Auction Design without Commitment

She wants to maximize her expected payoff subject to the constraint of not re-
designing the mechanism after observing the signal s from the information process-
ing device h. That is, of replacing the outcome g(s) with another mechanism in
Φ that generates her a higher expected payoff than g(s). Our task is to identify
conditions under which she will not do that.

Let the seller’s (pure) mechanism design strategy be captured by a choice rule
σ that specifies, for each prior belief p, a mechanism that the seller can commit
to under these beliefs:

σ : ∆(Θ) → Φ such that σ[p] ∈ VIC[p], for all p.           (3)

Then σ [p] represents the mechanism that the seller implements under p. Rule σ
represent in reduced form the dynamic mechanism selection strategy of the seller.

We now identify properties that the choice rule σ should satisfy. We argue
that sequential rationality of the seller, and the buyers’ knowledge of this, asks σ
to reflect internal consistency and maximization. To present these conditions, we
develop some concepts. We say that a mechanism g ◦ h ∈ Φ is (weakly) ex post
dominated by a mechanism φ ∈ Φ if there is a signal s ∈ h(supp(p)) such that

v(φ, p(s, h)) ≥ v(g(s)) and φ 6= 1g(s) under p(s, h).

That is, the seller weakly prefers φ over the recommended outcome g(s), given the
ex post information due to signal s. In such case, the original mechanism g ◦ h
may be subject to redesign. It is easy to see that in a typical scenario, there is no
veto-incentive compatible mechanism that is not ex post dominated.¹⁵ Thus the
seller is typically (weakly) tempted to redesign the mechanism.

Under prior p, denote by C^σ [p] the set of mechanisms that are not subject to
redesign under the hypothesis that σ is followed ex post:

C^σ [p]={g ◦ h ∈ VIC[p]:g ◦ h is not ex post dominated by σ[p(s, h)], for any s ∈ h(supp(p))} .
(4)

Hence, by the revelation principle, and under the hypothesis that the seller can
commit to the choice rule σ :

• a mechanism φ is truthfully playable if φ ∈ C^σ [p], since then it will not be
redesigned ex post, and

• a mechanism φ is not truthfully playable if φ 6∈ C^σ [p], since then it will be
redesigned ex post.

We now specify formally conditions that sequential rationality imposes on the
choice rule σ . The first condition requires consistency in the sense that employing
σ ex ante should not contradict σ being employed ex post.

Definition 2 (Consistency) Choice rule σ is consistent if σ[p] ∈ C^σ [p], for all
p.

¹⁵If 0 ∈supp(pi) for all i, then a veto-incentive compatible mechanism is not ex post dominated
only if extracts all surplus from the buyers. But full surplus extraction a la Cremes and McLean
(1984) is not possible under veto-incentive compatibility.

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