A.1 Proof of Lemma 1

First, we prove that $v>\underline {s}$ implies $\beta _B(v)\leq v$ .

When $k=0$ , that is, the transaction price is completely determined by the seller’s bid, a buyer with private value $v\geq \underline {s}$ will get

$$\begin{align*}\pi_B(b,v)=\int_{\underline{s}}^b (v-s)\,\mathrm{d} G_S(s)\end{align*}$$

from bidding b. Note that the integrand, $v-s$ , is strictly decreasing in s, thus

(A.1) $$ \begin{align} \int_{\underline{s}}^b (v-s)\,\mathrm{d} G_S(s)\leq \int_{\underline{s}}^{+\infty}\max\{v-s,0\} \,\mathrm{d} G_S(s). \end{align} $$

Since $v>\underline {s}$ , the equality in (A.1) holds if $b=v$ , and the equality holds for all $G_S$ only if $b=v$ . This implies that, when $k=0$ , the truthful strategy $\beta _B(v)=v$ is the unique (weakly) dominant strategy for the buyer.

When $k\in (0,1]$ , we shall show that it is better for the buyer with value $v>\underline {s}$ to bid her value v than any bid $b>v$ . Since $\underline {s}$ is the lower bound of the support of $G_S$ , $G_S(\underline {s})=0$ and $G_S(v)>0$ , then

$$ \begin{align*} \pi_B(v,v)-\pi_B(b,v) &= \int_{\underline{s}}^v [v-kv-(1-k)s]\,\mathrm{d} G_S(s)-\int_{\underline{s}}^b [v-kb-(1-k)s]\,\mathrm{d} G_S(s)\\ &= \begin{aligned}[t] \int_{\underline{s}}^v [v-kv-(1-k)s]\,\mathrm{d} G_S(s)-\int_{\underline{s}}^v [v-kb-(1-k)s]\,\mathrm{d} G_S(s)\\ -\int_v^b [v-kb-(1-k)s]\,\mathrm{d} G_S(s) \end{aligned}\\ &= \int_{\underline{s}}^v k(b-v)\,\mathrm{d} G_S(s)-\int_v^b [v-kb-(1-k)s]\,\mathrm{d} G_S(s)\\ &= k(b-v)G_S(v)+\int_v^b [kb+(1-k)s-v]\,\mathrm{d} G_S(s). \end{align*} $$

Since $b>v$ and $G_S(v)>0$ , the first term is positive and the second term

$$\begin{align*}\int_v^b [kb+(1-k)s-v]\,\mathrm{d} G_S(s) &\geq \int_v^b [kb+(1-k)v-v]\,\mathrm{d} G_S(s) \\ &=k(b-v)[G_S(b)-G_S(v)]\geq 0. \end{align*}$$

This completes the proof of $\beta _B(v)\leq v$ .

To see that $\beta _B(v)<v$ for $v>\underline {s}$ if $k>0$ , note that by (2.1),

$$ \begin{align*} \left.\frac{\partial\pi_B(b,v)}{\partial b}\right|{}_{b=v}=-kG_S(v)<0. \end{align*} $$

It implies that there exists $\Delta>0$ small enough such that $\pi _B(v-\Delta ,v)>\pi _B(v,v)$ , therefore, bidding the true value for the buyer with private value v is no longer optimal, that is, ${\beta _B(v)\neq v}$ . Since we have already shown that $\beta _B(v)\leq v$ , the desired result follows.

In an analogous way, the second conclusion can be proved by showing that the truthful bidding strategy is dominant when $k=1$ and is dominated by some $\tilde \beta _S(c)>c$ when $k\in [0,1)$ and $c<\overline {b}$ .

A.2 Proof of Lemma 2

First, note that when $k\in (0,1]$ , we can rewrite (3.1) and (3.3) together as

(A.2) $$ \begin{align} \Psi_k(p)=\int_{\underline{s}}^p \int_s^{\min\left(\frac{p-(1-k)s}{k},\overline{b}\right)} g_2(b,s)\,\mathrm{d} b\,\mathrm{d} s. \end{align} $$

Keep $p\in (\underline {s},\overline {b})$ fixed and define a function $\varphi $ as the inner integral in (A.2), that is,

(A.3) $$ \begin{align} \varphi(k,s)=\int_s^{\min\left(\frac{p-(1-k)s}{k},\overline{b}\right)} g_2(b,s)\,\mathrm{d} b,\quad k\in(0,1],\ s\in[\underline{s},p]. \end{align} $$

Since $g_2(b,s)$ is integrable, $\varphi $ is continuous in the upper limit of the integral. And since the upper limit, $\min \left (\frac {p-(1-k)s}{k},\overline {b}\right )$ , is continuous in k, $\varphi $ is continuous in k. Note that $g_2(b,s)>0$ because the integration interval is in support of G, and note that $\min \left (\frac {p-(1-k)s}{k},\overline {b}\right )\le \overline {b}$ , thus for any $k\in (0,1]$ ,

$$\begin{align*}0\le \varphi(k,s)\le \int_s^{\overline{b}} g_2(b,s)\,\mathrm{d} b\equiv \bar{\varphi}(s),\quad \forall\,s\in[\underline{s},p]. \end{align*}$$

Therefore, for any $k\in (0,1]$ , for any sequence $\{k_n\}$ in $(0,1]$ such that $k_n\to k$ as $n\to \infty $ , by continuity of $\varphi $ in k, we have $\tilde {\varphi }_n(s)\equiv \varphi (k_n,s)$ converges pointwise to $\tilde {\varphi }(s)\equiv \varphi (k,s)$ in $[\underline {s},p]$ . Since $\bar {\varphi }(s)$ is integrable, by the dominated convergence theorem, as $n\to \infty $ ,

$$\begin{align*}\int_{\underline{s}}^p \tilde{\varphi}_n(s)\,\mathrm{d} s\to\int_{\underline{s}}^p \tilde{\varphi}(s)\,\mathrm{d} s. \end{align*}$$

This implies $\Psi _{k_n}(p)\to \Psi _k(p)$ .

To see the (right) continuity at $k=0$ , we just need to rewrite (3.1) and (3.2) as

$$\begin{align*}\Psi_k(p) = 1-\int_p^{\overline{b}} \int_{\frac{p-kb}{1-k}}^b g_2(b,s)\,\mathrm{d} s\,\mathrm{d} b,\quad 0\le k<\frac{p-\underline{s}}{\overline{b}-\underline{s}} \end{align*}$$

and define

$$\begin{align*}\psi(k,b) = -\int_{\frac{p-kb}{1-k}}^b g(b,s)\,\mathrm{d} s,\quad k\in\left[0,\frac{p-\underline{s}}{\overline{b}-\underline{s}}\right),\ b\in[p,\overline{b}].\end{align*}$$

Then applying analogous argument, we have that $\psi $ is continuous in k so that for the sequence $\{k_n\}$ in $\left [0,\frac {p-\underline {s}}{\overline {b}-\underline {s}}\right )$ such that $k_n\to 0$ , the sequence $\{\tilde {\psi }_n(b)\equiv \psi (k_n,b)\}$ converges pointwise to $\tilde {\psi }(b)\equiv \psi (0,b)$ . Since $\{\tilde {\psi }_n(b)\}$ is dominated by $\bar {\psi }(b)\equiv \int _{\underline {s}}^b g(b,s)\,\mathrm {d} s$ , we can finally obtain $\Psi _{k_n}(p)\to \Psi _0(p)$ .

It remains to show the monotonicity of $\Psi _k(p)$ in k. Suppose $0\le k_1<k_2\le 1$ , then by (3.1)–(3.3):

1. (i) If $k_2<\frac {p-\underline {s}}{\overline {b}-\underline {s}}$ , then

   $$\begin{align*}\Psi_{k_1}(p)-\Psi_{k_2}(p) = \int_p^{\overline{b}}\int_{b-\frac{b-p}{1-k_2}}^{b-\frac{b-p}{1-k_1}} g_2(b,s)\,\mathrm{d} s\,\mathrm{d} b>0\end{align*}$$
   due to $\frac {b-p}{1-k_2}>\frac {b-p}{1-k_1}$ .

2. (ii) If $k_1\ge \frac {p-\underline {s}}{\overline {b}-\underline {s}}$ , then

   $$\begin{align*}\Psi_{k_1}(p)-\Psi_{k_2}(p) = \int_{\underline{s}}^p \int_{s+\frac{p-s}{k_2}}^{s+\frac{p-s}{k_1}} g_2(b,s)\,\mathrm{d} b\,\mathrm{d} s>0\end{align*}$$
   due to $\frac {p-s}{k_2}<\frac {p-s}{k_1}$ .

3. (iii) If $k_1<\frac {p-\underline {s}}{\overline {b}-\underline {s}}\le k_2$ , then

   $$\begin{align*}\Psi_{k_1}(p)-\Psi_{k_2}(p) &=\int_{\underline{s}}^p \int_{s+\frac{p-s}{k_2}}^{s+\frac{(p-s)(\overline{b} -\underline{s})}{p-\underline{s}}} g_2(b,s)\,\mathrm{d} b\,\mathrm{d} s \\ &\quad + \int_p^{\overline{b}} \int_{b-\frac{(b-p)(\overline{b}-\underline{s})}{\overline{b}-p}}^{b-\frac{b-p}{1-k_1}} g_2(b,s)\,\mathrm{d} s \,\mathrm{d} b>0,\end{align*}$$
   where the first term is nonnegative and the second is positive.

A.3 Proof of Lemma 3

First, we will establish the following two properties in bidding strategies: (M1) under Assumption A, any regular equilibrium strategies $\beta _B$ and $\beta _S$ are twice continuously differentiable in $[\underline {s},\overline {v}]$ and $[\underline {c},\overline {b}]$ , respectively and (M2) for any $v\in [\underline {s},\overline {v}]$ and any $c\in [\underline {c},\overline {b}]$ , $\beta ^{\prime }_B(v)\geq \epsilon _B>0$ and $\beta ^{\prime }_S(c)\geq \epsilon _S>0$ . To show (M1), we need to rewrite (3.5) and (3.6) as follows:

(A.4) $$ \begin{align} \beta_S'(c) &= \frac{f_C(c)\big[\beta_B^{-1}(\beta_S(c))-\beta_S(c)\big]}{k\cdot F_C(c)}, \end{align} $$

(A.5) $$ \begin{align} \beta_B'(v) &= \frac{f_V(v)\big[\beta_B(v)-\beta_S^{-1}(\beta_B(v))\big]}{(1-k)\cdot [1-F_V(v)]}. \end{align} $$

By definition, any pair of regular equilibrium strategies $\beta _B$ and $\beta _S$ is continuously differentiable in $[\underline s,\overline v]$ and $[\underline c, \overline b]$ , respectively (see Assumption C). Consequently, under Assumption A, (A.4) and (A.5) imply that $\beta _S'(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _B'(\mkern 2mu\cdot \mkern 2mu)$ are continuously differentiable in $[\underline c,\overline b]$ and $[\underline s,\overline v]$ , respectively. This further implies that $\beta _S$ and $\beta _B$ are twice continuously differentiable in $[\underline c,\overline b]$ and $[\underline s,\overline v]$ . This completes the proof of (M1).

Now, we establish (M2). By definition of regular equilibrium, the seller and buyer bidding strategies are continuously differentiable with positive derivative on $[\underline c,\overline b]$ and $[\underline s,\overline v]$ , respectively (see condition A2 of Assumption C), that is, $\beta _S'(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _B'(\mkern 2mu\cdot \mkern 2mu)$ are continuous and positive on $[\underline c,\overline b]$ and $[\underline s,\overline v]$ . By the extreme value theorem, $\beta _S'(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _B'(\mkern 2mu\cdot \mkern 2mu)$ have positive minimum and maximum in $[\underline c,\overline b]$ and $[\underline s,\overline v]$ , respectively. Therefore, the conclusion of (M2) follows.

By (A.8) and (A.9), conditions (M1) and (M2) imply that both $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)$ and $\eta (\mkern 2mu\cdot \mkern 2mu,G_B)$ are twice continuously differentiable on $[\underline {s},\overline {b}]$ . Note that

$$\begin{align*}g_B(b)=\frac{f_V(\beta_B^{-1}(b))}{\beta^{\prime}_B(\beta_B^{-1}(b))},\quad g_S(s)=\frac{f_C(\beta_S^{-1}(s))}{\beta^{\prime}_S(\beta_S^{-1}(s))}. \end{align*}$$

In addition, $f_V$ and $f_C$ are bounded away from 0 by Assumption A, and $\beta ^{\prime }_B$ and $\beta ^{\prime }_S$ are bounded away from 0 by (M2). The conclusion of part (i) then follows. Lastly, to prove part (ii), notice that (3.7) and (3.8) yield the following:

$$\begin{align*}g_S(s)=k\frac{G_S(s)}{\xi(s,G_S)-s},\quad g_B(b)=(1-k)\frac{1-G_B(b)}{b-\eta(b,G_B)}. \end{align*}$$

Since every term on the right-hand side is twice continuously differentiable, the desired conclusion follows.

A.4 Proof of Theorem 1

Let $\beta _B(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S(\mkern 2mu\cdot \mkern 2mu)$ be the respective regular equilibrium bidding strategies of the buyer and the seller that induce the bid distribution G.

By Condition A1 of Assumption C, the strictly increasing and continuous bidding strategies imply that the support of bid distribution is a rectangular region, namely, $[\underline {b},\overline {b}]\times [\underline {s},\overline {s}]$ with $\underline {b}=\beta _B(\underline {v})$ , $\overline {b}=\beta _B(\overline {v})$ , $\underline {s}=\beta _S(\underline {c}),$ and $\overline {s}=\beta _S(\overline {c})$ . To show that $\overline {b}\leq \overline {s}$ and $\underline {b}\leq \underline {s}$ , first suppose that $\overline {b}>\overline {s}$ , then any buyer bidding $b>\overline {s}$ will be strictly inferior to bidding only $\overline {s}$ . Because this does not make the buyer lose any trades, but the expected profit on each trade will increase by lowering the transaction price. This deviation is contradicted by the assumption that $(\beta _B,\beta _S)$ is an equilibrium. Applying a similar argument to the seller’s bidding $s<\underline {b}$ , we can prove the second conclusion $\underline {s}\geq \underline {b}$ . Then we show that $\underline {s}<\overline {b}$ . Suppose not, then: (i) If $\overline {b}\leq \underline {s}<\overline {v}$ , the buyer with value $\overline {v}$ will have an incentive to bid $\frac {\underline {s}+\overline {v}}{2}$ instead of $\overline {b}$ , because by bidding $\frac {\underline {s}+\overline {v}}{2}$ , he can get

$$\begin{align*}\pi\left(\frac{\underline{s}+\overline{v}}{2},\overline{v}\right) &= \int_{\underline{s}}^{\frac{\underline{s}+\overline{v}}{2}}\left[\overline{v}-k\frac{\underline{s}+\overline{v}}{2}-(1-k)s\right]\,\mathrm{d} G_S(s)\\ & = \frac{k}{2}(\overline{v}-\underline{s})+(1-k)\int_{\underline{s}}^{\frac{\underline{s}+\overline{v}}{2}}(\overline{v}-s)\,\mathrm{d} G_S(s)>0\end{align*}$$

while bidding $\overline {b}\leq \underline {s}$ gives him zero expected profit. This contradicts the equilibrium requirement. (ii) If $\underline {c}<\overline {b}\leq \underline {s}$ , then an analogous argument can show that bidding $\frac {\overline {b}+\underline {c}}{2}$ is a profitable deviation for the seller with value $\underline {c}$ , which also presents a contradiction to the equilibrium condition. (iii) If $\overline {b}\leq \underline {c}<\overline {v}\leq \underline {s}$ , then condition A3 of Assumption C is contradicted because it requires that $\underline {s}=\underline {c}<\overline {v}=\overline {b}$ . From the above, C1 holds.

Because V and C are independent and because $\beta _B(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S(\mkern 2mu\cdot \mkern 2mu)$ are deterministic functions, it follows that the bids, $B=\beta _B(V)$ and $S=\beta _S(C)$ , are also independent. More precisely, since $\beta _B(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S(\mkern 2mu\cdot \mkern 2mu)$ are continuous and strictly increasing, there exist inverse functions, $\beta _B^{-1}(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S^{-1}(\mkern 2mu\cdot \mkern 2mu)$ , which are also continuous and strictly increasing. Thus,

$$ \begin{align*} G(b,s) &= \Pr(\beta_B(V)\leq b, \beta_S(C)\leq s)\\ &= \Pr(V\leq \beta_B^{-1}(b), C\leq \beta_S^{-1}(s))\\ &= \Pr(V\leq \beta_B^{-1}(b))\Pr(C\leq \beta_S^{-1}(s))=F_V(\beta_B^{-1}(b))F_C(\beta_S^{-1}(s)). \end{align*} $$

Define

(A.6) $$ \begin{align} G_B(b)&=F_V(\beta_B^{-1}(b)) \end{align} $$

(A.7) $$ \begin{align} G_S(s)&=F_C(\beta_S^{-1}(s)) \end{align} $$

for every $b\in [\underline {b}, \overline {b}]$ and $s\in [\underline {s}, \overline {s}]$ . Since $\beta _B^{-1}(\mkern 2mu\cdot \mkern 2mu)$ is continuous and strictly increasing in $[\underline {b}, \overline {b}]=[\beta _B(\underline {v}), \beta _B(\overline {v})]$ , we have $G_B\in \mathscr {P}_{[\underline {b}, \overline {b}]}$ by (A.6) and the assumption $F_V\in \mathscr {P}_{[\underline {v}, \overline {v}]}$ . Similar argument can be applied to show $G_S\in \mathscr {P}_{[\underline {s}, \overline {s}]}$ . Now, we get C2.

In order to show C3 and C4, note that $G_B(\mkern 2mu\cdot \mkern 2mu)$ and $G_S(\mkern 2mu\cdot \mkern 2mu)$ defined in (A.6) and (A.7) must be the distributions of observed (equilibrium) bids of the buyer and the seller, respectively. Now, $\beta _B(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S(\mkern 2mu\cdot \mkern 2mu)$ must solve the set of first-order differential equations (3.5) and (3.6). Since (3.7) and (3.8) follow from (3.5) and (3.6), $\beta _B(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S(\mkern 2mu\cdot \mkern 2mu)$ must satisfy

$$\begin{align*}\xi(\beta_B(v),G_S)=v,\quad \eta(\beta_S(c),G_B)=c\end{align*}$$

for all $v\geq \underline {s}$ and all $c\leq \overline {b}$ . Noting that $\underline {s}=\beta _S(\underline {c})$ and $\overline {b}=\beta _B(\overline {v})$ and making the change of variable $v=\beta _B^{-1}(b)$ and $c=\beta _S^{-1}(s)$ , we obtain

(A.8) $$ \begin{align} \xi(b,G_S)=\beta_B^{-1}(b) \end{align} $$

(A.9) $$ \begin{align} \eta(s,G_B)=\beta_S^{-1}(s) \end{align} $$

for all $b,s\in [\underline {s},\overline {b}]$ . By condition A1 of Assumption C, both $\beta _B^{-1}(\mkern 2mu\cdot \mkern 2mu)$ and $\beta _S^{-1}(\mkern 2mu\cdot \mkern 2mu)$ are strictly increasing, and by condition A3 of Assumption C, $\beta _B(\mkern 2mu\cdot \mkern 2mu)$ is differentiable on $[\underline {s},\overline {v}]$ and so is $\beta _S(\mkern 2mu\cdot \mkern 2mu)$ on $[\underline {c},\overline {b}]$ . Thus, C3 and C4 follow from the fact that $\xi (\underline {s},G_S)=\underline {s}$ by (3.7), $\eta (\overline {b},G_B)=\overline {b}$ by (3.8), and $\overline {v}=\beta _B^{-1}(\overline {b})=\xi (\overline {b},G_S)$ , $\underline {c}=\beta _S^{-1}(\underline {s})=\eta (\underline {s},G_B)$ .

It remains to show C5 and C6. Given $b\in [\underline s,\overline {b}]$ , for the buyer with private value v such that $\beta _B(v)=b$ , bidding any $b'\in [\overline {b},\overline {s}]$ should not give him greater profit than bidding b because $\beta _B$ is the equilibrium bidding strategy for the buyer. That is,

$$ \begin{align*} & 0\geq \pi_B(b',v)-\pi_B(b,v) \\ & \quad = \int_{\underline{s}}^{b'} [v-kb'-(1-k)s]\,\mathrm{d} G_S(s) - \int_{\underline{s}}^{b}[v-kb-(1-k)s]\,\mathrm{d} G_S(s) \\ &\quad = v[G_S(b')-G_S(b)] -kb'G_S(b')+kbG_S(b)-(1-k)\int_b^{b'}s\,\mathrm{d} G_S(s)\\ &\quad = k(v-b')G_S(b')-k(v-b)G_S(b) \\ &\qquad + (1-k)\left[(v-b')G_S(b')-(v-b)G_S(b)+\int_b^{b'}G_S(s)\,\mathrm{d} s\right]\\ &\quad = (v-b')G_S(b')-(v-b)G_S(b)+(1-k)\int_b^{b'}G_S(s)\,\mathrm{d} s. \end{align*} $$

Because $v=\beta _B^{-1}(b)=\xi (b,G_S)$ by (A.8), replacing v by $\xi (b,G_S)$ in the above inequality will give (3.11). Similarly, for the seller with private value c such that $\beta _S(c)=s\in [\underline {s},\overline b]$ , using the argument that any deviation of bidding $s'\in [\underline {b},\underline {s}]$ would not be profitable, we can show that (3.12) must hold. This completes the proof of C6 and the theorem.

A.5 Proof of Theorem 2

We show the identification of $F_V$ and $F_C$ in two steps. In the first step, we construct a pair of $F_V$ and $F_C$ to rationalize the given G. In the second step, we show that this pair is unique.

Step 1. To show the sufficiency of C1–C4, define

(A.10) $$ \begin{align} F_V(v)=\begin{cases} G_B(v) & \text{if } v<\underline{s}\\ G_B(\xi^{-1}(v,G_S)) & \text{if } \underline{s}\leq v\leq \xi(\overline{b},G_S)\\ 1 & \text{if } v>\xi(\overline{b},G_S) \end{cases} \end{align} $$

(A.11) $$ \begin{align} F_C(c)=\begin{cases} 0 & \text{if } c<\eta(\underline{s},G_B)\\ G_S(\eta^{-1}(c,G_B)) & \text{if } \eta(\underline{s},G_B)\leq c\leq\overline{b}\\ G_S(c) & \text{if } c>\overline{b} \end{cases} \end{align} $$

and

$$\begin{align*}\underline{v}=\underline{b},\quad \overline{v}=\xi(\overline{b},G_S),\quad \underline{c}=\eta(\underline{s},G_B),\quad \overline{c}=\overline{s}.\end{align*}$$

Condition C1 guarantees that the functions $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)$ in (3.7) and $\eta (\mkern 2mu\cdot \mkern 2mu,G_S)$ in (3.8) are well defined. Since $\underline {b}$ is the lower end point of the support of $G_B$ , for all $v\leq \underline {v}=\underline {b}$ , $F_V(v)=0$ , and by definition, $F_V(v)=1$ for all $v>\overline {v}=\xi (\overline {b},G_S)$ . Moreover, because $F_V(\overline {v})=G_B(\xi ^{-1}(\xi (\overline {b},G_S),G_S))=G_B(\overline {b})=1$ , $F_V(\underline {s}) = G_B(\xi ^{-1}(\xi (\underline {s},G_S),G_S)) = G_B(\underline {s})$ , $G_B$ is continuous and strictly increasing on $[\underline {b},\overline {b}]$ by C2, and $\xi ^{-1}(\mkern 2mu\cdot \mkern 2mu,G_S)$ is continuous and strictly increasing on $[\xi (\underline {s},G_S),\xi (\overline {b},G_S)]$ by C3. Then $F_V(\mkern 2mu\cdot \mkern 2mu)$ defined by (A.10) is continuous and strictly increasing in $[\underline {b},\xi (\overline {b},G_S)]=[\underline {v},\overline {v}]$ . Therefore, $F_V$ is a valid absolutely continuous distribution with support $[\underline {v},\overline {v}]$ , that is, $F_V\in \mathscr {P}_{[\underline {v},\overline {v}]}$ as required. We can also show $F_C\in \mathscr {P}_{[\underline {c},\overline {c}]}$ in a similar way.

We shall show that the distributions $F_V$ and $F_C$ of the respective private values of the buyer and the seller can rationalize G in a sealed-bid k-double auction, that is, $G_B(b)=F_V(\beta _B^{-1}(b))$ in $[\underline {b},\overline {b}]$ and $G_S(s)=F_C(\beta _S^{-1}(s))$ on $[\underline {s},\overline {s}]$ for some regular equilibrium profile $(\beta _B,\beta _S)$ . By construction of $F_V$ and $F_C$ , we have

$$ \begin{align*} G_B(b) & =F_V(b) \mathbb{1}(\underline{b}\leq b<\underline{s})+F_V(\xi(b,G_S)) \mathbb{1}(\underline{s}\leq b\leq\overline{b})\\ & =F_V\left(b \mathbb{1}(\underline{b}\leq b<\underline{s})+\xi(b,G_S) \mathbb{1}(\underline{s}\leq b\leq\overline{b})\right) \end{align*} $$

for $b\in [\underline {b},\overline {b}]$ and

$$ \begin{align*} G_S(s) & =F_C(\eta(s,G_B)) \mathbb{1}(\underline{s}\leq s\leq \overline{b}) + F_C(s) \mathbb{1}(\overline{b}<s\leq\overline{s})\\ & =F_C\left(\eta(s,G_B) \mathbb{1}(\underline{s}\leq s\leq \overline{b}) + s \mathbb{1}(\overline{b}<s\leq\overline{s})\right) \end{align*} $$

for $s\in [\underline {s},\overline {s}]$ , where $\mathbb {1}(\mkern 2mu\cdot \mkern 2mu)$ is the indicator function. Define

$$ \begin{align*} \xi_*(b,G_S) &\equiv b \mathbb{1}(\underline{b}\leq b<\underline{s})+\xi(b,G_S)\mathbb{1}(\underline{s}\leq b\leq\overline{b}),\\ \eta_*(s,G_B) &\equiv \eta(s,G_B)\mathbb{1}(\underline{s}\leq s\leq \overline{b})+s \mathbb{1}(\overline{b}<s\leq\overline{s}), \end{align*} $$

then by C3 and C4, $\xi _*(\mkern 2mu\cdot \mkern 2mu,G_S)$ is continuous and strictly increasing on $[\underline {b},\overline {b}]$ and so is $\eta _*(\mkern 2mu\cdot \mkern 2mu,G_B)$ on $[\underline {s},\overline {s}]$ . Define bidding strategies

(A.12) $$ \begin{align} \beta_B(v) &= \begin{cases} v & \text{if } \underline{v}\leq v \leq \underline{s}\\ \xi^{-1}(v,G_S) & \text{if } \underline{s}<v\leq\overline{v} \end{cases} \end{align} $$

(A.13) $$ \begin{align} \beta_S(c) &= \begin{cases} \eta^{-1}(c,G_B) & \text{if } \underline{c}\leq c <\overline{b}\\ c & \text{if } \overline{b}\leq c\leq\overline{c} \end{cases} \end{align} $$

so that $\beta _B(\mkern 2mu\cdot \mkern 2mu)=\xi _*^{-1}(\mkern 2mu\cdot \mkern 2mu,G_S)$ and $\beta _S(\mkern 2mu\cdot \mkern 2mu)=\eta _*^{-1}(\mkern 2mu\cdot \mkern 2mu,G_B)$ . By construction of these strategies, A1–A3 in Assumption C are satisfied, and also $G_B(b)=F_V(\beta _B^{-1}(b))$ and $G_S(s)=F_C(\beta _S^{-1}(s))$ so that G is the induced bid distribution for $(F_V,F_C)$ defined in (A.10) and (A.11) by the strategy profile $(\beta _B,\beta _S)$ defined above. Thus, it remains to show that $(\beta _B,\beta _S)$ is indeed an equilibrium. We show that the optimal offer for the buyer with private value v is $\beta _B(v)$ . A similar argument shows that $\beta _S$ is optimal for the seller.

Obviously, if $v\leq \underline {s}$ , then the buyer cannot make an advantageous trade and bidding $\beta _B(v)=v$ achieves zero as the greatest possible expected profit. Suppose $v>\underline {s}$ , since $G_S$ is the induced seller’s bid distribution, for bid $b\in [\underline {s},\overline {b}]$ , by (2.1), we obtain

$$ \begin{align*} \frac{\partial\pi_B(b,s)}{\partial b} &= -k G_S(b) + (v-kb)g_S(b) - (1-k)bg_S(b) \\ &= g_S(b)\left[v-\left(b+k\frac{G_S(b)}{g_S(b)}\right)\right] = g_S(b)\left[v-\xi(b,G_S)\right]. \end{align*} $$

Because $g_S(b)$ is positive, the monotonicity of $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)$ by C3 implies that $\partial \pi _B(b,v)/\partial b>0$ for all $b<\xi ^{-1}(v,G_S)$ and $\partial \pi _B(b,v)/\partial b<0$ for all $b>\xi ^{-1}(v,G_S)$ . Therefore, $b=\xi ^{-1}(v,G_S)=\beta _B(v)$ is the unique maximizer of the expected profit of the buyer in $[\underline {s},\overline {b}]$ . Now, we show that the buyer would not want to choose a bid within $[\overline {b},\overline {s}]$ , either. Recall that we have already shown that C5 is equivalent to $\pi _B(b',v)\leq \pi _B(b,v)$ for any $v\in [\underline s,\xi (\overline {b},G_S)]$ and any $b'\in [\overline {b},\overline {s}]$ when $b=\xi ^{-1}(v,G_S)=\beta _B(v)$ in the proof of Theorem 1, so the buyer is not profitable to deviate from $b\in [\underline s,\overline b]$ to $b'\in [\overline b,\overline s]$ . Finally, given that $\overline {s}$ is the highest bid of the seller, any buyer’s bid greater than $\overline {s}$ will be dominated by $\overline {s}$ . Hence, $F_V$ and $F_C$ rationalize G in a sealed bid k-double auction.

Step 2. From the proof of Theorem 1, we know that $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)=\beta _B^{-1}(\mkern 2mu\cdot \mkern 2mu)$ and $\eta (\mkern 2mu\cdot \mkern 2mu,G_B)=\beta _S^{-1}(\mkern 2mu\cdot \mkern 2mu)$ on $[\underline {s},\overline {b}]$ when $F_V(\mkern 2mu\cdot \mkern 2mu)$ and $F_C(\mkern 2mu\cdot \mkern 2mu)$ exist. Since $F_V(\mkern 2mu\cdot \mkern 2mu)=G_B(\beta _B(\mkern 2mu\cdot \mkern 2mu))$ and $F_C(\mkern 2mu\cdot \mkern 2mu)=G_S(\beta _S(\mkern 2mu\cdot \mkern 2mu))$ , $F_V(\mkern 2mu\cdot \mkern 2mu)=G_B(\xi _*^{-1}(\mkern 2mu\cdot \mkern 2mu,G_S))$ and $F_C(\mkern 2mu\cdot \mkern 2mu)=G_S(\eta _*^{-1}(\mkern 2mu\cdot \mkern 2mu,G_B))$ . Because $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)$ is uniquely determined by $G_S(\mkern 2mu\cdot \mkern 2mu)$ and $\eta (\mkern 2mu\cdot \mkern 2mu,G_B)$ is uniquely determined by $G_B(\mkern 2mu\cdot \mkern 2mu)$ , it follows that $\xi _*(\mkern 2mu\cdot \mkern 2mu,G_S)$ and $\eta _*(\mkern 2mu\cdot \mkern 2mu,G_B)$ are uniquely determined by G. Hence, the private value distribution $(F_V,F_C)$ that rationalizes G is unique.

This therefore establishes the identification of $F_V$ and $F_C$ from any given $G\in {\mathscr P}_{\mathscr D}$ satisfying C1–C6.

A.6 Proof of Theorem 4

It is straightforward to see that conditions C3 and C4 are implied by C7 and C8. Then it suffices to show that C5 and C6 are implied by C7 and C8.

We shall only show that C7 (more precisely, the monotonicity of $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)$ ) implies C5. A similar argument can show that C8 implies C6. For the buyer with private value v, since

$$\begin{align*}\frac{\partial\pi_B(b,v)}{\partial b} = g_S(b)\left[v-\xi(b,G_S)\right],\end{align*}$$

the strict monotonicity of $\xi (\mkern 2mu\cdot \mkern 2mu,G_S)$ in $[\underline {s},\overline {s}]$ ensures that for any $b\in \left (\xi ^{-1}(v,G_S),\overline {s}\right ]$ , $\partial \pi _B(b,v)/\partial b<0$ . As a result, the expected profit of the buyer $\pi _B(b,v)$ is strictly decreasing in the buyer’s bid for all $b\in (\xi ^{-1}(v,G_S),\overline s]$ . For $b'\in [\overline {b},\overline {s}]$ and $b\in [\underline s,\overline b]$ , let $v=\xi (b,G_S)$ , then it follows from the above conclusion that

$$\begin{align*}b'\geq\overline{b}\geq b=\xi^{-1}(v,G_S)\quad\Rightarrow\quad \pi_B(b',v)\leq \pi_B(b,v), \end{align*}$$

which is equivalent to C5 as shown in the proof of Theorem 1.

A.7 Proof of Theorem 3

By Theorem 1, C1–C4 hold. Let $m' = \Pr (\underline {s}\leq S \leq B\leq \overline {b})$ . By the definition of $G_2$ , D1 is the direct corollary of C1. Using $g_2(b,s)=g(b,s)/m'$ and $g(b,s)=g_B(b)g_S(s)$ by C2, we have

$$ \begin{align*}g_{2}(b,s)g_{2}(b',s')=g_{2}(b,s')g_{2}(b',s)=\frac{g_{B}(b)g_{B}(b')g_{S}(s)g_{S}(s')}{m^{'2}}, \end{align*} $$

so D2 is satisfied. D3 and D4 are implied by C3 and C4, respectively.

A.8 Proof of Theorem 4

By condition D1 and Lemma 1, we have $\underline {c}\leq \underline {s}<\overline {b}\leq \overline {v}$ , namely, condition E1 is satisfied.

Notice that, by D3 and D4, (3.13) is equivalent to

(A.14) $$ \begin{align} \frac{F_V(\xi(b,G_{S}))-F_V(\underline{s})}{1-F_V(\underline{s})}= \frac{G_B(b)-G_B(\underline{s})}{1-G_B(\underline{s})} ,\quad \frac{F_C(\eta(s,G_{B}))}{F_C(\overline{b})}=\frac{G_S(s)}{G_S(\overline{b})} \end{align} $$

for $(b,s)\in [\underline {s},\overline {b}]^2$ .

Next, we establish condition E2 by showing (A.14). According to the proof of Theorem 2, G can only be rationalized by $(F_V,F_C)$ defined in (A.10) and (A.11) which implies

(A.15) $$ \begin{align} F_V(\xi(b,G_S))=G_B(b),\quad F_C(\eta(s,G_B))=G_S(s) \end{align} $$

for $\underline {s}=\xi ^{-1}(\underline {s},G_S)\leq b\leq \overline {b}$ and $\underline {s}\leq s\leq \eta ^{-1}(\overline {b},G_B)=\overline {b}$ . By (A.15) and using $\xi (\underline {s},G_{S})=\underline {s}$ , $\eta (\overline {b},G_{B})=\overline {b}$ , we have condition (A.14) to hold for all $F_V$ and $F_C$ . We therefore establish condition E2 and complete the entire proof.

A.9 Proof of Theorem 4

We will only show the identification of $\theta _B$ . The case of $\theta _S$ can be proved in a similar way.

Let ${\mathcal B}(v_j) \equiv \big [G_B(\xi ^{-1}(v_j,G_{S})) - G_{B}(\underline {s})\big ]\big /\big (1-G_B(\underline {s})\big ) $ and $\phi _j(\theta _{B}) \equiv {\mathcal Q}_B(\theta _{B};v_j) - {\mathcal B}(v_j)$ for $j = 1,\dots ,m$ . By Condition E.2 of Theorem 4, we have $\phi _j(\theta _{B}) = 0$ for all $\theta _B\in \Theta _B'$ and $j=1,\dots ,m$ . Notice that the Jacobian matrix of $(\phi _1(\theta _{B}),\dots ,\phi _m(\theta _{B}))$ w.r.t. $\theta _B$ is $J{\mathcal Q}_B(\theta _B;v^m)$ . The desired conclusion therefore follows from Theorem 7 of Rothenberg (Reference Rothenberg1971).

A.10 Proof of Lemma 5

Part (i) follows directly from Lemma 7 given that the bandwidths $h_B$ and $h_S$ satisfy Assumption G. We next show part (ii). We shall show the convergence rate of $\sup _i |\hat {V}_i-V_i|$ . The result for $\sup _i |\hat {C}_i-C_i|$ can be shown analogously.

It follows from the definition of $\xi (b,G_S)$ and (4.3) that

(A.16) $$ \begin{align} &\mathbb{1}(V_i\in[\underline{s},\overline{v}]) |\hat{V}_i-V_i| = \mathbb{1}(B_i\in [\hat{\underline{s}},\overline{b}])\cdot k\Big|\frac{\hat{G}_S(B_i)}{\hat{g}_S(B_i)}-\frac{G_S(B_i)}{g_S(B_i)}\Big| + \mathbb{1}(B_i\in [\underline{s},\hat{\underline{s}}])\cdot k\cdot\frac{G_S(B_i)}{g_S(B_i)} \nonumber \\ &= \begin{aligned} \mathbb{1}(B_i\in [\hat{\underline{s}},\overline{b}])\cdot k\Big|\frac{\hat{G}_S(B_i)-G_S(B_i)}{g_S(B_i)}-\frac{G_S(B_i)}{g_S(B_i)^2}\big[\hat{g}_S(B_i)-g_S(B_i) \big]\\ \vphantom{\frac{\hat{G}_S}{g_S}}+o_p\big(\hat{G}_S(B_i)-G_S(B_i)\big)+o_p\big(\hat{g}_S(B_i)-g_S(B_i)\big)\Big| + O_p(1/n) \end{aligned} \nonumber\\ &\leq \begin{aligned} \mathbb{1}(B_i\in [\hat{\underline{s}},\overline{b}])\Big\{\frac{|\hat{G}_S(B_i)-G_S(B_i)|}{g_S(B_i)}+\frac{G_S(B_i)}{g_S(B_i)^2}|\hat{g}_S(B_i)-g_S(B_i)| \\ \quad + o_p\big(|\hat{G}_S(B_i)-G_S(B_i)|\big)+o_p\big(|\hat{g}_S(B_i)-g_S(B_i)|\big)\Big\} + O_p(1/n) \end{aligned} \nonumber \\ &\leq \sup_{B_i\in [\hat{\underline{s}},\overline{b}]}\Big\{\frac{|\hat{G}_S(B_i)-G_S(B_i)|}{g_S(B_i)}+\frac{G_S(B_i)}{g_S(B_i)^2}|\hat{g}_S(B_i)-g_S(B_i)| \nonumber\\ &\quad + o_p\big(|\hat{G}_S(B_i)-G_S(B_i)|\big)+o_p\big(|\hat{g}_S(B_i)-g_S(B_i)|\big)\Big\} + O_p(1/n) \nonumber\\ &\leq \frac{\sup_{b\in [\hat{\underline{s}},\overline{b}]}|\hat{G}_S(b)-G_S(b)|}{\alpha_S}+\frac{1}{\alpha_S^2}\sup_{b\in [\hat{\underline{s}},\overline{b}]}|\hat{g}_S(b)-g_S(b)| \nonumber\\ &\quad + o_p\big(\sup_{b\in[\hat{\underline{s}},\overline{b}]}|\hat{G}_S(b)-G_S(b)|\big)+ o_p\big(\sup_{b\in[\hat{\underline{s}},\overline{b}]}|\hat{g}_S(b)-g_S(b)|\big) + O_p(1/n), \end{align} $$

where the second equality is obtained by Taylor expansion and $\mathbb {1}(B_i\in [\underline s,\hat {\underline s}])\cdot k\cdot \frac {G_S(B_i)}{g_S(B_i)}\leq k\cdot G_S(\hat {\underline s})/\alpha _S = O_p(1/n)$ , and the last inequality holds since, for any b, $g_S(b)\geq \alpha _S$ and $G_S(b)\leq 1$ . Then,

$$ \begin{align*} &\sup_i \mathbb{1}(V_i\in[\underline{s},\overline{v}])|\hat{V}_i-V_i| \leq \frac{\sup_{b\in[\underline{s},\overline{b}]}|\hat{G}_S(b)-G_S(b)|}{\alpha_S}+\frac{1}{\alpha_S^2}\sup_{b\in [\underline{s},\overline{b}]}|\hat{g}_S(b)-g_S(b)|\\ &+ o_p\big(\sup_{b\in[\underline{s},\overline{b}]}|\hat{G}_S(b)-G_S(b)|\big)+ o_p\big(\sup_{b\in[\underline{s},\overline{b}]}|\hat{g}_S(b)-g_S(b)|\big) + O_p(1/n) \end{align*} $$

given that $\underline s\leq \hat {\underline s}$ .

Given that (i) $\sup _{b\in [\underline {s},\overline {b}]}|\hat {G}_S(b)-G_S(b)|\leq \sup _{b\in \mathbb {R}}|\hat {G}_S(b)-G_S(b)|=O_p(\log n/\sqrt {n})$ and (ii) $\sup _{b\in [\underline {s},\overline {b}]}\vert \hat {g}_S(b) - g_S(b)\vert = O_p\big ((\log n /n)^{2/5}\big )$ from Part (i), it follows that

(A.17) $$ \begin{align} \sup_i \mathbb{1}(V_i\in[\underline{s},\overline{v}])|\hat{V}_i-V_i| = O_p\big((\log n / n)^{2/5}\big). \end{align} $$

By the regular equilibrium assumption, the buyer with private value $V_i<\underline {s}$ will bid $B_i=V_i$ . In this case, her estimated private value $\hat V_i = B_i$ by (4.3) given that $B_i=V_i<\underline s \leq \hat {\underline s}$ . This implies that $\mathbb {1}(V_i\in [\underline v,\underline s))\cdot \big \vert \hat V_i - V_i\big \vert = 0$ . Then we can extend the result in (A.17) to all $V_i\in [\underline {v},\overline {v}]$ so that

$$\begin{align*}\sup_i |\hat{V}_i-V_i| = \sup_i \mathbb{1}(V_i\in[\underline{s},\overline{v}])|\hat{V}_i-V_i| = O_p\big((\log n /n )^{2/5}\big).\end{align*}$$

This completes the proof.

A.11 Proof of Lemma 6

We shall only show the uniform convergence result of $|\hat {f}_V(\cdot )-f_V(\cdot )|$ . The result of $|\hat {f}_C(\cdot )-f_C(\cdot )|$ can be shown in a similar way and is therefore omitted.

Let $\mathscr {C}_V$ be a closed inner subset of $[\underline {v},\overline {v}]$ , and $\tilde {f}_V(\cdot )$ be the (infeasible) one-step local linear density estimator (i.e., $p=1$ ) which uses the unobserved true private values $V_i$ instead of $\hat {V}_i$ . According to Lemma 6, we can show that $\sup _{v\in [\underline v, \overline v]}|\tilde {f}_V(v)-f_V(v)|=O_p\left ((\log n/n)^{1/5}\right )$ given a bandwidth $h_V = \lambda _V(\log n/n)^{1/5}$ . Since $\hat {f}_V(v)-f_V(v)=[\hat {f}_V(v)-\tilde {f}_V(v)]+[\tilde {f}_V(v)-f_V(v)]$ , it remains to show that $\sup _{v\in \mathscr {C}_V} \big | \hat {f}_V(v)-\tilde {f}_V(v)\big | = O_p \big ((\log n/n)^{1/5}\big )$ .

Let $\mathscr {C}^{\prime }_V=\bigcup _{v\in \mathscr {C}_V}[v-\Delta ,v+\Delta ]$ and $\mathscr {C}^{\prime \prime }_V=\bigcup _{v\in \mathscr {C}^{\prime }_V}[v-\Delta ,v+\Delta ]$ for some $\Delta>0$ . By construction, $\mathscr {C}^{\prime }_V$ and $\mathscr {C}^{\prime \prime }_V$ are also closed, and $\mathscr {C}_V\subset \mathscr {C}^{\prime }_V\subset \mathscr {C}^{\prime \prime }_V$ . Since $\mathscr {C}_V$ is a closed inner subset of $[\underline {v},\overline {v}]$ , $\Delta $ can be chosen small enough such that $\mathscr {C}^{\prime \prime }_V\subset [\underline {v},\overline {v}]$ . Now, by part (ii) of Lemma 5, for $v\in \mathscr {C}_V$ and n large enough, $\hat {f}_V(v)$ uses at most observations $\hat {V}_i$ in $\mathscr {C}^{\prime }_V$ and for which $V_i$ is in $\mathscr {C}^{\prime \prime }_V$ . Because (i) for any $v\in \mathscr {C}_V$ , $\tilde {f}_V(v)$ uses at most $V_i$ in $\mathscr {C}^{\prime \prime }_V$ and (ii) both $\hat {f}_V(v)$ and $\tilde {f}_V(v)$ are first-order asymptotically equivalent (in convergence rate) to the standard kernel density estimator, we can then obtain the convergence rate by analyzing the latter as follows:

$$\begin{align*}\hat{f}_V(v)-\tilde{f}_V(v) = \frac{1}{nh_V}\sum_{i=1}^n \mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)\big[K_V\big(\frac{v-\hat{V}_i}{h_V}\big)-K_V\big(\frac{v-V_i}{h_V}\big)\big].\end{align*}$$

A second-order Taylor expansion gives the following:

$$ \begin{align*} \big|\hat{f}_V(v)-\tilde{f}_V(v)\big| &= \big|\frac{1}{nh_V}\sum_{i=1}^n\big[\mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)(\hat{V}_i-V_i)\cdot\frac{1}{h_V}K^{\prime}_V\big( \frac{v-V_i}{h_V}\big)\big]\\& \quad +\frac{1}{2nh_V}\sum_{i=1}^n\big[\mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)(\hat{V}_i-V_i)^2\cdot\frac{1}{h_V^2}K^{\prime\prime}_V \big(\frac{v-\tilde{V}_i}{h_V}\big)\big]\big|, \end{align*} $$

where $\tilde {V}_i$ is some point between $\hat {V}_i$ and $V_i$ . By triangular inequality,

(A.18) $$ \begin{align} \big|\hat{f}_V(v)-\tilde{f}_V(v)\big|&\leq \frac{1}{nh_V^2}\sum_{i=1}^n \mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)\big|\hat{V}_i-V_i\big|\cdot\big|K^{\prime}_V\big( \frac{v-V_i}{h_V}\big)\big| \nonumber\\ &+ \frac{1}{2nh_V^3}\sum_{i=1}^n \mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)\big(\hat{V}_i-V_i\big)^2\cdot\big|K^{\prime\prime}_V\big( \frac{v-\tilde{V}_i}{h_V}\big)\big|. \end{align} $$

Because $\left |K^{\prime \prime }_V\left (\frac {v-\tilde {V}_i}{h_V}\right )\right |\leq \sup _u |K^{\prime \prime }_V(u)|$ , the right-hand side of (A.18) is bounded by

$$\begin{align*} &\frac{1}{h_V}\sup_i \mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)\big|\hat{V}_i-V_i\big|\cdot\frac{1}{nh_V}\sum_{i=1}^n \big| K^{\prime}_V\big( \frac{v-V_i}{h_V} \big) \nonumber \\ &\quad \big|+\frac{1}{2h_V^3}\sup_i \mathbb{1}(V_i\in\mathscr{C}^{\prime\prime}_V)\big|\hat{V}_i-V_i\big|^2\cdot \sup_u |K^{\prime\prime}_V(u)|.\end{align*}$$

By part (ii) of Lemma 5 and Assumption G,

(A.19) $$ \begin{align} &\big|\hat{f}_V(v)-\tilde{f}_V(v)\big| \nonumber\\ &\leq O_p\big((\log n / n)^{1/5}\big)\cdot\frac{1}{nh_V} \sum_{i=1}^n \big| K^{\prime}_V\big( \frac{v-V_i}{h_V} \big)\big|+O_p\big( (\log n / n)^{1/5}\big)\cdot\sup_u|K^{\prime\prime}_V(u)|. \end{align} $$

It can be shown that $\frac {1}{nh_V}\sum _{i=1}^n \big | K^{\prime }_V\big ( \frac {v-V_i}{h_V} \big )\big |$ converges uniformly to $f_V(v)\int _{-\infty }^\infty |K^{\prime }_V(u)|\,\mathrm {d} u$ thus being bounded uniformly. Moreover, $\sup _u|K^{\prime \prime }_V(u)|<\infty $ by Assumption F. Then it follows that $\sup _{v\in \mathscr {C}_V}|\hat {f}_V(v)-\tilde {f}_V(v)|=O_p\big ((\log n/n)^{1/5}\big )$ . The desired conclusion therefore follows.

A.12 Auxiliary Lemmas

We first define the local polynomial density estimator as given in Section 4.1. For a given random sample of $\{Z_1,Z_2,\dots ,Z_m\}$ from a distribution with a density of $g_Z(\cdot )$ on $[\underline z,\overline z]$ , the boundary-adaptive local polynomial density estimator (with a polynomial of order p) is defined as $\hat g_Z(z) = \hat {\theta }_2(z),$ where $\hat \theta (z) = \text {argmin}_{\theta \in \mathbb {R}^{p+1}} \sum _{i=1}^m \big [ \hat {G}_Z(Z_i) - \theta _1 - \theta _2\cdot (Z_i-z) - \dots - \theta _{p+1}\cdot (Z_i-z)^p \big ]^2\cdot K\big ((Z_i-z)/h\big )$ with $\hat G_Z(z) = (1/m)\sum _{i=1}^m \mathbf {1} (Z_i\leq z)$ , $K(\cdot )$ being a kernel function and h being a bandwidth. The following lemma characterizes the uniform convergence rate of $\hat g_Z(\cdot )$ on $[\underline z,\overline z]$ . It is a corollary of Theorem 1 in Cattaneo et al. (Reference Cattaneo, Chandak, Jansson and Ma2023) when there is no conditional covariate.

Proof. Our local polynomial density estimator (for unconditional density) can be viewed as a special case of the Cattaneo et al.’s (Reference Cattaneo, Chandak, Jansson and Ma2023) local polynomial one for conditional density, namely, $\mathfrak {p} = p$ , $\vartheta = 0$ , $d=0$ , and $\hat F(y_i|\mathbf {x})$ being replaced by the empirical CDF $\hat F(y_i) = (1/n)\sum _{j=1}^n \mathbf {1}(y_j \leq y_i)$ in their case.

We can then follow the technique used in the proofs of their Lemma 1 and Theorem 1 (i.e., discretization, union bound, and Bernstein’s inequality), and establish that

$$ \begin{align*} \sup_{z\in[\underline z,\overline z]} \big\vert \hat g_Z(z) - g_Z(z) \big\vert = O_p \Big( h^p + \sqrt{log(n)/ (nh)} \Big).\\[-41pt] \end{align*} $$

The second auxiliary lemma is about the convergence rates of $\hat g_S(\cdot )$ and $\hat g_B(\cdot )$ on the interval of $[\underline s,\overline b]$ .

Proof. We only show part (i). Part (ii) can be shown similarly, and is therefore omitted. Consider an infeasible estimator $\tilde g_S$ that uses the true $\overline b$ instead of its estimator $\hat {\overline b}$ to define the subsample as $\{S_i:S_i\leq \overline b\}$ with size of $\tilde {n}^-_S$ . Thus, $\tilde g_S(s) = \frac {\tilde {n}^-_S}{n}\cdot \tilde {g}_S^-(s),$ where $\tilde {g}_S^-(\cdot )$ is a local quadratic density estimator (i.e., $p=2$ ) from the subsample of $\{S_i: S_i\leq \overline b\}$ . By Lemma 6, the convergence rate of $\tilde g_S^-(\cdot )$ on $[\underline s,\overline b]$ is given by

$$ \begin{align*} \sup_{s\in[\underline s, \overline b]} \vert \tilde g_S^-(s) - g_S^-(s) \vert = O_p\Big(h_S^2 + \sqrt{log(n)/ (nh_S)}\Big), \end{align*} $$

where $g_S^-(\cdot )$ is the conditional density of sellers’ bids conditional on $S\leq \overline b$ , that is, $g_S^-(s) = g_S(s) / G_S(\overline b)$ for any $s\in [\underline s,\overline b]$ . This implies that

(A.20) $$ \begin{align} \sup_{s\in[\underline s, \overline b]} \vert \tilde g_S(s) - g_S(s) \vert = \sup_{s\in[\underline s, \overline b]} \Big\vert \frac{\tilde{n}^-_S}{n}\cdot \tilde g_S^-(s) - G_S(\overline b)\cdot g_S^-(s) \Big\vert = O_p\Big(h_S^2 + \sqrt{log(n)/ (nh_S)}\Big), \end{align} $$

where $ \big \vert \frac {\tilde {n}^-_S}{n} - G_S(\overline b)\big \vert = \big \vert \hat G_S(\overline b) - G_S(\overline b)\big \vert = O_p\big (log(n)/\sqrt {n}\big )$ with $\hat G_S(\cdot )$ being the empirical CDF of sellers’ bids.

We next study the difference between $\hat g_S$ and $\tilde g_S$ . It involves two differences: $\hat G_S(\hat {\overline b}) - \hat {G}_S(\overline b)$ and $\hat g_S^-(\cdot ) - \tilde g_S^-(\cdot )$ . The first difference can be characterized as follows:

(A.21) $$ \begin{align} \big\vert\hat G_S(\hat{\overline b}) - \hat{G}_S(\overline b)\big\vert & \leq \big\vert\hat G_S(\hat{\overline b}) - G_S(\hat{\overline b})\big\vert + \big\vert G_S(\hat{\overline b}) - G_S(\overline b)\big\vert + \big\vert G_S(\overline b) - \hat{G}_S(\overline b)\big\vert\nonumber\\ &\leq 2\cdot \sup_{s\in [\underline s,\overline b] } \big\vert \hat G_S(s) - G_S(s)\big\vert + \big\vert G_S(\hat{\overline b}) - G_S(\overline b)\big\vert \nonumber\\ & = O_p (log(n)/\sqrt{n}) + g_S(\tilde {\overline b})\cdot\big\vert \hat{\overline b} - \overline b\big\vert\nonumber \\ & = O_p (log(n)/\sqrt{n}), \end{align} $$

where $\tilde {\overline b} \in [\hat {\overline b}, \overline b]$ , the first equality holds due to $\sup _{s\in [\underline s,\overline b] } \big \vert \hat G_S(s) - G_S(s)\big \vert = O_p (log(n)/\sqrt {n})$ and Taylor expansion, and the second equality is derived by $\sup _{s\in [\underline s,\overline b]} g_S(s) \leq \overline {g}_S$ and $\vert \hat {\overline b} - \overline b\big \vert = O_p(1/n)$ . In addition, the second difference is given by

(A.22) $$ \begin{align} \sup_{s\in[\underline s,\overline b]}\big\vert\hat g_S^-(s) - \tilde g_S^-(s)\big\vert = \sup_{s\in[\hat{\overline b} - h_S,\overline b]}\big\vert\hat g_S^-(s) - \tilde g_S^-(s)\big\vert = o_p\Big(h_S^2 + \sqrt{log(n)/ (nh_S)}\Big), \end{align} $$

where the first equality holds because of $\hat g_S^-(s) = \tilde g_S^-(s)$ for all $s\in [\underline s,\hat {\overline b} - h_S]$ , and the second equality is obtained due to three facts: (i) there is a uniform convergence of the criterion functions used for obtaining the density estimators, which can be written as

$$ \begin{align*}\hat{Q}(s) = \frac{1}{n}\sum_{i=1}^n\big[ \hat{G}_S (S_i) - \theta_1 - \theta_2\cdot(S_i-s) - \theta_3(S_i-s)^2 \big]^2\cdot\mathbb{1} (S_i\leq \hat{\overline{b}}) \cdot K(\frac{S_i-s}{h_s})\end{align*} $$

for estimating $\hat {g}_S^-(s)$ , and

$$ \begin{align*}\tilde{Q}(s) = \frac{1}{n}\sum_{i=1}^n\big[ \tilde{G}_S (S_i) - \theta_1 - \theta_2\cdot(S_i-s) - \theta_3(S_i-s)^2 \big]^2\cdot\mathbb{1} (S_i\leq \overline{b}) \cdot K(\frac{S_i-s}{h_s})\end{align*} $$

for estimating $\tilde {g}_S^-(s)$ with $\hat { G}_S(s) = (1/n)\cdot \sum _{i=1}^n \mathbb {1} (S_i\leq s)\cdot \mathbb {1}(S_i\leq \hat {\overline {b} }) $ and $\tilde {G}_S(s)= (1/n)\cdot \sum _{i=1}^n \mathbb {1} (S_i\leq s)\cdot \mathbb {1}(S_i\leq \overline {b} )$ ; (ii) $\hat {g}_S^-(s)$ for any $s\in [\hat {\overline b} - h_S,\overline b]$ uses all data in the interval of $[s-h_S,\hat {\overline b}]$ which consists of a rough fraction no less than $G_S(\hat {\overline b}) - G_S(\overline b - h_S) = O_p(h_S)$ in the whole sample; and (iii) $\tilde {g}_S^-(s)$ for any $s\in [\hat {\overline b} - h_S,\overline b]$ uses all data in the interval of $[s-h_S,\overline b]$ , namely, uses some additional observations (relative to $\hat {g}_S^-(s)$ ) with a rough fraction no more than $G_S(\overline b) - G_S(\hat {\overline b}) = O_p(1/n)$ in the whole sample.

Therefore, the difference between $\hat g_S$ and $\tilde g_S$ can be bounded as

(A.23) $$ \begin{align} \sup_{s\in[\underline s,\overline b]} \big\vert \hat g_S(s) - \tilde g_S(s)\big\vert = \sup_{s\in[\underline s,\overline b]} \big\vert \hat G_S(\hat{\overline b})\cdot\hat g_S^-(s) - \hat G_S(\overline b)\cdot \tilde{g}_S^-(s)\big\vert = o_p\Big(h_S^2 + \sqrt{log(n)/ (nh_S)}\Big), \end{align} $$

where the second equality comes from (A.21) and (A.22).

In summary, we have

$$ \begin{align*} \sup_{s\in[\underline s,\overline b]} \big\vert \hat g_S(s) - g_S(s)\big\vert &\leq \sup_{s\in[\underline s,\overline b]} \big\vert \hat g_S(s) - \tilde{g}_S(s)\big\vert + \sup_{s\in[\underline s,\overline b]} \big\vert \tilde g_S(s) - g_S(s)\big\vert \\ &= O_p\Big(h_S^2 + \sqrt{log(n)/ (nh_S)}\Big), \end{align*} $$

where the last equality comes from (A.20) and (A.23). The desired conclusion therefore follows.