Engine · Chapter 3, Bargaining, Commitment, and Repeated Cooperation

The forgiving trigger

How patient must two sides be for cooperation to survive noisy monitoring?

Monitoring

p: good signal when both cooperate % 90

q: good signal when one defects % 50

δ: patience % 90 the weight on the future

Stage payoffs (T > R > P > S)

R: both cooperate 3

T: defect alone 5

P: both defect 1

S: cooperate alone 0

The book's scenarios

Each player's value under the best forgiving trigger

\frac{5}{2}

≈ 2.5000

against $R = 3$ under perfect monitoring: the price of noise is $\frac{1}{2}$ , and it does not depend on patience.

The trigger

Informativeness margin $K$	$\frac{3}{5}$
Patience needed $δ_{m i n}$	$\frac{10}{13}$
Punishment probability $π^{*}$ at δ = 9/10	$\frac{10}{27}$
Chance of an unjust punishment per period	$\frac{1}{27}$
Expected cooperative periods before it	$27$

D The book certifies $K = \frac{3}{5}$ , $δ_{m i n} = \frac{10}{13}$ , $π^{*} = \frac{10}{27}$ , $V^{*} = \frac{5}{2}$ . Agreement, exactly.

V*, the value of the best forgiving trigger (left axis) · δ_min, the patience it needs (right axis), at p = 9/10. Two levers: at the margin, raising p is worth 25/4 per unit and lowering q is worth 5/4, a ratio of 5 = (1 − q)/(1 − p), whatever the payoffs.

The formulas

$K = (p - q) (R - P) - (1 - p) (T - R) = \frac{3}{5}, δ_{m i n} = \frac{T - R}{T - R + K} = \frac{10}{13}$

$π^{*} = \frac{( 1 - δ ) ( T - R )}{δ K} = \frac{10}{27}, V^{*} = R - \frac{( 1 - p ) ( T - R )}{p - q} = \frac{5}{2}$

The value satisfies $V = (1 - δ) R + δ [V - (1 - p) π (V - P)]$ ; a one-shot defection gains $(1 - δ) (T - R)$ and raises the chance of punishment by $(p - q) π$ ; the smallest punishment rate that deters is best, since punishment also falls on the innocent.

The model it runs

Model B4 · p. 132

What does noisy monitoring cost, and how should punishments be calibrated?

1Decision-makers: two players
2Time: infinitely repeated; discount $δ$
3Information: imperfect public monitoring: only the signal and the lottery are seen
4Commitment: none
5Payoffs: as in Model B3
6Strategies: the class of trigger profiles with forgiveness, indexed by $π \in [0, 1]$

Analysis contract

(i) concept: perfect public equilibrium within the stated class
(ii) deviations: one-shot deviations from cooperation (mutual defection is a stage equilibrium)
(iii) beliefs: public signals only
(iv) selection: the incentive-compatible $π$ with the highest value
(v) attainment: attained; exact; optimality is within the class only