Folk Theorem and Equilibrium Payoffs

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# Folk Theorem and Equilibrium Payoffs ## 🎯 Learning Objectives - Define **feasible payoffs** and the **individually rational payoff set** - State the **Folk Theorem** for infinitely repeated games - Characterise the set of SPNE payoffs - Design trigger strategies to achieve desired outcomes - Understand the role o...

Folk Theorem and Equilibrium Payoffs

🎯 Learning Objectives

Define feasible payoffs and the individually rational payoff set
State the Folk Theorem for infinitely repeated games
Characterise the set of SPNE payoffs
Design trigger strategies to achieve desired outcomes
Understand the role of min-max payoffs

7.1 Intuition: Anything Goes?

The Folk Theorem says: if players are patient enough, any feasible, individually rational payoff can be sustained as an equilibrium in an infinitely repeated game.

🔑 Key Insight: When the shadow of the future is long enough, nearly any outcome — even very "cooperative" ones — can be sustained by the threat of reverting to a punishment that hurts all players. The name: "Folk" because it was part of game theory's oral tradition before being formally published.

7.2 Feasible Payoffs

A payoff vector

(v_1, v_2)

is feasible if it can be expressed as a convex combination of the pure-strategy payoff vectors of the stage game. Example (Prisoner's Dilemma): Payoff vectors: (C,C)=(3,3), (C,D)=(0,5), (D,C)=(5,0), (D,D)=(1,1) Feasible set = convex hull of these four points. Any point inside this polygon can be achieved by correlating or mixing strategies. (Diagram)

7.3 Individually Rational Payoffs

Player

i

's min-max payoff is the lowest payoff others can force on player

i

v_i^{\min} = \min_{s_{-i}} \max_{s_i} u_i(s_i, s_{-i})

A payoff vector is individually rational if

v_i \geq v_i^{\min}

for all

i

. Prisoner's Dilemma min-max: To min-max Player 1: Player 2 chooses Silence → Player 1's max response = 0 (Confess) → min-max = 0.

Why Min-Max Matters

In a repeated game, the worst that can happen to player

i

in equilibrium is the min-max payoff — because

i

can always guarantee at least that by playing their max-min strategy (best response to the worst-case opponent choice).

7.4 The Folk Theorem

Theorem (Friedman, 1971): Consider an infinitely repeated game with discount factor

\delta

. If the stage game has at least one Nash equilibrium with payoffs

(e_1, e_2)

, then for any feasible payoff vector

(v_1, v_2) > (e_1, e_2)

element-wise, there exists

\underline{\delta} < 1

such that for all

\delta > \underline{\delta}

(v_1, v_2)

can be supported as a SPNE payoff.

Stronger Version (Fudenberg-Maskin, 1986)

Any feasible, individually rational payoff vector can be supported as a SPNE if

\delta

is sufficiently close to 1. (Diagram)

7.5 Trigger Strategy Design

Standard trigger strategy to support payoff

(v_1, v_2)

Cooperation phase: Play the strategies that yield $(v_1, v_2)$
Punishment phase: If anyone deviates, play stage-game Nash equilibrium forever (Grim Trigger)

Condition for Sustainability

For player

i

v_i \geq (1-\delta)d_i + \delta e_i

Where:

$v_i$ = cooperative payoff
$d_i$ = best one-period deviation payoff
$e_i$ = stage-game Nash equilibrium payoff (punishment) Rearranging:

\delta \geq \frac{d_i - v_i}{d_i - e_i}

Example: Supporting (3,3) in Prisoner's Dilemma

v_i = 3

d_i = 5

e_i = 1

\delta \geq \frac{5 - 3}{5 - 1} = \frac{2}{4} = 0.5

Matches our earlier calculation!

7.6 Optimal Punishments

Grim Trigger (revert to stage-game NE) is simple but may not be the harshest punishment. To support more cooperative payoffs, we might need harsher punishments. Min-max punishment: Instead of reverting to NE, players choose strategies that keep the deviator at their min-max level. Nash reversion (Friedman): Revert to stage-game NE. Min-max reversion (Fudenberg-Maskin): Revert to min-max punishment. The min-max punishment can support a larger set of payoffs because it's harsher.

📊 Formula Summary

Concept	Formula
Min-max payoff	$v_i^{\min} = \min_{s_{-i}} \max_{s_i} u_i(s_i, s_{-i})$
Trigger condition	$\delta \geq \frac{d_i - v_i}{d_i - e_i}$
Discounted sum	$U_i = \sum_{t=1}^{\infty} \delta^{t-1} u_i$

✅ Practice Questions

Q1: In a stage game where NE payoff is (2, 2) and feasible cooperative payoff is (4, 4), with deviation payoff 6. Find the minimum

\delta

Solution

$\delta \geq \frac{6 - 4}{6 - 2} = \frac{2}{4} = 0.5$

So $\delta \geq 0.5$ is required to sustain cooperation. Q2: True or False: The Folk Theorem says every feasible payoff can be sustained for sufficiently patient players. Solution

False. It says every feasible AND individually rational payoff can be sustained. Payoffs below a player's min-max are not individually rational — that player would prefer to min-max rather than accept such an outcome. Q3: Calculate the min-max payoff for Player 1 in:

\begin{array}{c|cc} & L & R \\ \hline U & (3, 2) & (0, 0) \\ D & (1, 0) & (2, 1) \end{array}

Solution

For each P2 strategy:

If P2 plays L: P1's max response = U (3)

If P2 plays R: P1's max response = D (2)

Player 2 chooses between L (gives P1 at most 3) and R (gives P1 at most 2). Since P2 wants to minimise P1's max, P2 chooses R.

Min-max = 2 (P1 gets 2 by playing D when P2 plays R). Join Discord PreviousRepeated Games ExtensionsNextBayesian Games