the n ≥ 1 case φE [p] is still the unique feasible mechanism when the seller cannot
commit to not sell to the buyer who values the good more than the buyer with the
second highest valuation. This suggests a generalization of the Coase conjecture:
Without external commitment devices, the seller can commit only to the English
auction.
5 Discussion
Mechanism design requires commitment since at the ex post stage, when the mech-
anism has produced information needed for choosing the output, the seller may
want to change the rules of the game and implement a new mechanism. We have
studied auction mechanisms that the seller can commit to implement. Two con-
ditions reflecting sequential rationality of the seller have been imposed on the
feasible mechanism selection rule. The conditions are internal consistency and
an optimality condition called one-deviation property. We show that the unique
mechanism that satisfies the restrictions (and a stationarity condition) is a version
of the traditional English auction.
At the heart of the analysis is the argument that a sequentially rational seller
can always commit to the English auction when her choices are stationary. This
idea can be seen as a generalization of the Coase conjecture (e.g. Fudenberg et al.,
1985; Gul et al., 1986). In the one buyer case, the seller cannot commit (under
stationary strategies) to not to sell the good to the buyer with strictly positive
valuation. In the multiple buyers case, the seller cannot commit not to sell the
good to the buyer with the highest valuation. The reason is that the seller can
always commit to the English auction and hence she cannot commit to mechanisms
that are ex post dominated by the English auction. Our main result is that this
constraint is very severe: only versions of the English auction satisfy it.21
One may wonder whether the ex post domination criterion in the definition
of one-deviation property is needlessly strong. A natural weaker candidate would
be to demand strict payoff dominance. Strict domination would, however, be in
conflict with our basic assumption that the seller’s mechanism selection rule is
dependent only on the prior p. To see this, consider the n =2case and supp(p) =
{0} × {0, 1}. With the tie-breaking rule w that allocates the good under θ =(0, 0)
to buyer 2, mechanism φE [p] would always sell to buyer 2 with price 0. With strict
domination criterion, a procedure that sells to 1 under θ =(0, 0) with price 0 and
to 2 under θ =(0, 1) with price 1 would be not be strictly ex post dominated. And
selling to 1 under θ =(0, 0) with price 0 would be in conflict with φE [q] where
prior q is degenerate on θ =(0, 0). Combining strict dominance with sequential
rationality would therefore require history dependent choices, and the mechanism
selection rule σ would no longer be definable as a function of p alone. However,
while this seems to be technically burdensome, an analogue of Theorem 1 should
hold with a history dependent tie-breaking rule. I conjecture that the English
21 Milgrom (1987) argues that the core implements the efficient allocation under complete
information.
15