A repeated game is a special type of game in which the same structured game is played multiple times, even infinitely. Each individual game is called a "stage game." In each stage game, players may act simultaneously or sequentially. Since the history of other players' past actions is observable, in a repeated game, each player can choose strategies at each stage that depend on the past behavior of other players.
For example, when you go to a vegetable market to buy a watermelon and hesitate because you're worried about whether it's ripe, the vendor might say, "I run this fruit stall—would I sell you an unripe watermelon? If there's a problem, you can come back to me." Here, the vendor emphasizes that they "run this fruit stall" to sell watermelons, which reassures you to make the purchase. In game theory terms, the vendor's statement means, "I am engaged in a repeated game with you." This is an example of how formal or informal agreements in long-term economic cooperation and competition can influence behavior even without legally binding contracts.
As the name suggests, a repeated game refers to the same structured game being played many times, with each individual game called a "stage game." Repeated games are an important concept in dynamic games and can involve either complete or incomplete information. When a game is played only once, each player cares only about the immediate payoff. However, if the game is repeated multiple times, players may sacrifice short-term gains for long-term benefits, leading to different equilibrium strategies. Thus, the number of repetitions affects the equilibrium outcome of the game.
In a repeated game, the conditions, rules, and content of each stage game are identical. However, because long-term interests exist, players must consider avoiding actions that could provoke retaliation or恶性竞争 from other players in future stages. Unlike in a one-shot static game, players cannot disregard others' interests entirely. Sometimes, one party's cooperative gesture may encourage others to adopt cooperative attitudes in subsequent stages, thereby achieving shared long-term benefits.
Here are two important definitions:
Definition 1: Credibility refers to whether the first-moving player in a dynamic game should believe that the later-moving player will take actions favorable or unfavorable to them.
Definition 2: If the strategies of all players in a dynamic game form an equilibrium in both the game itself and all its subgames, the strategy profile is said to be subgame perfect.
In repeated games, credibility remains crucial—subgame perfection is still a key criterion for judging the stability and reliability of equilibria. Due to the constraining effect of long-term interests on short-term behavior, some threats or promises that are unfeasible in one-shot games become credible in repeated games, leading to a wider range of possible equilibrium outcomes.
