An Introduction to Repeated Games in Game Theory

Imagine you’re running a business and face a strategic decision. Should you undercut your competitor’s prices today, even if it might trigger a price war tomorrow? Or should you maintain higher prices in hopes your competitor does the same? This dilemma captures something fundamental about human interaction: when we know we’ll face the same situation again and again, our choices today shape what happens in the future. Welcome to the world of repeated games, where the shadow of tomorrow transforms how we play today.

In game theory, repeated games examine what happens when players face the same strategic situation multiple times. Unlike one-shot games where you make a single decision and move on, repeated games capture the reality that most business relationships, political negotiations, and social interactions occur over extended periods. This temporal dimension fundamentally changes the strategic landscape, opening doors to cooperation that would seem impossible in isolated encounters.

The paradox of one-time versus ongoing interaction 🔗

Consider the classic Prisoner’s Dilemma. Two suspects are interrogated separately. Each can either cooperate with their partner by staying silent or defect by betraying them. The individual incentive is clear: betrayal pays better regardless of what the other person does. Yet if both betray, they end up worse off than if both had stayed silent. In a single encounter, betrayal appears rational. The Nash equilibrium points toward mutual defection, even though mutual cooperation would benefit everyone.

But something remarkable happens when this game repeats. Research dating back to the 1950s showed that when players knew they would meet again, cooperation often emerged spontaneously. The reason? Future consequences suddenly matter. Betray your partner today, and they’ll remember tomorrow. This simple insight transforms the entire game.

Understanding the stage game 🔗

At the heart of every repeated game sits what we call the stage game. This is simply the basic game that gets played in each round. Think of it as the fundamental building block. In our business pricing example, the stage game might involve two firms simultaneously setting prices. The stage game is usually one of the well-studied two-person games from game theory, whether it’s a Prisoner’s Dilemma, a coordination game, or something else entirely.

The crucial difference between playing the stage game once versus repeatedly lies in how players calculate their overall payoffs. In a repeated game, your total payoff isn’t just what you earn in a single round. It’s the accumulated sum across all rounds, adjusted for the timing of those rewards.

The role of discounting in repeated games 🔗

Here’s where patience enters the picture. Game theorists use something called a discount factor, typically denoted by the Greek letter delta (δ), which represents how much players value future payoffs compared to immediate ones. If you strongly prefer a reward today over the same reward next year, you have a low discount factor. If you’re willing to wait for future benefits, your discount factor is high.

Why does this matter? Because the discount factor determines whether cooperation can survive. Imagine you’re tempted to cheat your business partner today for a quick gain. But you know that cheating triggers punishment in all future rounds. Whether cheating is worth it depends on how heavily you discount the future. If players are sufficiently patient-meaning their discount factor is high enough-they’ll resist the temptation to defect because the long-term losses outweigh the short-term gains.

How trigger strategies enforce cooperation 🔗

So how exactly do players sustain cooperation in repeated games? The answer lies in carefully designed strategies that use threats and promises about future behavior. The most famous of these is called the grim trigger strategy.

The grim trigger strategy works like this: start by cooperating, and continue cooperating as long as everyone else does the same. But if anyone defects even once, switch to defecting forever. It’s unforgiving, hence the name “grim.” Yet this harsh simplicity makes it effective. Everyone knows that a single betrayal triggers permanent retaliation, making defection extremely costly.

Does this work in practice? For sufficiently patient players, yes. Consider our business example again. Suppose both firms could earn substantial profits by maintaining high prices. Each firm is tempted to undercut slightly and steal all the customers. But under grim trigger, any price cut leads both firms to drop prices to marginal cost forever after. If firms value their long-term profits enough, they’ll maintain the cooperative high prices. The mathematics works out such that cooperation becomes a subgame perfect equilibrium when players place sufficient value on the future.

Beyond grim trigger: more forgiving approaches 🔗

While grim trigger is theoretically elegant, it’s also extremely harsh. A single mistake, even an accidental one, destroys cooperation permanently. Real-world interactions often employ more forgiving strategies. One popular alternative is tit-for-tat: cooperate initially, then mirror whatever your opponent did in the previous round. Defect, and I’ll defect next time. But return to cooperation, and so will I.

There are also strategies with temporary punishment phases. Defect once, and everyone punishes you for a fixed number of rounds before returning to cooperation. These approaches recognize that maintaining long-term relationships often requires some forgiveness and the possibility of redemption.

The Folk Theorem and the multiplicity of equilibria 🔗

Now we arrive at one of game theory’s most surprising and powerful results. The Folk Theorem states that in infinitely repeated games, if players are patient enough, virtually any outcome that gives players more than their worst-case payoff can be sustained as an equilibrium. This isn’t just about sustaining cooperation. It’s about sustaining almost any pattern of behavior.

Why is it called the Folk Theorem? Because this result was widely known among game theorists in the 1950s and discussed informally, yet nobody had published a formal proof. When economist James Friedman finally published it in 1971, the “folklore” name stuck.

The theorem’s implications are profound. It tells us that repeated interaction can result in virtually any average payoff in equilibrium, as long as the payoff is both individually rational and feasible. Individually rational means each player gets at least what they could guarantee themselves by playing their minimax strategy. Feasible means the payoff profile can be achieved through some combination of actions in the stage game.

What the Folk Theorem really means 🔗

Think about what this implies. In a one-shot game, we often find a unique Nash equilibrium-a single predicted outcome. But in infinitely repeated games with patient players, we have a vast multiplicity of equilibria. The same stage game repeated over time can support wildly different patterns of behavior, all stable.

This multiplicity has a subtle but important consequence. The Folk Theorem isn’t a predictive theory. As MIT economist Franklin Fisher noted, the Folk Theorem doesn’t tell us what players will actually do. Instead, it tells us what’s possible. To predict actual behavior, we need to understand the social context, norms, communication channels, and shared understandings that players bring to the game.

From theory to reality: applications of repeated games 🔗

Repeated games aren’t just abstract theory. They help us understand numerous real-world phenomena. Consider international trade agreements. Countries can’t enforce treaties through legal mechanisms the way contracts work domestically. Yet trade agreements often hold because countries interact repeatedly. Violate a trade deal today, and your trading partners might retaliate in future negotiations. The threat of future punishment sustains cooperation.

Or think about cartels and collusion. Antitrust laws prohibit firms from explicitly agreeing to fix prices. Yet in industries with few competitors who interact repeatedly, firms can sometimes sustain cooperative high prices without explicit communication, using trigger strategies that punish price-cutting with future retaliation. Understanding repeated games helps regulators identify and prevent such tacit collusion.

Even social norms can be understood through this lens. Why do people tip in restaurants, even when traveling to cities they’ll never visit again? Part of the answer involves repeated games within communities. Your reputation for tipping or not tipping spreads. Fail to tip, and you might face social sanctions in future interactions with others who heard about your behavior.

The limits of cooperation: finite horizons and the end game problem 🔗

There’s a catch to everything we’ve discussed. If a game has a known, finite number of rounds and the stage game has a unique Nash equilibrium, then cooperation unravels completely. This happens through backward induction. In the last round, there’s no tomorrow, so players revert to the one-shot Nash equilibrium. Knowing the last round will be uncooperative, the second-to-last round also becomes uncooperative, and so on backward to the very first round.

This “end game problem” appears in various contexts. In laboratory experiments, cooperation often breaks down as the final round approaches. In business, firms approaching bankruptcy or exit might abandon cooperative strategies. The lesson is clear: cooperation requires either an indefinite time horizon or at least uncertainty about when interactions will end.

Patience as the foundation of cooperation 🔗

Stepping back, the core insight of repeated games is beautifully simple yet profound. Cooperation among self-interested individuals becomes possible when the shadow of the future is long enough. Patient players-those who value future payoffs sufficiently-can sustain outcomes that would collapse in one-shot interactions. The discount factor, seemingly just a technical parameter, captures something essential about human cooperation: our ability to sacrifice today for tomorrow.

This framework helps explain why cooperation is easier in some contexts than others. Stable business relationships, long-term political alliances, and tightly-knit communities all share a common feature: the participants expect to interact repeatedly over extended periods. This expectation creates the conditions where cooperation can flourish, sustained not by altruism but by the enlightened self-interest of players who understand that today’s choices shape tomorrow’s possibilities.

What do you think? In your own experience, have you noticed how ongoing relationships change the way people behave compared to one-time encounters? Can you identify situations where the “shadow of the future” has encouraged cooperation that wouldn’t exist otherwise?

References

1. https://en.wikipedia.org/wiki/Repeated_game
2. https://en.wikipedia.org/wiki/Prisoner's_dilemma
3. https://nonzerosum.games/discountfactor.html
4. https://en.wikipedia.org/wiki/Folk_theorem_(game_theory)
5. https://en.wikipedia.org/wiki/Grim_trigger
6. https://gametheory101.com/courses/game-theory-101/grim-trigger-in-the-repeated-prisoners-dilemma/