Lecture 21: Repeated Games - Cooperation vs. the End Game

What this lecture covers

The lecture opens the course's unit on repeated games by asking whether playing a game more than once can sustain cooperative behavior without contracts or outside enforcement. Using in-class demonstrations of Prisoners' Dilemma played a known, finite number of times, it shows that cooperation unravels from the last period backward: since there is no future in the final round, players defect, which removes any incentive to cooperate in the round before, and so on to the very first move.

The lecture then modifies the setup in two ways to recover cooperation: first by using a stage game with multiple Nash equilibria that can serve as reward and punishment even over a known finite horizon, and second by introducing an uncertain stopping rule (a repeated coin toss), which removes the fixed last period and lets students discover the grim trigger strategy. By the end, you should be able to explain why finite, known-length repeated games force defection, and set up the comparison between the temptation to cheat today and the discounted value of future rewards and punishments.

Key ideas

• Unraveling from the back: in a finitely repeated Prisoners' Dilemma, backward induction forces defection in the last period, which removes any incentive to cooperate earlier, so the whole game unravels to all-defect.
• Stage game: the single-period game that gets repeated; payoffs from earlier periods are sunk and don't affect current incentives.
• Multiple equilibria as leverage: if a stage game has more than one Nash equilibrium, players can promise a better one as a reward and threaten a worse one as a punishment, sustaining cooperation even in early periods of a finite game.
• Sub-game perfect equilibrium check: a proposed strategy must induce Nash behavior in every sub-game, including all the ones that never occur along the equilibrium path.
• Renegotiation problem: if players can communicate between periods, they may agree to abandon a costly punishment and jump straight to a better equilibrium, which then removes the original incentive to cooperate in the first place.
• Lame duck / end effects: known end points (retirement, term limits, a relationship you know will end) undermine cooperation because there is no future left to reward or punish.
• Grim trigger strategy: cooperate as long as no one has ever defected; if anyone defects even once, defect forever afterward.
• Discounting with delta: future payoffs are weighted by delta < 1 mainly because the game might end before the next round is played, not only because of impatience.

Walkthrough

Setting up the puzzle: does repetition rescue Prisoners' Dilemma? (0:01)

The lecture reintroduces Prisoners' Dilemma with cooperate/defect payoffs and asks two pairs of students to play it two or three times in front of the class. Students explain their choices in terms of building reputation or reacting to the other player's previous move, but the lecture points out that in the very last round, since there is nothing left to gain from future cooperation, defection is a dominant strategy. That logic then rolls backward: if defection is certain in the last round, there's no reason to cooperate in the second-to-last round, and the argument repeats all the way to round one, regardless of how many rounds are played.

The lame duck effect and why finite futures fail (18:22)

The lecture generalizes the unraveling result: it helps to have a future, but a known end point destroys the incentive value of that future as the end approaches. Examples include lame duck presidents and CEOs, employees nearing a known retirement date, and short relationships, all of which lose the implicit incentive that comes from expecting future interaction.

Sustaining cooperation with multiple equilibria (22:36)

A new three-strategy stage game is introduced with two pure-strategy Nash equilibria, one better for both players than the other. Played twice, the class checks a strategy that plays the cooperative outcome in period one, then switches to the good equilibrium if cooperation held or the worse equilibrium if it didn't. Verifying this is sub-game perfect requires checking every one of the nine possible second-period sub-games, and comparing the one-period temptation to defect against the discounted difference between the reward and punishment equilibria available tomorrow.

Renegotiation and its real-world echoes (45:09)

A student's objection that "why punish ourselves once someone has already cheated" motivates a discussion of renegotiation: if players can talk between periods and agree to skip the punishment because it's mutually costly, the threat loses its force, and the original cooperative equilibrium falls apart. The lecture connects this to nineteenth-century bankruptcy law cycles and to modern debates over financial bailouts, framing it as a trade-off between ex-ante efficiency (keeping incentives intact) and ex-post efficiency (doing what's best once bad outcomes have already occurred).

Removing the known end point: the coin-toss game (53:45)

The class replays Prisoners' Dilemma, but now the game continues after each round unless two coin tosses both come up heads, giving roughly a 0.75 probability of continuing each period. Because there is no fixed last period, students playing this version discover on their own that starting with cooperation and reverting to permanent defection after any betrayal, the grim trigger strategy, can sustain cooperation for as long as nobody breaks it.

Setting up the equilibrium condition (1:08:06)

The lecture closes by setting up the formal check for whether grim trigger is an equilibrium: compare the one-period temptation to defect against the discounted value of cooperating forever minus the discounted value of defecting forever, using delta as the probability the game continues each period. The value of perpetual defection is computed as zero, and the value of perpetual cooperation is left as an exercise to complete before the next class.

Before you watch

• Review the definitions of Nash equilibrium and sub-game perfect equilibrium from earlier in the course.
• Recall the war-of-attrition lecture's technique of splitting payoffs into sunk, stage, and continuation payoffs from future equilibrium play.
• Be comfortable with basic backward induction, since the unraveling argument depends on it.

Check your understanding

1. Why does backward induction force defection in every period of a Prisoners' Dilemma with a known, finite number of repetitions?
2. In the three-strategy stage game with two Nash equilibria, why does the proposed strategy still count as a valid strategy even though most of its nine second-period instructions are never reached in equilibrium?
3. What is the renegotiation problem, and why does it threaten to undo a cooperative equilibrium that relies on a costly punishment?
4. What changes about the strategic structure of the game once the number of repetitions is uncertain rather than fixed, and why does this open the door to the grim trigger strategy?
5. Why is delta less than 1 in this setting, and what is its main real-world interpretation here compared to ordinary impatience?