Repeated Game
What is Repeated Game?
A repeated game is the same game played again and again by the same players, so cheating today can be punished later and cooperation becomes possible.
In a one-shot game, players only weigh today's payoff, so a tempting defection carries no cost. Repeat the game and each player's move becomes a signal about how the rest of the relationship will go, which lets strategies such as tit-for-tat or a grim trigger punish a cheat in every later round. Cooperation holds when the stream of future losses from being punished outweighs the one-time gain from cheating, so it depends on players being patient and on the game not having a known final round. This is how oligopolists sustain high prices without a written agreement, and it explains why the same firms behave differently in a one-time deal than in an ongoing supply relationship.
Repeated Game: a worked example
Two suppliers hold a high price and each earns $9,000 a period. Undercutting wins the whole market for one period, worth $12,000, after which the rival matches the low price and both settle at $6,000 forever. Compare five periods of each path. Holding the price pays 5 × $9,000 = $45,000, while cheating pays $12,000 plus 4 × $6,000 = $36,000, so cooperation wins by $9,000. Run the same comparison with only two periods left and the paths tie at $18,000 each, which shows why cooperation collapses as the end of the relationship comes into view.
The mistake students make with repeated game
A widespread belief is that simply repeating a prisoner's dilemma guarantees cooperation. It does not when the number of rounds is known in advance. In the final round there is no future left to protect, so both defect; knowing that, the round before it carries no threat either, and the reasoning unravels all the way back to the first round. Cooperation needs an open-ended horizon, or at least real uncertainty about when the game ends.
Repeated Game questions
Why does cooperation survive in a repeated game but not a one-shot game?
Cooperation survives because a repeated game lets the other player punish you later, which a one-shot game does not. The gain from cheating arrives once, while the punishment can continue for every round that follows. When the future is worth enough, the arithmetic favors keeping the agreement.
What happens if both players know the exact number of rounds?
Cooperation unravels when the number of rounds is known, because the last round is effectively a one-shot game. Both players defect there, which removes any reason to cooperate in the round before, and the logic works its way back to the very first round. This is why an unknown or open-ended ending matters so much.
Is tacit collusion a repeated game?
Yes, tacit collusion is the repeated-game outcome in which rival firms keep prices high without any explicit agreement. Each firm holds its price because undercutting would start a price war that costs more than the short-lived gain. No contract is needed, which is part of why it is hard for regulators to prove.
Formula / Example
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