Auction Design without Commitment

Finally we check the case of random h. Note that even if g ◦ h is random, it has
to allocate the good to the same buyer as φ^E [p] does. Thus randomness of g ◦ h
may only concern the monetary transfer m. The proof, which is by induction,
proceeds along the above lines. It is easy to verify that (11) holds also for any
randomly generated monetary transfer m under θ_i^k ,θ_-i . Moreover, (16) needs to
hold for the expected transfer m under (θ^k,θ_-i) . Again, this just means that (11)
holds as equality for each m, thus m^E (i, θ-i, p) is implemented with probability
one under all θ_i^k, θ-i ∈supp(p). This completes the proof.

B Appendix

Non-stationarity of σ^λ

Let n =1and assume the "no gap" -case 0 ∈ Θ 6= {0}. We construct a seller’s
choice function σ^λ that allows the seller to commit to any price λ ∈ Θ. Define a
take-it-or-leave-it offer

_φλ(_θ)= ^½ (1,λ), ifθ≥λ,
^φ (^θ)= (0, 0), ifθ<λ.

That is, "sell with price λ to any type θ at least λ and do not sell to anyone else".
Define σ^λ such that

_σλ[_p] =   φ^λ, if0,λ∈ supp(p),

I 1(₁,θ(p)), otherwise.

We claim that σ^λ satisfies the one-deviation property.

(i) If 0,λ ∈supp(p), then {(1, λ), (0, 0)} = σ^λ [p]. Now:

- 0 < λ = θ(p((1 ,λ) ,σ^λ [p])) and thus σ^λ [p((1 ,λ) ,σ^λ [p])] = 1(₁,_λ) , and

- 0 = θ(p((0, 0),σ^λ[p])) and thus σ^λ[p((0, 0),σ^λ[p])] = 1(o,o).

(ii) If 0 ∈supp(p) and/or λ ∈supp(p), then σ^λ(p) = 1(₁,θ(_p)) and σ^λ(p) = 1(₀,₀),
respectively.

Since constant mechanisms do not affect beliefs, σ^λ satisfies the one-deviation
property. Thus, the seller can commit to any price λ ∈supp(p).

We now argue that σ^λ is not stationary. To see this, let 0,λ ∈supp(p). Find
the degenerate prior 1₀ such that supp(1₀) = {0} 63 λ. Then supp(1₀) ⊂supp(p).
However, σ^λ [10] = 1(1,0) and v((1, 0)) = v((0, 0)) = 0, violating stationarity.

This result is analogous to Ausubel and Deneckere (1989), who show that any
price can be supported in equilibrium in the one sided offers bargaining game
when the discount factor δ tends to 1. Strategies needed for these equilibria are
complicated, i.e. non-stationary.

References

[1] Ausubel, L. and Cramton, P. (1999), The Optimality of Being Efficient,
working paper, University of Maryland

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