Moreover, if w(θ) = 0, then θj =0for all j ∈ N .
Given a prior distribution p and a tie-breaking rule w, we now construct a
deterministic mechanism φE (∙) = gE (hE(∙)} , an English auction.17 Label the
elements of a subset of the signal set SE ⊂ S by
SE = {(w(θ),θ-w(θ)): θ ∈ Θ},
and define a deterministic information processing device hE : Θ → SE such that
hE(θ) = (w(θ),θ-w(θ)), for all θ ∈ Θ.
In order to specify the implementation device gE, construct the winner’s money
transfer rule as follows: for any i ∈ N,
mE(i, θ-i, p) = min{θ0i : (θ0i , θ-i) ∈ w-1(i) ∩ supp(p)}, if θ ∈ w-1(i). (6)
That is, mE (i, θ-i ,p) is the smallest possible valuation of i given the information
that (i) i is the winner, (ii) the other buyers’ types are θ-i . The implementation
device gE : SE → X now satisfies, for each buyer j ∈ N , for all (i, θ-i) ∈ SE,
E(i θ ) (1,mE(i, θ-i, p)), if j = i,
(7)
gj (i, θ-i) = (0, 0), ifj 6=i.
That is, φE(θ) = gE(hE(θ)) allocates the good to the winner i = w(θ) who pays
a price equal to his least possible valuation (i) given the other players’ types, (ii)
the fact that he is the winner, and (iii) p.18 The corresponding payoffs are
Ui (φE(θ),θi¢ = J
θi - mE (i, θ-i ,p), if θ ∈ w-1 (i) ∩ supp(p),
0, if θ 6∈ w-1 (i) ∩ supp(p).
By construction, φE is efficient and the price paid by the winner is less than
or equal to his valuation, and at least as high as the other buyers’ valuations.19
Moreover, since the winner i becomes publicly known with the signal s =(i, θ-i),
the posterior belief p((i, θ-i),hE) satisfies
supp(p((i, θ-i), hE)) ⊆ w-1 (i). (8)
Since the mechanism is straightforward, i.e., there is a bijection between the do-
main and range of gE, we may denote the posterior p(s, hE) by p(x, φE).
Mechanism φE has the familiar pivotal structure: a buyer’s payment - and
hence his payoff - is independent of his announcement as long as he wins (or
loses). The impact of lying on his payoff cannot be positive since it either induces
the buyer to win when he would like to lose or to lose when he would like to win.
Since truthtelling forms an equilibrium,
φE ∈ VIC[p].
17We relax w from the description of the English auction φE for notational simplicity.
18Note that φE (p) reveals only the winner’s identity and the other players’ valuations. Hence
it cannot be interpreted as the Vickrey (second-price) auction which asks all buyers to reveal
their valuations.
19When the valuations are correlated, there may be a gap between this and the second highest
valuation
11