Repeated Games and the Folk Theorem

The Shadow of the Future Changes Everything

When two agents interact only once, the prisoner’s dilemma leads to mutual defection. But in real business relationships, companies meet over and over again—quarterly negotiations, repeat purchases, long-term contracts. In such conditions, the “shadow of the future”—the expectation of repeated interactions—can sustain cooperation.

Repeated games are a model that formalizes this mechanism. They explain why honesty can be strategically profitable and how tacit collusion arises without explicit agreements.

Structure of the Repeated Game

Standard game G (stage game) is played in periods t = 1, 2, 3, ... Player i discounts the future with factor δᵢ ∈ (0, 1): the payoff in the next period is worth δ times less today. The total (normalized) payoff:

Vᵢ = (1−δ) Σ_{t=1}^∞ δ^{t−1} · uᵢ(aᵗ)

Normalization by (1−δ) makes Vᵢ comparable to the payoff in a single stage. When δ → 1: the player is patient, highly values the future. When δ → 0: “lives only for today,” the future does not matter.

A strategy in a repeated game is a function from the full history to the current action: sᵢ: ∪_{t≥0} Aᵗ → Aᵢ.

Key Strategies and a Numerical Example

Prisoner’s dilemma:

Payoffs: a = 3 (mutual cooperation), b = 5 (defection when the opponent cooperates), d = 1 (mutual defection).

Grim trigger (strict trigger): Start with C. If the opponent defects—switch to D forever.

Checking stability: With mutual cooperation, each receives V = 3 (normalized). Gain from unilateral defection today: 5 − 3 = 2. Future losses (penalty): (3 − 1)·δ/(1−δ) = 2δ/(1−δ) (infinite stream, reduction from 3 to 1).

Cooperation is stable if: 2 ≤ 2δ/(1−δ) → δ ≥ 1/2.

When δ ≥ 1/2: cooperation is sustained as an SPNE with the grim trigger strategy. When δ = 0.6: cooperation is profitable. When δ = 0.4: no, the short-term gain from defection outweighs future losses.

Tit-for-tat: Start with C, then copy the opponent’s action from the previous period. Axelrod (1984) conducted computer tournaments: tit-for-tat beat expert strategies due to four properties: kindness (starts with cooperation), reciprocity (punishes defection), forgiveness (returns to cooperation), clarity (predictable).

The Folk Theorem

Theorem (Friedman, 1971; Aumann–Shapley, 1976): Let V* be the set of feasible and individually rational payoff vectors: v ∈ V* if and only if v is attainable with mixed strategies and vᵢ ≥ min_{s₋ᵢ} max_{sᵢ} uᵢ (minimax). Then for any v ∈ int(V*), there exists δ̄ < 1 such that for δ ≥ δ̄, the payoff vector v is supported as an SPNE of the infinitely repeated game.

Practical meaning: For sufficiently patient players, almost any jointly profitable outcome can be achieved in equilibrium—cooperation “emerges” from pure self-interest. This explains: long-term business relationships; reputational mechanisms; tacit collusion in oligopolies.

Oligopoly and Tacit Collusion

The folk theorem explains why airlines, banks, or oil companies can maintain high prices without explicit collusion. Conditions that facilitate collusion: small number of firms (easier coordination and punishment); transparent pricing (observable deviations); stable demand (low discounting); high entry barriers.

Antitrust authorities monitor precisely these structural signs, and not just direct evidence of collusion.

Experimental data on cooperation: Axelrod (1984) conducted computer tournaments of the repeated prisoner’s dilemma—the tit-for-tat strategy (Anatol Rapoport) defeated 14 expert strategies. Real-world observations in organizations: staff rotation destroys repeated interaction and lowers cooperation. Stable teams with a long time horizon perform more effectively—a high δ turns cooperation into an equilibrium even without a formal contract.

Repeated Games and Institutional Trust

The folk theorem explains why stable social institutions can arise without external enforcement. The reputational mechanism works as follows: a firm that cheats a buyer today loses him forever—the loss of future revenue exceeds the gain from a one-time fraud if δ is sufficiently high. That is why long-term brands value reputation: “Reputation is worth more than a one-off contract.” In international relations, countries honor trade agreements not from altruism but because they value future access to the markets. In labor relations, the phenomenon of “gift exchange” (Akerlof, 1984) describes how workers are more productive than required by contract—in response to generous pay, as in an implicit repeated game. Limitations of the folk theorem: with a finite horizon, cooperation collapses by backward induction (in the last period there is no future punishment → all defect → roll back one period before the end). Therefore, an indefinite horizon of interaction is fundamentally important to sustain cooperation.

Stability of Cooperation in International Relations and Trade

The folk theorem is confirmed by numerous real-world examples. OPEC cartel agreements, arms limitation treaties, WTO international trade regimes—all of them represent equilibria of repeated games. Cartel stability is inversely proportional to the interest rate (the higher the rate, the stronger the short-term temptation to deviate) and directly proportional to the frequency of monitoring of participants’ behavior. That is why OPEC countries periodically accuse each other of violating quotas: secret price reductions are “defections” in a repeated game. International climate negotiations are also modeled as a repeated game with a free-rider problem: each country has an incentive to deviate from emissions reduction, shifting the burden to others. Frameworks such as “pressure and reward” (emissions trading system, cross-border carbon tax) transform a one-shot game into a set of repeated interactions with monitoring and punishment. WTO trade agreements apply the principle of “reciprocity”: a country that violates tariff commitments is subject to retaliatory tariffs by affected countries—this is the punishment mechanism in the spirit of grim trigger.

Assignment: (a) For the prisoner’s dilemma (a=3, b=5, d=1): at δ=0.6 and grim trigger, is cooperation an SPNE? Calculate the threshold δ. (b) For the tit-for-tat strategy: show that cooperation is sustained at δ ≥ (b−a)/(b−d). Is the threshold the same as for grim trigger and tit-for-tat? (c) What if b = 4 (lower profit from defection)? How does this affect the stability of cooperation?