Engine · Chapter 6, Leadership and Rules: Inspection

Inspection with a noisy signal

How often must an inspector check to deter, and what is the burden of deterrence?

Payoffs

a: loss, undetected 10 in tenths; the inspector's loss from an undetected violation

b: loss, detected 5 the inspector's loss even when it detects (b < a)

c: penalty 10 the inspectee's loss when caught

d: gain 10 the inspectee's gain when not caught

The signal

β = P(fail | violation) % 80

α = P(fail | legal) % 20

The book's scenarios

One game, four questions: the inspector's value

	Without the signal	With the signal (β, α)
No commitment (Nash)	$- \frac{2}{3}$ ≈ -0.6667	$- \frac{1}{3}$ ≈ -0.3333
Commitment (the inspector announces first)	$- \frac{1}{3}$ ≈ -0.3333	$- \frac{1}{9}$ ≈ -0.1111

Down a column: what commitment is worth. Across a row: what the signal is worth. In the book's base case the two are equal without the other, and together they cut the loss from $\frac{2}{3}$ to $\frac{1}{9}$ .

D The book's Table 0.1 and Chapter 6 certify $- \frac{2}{3}$ , $- \frac{1}{3}$ , $- \frac{1}{3}$ , $- \frac{1}{9}$ . Agreement, exactly.

What happens in each regime

Regime	Inspector	Inspectee	No violation	Detected	Undetected	How
Nash, no signal	$- \frac{2}{3}$	$\frac{1}{3}$	$\frac{2}{9}$	$\frac{2}{9}$	$\frac{5}{9}$	inspect with probability 1/3, violate with probability 2/3
Commit to a probability (Γ₁)	$- \frac{1}{3}$	$\frac{1}{3}$	$\frac{2}{3}$	$0$	$\frac{1}{3}$	announce 1/3; the inspectee stays legal
Nash, with the signal	$- \frac{1}{3}$	$\frac{1}{9}$	mixed play; see the chapter's table			rules (never, flag-only, clear-only, always) with weights 4/9, 5/9, 0, 0; violate with probability 1/3
Commit to a rule (Γ₂)	$- \frac{1}{9}$	$\frac{1}{9}$	$\frac{8}{9}$	$0$	$\frac{1}{9}$	review 5/9 of failed checks and 0 of passed ones; the inspectee stays legal

The least-burden deterring rule (Γ₂)

Review $\frac{5}{9}$ of failed checks and $0$ of passed ones. The burden on compliant inspectees, the probability that a legal shipment is reviewed, is $\frac{1}{9}$ ; the inspector's value is $- a p_{L} = - \frac{1}{9}$ .

Gain share $θ = d / (c + d) = \frac{1}{2}$ . β ≥ θ(1 − α): partial review of failed checks suffices.

Against the rule “review every failed check”

That rule carries a burden of $\frac{1}{5}$ , $\frac{9}{5}$ times the least burden, and deters gain shares up to $1$ . Against it a violator expects $- \frac{3}{5}$ and a compliant inspectee $\frac{1}{5}$ .

least burden of a deterring rule, as the inspectee's gain share varies · the rule “review every failed check”: burden 1/5, deters up to θ = 1. Where the solid line is below the dashed one, a partial rule deters at lower cost to compliant inspectees.

The model it runs

Model N · p. 255

What should the inspector announce, and how much of the gain is commitment and how much information?

1Decision-makers: inspector (leader in $Γ_{1}$ , $Γ_{2}$ ) and inspectee (follower)
2Time: two events; event 2 at its credible continuation
3Information: no signal, or a signal $(β, α)$ about the inspectee's current action before the inspection decision; the inspectee sees neither the signal nor the draw
4Commitment: $Γ_{0}$ ; $Γ_{1}$ : announced inspection probability; $Γ_{2}$ : announced rule $(r_{f}, r_{c})$ on the signal
5Payoffs: non-zero-sum: $(0, 0)$ , $(- b, - c)$ , $(- a, d)$ ; $b < a$ , $c, d > 0$ , $a c + b d > 0$
6Strategies: mixtures in event 1; rules $(r_{f}, r_{c}) \in [0, 1]^{2}$

Analysis contract

(i) concept: Nash equilibrium in $Γ_{0}$ (with and without the signal); leader values $V_{r}^{c}$ in $Γ_{1}$ , $Γ_{2}$ (Definition 0.9 (p. 10))
(ii) deviations: unilateral deviations (Nash); follower best replies to the whole announcement (commitment); event 2 fixed at its credible continuation
(iii) beliefs: none beyond the signal law; the follower conditions only on the announcement
(iv) selection: both tie conventions, $(L)$ and $(P)$ , reported
(v) attainment: for $r = 1, 2$ : $V_{r}^{L}$ attained; $V_{r}^{P}$ a supremum, attained only where a strictly deterring optimal rule exists; exact arithmetic