Auction Design without Commitment

To economize on notation, write p(∙ : s, h) = p(s, h). Since the signals are pub-
lic, p(s, h) ∈ ∆(Θ) for all s ∈ h(supp(p)). By the definition of the support,
supp(p(s, h)) ⊆supp(p).

The mechanism g ◦ h is constant under p if h(supp(p)) is singleton. A constant
mechanism implementing outcome x is denoted by

1x ∈ Φ.

A constant mechanism does not affect the beliefs and always implements the same
outcome. The two mechanisms (goh) and (g⁰oh⁰) are outcome equivalent under p if
they induce the same outcome function: (g o h)(θ) = (g⁰ ◦ h⁰)(θ), for all θ ∈supp(p).
Finally, if the information provided by the mechanism g ◦ h is not finer than what
is necessary to implement the outcome, i.e., if g(s) = g(s⁰) implies s = s⁰ for all
s, s⁰ ∈ h(supp(p)), then we may write p(s, h)=p(g(s),go h).

Buyer θ_i⅛ and the seller’s payoffs from allocation x = (a, m) are, respectively,

ui(x, θi) = θiai - mi,
v(x) = ^P mi.

i∈N

3 Solution

The seller’s problem is that she cannot commit to the implementation device
g once the signal s has been produced by the information processing device h.
Rather, she may want to design a new mechanism under her post-signal belief.
In this section, we identify conditions that the mechanism needs to satisfy if the
seller is about to commit to it.

We appeal to the revelation principle by assuming that the mechanism is played
truthfully if and only if (i) the appropriate incentive and participation constraints
of the buyers are satisfied, (ii) the seller can commit to the mechanism. The
latter requires that the seller cannot commit to implementing any more profitable
mechanism given the post-signal beliefs. This means that the potentially more
profitable mechanisms meeting (i) need to be blocked by a yet third layer of
mechanisms which, in turn, the seller can commit to. The self-referential nature of
the blocking-relationship between the mechanisms means that there is no recursive
way to identify the feasible mechanisms. Indeed, the mechanism selection needs to
be solved for all cases simultaneously. Thus to solve (ii), novel modeling techniques
needs to be developed.

Buyers’ incentives We assume that the buyers can exit any point of the
game. Thus any implementable mechanism g o h must be ex post individually
rational (EXP-IR):¹²

u_i(g(s),θ_i) ≥ 0, for all s ∈ h(θ), for all θ ∈ supp(p), for all i ∈ N.

¹² Interim individual rationality requires that participation be weakly profitable before the
output has been realized. Ex post constraint has been analysed e.g. by Forges (1993, 1998) and
Gresik (1991, 1996).

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