Repeated games and the roots of cooperation

Cooperation with no contract, no court, no collateral

Twelve women sit in a circle in a Nairobi estate, a cash box and a ledger on the table. Each lays down the same sum; this month the pot goes to one of them, next month to another. No contract binds them, no court will hear a complaint, no collateral has been pledged — a self-interested member should take the pot on her turn and never pay in again. Yet chamas in Kenya, stokvels in South Africa, susu in Ghana and tontines across West Africa move enormous sums and rarely collapse. Why does cooperation hold when defection so plainly pays? Because the members expect to meet again — and a game played once and a game played over and over are different games. That difference drives nearly all the cooperation you will ever observe.

Finite, infinite, and indefinite horizons

A repeated game plays a stage game — here the Prisoner's Dilemma — in periods t = 0, 1, 2, … The payoffs never change; what changes is the strategic object. You no longer choose an action but a strategy: a rule mapping the whole history of play into today's move. Because today can depend on yesterday, you can reward cooperation and punish betrayal. Two horizons matter. A finitely repeated game ends at a known, fixed period. An indefinitely repeated game ends eventually, but no one knows when — each period continues only with some probability. That gap, between a known end and an uncertain one, decides everything.

The finite-horizon trap

Suppose the game is repeated a known number of times, and both players know it. Reason from the last period back. In the final round there is no future to protect, so it is a one-shot dilemma: both defect. Step back one round. Both already know the last round will be mutual defection whatever happens now, so cooperating today buys nothing — defect. The same logic bites in every earlier round, back to the first. Cooperation unravels entirely: the unique subgame-perfect equilibrium of a known, finitely repeated Prisoner's Dilemma is to defect in every period.

Discounting and the factor δ

Remove the known endpoint and the cascade never starts — there is no last round to reason back from. To weigh payoffs across time, use the discount factor δ ∈ [0, 1): a shilling next period is worth δ shillings now. Two forces load onto δ. One is patience — how much you value the future for its own sake. The other is continuation — the probability the relationship survives to be played again. If your time-preference factor is β and the group lasts each period with probability p, then δ = β × p. Impatience and fragility pull the same way: both shrink δ and make cooperation harder.

Deriving the cooperation threshold

Now the central result: the exact patience cooperation requires. Label the four payoffs — T for temptation (you defect, they cooperate), R for reward (both cooperate), P for punishment (both defect), S for the sucker's payoff (you cooperate, they defect) — with T > R > P > S. Consider the grim-trigger strategy: cooperate while no one has ever defected, and the instant anyone defects, defect forever after. It is the harshest punishment imaginable — and exactly how a savings circle treats a member who fails to pay: expulsion, no way back.

                     Player 2
                   C          D
              +----------+----------+
   Player 1 C |  R , R   |  S , T   |
            D |  T , S   |  P , P   |
              +----------+----------+

   Ordering:  T > R > P > S
   (temptation > reward > punishment > sucker)

To test whether both playing grim trigger is subgame-perfect, use the one-shot-deviation principle: a profile is subgame-perfect if and only if no player gains by deviating in a single period and then reverting. Sit on the cooperative path. Cooperate forever and you earn R every period. Or defect now: the opponent still cooperates today, so you seize T, but the trigger fires and from next period both defect forever at P apiece. Cooperation survives when R/(1−δ) ≥ T + δP/(1−δ). The punishment is credible — mutual defection is itself a stage-game Nash equilibrium, so playing it forever is an equilibrium of the continuation game, which makes grim trigger subgame-perfect, not merely Nash.

Cooperate forever (geometric series, common ratio δ):
    V(C) = R + δR + δ²R + ⋯ = R / (1 − δ)

Defect once, then punished forever:
    V(D) = T + δP + δ²P + ⋯ = T + δP / (1 − δ)

No profitable one-shot deviation  ⇔  V(C) ≥ V(D):
    R / (1 − δ)  ≥  T + δP / (1 − δ)

Multiply through by (1 − δ) > 0:
    R  ≥  T(1 − δ) + δP
    R  ≥  T − δT + δP
    R − T  ≥  − δ(T − P)

Multiply by −1 (flips ≥ to ≤), then divide by (T − P) > 0:
    T − R  ≤  δ(T − P)
    δ  ≥  (T − R) / (T − P)

Read the threshold δ ≥ (T − R)/(T − P). The numerator T − R is the immediate prize for betrayal — what you pocket today by defecting instead of taking the reward. The denominator T − P is the whole stake — the distance between the best one-shot grab and the grim life you inhabit ever after. Cooperation holds exactly when patience is at least the ratio of temptation to stake. Shrink the temptation, harden the punishment, or make players more patient, and cooperation gets easier.

The chama as a repeated game

Put the opening circle under the microscope. A rotating savings and credit association — chama, stokvel, susu, tontine, esusu — works like this: n members each pay a fixed amount c into a pot every month, and the whole pot goes to one member, the turn rotating so that over n months everyone collects once. It converts a trickle of contributions into a usable lump sum — school fees, restocking a stall, a hospital bill — and runs on nothing but members' willingness to keep paying after they have been paid out. Model each monthly choice as a move in a repeated game: contribute (cooperate) or walk away (defect), a Prisoner's Dilemma against the rest of the group.

• R, the reward: you contribute and the circle keeps turning, so you keep the device — the lump sum on your turn, the discipline of forced saving, the insurance of a group that rallies in a crisis. This is why you joined.
• T, the temptation: take the pot on your month and never pay in again — a one-off windfall banked at everyone else's expense.
• P, the punishment: default and you are expelled and sanctioned — shut out of this circle and, by reputation, the next, and marked in church, market and street.
• S, the sucker's payoff: you keep paying into a circle others have already abandoned.

The result transfers verbatim: contribution is sustainable under grim-trigger exclusion if and only if δ ≥ (T − R)/(T − P). Each term now bites. δ is whether members expect the group to survive and weigh that future heavily — neighbours bound for years have δ near one; strangers who may scatter have a low δ. The temptation T − R is roughly the pot net of a month's benefit. And the stake T − P is where social collateral does its work.

The last-slot problem

One arrangement breaks even honest, patient people: a fixed cycle everyone knows will end. Suppose the circle runs exactly n months, then disbands. Reason backwards. In the final month, every member but that month's recipient has already collected; with the group dissolving after, paying in once more is a pure outflow protecting no future, so they walk — and the holder of the last slot is left with an empty pot. Knowing this, no one funds the last month; knowing that, members due earlier watch their own future evaporate and stop paying too. The cycle unravels from the end, exactly as backward induction warns, and in practice never even starts. Nobody wants to be last, because the last slot is where the finite horizon bites. Real associations defuse this in ways that map cleanly onto the theory:

1. Keep the horizon open. Most circles roll into a fresh cycle when one ends, so there is never a known final round; the end recedes into probability, and contribution holds whenever δ = p is large enough. This is the single most powerful fix.
2. Order the rotation by trust. Seat anchored, well-known members early and make newer or riskier ones wait, so a member with little reputation to lose must first pay in for months — posting a bond of contributions — before ever holding the pot.
3. Price the reputation. A defaulter's name travels the same church, market and family networks that vet them for the next circle, the next loan, the next job — turning a one-time betrayal into a lasting cost that pushes P down.
4. Post social collateral and guarantors. Members are admitted on an existing member's word, and that sponsor shares the shame of a default; elders may hold reserves or levy fines. The circle borrows enforcement from the community around it.

The Folk Theorem

Grim-trigger cooperation is one equilibrium of the repeated game; the Folk Theorem — so named because the idea circulated among game theorists before anyone claimed it — says it is far from the only one. In standard form: in an infinitely repeated game with sufficiently patient players (δ close to 1), any feasible payoff profile giving every player strictly more than their minmax payoff — the worst the others can force on them — can be sustained as a subgame-perfect equilibrium. The intuition is your derivation generalised: when the future weighs heavily, the threat of reverting to punishment swamps the gain from any single deviation, so a vast range of behaviour can be sustained. That is the promise and the warning. The promise: efficient cooperation without contracts or courts. The warning: so is much else — patience makes cooperation possible alongside many other equilibria, some mediocre, some collusive, and never guarantees the good one. Which one a society reaches is the work of institutions, norms and history.

Strategies, reputation, and the reach of repetition

Grim trigger is unforgiving by design, which makes it brittle: one mistake — a payment lost in transit, a turn misremembered — dooms the relationship forever. Is there a better rule? Around 1980 Robert Axelrod ran two computer tournaments, inviting scholars to submit strategies for a repeated Prisoner's Dilemma. Both times the winner was among the simplest entries: TIT FOR TAT, submitted by Anatol Rapoport — cooperate first, then copy whatever the opponent did last period. Axelrod drew from the results the qualities that thrive: be nice (never defect first), provocable (punish promptly), forgiving (return to cooperation once the other does), and clear (simple enough that others learn to trust you).

Forgiveness and clarity point to a deeper force: reputation. The unravelling result assumed everyone knew they faced coldly rational opponents. Relax that. If there is even a small chance a player is a committed type who always cooperates, a rational player may find it worth mimicking that type — cooperating to build a reputation — even under a known finite horizon. A little cultivated uncertainty about types can restore the cooperation backward induction had destroyed. This is why reputation is the load-bearing wall of informal economies: it lets people act as though the relationship were open-ended even when, strictly, it is not.

The same machine runs far beyond the savings circle. Cross-border trader networks along African frontiers move goods on credit through channels no formal court reaches, and honest dealing holds because the trading community remembers: cheat one supplier and word runs the corridor, closing the network to you. The medieval Maghribi traders enforced long-distance contracts the same way, by collective exclusion of any agent who cheated. And the identical arithmetic, turned to darker ends, holds cartels together: firms colluding to keep prices high play a repeated Prisoner's Dilemma, deterring any undercut with the grim threat of a price war. A cartel is stable for the very reason a chama is — δ ≥ (T − R)/(T − P) holds.