Given p, buyer θi's interim payoff from a mechanism g ◦ h is
PPp (θ) Ui(g(s),θi)h (s : θ).
By the revelation principle (cf. Myerson, 1979), an implementable mechanism
must be incentive compatible. A mechanism g ◦ h is incentive compatible (IC) if,
for all θi ,θ0i ∈ Θi , for all i ∈ N,
PPp (θ) Ui(g(s),θi)[h (s : θ) - h (s : θ-i,θɔ] ≥ 0.
θ-i s
However, incentive compatibility and ex post individual rationality are not inde-
pendent conditions: veto right might be exercised at the off-equilibrium nodes.
The following simple extension of incentive compatibility resolves the problem by
allowing i veto the outcome also after his untruthful announcements.13
Definition 1 (VETO-IC) Given p, a mechanism g ◦ h ∈ Φ is veto-incentive
compatible if, for all θi ,θ0i ∈ Θi , for all i ∈ N,
P P p (θ) [ui(g(s), θi)h (s : θ) - max{ui(g(s), θi), 0}h (s : θ-i, θ0i)] ≥ 0. (2)
θ-i s
Veto-incentive compatibility requires that truthful reporting forms a Bayes-
Nash equilibrium even if vetoing is possible after untruthful announcement. Any
implementable mechanism must thus be veto-incentive compatible. For any p,
denote the set of veto-incentive compatible mechanisms by
VIC[p] ⊂ Φ.
It is easy to see that any veto-incentive compatible mechanism is incentive com-
patible and ex post individually rational (but not vice versa).14
Truthful announcements form a Bayes-Nash equilibrium in a veto-incentive
compatible mechanism φ = g ◦ h if the seller can commit to following g after h
has performed its information processing task, i.e., produced its signal s. Thus
a mechanism maximizing the seller’s payoff subject to the veto-incentive com-
patibility could be interpreted as the seller’s full commitment benchmark. Since
veto-incentive compatibility concerns only the payoffs, any signal structure - even
a one that fully reveals the buyers’ types - is consistent with veto-incentive com-
patibility. However, while signals do not affect anyone’s payoff directly, they may
do so indirectly, via seller’s behavior at the ex post stage.
Seller’s incentives The seller’s expected payoff from a mechanism φ = g ◦ h
is
v(φ,p) = P Pp(θ)v(g(s))h (g(s) : θ).
θs
13 Veto-incentive compatibility is due to Forges (1998), and is closely related to IC* of Matthews
and Postlewaite (1989).
14 Choose θi = θ0i in (2). We only need EXP-IR and IC in the remainder of the paper.