Week 10: Repeated Games & Trust

Why the same one-shot game plays out completely differently once you expect to play it again tomorrow

1 May 2028

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The prisoner’s dilemma has one Nash equilibrium played once: defect. Played an unknown number of times against the same partner, the folk theorem says almost any outcome — including sustained cooperation — can be an equilibrium too, provided both sides value the future enough and can punish defection when it happens. Trust, on this account, isn’t a feeling; it’s what a repeated game looks like from the inside when the shadow of the future is long enough to make cooperation self-enforcing.

Tit-for-tat’s tournament success is the concrete evidence: a strategy that cooperates first, then simply mirrors what it was shown, beats far more elaborate ones — not because it’s clever, but because it’s legible. It’s instantly readable (your partner knows exactly what triggers what), quick to forgive (one bad round costs exactly one round of retaliation, not the whole relationship), and impossible to exploit for more than a single round at a time. Every one of those properties is about how the strategy behaves under a partner’s mistakes, not how it behaves against a perfectly rational opponent — which is the more realistic test, since real partners do have bad weeks.

That last point matters more than it looks. Swap tit-for-tat’s forgiveness for a harsher rule — defect forever after the first defection, no matter what happens next — and the exact same one-off bad month that tit-for-tat absorbs in a single round of mutual sulking instead ends the relationship permanently. The folk theorem doesn’t pick between these; both “forgiving cooperation” and “grim, permanent retaliation” are equilibria once the shadow of the future is long enough. Which one two real people actually play is a fact about the strategies they carry into the relationship, not a fact the model derives for you.

This is also where week 9’s negotiated split stops being an abstract split and starts being something that has to survive Tuesday, and the Tuesday after that. A split isn’t self-enforcing the moment it’s signed — nothing stops either side from quietly taking more the instant they believe it’s the last round. What actually holds a negotiated split together over time is this week’s shadow of the future: the expectation of enough future rounds that breaking the deal now costs more than whatever it gains.

Priya and Theo, several months into whatever week 6 started, supply this week’s running case: Theo has a standing habit of forgetting to text when he’s running late, and Priya has settled — without ever deciding to, exactly — into a functioning tit-for-tat, one noticeably cool reply after a missed heads-up, full warmth restored the instant the next one arrives on time. Neither of them has read Axelrod. Both of them are, completely unknowingly, playing the equilibrium the folk theorem predicts should be stable.

Outline

One thing this framework won’t explain

The folk theorem’s proof is also its limitation: it shows cooperation can be an equilibrium once the shadow of the future is long enough, but so can almost anything else two sides can credibly threaten each other into, including permanent mutual retaliation over a single misread bad month. The model doesn’t say which equilibrium two real people actually settle into — it only says the set of possible ones is enormous.

That’s week 7’s gap again, one level up. Week 7 couldn’t say who breaks a standoff on a single night; this week, the same missing ingredient decides which of many possible relationships two people end up spending years inside — forgiving or grim, cooperative or locked in retaliation, with the same starting conditions each time. Whatever actually picks one of these in practice (a couple’s own history, an explicit conversation, plain personality) is, once again, coming from outside the game.

Further reading

Teaching team