Auction Design without Commitment

Let x allocate the good to player i. By stationarity, since g ◦ h ex post dominates
1_x, g ◦ h cannot be a constant mechanism. By IC there are θ⁰ ∈supp(p(x, φE[p])),
s ∈ h(θ⁰), and j 6= i such that θ⁰_i = θ⁰_j, and such that g(s) allocates the good to j.
By Lemma 1, 1_g(_s) = σ[p(x, φ^E [p])(s, h)]. Since

1x ∈ VIC[p(x,φ^E[p])] ⊆ VIC[p(x,φ^E[p])(s,h))],

it follows by Lemma 2 that v(g(s)) ≥ v(x).

By (8), supp(p(x, φ^E [p])) ⊆ Y_i and, by the definition of support,
supp(p(x, φ^E[p])(s, h)) ⊆ supp(p(x, φ^E [p])).

Thus, by the construction of w, EXP-IR, and Lemma 1, σ[1_θ0]=1x. However, since
g(s) ∈ h(θ⁰), also supp(1_θ0 ) ⊆supp(p(x, φ^E [p])(s, h)). This implies, by stationarity,
that g(s) = x, violating the assertation that x allocates the good to i and g(s) to
^j = i. ■

Now we prove that if a stationary σ is consistent and meets the one-deviation
property relative to C, then there is a tie-breaking rule w such that no element of
C [p] is ex post dominated by φ^E for any q.

Lemma 6 Let a stationary choice rule σ satisfy consistency and the one-deviation
property. Then C^σ [p] ⊆ C^φE [p], for all p, for some tie-breaking rule w.

Proof. Let g◦ h ∈ C^σ[p] and s ∈ h(supp(p)). Denote x = g(s) and q = p(s, h).
By Lemma 1, σ[q] = 1_x. Identify w as in Lemma 5. It suffices for us to show
that 1x is not ex post dominated by φ^E [q]. Suppose, to the contrary, that it
is. By Lemma 5, φ^E [q] ∈ C^σ [q]. By the definition of one-deviation property,
v(σ[q], q) ≥ v(φ^E [q],q). Thus v(x) ≥ v(φ^E[q], q).

Take any y ∈ φ^E[q](supp(q)). Then supp(q(y, φ^E [q])) ⊆supp(q) and, hence,

x ∈ {x⁰ : 1x0 ∈ VIC[q]} ⊆ {x⁰ : 1x0 ∈ V IC[q(y, φ^E[p])]}.

Since, by Lemma 1, σ[q(y, φ^E[q])] = 1_y and 1_x ∈ V IC[q(y, φ^E[q])], it follows
by Lemma 2 that v(x) ≤ v(y). Since y was arbitrary, and v(x) ≥ v(φ^E[q],q),
the inequality must hold as equality. But then, since supp(q(y, φ^E [q])) ⊆supp(p),
stationarity implies that x = y. Thus φ^E [q]=1x , which contradicts the hypothesis
that φE [q] ex post dominates 1_x. ■

By Lemma 6, a stationary and consistent σ that meets the one-deviation prop-
erty is not ex post dominated by some English auction. Hence σ[p] cannot allocate
the good to anyone but the buyer with the highest valuation. Therefore, σ [p] must
be efficient. Another way to put this is that commitment inability of the seller
leads to an efficient allocation, as suggested by the ”Coase theorem".

Since σ [p] is efficient, and the lowest type of a buyer earns zero payoff, the
revenue equivalence theorem implies that φ^E [p] is the (generically) unique im-
plementable mechanism if the buyers’ valuations are independent. However, we
can say more: by Lemma 4, if the seller is unable to commit, the uniqueness of

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