Auction Design without Commitment



the implementable mechanisms is a general phenomenon and holds for any prior
distribution.

For an illustrative example, let N = {1, 2} and Θ =supp(p) = {5, 10}2. Let,
say,
w(10, 10) = w(10, 5) = w(5, 5) = 1 and w(5, 10) = 2. Take φ Cσ[p]. Since
1x is not ex post dominated by φE [p(x, φ)] for any x φ(supp(p)), φ(θ) allocates
the good to buyer
1 under all θ ∈ {(5, 5), (10, 5), (10, 10)}. Transfers from 1 under
θ =(5, 5) and θ = (10, 10) are 5 and 10, respectively. By incentive compatibility,
transfer from
1 under φ(10, 5) is 5. Since only φ(5, 10) allocates the the good to
2, 2s type θ2 =10is then revealed. Hence his transfer must be 10 which means
that
φ = φE [p] under p.

Now we are ready to state our main result.

Theorem 1 1. Choice rule φE[] is consistent and satisfies the one-deviation prop-
erty, for any tie-breaking rule
w.

2. If a stationary choice rule σ is consistent and satisfies the one-deviation
property, then there is a tie-breaking rule
w such that σ [p] is outcome equivalent
to
φE [p], for all p.

Proof. 1. Consistency follows from Lemma 3. Since, by Lemma 4 any φ
CφE [p] agrees with φE [p] on X and hence induces the same payoff as φE [p], it
follows that
φE [p] maximizes v on CφE [p] under p. Thus the one-deviation property
is implied.

2. Let stationary rule σ be consistent and satisfy the one-deviation property.
By Lemma 5, there is
w and Cσ such that φE Cσ. By Lemma 6, Cσ CφE . By
construction,
σ Cσ . Thus, by Lemma 4, σ [p] is outcome equivalent to φE [p], for
all
p. ■

That is, the seller can commit to the English auction provided that she does
that consistently, under all scenarios. Moreover, the payoff structure of every
feasible auction coincides that of the English auction (defined for some tie breaking
rule
w). The only difference of a committable mechanism and the English auction
may concern additional, payoff irrelevant information on the player’s valuations.

It is interesting that while full surplus extraction is feasible under full commit-
ment under almost all
p (Crémer and McLean, 1988), only the English auction is
feasible without commitment.20

The generalized Coase conjecture With the stationarity assumption the
Coase conjecture can now also be verified in the "no gap" case. By Theorem 1.2., if
σ is stationary, consistent, and meets the one-deviation property, then σ [p]=φE [p]
which always allocates the good to the buyer θ>0 with price min w-1(1)supp(p).

To conclude, if one accepts the Coase conjecture, that the mechanism φE [p] is
the unique feasible mechanism in the
n =1 case when the seller cannot commit
to not sell to the buyer who values the good more than she does, then there is
no reason not to accept also a more general version of the claim saying that in

20See also McAfee and Reny (1992).

14



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