Auction Design without Commitment

The second condition implies optimality. Given σ and p, the seller should
choose a mechanism that maximizes her payoff in the set C^σ [p].

Definition 3 (One-Deviation Property) Choice rule σ satisfies the one-deviation
property if v(σ[p],p) ≥ v(φ, p), for all φ ∈ C^σ [p], for all p.

Under the hypothesis that σ can be committed to in the future, the seller
does not want to change σ under any current prior p. Without the one-deviation
property σ could not be convincingly committed to. One-deviation property is in
line with standard equilibrium reasoning. Indeed it is often drawn as a consequence
of it.

Now we state two straightforward but important implications of consistency
and the one-deviation property. First, the seller always implements the outcome
of a mechanism that she can commit to.

Lemma 1 Let σ be consistent and satisfy the one-deviation property. Then g◦h ∈
C^σ [p] implies that σ[p(s, h)] = 1_g(_s) , for all s ∈ h(supp(p)).

Proof. Take any s ∈ h(supp(p)). By consistency, g ◦ h is not ex post domi-
nated by σ[p(s, h)] under p. By the definition of ex post dominance, 1g(s) is not
ex post dominated by σ[p(s, h)]. Hence either v(g(s)) > v(σ[p(s, h)], p(s, h)) or
v(g(s)) = v(σ[p(s, h)], p(s, h)) and σ[p(s, h)] = 1_g(_s) . By the one-deviation prop-
erty, v(σ[p(s, h)], p(s, h)) ≥ v(g(s)). Hence it must be the case that σ[p(s, h)] =
¹g(s). ^и

In particular, the choice rule σ is idempotent in the following sense: if σ[p] =
g ◦ h, then σ[p(s, h)] = 1_g(_s), for all s ∈ h(supp(p)). That is, running σ twice in a
row rather than once will not affect the outcome.

Second, if the seller can commit to implementing an outcome, then that out-
come must maximize her payoff in the class of individually rational outcomes.

Lemma 2 Let σ satisfy the one-deviation property. Then σ[p] = 1_x implies that
v(x) ≥ v(y), for all 1_y ∈ VIC[p].

Proof. Let σ[p] = 1_x, v(x) < v(y), and 1_y ∈ VIC[p]. Since σ[p] ∈ VIC[p],
and since neither 1x nor 1y are ex post dominated by 1x, we have 1x, 1y ∈ C^σ [p].
But this violates the one-deviation property. ■

We now check that our solution is consistent with the standard bargaining
theory.

The Coase conjecture The Coase conjecture, which pertains to our n =1
case, argues that when the seller is unable to commit not to sell the good, the
buyer is able to extract all the surplus. That is, the outcome of the one-sided
bargaining game is to sell the good with price θ(p), the minimal possible valuation
θ in the support of p. The Coase conjecture has been extensively studied in the

<<   10   11   12   13   14   15   16   >>

More intriguing information