Quiz 2

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 (v1,v2)(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 ii's min-max payoff is the lowest payoff others can force on player ii:
vimin⁡=min⁡s−imax⁡siui(si,s−i)v_i^{\min} = \min_{s_{-i}} \max_{s_i} u_i(s_i, s_{-i})
A payoff vector is individually rational if vi≥vimin⁡v_i \geq v_i^{\min} for all ii. 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 ii in equilibrium is the min-max payoff — because ii 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 δ\delta. If the stage game has at least one Nash equilibrium with payoffs (e1,e2)(e_1, e_2), then for any feasible payoff vector (v1,v2)>(e1,e2)(v_1, v_2) > (e_1, e_2) element-wise, there exists δ‾<1\underline{\delta} < 1 such that for all δ>δ‾\delta > \underline{\delta}, (v1,v2)(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 δ\delta is sufficiently close to 1. (Diagram)

7.5 Trigger Strategy Design

Standard trigger strategy to support payoff (v1,v2)(v_1, v_2):
  1. Cooperation phase: Play the strategies that yield (v1,v2)(v_1, v_2)
  2. Punishment phase: If anyone deviates, play stage-game Nash equilibrium forever (Grim Trigger)

Condition for Sustainability

For player ii:
vi≥(1−δ)di+δeiv_i \geq (1-\delta)d_i + \delta e_i
Where:
  • viv_i = cooperative payoff
  • did_i = best one-period deviation payoff
  • eie_i = stage-game Nash equilibrium payoff (punishment) Rearranging:
δ≥di−vidi−ei\delta \geq \frac{d_i - v_i}{d_i - e_i}

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

vi=3v_i = 3, di=5d_i = 5, ei=1e_i = 1:
δ≥5−35−1=24=0.5\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

ConceptFormula
Min-max payoffvimin⁡=min⁡s−imax⁡siui(si,s−i)v_i^{\min} = \min_{s_{-i}} \max_{s_i} u_i(s_i, s_{-i})
Trigger conditionδ≥di−vidi−ei\delta \geq \frac{d_i - v_i}{d_i - e_i}
Discounted sumUi=∑t=1∞δt−1uiU_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 δ\delta.
Solution
δ≥6−46−2=24=0.5\delta \geq \frac{6 - 4}{6 - 2} = \frac{2}{4} = 0.5
So δ≥0.5\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:
LRU(3,2)(0,0)D(1,0)(2,1)\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
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