non-cooperative bargaining literature, and verified in the so called "gap" case
θ(p) > 0 e.g. by Fudenberg et al. (1985) and Gul et al. (1986).
The next proposition shows that the result can be derived also in our set up,
without going into the details of the bargaining process. Thus consistency and the
one-deviation property do capture the key aspects of sequential rationality.
Remark 1 (Gap-case) Let n =1. Let σ be a consistent choice function meeting
the one-deviation restriction. Then σ[p] = 1(ι,θ(p)), for all P such that θ(p) > 0.
That is, any σ [p] sells the good to the buyer with the price equal to his min-
imal possible valuation. To see this, note that by Lemma 1, σ[p(s, h)] = 1g(s) ,
for all s ∈ h(supp(p)). By Lemma 2, g(s) maximizes v in the class of constant,
individually rational mechanisms under p(s,h). Since θ(p(s,h)) > 0 we have
g(s) = (1,θ(p(s, h))). But by imitating θ = θ(p) > 0, any θ0 ∈supp(p) can guaran-
tee to be able to buy the good with price θ(p). Hence by incentive compatibility,
g(h(θ)) = (1,θ(p)), for all θ ∈supp(p).
However, it is also well known that in the "no gap" -case, θ(p) = 0, other more
complex equilibria can be constructed (see e.g. Ausubel and Deneckere, 1989).
To avoid them, the literature often focusses on simple "stationary" equilibria (see
e.g. Ausubel et al., 2001).
The problem with multiplicity of complex solutions applies also in our case
when θ(p) = 0. It can be shown that for any λ ∈ Θ there is a choice rule σλ that
is consistent and meets the one-deviation property, and sells to types θ ≥ λ and
never to types θ<λof the buyer given the prior p (see Appendix B for precise
exposition). However, all constructed σλ are complex, and require the seller to
condition σλ on seemingly superficial information. To remove these complexities,
our final restriction imposes a degree of simplicity on choice rules. It demands
that the implemented outcome is not conditioned on information that does not
provide more profitable transaction opportunities.
Definition 4 (Stationarity) A choice rule σ is stationary if σ[p]=1x,σ[p0]=
1x0 , v(x) ≥ v (x0), and supp(p0) ⊆ supp(p) imply x = x0 .
That is, signals that do not allow the seller to implement a more profitable
choice do not affect the seller’s choice. For example, the choice rule σλ above fails
stationarity.16 The next section characterizes the inducable stationary choice rules
in the general n ≥ 1 case.
4 Results
The English auction φE The tie-breaking rule w always selects one of the
players with the highest valuation:
w(θ) ∈ arg max θj , for all θ ∈ Θ. (5)
j∈N∪{0} j
16For an analogous restriction, see Ausubel and Deneckere (1992).
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