Repeated Games: Why Cooperation Emerges

In 1984, political scientist Robert Axelrod ran a tournament that changed how we understand cooperation.

He invited experts in game theory, economics, and computer science to submit strategies for an iterated prisoner's dilemma. Each strategy would play against every other strategy, round after round. The winner would be whoever accumulated the most points.

The submissions ranged from sophisticated to simple, from cooperative to ruthless. The winning strategy was the simplest one: Tit for Tat. Cooperate on the first move. After that, do whatever your opponent did last time.

The result surprised even Axelrod. Tit for Tat wasn't trying to beat its opponents. It never "won" a single match. But it accumulated the highest total score because it elicited cooperation from cooperators and avoided exploitation by defectors.

The tournament revealed something profound about trust: cooperation can emerge without trust, as a strategic equilibrium in repeated interactions.

The One-Shot Problem

The prisoner's dilemma is famous for a reason: it captures a fundamental tension in social life.

Two players each choose to cooperate or defect. If both cooperate, both do well. If both defect, both do poorly. But if one cooperates and one defects, the defector does best and the cooperator does worst.

In a single interaction, defection dominates. Whatever the other player does, you're better off defecting. If they cooperate, you gain by exploiting them. If they defect, at least you don't get exploited. Defect, defect—the logic is inexorable.

This creates the dilemma. Both players would be better off if both cooperated. But individual rationality leads to mutual defection. The rational outcome is collectively worse than the cooperative outcome.

This seems to doom cooperation. And in one-shot interactions, it mostly does. If you'll never see someone again, if there's no reputation, no enforcement, no memory—defection is the rational strategy.

But most interactions aren't one-shot. Most relationships are repeated. And repetition changes everything.

The Shadow of the Future

When interactions repeat, the future casts a shadow over the present.

If you defect today, your partner can punish you tomorrow. If you cooperate today, you build a relationship that pays off later. The single-shot calculation (defect always beats cooperate) becomes clouded by long-run considerations.

Game theorists formalize this with the "discount factor"—how much you value future payoffs relative to present ones. If you care about the future (high discount factor), cooperation becomes sustainable. If you don't care (low discount factor), you're back to one-shot logic.

The folk theorem of repeated games proves this formally: when players care enough about the future, almost any outcome can be sustained as an equilibrium—including full cooperation.

"Almost any" is important. The folk theorem is permissive. It says cooperation can emerge, not that it will emerge. Mutual defection is also an equilibrium. So is alternating cooperation and defection. The repeated game has many equilibria; which one you land in depends on how players coordinate.

This explains why trust matters even when cooperation is self-enforcing. Players need to coordinate on the cooperative equilibrium rather than the defection equilibrium. Trust isn't just a feeling—it's a coordination device.

Tit for Tat and Its Lessons

Axelrod's tournament revealed which strategies succeed in repeated games.

Tit for Tat won because it had four properties:

Nice: It started by cooperating. This invited cooperation from cooperative partners.

Retaliatory: It punished defection immediately. This deterred exploitation.

Forgiving: It returned to cooperation after a single punishment. This prevented spirals of mutual defection.

Clear: Its strategy was transparent. Partners could predict its behavior and coordinate accordingly.

These properties aren't obvious. You might think ruthless strategies would dominate. They didn't. Ruthless strategies triggered retaliation and got locked in mutual defection. Nice strategies that were too forgiving got exploited. Tit for Tat hit a sweet spot: nice enough to elicit cooperation, tough enough to deter defection, forgiving enough to escape conflicts.

Subsequent tournaments found variations that did even better:

Tit for Two Tats: Retaliate only after two consecutive defections. More forgiving, less prone to cycles.

Generous Tit for Tat: Sometimes cooperate even after defection. Adds noise-tolerance.

Win-Stay, Lose-Shift: If the outcome was good, repeat your action. If bad, switch. Simple adaptive rule.

The common theme: strategies that balance cooperation and protection outperform those that lean too far in either direction.

Why Cooperation Is Fragile

The folk theorem says cooperation can be sustained. But actually achieving it is harder than the theorem suggests.

Multiple equilibria: There are many equilibria, and players might coordinate on different ones. If I expect you to defect, I should defect. If you expect me to defect, you should defect. The expectation becomes self-fulfilling. Getting to cooperation requires both players expecting cooperation simultaneously.

Noise: Real interactions are noisy. Sometimes cooperation looks like defection. Sometimes defection is misperceived as cooperation. In noisy environments, Tit for Tat can trigger spirals—one misperceived defection leads to retaliation, which looks like defection, which leads to more retaliation.

Ending problem: If the interaction has a known endpoint, backward induction kicks in. On the last round, defect (it's a one-shot game). But then on the second-to-last round, defect (the last round is determined). The logic unravels all the way back. Cooperation requires that players don't know when the game ends.

Incomplete information: You don't always know who you're playing with. Is this a Tit for Tat player or an Always Defect player? Early moves are partly about learning your opponent's type, which creates risk. You might cooperate to test cooperation, but that makes you vulnerable if your opponent is a defector.

Coalition formation: In multi-player settings, subgroups can form coalitions that exploit others. The two-player logic doesn't scale cleanly.

These difficulties explain why cooperation is so precarious. The theoretical possibility doesn't guarantee the practical reality.

Institutions as Coordination Devices

This is where institutions come in.

Institutions help players coordinate on cooperative equilibria. They do this by:

Signaling. Joining an institution signals your type. If you're a member of a trade association with a code of conduct, you're probably a cooperator. The institution serves as a sorting mechanism.

Monitoring. Institutions can monitor behavior that individuals can't. A credit bureau tracks payment history across many transactions. A professional licensing board monitors standards. This makes defection more visible.

Punishing. Institutions can coordinate punishment that individuals can't. If everyone in a community knows that defectors are punished, each individual's punishment is credible. Collective action problems in punishment are solved by institutional coordination.

Setting expectations. Institutions create focal points—common knowledge about what behavior to expect. "In this community, we cooperate" is a shared expectation that makes cooperation self-fulfilling.

Extending time horizons. Institutions create continuity. Even if individual relationships end, the institution persists. You cooperate today because your institutional reputation affects future institutional interactions.

The math of repeated games shows that cooperation is possible. Institutions make it likely.

Trust as Equilibrium Selection

This framework reframes trust.

In the repeated game setting, trust isn't a warm feeling. Trust is a belief about which equilibrium you're playing. If you trust your partner, you believe you're both playing cooperate-cooperate. If you distrust, you believe you're in defect-defect.

Building trust is moving from a defection equilibrium to a cooperation equilibrium. That requires coordinated expectation change—both parties must come to expect cooperation simultaneously.

Destroying trust is moving from cooperation to defection. This can happen suddenly—one betrayal can flip expectations. The asymmetry matters: building trust requires many rounds of reinforced expectations; destroying it requires one disconfirming event.

This explains why trust is so hard to rebuild. Once you're in a defection equilibrium, every move looks like defection. Even cooperative overtures are suspected of being manipulation. The equilibrium is self-reinforcing.

Breaking out requires either: - A credible commitment that changes the game structure - A third party that guarantees behavior - Sufficient history that outlier events can be forgiven - Willingness to cooperate despite expected defection (risky, but can shift equilibrium)

The Temporal Structure of Trust

The repeated-games framework reveals why trust has a particular temporal structure.

Trust requires a future. Without ongoing interaction, there's no shadow of the future, no strategic reason to cooperate. This is why trust is so hard in anonymous, one-shot contexts. It's also why building relationships matters—the relationship creates the future that sustains cooperation.

Trust requires patience. If you discount the future heavily—if you need rewards now—cooperation breaks down. This is why desperation undermines trust. It's also why stability matters: people who expect to be around cooperate differently than people expecting to exit.

Trust requires memory. If past defection is forgotten, punishment can't work. This is why reputation matters. It's also why transparent track records enable trust with strangers—you import memory from other relationships.

Trust requires uncertainty about ending. If you know exactly when the relationship ends, backward induction collapses cooperation. This is why open-ended relationships support trust better than term-limited ones.

These structural features aren't about feelings. They're about the strategic logic of repeated interaction. Trust emerges when the structure supports it.

Applications: From Trenches to Trade Wars

The repeated-games framework explains phenomena across many domains.

WWI trenches. Axelrod documented how opposing soldiers in WWI trenches developed tacit cooperation—"live and let live" systems where each side refrained from killing the other. The soldiers were in a repeated game. Defection (shooting to kill) would be punished. Cooperation (shooting to miss, or on predictable schedules) was rewarded with reciprocal restraint. Commanders had to rotate units to break the cooperative equilibria that kept emerging.

Trade agreements. International trade is a repeated game between countries. Tariff wars are mutual defection. Free trade is mutual cooperation. Trade agreements work by making the cooperation equilibrium focal—creating shared expectations that cooperation will be reciprocated. Trade wars happen when one party defects and triggers retaliation spirals.

Business relationships. Long-term supplier relationships function like Tit for Tat. Good performance is rewarded with continued business. Poor performance triggers renegotiation or replacement. The shadow of the future disciplines behavior on both sides.

Arms races. Countries in security dilemmas face repeated prisoner's dilemmas. Each side is better off armed, but both armed is worse than both disarmed. Arms control agreements try to shift to cooperative equilibrium. They often fail because verification is hard (the monitoring problem) and commitment is weak (states can defect faster than retaliation arrives).

Online communities. Reputation systems in online marketplaces implement repeated-game logic. Your past behavior is visible. Good ratings lead to more transactions. Bad ratings lead to exclusion. The system makes defection costly and cooperation attractive—even among strangers who will never meet in person.

In each case, the key question is: what's the structure of repetition, monitoring, and punishment that makes cooperation rational?

The Takeaway

Repeated games explain why cooperation emerges: the shadow of the future changes incentives. When players care about tomorrow, cooperation can be a stable equilibrium today.

But cooperation is fragile. Multiple equilibria exist. Noise creates problems. Endings unravel cooperation. Moving from defection to cooperation is hard; moving from cooperation to defection is easy.

Trust is equilibrium selection. Trusting someone means believing you're both playing the cooperative equilibrium. Building trust is coordinating on that equilibrium. Destroying trust is flipping to the defection equilibrium.

Institutions help by signaling types, monitoring behavior, coordinating punishment, and creating the expectation structures that make cooperation self-fulfilling.

Further Reading

- Axelrod, R. (1984). The Evolution of Cooperation. Basic Books. - Fudenberg, D., & Maskin, E. (1986). "The folk theorem in repeated games." Econometrica. - Nowak, M. A. (2006). "Five rules for the evolution of cooperation." Science.

This is Part 5 of the Economics of Trust series. Next: "Social Capital: Bowling Alone"