Infinitely Repeated Games

Game Theory · Axiom Academy

LESSON Infinitely Repeated Games In an infinitely repeated game, there is no known final period. Players interact indefinitely, or with some probability of continuation each period. Why this matters: Without a known end date, backward induction breaks down. There's no "last period" from which to work backward, fundamentally changing what strategies are sustainable. We model infinite repetition using a discount factor δ (delta), which can be interpreted as: Time preference: Players value payoffs today more than identical payoffs tomorrow Survival probability: Each period, there's a probability δ that the game continues for another round 3. Payoff Streams and Present Values A strategy in an infinitely repeated game generates a stream of payoffs: (π₁, π₂, π₃, ...). The present discounted value aggregates these payoffs: If the same payoff π is received every period, this simplifies to a geometric series: 4. Strategies that Condition on History The power of infinitely repeated games comes from strategies that respond to past behavior. Players can: Reward cooperation: Continue cooperating if others have cooperated Punish defection: Revert to non-cooperative play if anyone deviates This threat of future punishment can sustain cooperation even when the one-shot Nash equilibrium would be defection. 5. Infinite Game Structure Visualization The structure of an infinitely repeated game creates a "shadow of the future" that makes current actions consequential for all future periods.

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