What is a Repeated Game?

A repeated game is a special form of game theory in which the same basic game (called a stage game) is played multiple times, potentially infinitely. In each stage game, players may act simultaneously or sequentially. Since players can observe the history of past actions, their strategies in a repeated game may depend on the previous behavior of other players.

For example, when buying a watermelon at a market, if you hesitate because you’re unsure whether it’s ripe, the vendor might say: "I run this fruit stall—why would I sell you an unripe melon? If there’s a problem, you can come back to me." By emphasizing that they operate a long-term business, the vendor reassures you, making you more likely to buy. In game theory terms, this statement implies: "We are engaged in a repeated game." This is an example of how formal or informal agreements in long-term economic interactions can influence behavior, even without legally binding contracts.

As the name suggests, a repeated game involves the same game structure played over multiple rounds (stage games). It is a key concept in dynamic games and can involve either complete or incomplete information. In a one-shot game, players care only about immediate payoffs. However, in repeated games, players may sacrifice short-term gains for long-term benefits, leading to different equilibrium strategies. Thus, the number of repetitions affects the game’s equilibrium outcome.

In repeated games, the rules and conditions remain the same in each stage, but the presence of long-term interests means players must consider whether their actions might trigger retaliation or competition in future rounds. Unlike one-shot static games, players cannot act purely selfishly. Sometimes, a cooperative gesture by one player can encourage mutual cooperation in later stages, maximizing collective long-term gains.

Two Key Definitions:

Definition 1: Credibility
In dynamic games, credibility refers to whether the first-moving player can trust the subsequent player(s) to act in their favor (or disfavor).

Definition 2: Subgame Perfectness
A strategy profile is subgame perfect if it forms a Nash equilibrium in every subgame of the original dynamic game.

In repeated games, credibility is critical, and subgame perfection remains a key criterion for stable equilibria. Due to the influence of long-term incentives, threats or promises that are non-credible in one-shot games may become credible in repeated interactions, expanding the range of possible equilibrium outcomes.