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 = (θi)i∈N , and Θ-i = ×j6=iΘi 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, ..., an). A money transfer from buyer i to the seller is denoted by mi ∈ R+
and m = (m1, ..., mn) 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
p(θ : s, h)=
p(θ)h (s : θ)
Pθ∈Θp(θ)h (s : θ).
(1)
10 Hence countable and without accumulation points. This assumption is for expositional
simplicity.
11That is, pi(θi) = Pθ-i p(θi, θ-i).