Auction Design without Commitment

concludes with discussion.

2Setup

There is a seller of a single indivisible good and a set N = {1, ..., n} of buyers.
Seller’s publicly known valuation of the good is 0.Buyeri’s privately known
valuation θi is drawn from a discrete set Θi ⊆ R+.¹⁰ Write Θ = ×i∈N Θi with
a typical element θ = (θ_i)_i∈N , and Θ_-i = ×j6_=iΘ_i with a typical element θ_-i =
(θj)j6_=i. ¹¹ Denote by ∆(Θ) the set of probability distributions p over Θ, and by
pi the ith marginal distribution of p.

The set of allocations of the good is A = {(a1, ..., an) ∈ {0, 1}ⁿ : a1 + ... + an ≤
1}, where ai =1if the good is allocated to i and ai = 0 otherwise. Write
a = (a₁, ..., a_n). A money transfer from buyer i to the seller is denoted by m_i ∈ R₊
and m = (m₁, ..., m_n) is a profile of transfers. The set of all outcomes x =(a, m)
is then X = A × Rⁿ₊ .

Now we define a mechanism. A mechanism does two things: processes infor-
mation and implements an outcome. We separate these tasks. A mechanism is
a composite function φ = g ◦ h, consisting of an information processing device h
and an implementation device g such that

h : Θ → ∆(S) and g : S → X,

where ∆(S) is the set of probability distributions over S, an open subset of an
Euclidean space. That is, the information processing device h generates, after
receiving the buyers’ messages, a public signal s ∈ S. The signal s is the only
information anyone - including the seller - obtains from h. The outcome function
g then implements an outcome x ∈ X conditional on the realized signal s.

Thus the mechanism φ = g ◦ h is a composite function

g ◦ h : Θ → ∆(X),

where ∆(X) is the set of probability distributions over X. Letting H = {h : Θ →
∆(S)} and G = {g : S → X} denote the sets of information processing devices
and implementation devices, respectively, the set of all composite mechanisms is

Φ = {g ◦ h : Θ → ∆(X) such that g ∈ G and h ∈ H}.

The support of distribution p is denoted by supp(p). Also write h(θ) = {s :
h(s : θ) > 0} and h(supp(p)) = {s : h(s : θ) > 0 and θ ∈supp(p)}. Given p,a
signal s ∈ h(supp(p)) of the information processing device h induces a posterior

¹⁰ Hence countable and without accumulation points. This assumption is for expositional
simplicity.

¹¹That is, pi(θi) = ^P_θ-i p(θi, θ-i).

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