Folk Theorem: Payoff Normalization and Punishment Strategies

The Folk Theorem states that in infinitely repeated games any outcome that is feasible and individually rational can be sustained as a subgame perfect Nash equilibrium (SPNE) when players are sufficiently patient. The term “folk” reflects the fact that game theorists intuitively recognized this result in the 1950s and 1960s before formal proofs appeared in the 1980s, notably through Drew Fudenberg’s work.

Geometric Representation of Payoffs

A game’s feasible set contains all payoff vectors that can be achieved by some mixture of action profiles, while the individually rational set consists of payoffs that give each player at least their minmax value. Plotting these sets reveals the region where the Folk Theorem can operate.

Standard discounted sums (u_0+\delta u_1+\delta^2 u_2+\dots) change with the discount factor (\delta). To keep payoff values comparable across different (\delta), the lecture introduces average discounted payoffs:

[ (1-\delta)\sum_{t=0}^{\infty}\delta^{t}u_t . ]

If a player receives a constant payoff (u) every period, this average equals (u) exactly, eliminating the distortion caused by (\delta).

The Folk Theorem Template

The general construction picks a feasible payoff vector (v) and a discount factor (\delta) close to one. Players follow a target action profile (a^{*}) that yields (v) as long as no deviation occurs. The “patience” condition—(\delta) sufficiently near 1—ensures that the present value of future punishments outweighs any short‑run gain from deviating.

Versions of the Folk Theorem

Nash Reversion

Assume a stage‑game Nash equilibrium exists in which every player receives a payoff strictly lower than their target (v_i). The strategy plays (a^{*}) until a deviation is observed; then the game reverts forever to that Nash equilibrium. Because the Nash payoff is lower, the threat of permanent punishment deters one‑shot deviations when (\delta) is high enough.

Individualized Nash Reversion

For each player (i) suppose there is a Nash equilibrium (a^{NE,i}) that gives (i) a payoff below (v_i). If player (i) deviates, the group switches to the specific equilibrium (a^{NE,i}) that punishes (i) most effectively. Tailoring the punishment strengthens the incentive to obey the target profile.

Pure Minmax Folk Theorem

Let (\underline{v}_i) denote player (i)’s pure minmax value—the lowest payoff that the others can force while (i) best‑responds. If (v_i>\underline{v}_i) for all players, the construction proceeds as follows: after a deviation, the non‑deviators impose the harshest possible punishment (driving the deviator down to (\underline{v}_i)) for a finite number of periods, then reward the punishers to ensure they carry out the punishment. This “minmaxing” strategy expands the set of enforceable outcomes beyond what simple Nash reversion can achieve.

Mechanisms and Explanations

One‑Shot Deviation Principle provides a test for SPNE: a strategy profile is an SPNE if no player can profit by deviating at any single history, assuming everyone else follows the prescribed strategy thereafter.

The Nash Reversion Mechanism relies on the fact that the threatened Nash payoff is strictly lower than the target payoff; with a high (\delta), the discounted loss from future punishment outweighs any immediate gain.

Minmaxing involves all players except the deviator choosing actions that minimize the deviator’s payoff, while the deviator best‑responds to those actions. The resulting payoff (\underline{v}_i) serves as the baseline for the harshest punishment.

Hard Facts and Numbers

• Drew Fudenberg formalized a version of the Folk Theorem in 1986.
• In the Prisoner’s Dilemma, cooperation can be sustained as an SPNE when (\delta \geq \tfrac{1}{3}).
• The average discounted payoff formula is ((1-\delta)\sum_{t=0}^{\infty}\delta^{t}u_t).

Quotable Insights

“In infinitely repeated games, we often say anything can happen if the players are sufficiently patient.”

“It’s not very informative, folk theorem. What it refers to is the fact that this was intuited by game theorists in the 1950s and the 1960s.”

“The issue is that we’re missing a normalization. What we really want to keep track of is the average payoff.”

“The threat of punishing players if they deviate to discipline their behavior and prevent them from deviating today from a star.”

“The trick is actually to reward the punisher for carrying out the punishment.”