How do you choose the optimal trigger strategy for repeated games?

2 How do trigger strategies sustain cooperation?

One of the main reasons to use trigger strategies in repeated games is to sustain cooperation, which means achieving a mutually beneficial outcome that would not be possible in a one-shot game. For example, consider the prisoner's dilemma, where two prisoners can either confess or remain silent, and their payoffs depend on what the other prisoner does. In a one-shot game, the dominant strategy for both prisoners is to confess, even though they would both be better off if they both remained silent. However, in a repeated game, they can use trigger strategies to cooperate and achieve the higher payoff. For instance, they can use the tit-for-tat strategy, which means that they will remain silent as long as the other prisoner does the same, and confess only if the other prisoner betrays them. This way, they can establish trust and avoid the temptation to defect.

3 How do trigger strategies deter defection?

Another reason to use trigger strategies in repeated games is to deter defection, which means preventing the other players from deviating from a desired outcome. For example, consider a duopoly market, where two firms can either charge a high price or a low price, and their profits depend on what the other firm does. In a one-shot game, the dominant strategy for both firms is to charge a low price, even though they would both be better off if they both charged a high price. However, in a repeated game, they can use trigger strategies to deter defection and maintain the high price. For instance, they can use the grim trigger strategy, which means that they will charge a high price as long as the other firm does the same, and charge a low price forever if the other firm undercuts them. This way, they can create a credible threat and discourage the other firm from cheating.

4 How to select the optimal trigger strategy?

The optimal trigger strategy for a repeated game depends on several factors, such as the payoffs of the players, the discount factor, and the probability of continuation. The payoffs of the players determine the potential gains and losses from cooperation or defection. The discount factor measures how much the players value the future relative to the present. The probability of continuation measures how likely the game will continue in the next period. Generally, the optimal trigger strategy is the one that maximizes the expected payoff of the player, given the constraints and expectations of the game. For example, if the payoffs are high enough, the discount factor is close to one, and the probability of continuation is high, then a grim trigger strategy may be optimal, because it can ensure a stable and efficient outcome. However, if the payoffs are low, the discount factor is close to zero, or the probability of continuation is low, then a tit-for-tat strategy may be optimal, because it can adapt to changing situations and avoid unnecessary punishments.