Auction Design without Commitment

Given p, buyer θ_i'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 θ_i ,θ⁰_i ∈ Θ_i , 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 ,θ⁰_i ∈ 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, θ⁰_i)] ≥ 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 = θ⁰_i in (2). We only need EXP-IR and IC in the remainder of the paper.

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