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

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Repeated games are situations where the same players interact over multiple periods, and each player's decision depends on the past actions of others. How can you choose the best strategy to maximize your payoff in such scenarios? One possible answer is to use a trigger strategy, which is a rule that specifies how to react to the other players' choices. In this article, you will learn what trigger strategies are, how they can sustain cooperation or deter defection, and how to select the optimal trigger strategy for different types of repeated games.

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1 What are trigger strategies?

A trigger strategy is a strategy that tells you what to do in each period of a repeated game, based on the history of the game so far. For example, a trigger strategy could be: "Play cooperate in the first period, and then play whatever the other player played in the previous period." This is also known as a tit-for-tat strategy. Another trigger strategy could be: "Play cooperate in every period, unless the other player defects, in which case play defect forever." This is also known as a grim trigger strategy. Trigger strategies can be used to enforce or punish certain behaviors in repeated games, by creating incentives or threats for the other players.

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Dr Djamila Amimer

Helping Businesses Unlock AI Potential | CEO & Founder | Top 10 Global Thought Leaders on AI, Predictive Analysis and Emerging Tech. | AI Speaker | AI Author| LinkedIn Top AI Voice
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Worth defining trigger strategies in the context of game theory. Trigger strategies are defined as :"any class of strategy that is employed in a repeated non-cooperative game. A player using a trigger strategy will initially cooperated but punishes opponent if defection is detected".

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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.

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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.

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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.

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5 What are some examples of trigger strategies in real life?

Trigger strategies are not only useful for theoretical models, but also for real-life situations where repeated interactions occur. For example, in the domains of trade, diplomacy, politics, and social norms, trigger strategies can be used to explain or improve the behavior of individuals, groups, or countries. To illustrate this concept, consider the following: in trade agreements, countries may use a tit-for-tat strategy to enforce tariffs or a grim trigger strategy to withdraw from an agreement and impose sanctions; in diplomacy, countries may use a tit-for-tat strategy to respond to an attack or a grim trigger strategy to cut off diplomatic ties; in politics, parties or candidates may use a tit-for-tat strategy to counterattack or a grim trigger strategy to expose and condemn an opponent; and in social norms, individuals or groups may use a tit-for-tat strategy to react similarly or a grim trigger strategy to ostracize or punish someone who has violated a norm.

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Adam DeJans Jr.

Decision Science Leader @ Toyota | Drives Billion-Dollar Decisions | Optimization Strategist for Business Excellence | Author
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Here is a concrete example I like to give: In international relations, a tit-for-tat strategy can often govern the interactions between neighboring countries, A and B. For instance, when A initiates a favorable trade agreement, B responds in kind with a beneficial arrangement of its own. Conversely, when A imposes trade restrictions, B retaliates with similar restrictions. This strategy fosters a cooperative relationship, promoting a balance wherein positive actions are reciprocated and deviations from cooperation are met with equal retaliation, encouraging a harmonious coexistence.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Game Theory

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

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1 What are trigger strategies?

2 How do trigger strategies sustain cooperation?

3 How do trigger strategies deter defection?

4 How to select the optimal trigger strategy?

5 What are some examples of trigger strategies in real life?

6 Here’s what else to consider

Game Theory

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