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Beyond A-level · 6 of 14

Repeated games and cooperation

Say why repeating a game can sustain cooperation, and when it breaks down

A one-off prisoner's dilemma predicts that every cartel collapses

A-level oligopoly theory says firms are tempted to collude and tempted to cheat. Played once, the cartel game is a prisoner's dilemma: whatever its rival does, each firm earns more by cheating on the agreed price, so both cheat and each earns £3 million instead of £5 million. The way out of a prisoner's dilemma is to find a way to penalise those who do not cooperate. A cartel cannot sign a contract a court will enforce, so the penalty has to come from somewhere else: the future.

One round of the cartel game: profits in £ million, Firm A firstA two-by-two payoff matrix. Rows are Firm A holding to the agreed price or cheating; columns are Firm B doing the same. Both hold: 5 and 5. A holds, B cheats: 2 and 6. A cheats, B holds: 6 and 2. Both cheat: 3 and 3.B: HoldB: CheatA: HoldA: Cheat5, 52, 66, 23, 3
Each cell gives Firm A's profit, then Firm B's, in £ million, for one round. Illustrative figures.

A known last round unravels cooperation all the way back to the first

Suppose the two firms know they will play exactly ten rounds, perhaps because a licence runs out. In round ten nothing follows, so it is a one-off game and both cheat. In round nine, both know round ten will be cheating whatever happens now, so good behaviour in round nine buys nothing, and both cheat. The same reasoning runs back to round one. Working from the end of a game to the start like this is backward induction, and with a known finite horizon it predicts cheating from the start.