The Folk Theorem and Cooperation in Repeated Games

1. The Core Idea: "Anything Goes"

The name "Folk Theorem" is unusual. It doesn't credit a single person because the core idea was part of the "folklore" among game theorists in the 1950s before it was formally published. Many different researchers understood it intuitively and developed various versions of it.

The central message is powerful: in an infinitely repeated game, the grim reality of a one-shot equilibrium (like mutual defection in the Prisoner's Dilemma) is not the only possible outcome. If players are sufficiently patient, a vast range of outcomes—essentially any outcome in the FIR set we defined last lesson—can be sustained as an equilibrium.

2. A First Look: The Subgame Perfect Folk Theorem

Let's start with a very common and intuitive version of the theorem, sometimes called Friedman's Folk Theorem. It relies on a concept you're already familiar with: using the threat of reverting to a stage-game Nash Equilibrium to enforce cooperation.

Friedman's Folk Theorem: Let $e^{NE}$ be the payoff vector from a Nash Equilibrium of the stage game. Any feasible payoff vector $v$ where every player gets a strictly better payoff than in that Nash Equilibrium (i.e., $v_i>e_i^{NE}$ for all players $i$) can be supported as a Subgame Perfect Nash Equilibrium (SPNE) of the infinitely repeated game, provided the discount factor $\delta$ is sufficiently high.

The strategy that supports this is a direct generalization of the grim trigger:

1. Phase I (Cooperation): Play the sequence of actions that generates the target average payoff $v$.
2. Phase II (Punishment): If any player deviates, immediately and forever revert to playing the stage-game Nash Equilibrium strategy profile.

The punishment is credible because, once in the punishment phase, no player has an incentive to deviate from playing the Nash Equilibrium. It is an equilibrium of that subgame. Because the threat is credible, the equilibrium is subgame perfect.

As long as players are patient enough ($\delta$ is close to 1), the one-time gain from cheating is outweighed by the infinite stream of losses from triggering the punishment phase.

3. A Deeper Dive: The Nash Folk Theorem and Its Limits

Friedman's theorem is powerful, but it's limited to payoffs that are better than a stage-game NE. What about other payoffs in the FIR set? The Nash Folk Theorem makes a more general claim.

The Nash Folk Theorem: Any feasible payoff vector $v$ that is strictly individually rational (i.e., $v_i>{\underline{v}}_i$ for all players $i$, where ${\underline{v}}_i$ is the min-max payoff) can be supported as a Nash Equilibrium of the infinitely repeated game, provided $\delta$ is sufficiently high.

This is a stronger statement because the min-max value ${\underline{v}}_i$ is the absolute lowest payoff a rational player will accept, and it can be lower than the NE payoff. The proof for this theorem uses a more severe punishment: if a player deviates, all other players coordinate to play a strategy that holds the deviant down to their min-max payoff, forever.

However, this raises a critical issue that gets to the heart of your interest in strategic planning.

The issue is that forcing an opponent down to their min-max payoff can be extremely costly for the punishers. If carrying out the punishment hurts you more than letting the deviation slide, is your threat to punish truly credible? In many cases, it is not. This means that while the outcome is a Nash Equilibrium, it may not be a Subgame Perfect Nash Equilibrium. A rational player, anticipating that your threat is empty, might deviate anyway.

For completeness, you should know that more advanced theorems (like the Fudenberg-Maskin Folk Theorem) show that under certain technical conditions, it is possible to construct clever punishment-and-reward schemes that do make these min-max punishments subgame perfect.

4. Interpreting the Folk Theorem: Chaos or Cooperation?

So, what are the implications of the fact that almost any FIR payoff can be an equilibrium?

1. The "Shadow of the Future" Enables Cooperation: The theorem's primary message is optimistic. It provides a formal basis for how long-term interaction enables outcomes that are impossible in the short term. The threat of future punishment or the promise of future reward can discipline behavior and sustain cooperation. The key variable is patience ($\delta$).

2. An Explosion of Equilibria: The Folk Theorem is sometimes called an "embarrassment of riches." By showing that so many outcomes are possible equilibria, it makes the theory less predictive. If (2,2), (2.5, 1.5), and (1.5, 2.5) are all potential stable outcomes in a repeated Prisoner's Dilemma, which one will players choose? Game theory alone cannot answer this.

3. The Importance of Coordination and Norms: This ambiguity highlights that other factors must be at play in selecting an equilibrium. Social norms, culture, history, and explicit communication can act as coordination devices, helping players converge on one of the many possible cooperative equilibria.

This video provides a great visual summary of the vast set of outcomes the Folk Theorem allows.

This connects back to your goal of "meaningful planning in a chaotic world." The Folk Theorem suggests that in ongoing relationships, the future is not pre-determined by a single, bleak equilibrium. Instead, there is a wide range of possible stable futures. The challenge shifts from simply finding the "best" move to actively shaping expectations and coordinating with others to land on a favorable equilibrium within that vast set.

Conclusion

This lesson, and this module, concludes with one of the most significant results in game theory. We moved from the strict confines of one-shot games to the wide-open possibilities of repeated interaction.

Key Takeaways:

• The Folk Theorem asserts that in an infinitely repeated game with sufficiently patient players, almost any feasible and individually rational payoff can be sustained as an equilibrium outcome.
• The credibility of the supporting strategy depends on the punishment. Using reversion to a stage-game NE as punishment leads to a Subgame Perfect Nash Equilibrium (Friedman's Theorem). Using a more severe min-max punishment may result in a Nash Equilibrium that is not subgame perfect because the threat may not be credible.
• The main implication of the Folk Theorem is that long-term interaction opens a vast space for cooperation. However, it also creates a coordination problem, as players must find a way to agree on which of the many possible cooperative equilibria to aim for.

In all our analysis so far, we have assumed that players have complete information—they know all the rules, and crucially, all the payoffs for all other players. In the next module, we will relax this powerful assumption and enter the fascinating world of incomplete information, where players must make decisions without knowing the true "type" or preferences of their opponents.