Repeated Games & the Folk Theorem
Repeated Games & the Folk Theorem
When a stage game is played repeatedly, cooperation can be sustained by future punishment threats — even when defection is dominant in the one-shot game. The Folk Theorem characterizes the set of payoff vectors achievable as Nash (or subgame perfect) equilibrium outcomes of infinitely repeated games: roughly, any individually rational feasible payoff can be sustained when players are sufficiently patient. This is the theoretical foundation for long-run cooperation in economics, law, and political science.
Feasible and Individually Rational Payoffs
Let $V=\text{co}\{u(s):s\in S\}$ be the convex hull of all stage-game payoffs (feasible set). The minmax value for player $i$ is $\underline{v}_i=\min_{s_{-i}}\max_{s_i}u_i(s_i,s_{-i})$, the lowest payoff that opponents can hold player $i$ down to. A payoff vector $v\in V$ is individually rational if $v_i\geq\underline{v}_i$ for all $i$. The set $V^*=\{v\in V:v_i\geq\underline{v}_i\,\forall i\}$ is the intersection of feasible and individually rational payoffs.
Folk Theorem (Fudenberg-Maskin 1986)
In the infinitely repeated game with discount factor $\delta$, any payoff $v\in V^*$ with $v_i>\underline{v}_i$ for all $i$ (strict individual rationality) is achievable as a subgame perfect equilibrium payoff for sufficiently high $\delta<1$. The construction uses grim-trigger strategies: play the cooperative action; if any deviation occurs, revert permanently to the minimax strategy. Higher $\delta$ makes future punishments more effective relative to one-shot gains from deviation.
Example 1
Sustain cooperation in the infinitely repeated Prisoner's Dilemma (stage payoffs: (C,C)=(3,3),(D,D)=(1,1),(C,D)=(0,5)) using grim trigger. Find the critical discount factor.
Solution: Grim trigger: play C; if any deviation, play D forever. On path: $V_C=3+\delta\cdot 3+\delta^2\cdot 3+\cdots=3/(1-\delta)$. Deviation: play D this period (get 5), then receive (D,D)=(1,1) forever: $V_D=5+\delta\cdot 1/(1-\delta)$. Cooperate iff $V_C\geq V_D$: $3/(1-\delta)\geq 5+\delta/(1-\delta)$; $3\geq 5(1-\delta)+\delta=5-4\delta$; $4\delta\geq 2$; $\delta\geq 1/2$. So cooperation is SPE for $\delta\geq 1/2$.
Example 2
Characterize the Folk Theorem payoffs for the Battle of Sexes repeated game: (O,O)=(2,1),(O,F)=(0,0),(F,O)=(0,0),(F,F)=(1,2).
Solution: Minmax values: $\underline{v}_1=\min_{s_2}\max_{s_1}u_1=\min(2,0)=?$ — actually player 2 minmaxes player 1 by playing F with prob $q$ to minimize player 1's max payoff. $\max_{s_1}u_1=(2,0)$ if P2 plays O, or $(0,1)$ if P2 plays F. Min over P2: mix $q$ s.t. $2(1-q)=q$, $q=2/3$, minmax $=2/3$. Similarly $\underline{v}_2=1/3$. The Folk Theorem set $V^*$ is the convex hull of $\{(2,1),(1,2)\}$ intersected with $v_1\geq 2/3, v_2\geq 1/3$, a line segment from $(2,1)$ to $(1,2)$. Both cooperative outcomes are sustainable for high $\delta$.
Practice
- In the infinitely repeated Cournot duopoly (stage game: each firm chooses $q_i$, profit $\pi_i=(a-q_1-q_2)q_i$), find the discount factor needed to sustain the monopoly outcome using grim trigger.
- Explain why the Folk Theorem fails for finitely repeated games when the stage game has a unique Nash equilibrium.
- State and prove the 'easy direction' of the Folk Theorem: show any SPE payoff must be in $V^*$.
- In a three-player game, how does the Folk Theorem change? What additional conditions on the dimensionality of $V^*$ are needed?
Show Answer Key
1. Cournot NE: $q_i=a/3$, $\pi_i^{NE}=a^2/9$. Monopoly: total $Q=a/2$, each produces $a/4$, $\pi_i^M=a^2/8$. Grim trigger: deviate optimally to $q_i=(a-a/4)/2=3a/8$, getting $\pi^D=9a^2/64$. Sustain cooperation if $\frac{\delta}{1-\delta}(\pi^M-\pi^{NE})\ge\pi^D-\pi^M$. Solving: $\delta\ge9/17\approx0.529$.
2. With a unique stage-game NE, the last period must play NE (no future punishment). By backward induction, every period plays NE. The Folk Theorem requires infinite horizons to sustain cooperation, because finite games unravel from the last period.
3. If $v\in V^*$ (set of individually rational, feasible payoffs) is an SPE payoff, then $v_i\ge\underline{v}_i$ (minimax value) for each player $i$. Proof: in any SPE, player $i$ can guarantee at least $\underline{v}_i$ in every period by playing a minimax strategy, so the average payoff $\ge\underline{v}_i$.
4. With 3+ players, the Folk Theorem requires $\dim(V^*)\ge|N|-1$ (where $|N|$ is the number of players). This full-dimensionality condition ensures that each player can be punished independently. If $V^*$ is lower-dimensional, some players cannot be differentially punished, and the Folk Theorem may fail.