes the network by stopping the interaction and leaving the network as soon as the user finds out the opponent defects.[8] Antoniou et al. explains that “given such a strategy, the network has a stronger incentive to keep the promise given for a certain quality, since it faces the threat of losing its customer forever.”[7]

Tit for tat and grim trigger strategies are similar in nature in that both are trigger strategy where a player refuses to defect first if he has the ability to punish the opponent for defecting. The difference, however, is that grim trigger seeks maximal punishment for a single defection while tit for tat is more forgiving, offering one punishment for each defection.[9]

• Folk theorem (game theory)
• Mutually assured destruction
• Repeated game
• Trigger strategy
• Tit for tat