Infinitely Repeated Games

An infinitely repeated game is a scenario where players repeatedly engage in the same game without a predetermined end. Examples include firms continuously adjusting prices or countries consistently negotiating trade terms over many years. Players recognize that short-term gains from non-cooperative actions are less valuable than long-term losses, making mutual cooperation a rational strategy that benefits all parties over time.

Tit-for-tat involves starting with a cooperative action, such as setting a high price, then matching the opponent's previous action in each subsequent round. If one player lowers the price, the other follows suit in the next round but returns to cooperation when the opponent does. This strategy encourages mutual cooperation while punishing defection, creating sustained high profits for both firms.

Firms maintain high prices because they recognize that mutual cooperation yields better long-term benefits than short-term gains from price cutting. When both firms keep prices high, they enjoy sustained high profits. The threat of retaliation—where the opponent matches a price cut—makes defection unprofitable, so maintaining high prices becomes the rational equilibrium strategy for maximizing long-term payoffs.

In infinitely repeated games, players interact repeatedly without a predetermined end, allowing them to build reputation and enforce cooperation through retaliation. In contrast, nash equilibrium in one period games involves single interactions where players cannot punish future defection. The repeated nature enables sustained cooperation, whereas one-time games often result in non-cooperative outcomes due to lack of future consequences.

Infinitely repeated games model long-term business and international interactions. Firms use pricing strategies to maintain market stability, while countries apply cooperation principles to trade negotiations and environmental agreements. In environmental agreements, nations agree to reduce emissions and maintain cooperation to avoid long-term detrimental effects, demonstrating how repeated interaction incentivizes mutual benefit over individual short-term gains.

When a player deviates by lowering prices or breaking cooperation, the opponent retaliates in the next round by matching that action, creating a profit disadvantage for the deviator. However, retaliation is temporary; when the deviator returns to cooperation, the opponent forgives and resumes high prices. This cycle of retaliation and forgiveness reinforces the incentive to maintain cooperation rather than pursue short-term gains.

Mutual cooperation emerges because players recognize that the present value of long-term cooperative profits exceeds the one-time gain from defection plus subsequent losses from retaliation. The infinite time horizon makes future payoffs significant, and the threat of punishment deters deviation. Both players rationally choose to maintain high prices or other cooperative actions, achieving a stable equilibrium that maximizes collective and individual long-term benefits.