On the existence of efficient multilateral trading mechanisms with interdependent values

Abstract

This paper studies multilateral trading problems in which agents’ valuations for items are interdependent. Assuming that each agent’s information has a greater marginal effect on her own valuation than on the other agents’ valuations, the paper identifies a necessary and sufficient condition for the existence of trading mechanisms satisfying efficiency, ex-post incentive compatibility, ex-post individual rationality, and ex-post budget balance. The paper presents a trading mechanism that satisfies the four properties when the necessary and sufficient condition holds and shows that this mechanism maximizes the ex-post budget surplus among all efficient, ex-post incentive compatible, and ex-post individually rational trading mechanisms. The paper examines an environment where each agent can possess at most one unit of an item, and her information about the item is one-dimensional. It then extends the results to two general environments: the multiple units environment and the multidimensional information environment.

Appendices

Appendix 1: Stochastic allocation

The definition of EF of the mechanism $(\pi , \tau )$ with stochastic allocation is as follows: For each $t\in T$, $b_i\in B$, and $s_j\in S$, if $v_{(m)}(t)<v_{(m+1)}(t)$,

$$\begin{aligned} \pi _{b_i}(t)= {\left\{ \begin{array}{ll} 1 &{}\quad \text {if } v_{b_i}(t)\ge v_{(m+1)}(t),\\ 0 &{}\quad \text {if } v_{b_i}(t) < v_{(m+1)}(t), \end{array}\right. } \quad \pi _{s_j}(t)= {\left\{ \begin{array}{ll} 1 &{}\quad \text {if } v_{s_j}(t)\le v_{(m)}(t),\\ 0 &{}\quad \text {if } v_{s_j}(t) > v_{(m)}(t). \end{array}\right. } \end{aligned}$$

If, however, $v_{(m)}(t)=v_{(m+1)}(t)$,

$$\begin{aligned} \pi _{b_i}(t)\in {\left\{ \begin{array}{ll} \{1\} &{}\quad \text {if } v_{b_i}(t)> v_{(m+1)}(t),\\ {[}0,1] &{}\quad \text {if } v_{b_i}(t)= v_{(m+1)}(t),\\ \{0\} &{}\quad \text {if } v_{b_i}(t)< v_{(m+1)}(t), \end{array}\right. } \quad \pi _{s_j}(t)\in {\left\{ \begin{array}{ll} \{1\} &{}\quad \text {if } v_{s_j}(t)< v_{(m)}(t),\\ {[}0,1] &{}\quad \text {if } v_{s_j}(t)= v_{(m)}(t),\\ \{0\} &{}\quad \text {if } v_{s_j}(t) > v_{(m)}(t), \end{array}\right. } \end{aligned}$$

satisfying $\sum \nolimits _{b_i\in B}\pi _{b_i}(t)=\sum \nolimits _{s_j\in S}\pi _{s_j}(t)$.

The following theorem provides the necessary and sufficient condition for the existence of mechanisms satisfying the four properties if stochastic allocation is allowed. Note that the condition in the theorem is equivalent to (2) if we restrict attention to deterministic allocation rules.

Theorem 1′

(Stochastic allocation) Suppose that (A1)–(A4) hold. There exists a multilateral trading mechanism satisfying EF, EPIC, EPIR, and EPBB if and only if for some EF allocation rule $\pi$,

$$\begin{aligned} \sum _{b_i\in \{b_i\in B\,|\,\pi _{b_i}(t)>0\}}v_{b_i}\big ({\hat{t}}_{b_i}(t_{-b_i}), t_{-b_i}\big )\pi _{b_i}(t)\ge \sum _{s_j\in \{s_j\in S\,|\,\pi _{s_j}(t)>0\}}v_{s_j}\big ({\hat{t}}_{s_j}(t_{-s_j}), t_{-s_j}\big )\pi _{s_j}(t) \end{aligned}$$

for each $t\in T$.

Proof

$\Rightarrow$: Suppose that $(\pi , \tau )$ satisfies EF, EPIC, EPIR, and EPBB. Consider $t\in T$ and the non-trivial case that $\{b_i\in B\,|\,\pi _{b_i}(t)>0\}\ne \emptyset$ and $\{s_j\in S\,|\,\pi _{s_j}(t)>0\}\ne \emptyset$. Each $b_i\in B$ falls under one of the following cases: (i) $\pi _{b_i}(t)=1$ and $t_{b_i}>{\hat{t}}_{b_i}(t_{-b_i})$, (ii) $\pi _{b_i}(t)\in (0,1]$ and $t_{b_i}={\hat{t}}_{b_i}(t_{-b_i})$, or (iii) $\pi _{b_i}(t)=0$. If we repeat the argument in the proof of Theorem 1 and examine $\tau _{b_i}(t)$ in each case, we have

$$\begin{aligned} \sum _{b_i\in \{b_i\in B\,|\,\pi _{b_i}(t)>0\}}v_{b_i}\big ({\hat{t}}_{b_i}(t_{-b_i}), t_{-b_i}\big )\pi _{b_i}(t)\ge \sum _{b_i\in B}\tau _{b_i}(t). \end{aligned}$$

I omit the remaining part of the proof because it is the same as the proof of Theorem 1.

$\Leftarrow :$ Define the transfer rule $\tau$ as follows: for $b_i\in B$, $s_j\in S$, and $t\in T$,

$$\begin{aligned} \begin{aligned} \tau _{b_i}(t)&= {\left\{ \begin{array}{ll} v_{b_i}\big ({\hat{t}}_{b_i}(t_{-b_i}), t_{-b_i}\big )\pi _{b_i}(t) &{}\quad \text {if } \pi _{b_i}(t)>0,\\ 0 &{}\quad \text {if} \pi _{b_i}(t)=0,\\ \end{array}\right. } \\ \tau _{s_j}(t)&= {\left\{ \begin{array}{ll} v_{s_j}\big ({\hat{t}}_{s_j}(t_{-s_j}), t_{-s_j}\big )\pi _{s_j}(t) &{}\quad \text {if } \pi _{s_j}(t)>0,\\ 0 &{}\quad \text {if } \pi _{s_j}(t)=0.\\ \end{array}\right. } \end{aligned} \end{aligned}$$

$(\pi , \tau )$ satisfies EF, EPIC, EPIR, and EPBB by the same argument as in the proof of Theorem 1. $\square$

Appendix 2: Assumptions under private values

Proposition 1

Suppose that $v_k(t)=t_k$ for all $t\in T$ and $k\in B\cup S$. Then, (A4) holds if and only if

$$\begin{aligned} \bigcap _{k\in B\cup S}({\underline{t}}_k,{\overline{t}}_k)\ne \emptyset . \end{aligned}$$

Proof

$\Rightarrow$: Suppose to the contrary that $\bigcap _{k\in B\cup S}\,({\underline{t}}_k,{\overline{t}}_k)=\emptyset .$ Let $k_1=\textrm{argmin}_{k\in B\cup S}\,\,{\overline{t}}_k$ and $k_2=\textrm{argmax}_{k\in B\cup S}\,\,{\underline{t}}_k$. Then, ${\overline{t}}_{k_1}\le {\underline{t}}_{k_2}$. Thus, $t_{k_1}<t_{k_2}$ for any $t_{k_1}\in ({\underline{t}}_{k_1},{\overline{t}}_{k_1})$ and $t_{k_2}\in ({\underline{t}}_{k_2},{\overline{t}}_{k_2})$. Therefore, (A4) does not hold.

$\Leftarrow$: There exist $x\in \bigcap _{k\in B\cup S}\,({\underline{t}}_k,{\overline{t}}_k)$ and $\epsilon >0$ such that $(x-\epsilon , x+\epsilon ) \subset ({\underline{t}}_k,{\overline{t}}_k)$ for all $k\in B\cup S$. Let $\sigma$ be a permutation of $B\cup S$. Consider $(\epsilon _k)_{k\in B\cup S}$ satisfying $0<\epsilon _{b_1}<\epsilon _{b_2}<\cdots<\epsilon _{b_m}<\epsilon _{s_1}<\cdots<\epsilon _{s_n}<\epsilon$. Define $t_{\sigma (k)}=x+\epsilon _k$ for $k\in B\cup S$. Then, $t_{\sigma (k)}\in ({\underline{t}}_{\sigma (k)},{\overline{t}}_{\sigma (k)})$ for all $k\in B\cup S$ and $t_{\sigma (b_1)}<\cdots<t_{\sigma (b_m)}<t_{\sigma (s_1)}<\cdots <t_{\sigma (s_n)}$. $\square$

Appendix 3: Multiple units

1.1 Definitions of EPIC, EPIR, EPBB, and EF

A mechanism $(\pi , \tau )$ is EPIC if

$$\begin{aligned}{} & {} \displaystyle \sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^q(t)-\tau _{b_i}(t) \ge \displaystyle \sum _{q=1}^{\pi _{b_i}(t'_{b_i},t_{-b_i})}v_{b_i}^q(t)-\tau _{b_i}(t'_{b_i},t_{-b_i})\quad \hbox {for each}\,t'_{b_i}, t,\,\hbox {and} \,b_i\in B,{\hbox { and}}\\{} & {} \tau _{s_j}(t)-\displaystyle \sum _{q=1}^{\pi _{s_j}(t)}v_{s_j}^q(t) \ge \tau _{s_j}(t'_{s_j},t_{-s_j})-\displaystyle \sum _{q=1}^{\pi _{s_j}(t'_{s_j},t_{-s_j})}v_{s_j}^q(t)\quad \hbox {for each} \,t'_{s_j}, t,\,\hbox {and} \,s_j\in S. \end{aligned}$$

A mechanism $(\pi , \tau )$ is EPIR if

$$\begin{aligned}{} & {} \displaystyle \sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^q(t)-\tau _{b_i}(t) \ge 0\quad \hbox {for each} \,t{\hbox { and }}b_i\in B,\,\hbox {and}\\{} & {} \tau _{s_j}(t)-\displaystyle \sum _{q=1}^{\pi _{s_j}(t)}v_{s_j}^q(t) \ge 0\quad \hbox {for each} \,t\,{\hbox {and}} \,s_j\in S. \end{aligned}$$

A mechanism $(\pi , \tau )$ satisfies EPBB if

$$\begin{aligned} \displaystyle \sum _{b_i\in B}\tau _{b_i}(t) \ge \sum _{s_j\in S}\tau _{s_j}(t)\quad \hbox {for each} \,t. \end{aligned}$$

A mechanism $(\pi , \tau )$ is EF if it allocates the nQ items to the nQ highest marginal values or costs: For each $t\in T$, we order the agents’ marginal values and costs.

$$\begin{aligned} v_{(1)}(t)\le v_{(2)}(t)\le \ldots \le v_{((m+n)Q)}(t). \end{aligned}$$

If $v_{(mQ)}(t) < v_{(mQ+1)}(t)$, for $b_i\in B$ and $s_j\in S$,

$$\begin{aligned} \pi _{b_i}(t)=\big |\{q\,|\,v_{b_i}^q(t)\ge v_{(mQ+1)}(t)\} \big |\quad \hbox {and} \quad \pi _{s_j}(t)=\big |\{q\,|\,v_{s_j}^q(t)\le v_{(mQ)}(t)\}\big |. \end{aligned}$$

If $v_{(mQ)}(t) = v_{(mQ+1)}(t)$, for $b_i\in B$ and $s_j\in S$,

$$\begin{aligned}{} & {} \pi _{b_i}(t)\in \Big [\,\big |\{q\,|\,v_{b_i}^q(t)> v_{(mQ+1)}(t)\}\big |,\,\big |\{q\,|\,v_{b_i}^q(t)\ge v_{(mQ+1)}(t)\}\big |\,\Big ],\,\hbox {and}\\{} & {} \pi _{s_j}(t)\in \Big [\,\big |\{q\,|\,v_{s_j}^q(t) < v_{(mQ)}(t)\}\big |,\,\big |\{q\,|\,v_{s_j}^q(t)\le v_{(mQ)}(t)\}\big |\,\Big ] \end{aligned}$$

satisfying $\sum \nolimits _{b_i\in B}\pi _{b_i}(t) = \sum \nolimits _{s_j \in S}\pi _{s_j}(t).$

1.2 Proof of Theorem 5

The following lemma derives the necessary condition for EPIC when agents can trade multiple units. I omit the proof as it is similar to the proof of Lemma 1.

Lemma 2

Suppose that a mechanism $(\pi , \tau )$ satisfies EPIC. Then:

• (i) for any $b_i$ and $t_{-b_i}$, $\tau _{b_i}(t_{b_i},t_{-b_i}) = \tau _{b_i}(t_{b_i}',t_{-b_i})\,\,\,$ if $\,\pi _{b_i}(t_{b_i},t_{-b_i})=\pi _{b_i}(t_{b_i}',t_{-b_i})$, and

• (ii) for any $s_j$ and $t_{-s_j}$, $\tau _{s_j}(t_{s_j},t_{-s_j}) = \tau _{s_j}(t_{s_j}',t_{-s_j})\,\,\,$ if $\,\pi _{s_j}(t_{s_j},t_{-s_j})=\pi _{s_j}(t_{s_j}',t_{-s_j})$.

Proof of Theorem 5

$\Rightarrow$: Suppose that a mechanism $(\pi , \tau )$ satisfies EF, EPIC, EPIR, and EPBB. Consider $t\in T$. Each buyer $b_i$ falls under one of the following two cases: (i) $\pi _{b_i}(t)>0$ or (ii) $\pi _{b_i}(t)=0$. We examine $\tau _{b_i}(t)$ in each case.

(i) $\pi _{b_i}(t)>0$: For $q=1,\ldots ,\pi _{b_i}(t)$, the definition of ${\hat{t}}_{b_i}^q(t_{-b_i})$ and the decreasing marginal values, (A4$^{\prime }$), imply $v_{b_i}^q({\hat{t}}_{b_i}^q(t_{-b_i}), t_{-b_i}) = v_{(mQ+1)}({\hat{t}}_{b_i}^q(t_{-b_i}),t_{-b_i})$ and $v_{b_i}^{q+1}({\hat{t}}_{b_i}^q(t_{-b_i}), t_{-b_i}) < v_{(mQ+1)}({\hat{t}}_{b_i}^q(t_{-b_i}),t_{-b_i})$.⁸ For sufficiently small $\epsilon >0$, the single-crossing condition, (A3$^{\prime }$), implies $v_{b_i}^q({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon , t_{-b_i}) > v_{(mQ+1)}({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon ,t_{-b_i})$ or $v_{b_i}^q({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon , t_{-b_i}) = v_{(mQ+1)}({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon ,t_{-b_i}) > v_{(mQ)}({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon ,t_{-b_i})$, and the continuity of $v_k^q$, (A1$^{\prime }$), implies $v_{b_i}^{q+1}({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon , t_{-b_i}) < v_{(mQ+1)}({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon ,t_{-b_i})$. Thus, we have

$$\begin{aligned} \pi _{b_i}({\hat{t}}_{b_i}^q(t_{-b_i})+\epsilon ,t_{-b_i})=q \end{aligned}$$

(18)

because $\pi$ is an efficient allocation rule. Consider sufficiently small $\epsilon >0$ such that (18) holds for all $q=1,\ldots ,\pi _{b_i}(t)$. EPIR at $({\hat{t}}_{b_i}^1(t_{-b_i})+\epsilon ,t_{-b_i})$ implies

$$\begin{aligned} \sum _{q'=1}^{\pi _{b_i}({\hat{t}}_{b_i}^1(t_{-b_i})+\epsilon ,t_{-b_i})}v_{b_i}^{q'}({\hat{t}}_{b_i}^1(t_{-b_i})+\epsilon ,t_{-b_i})-\tau _{b_i}({\hat{t}}_{b_i}^1(t_{-b_i})+\epsilon ,t_{-b_i})\ge 0. \end{aligned}$$

(19)

Plugging (18) into (19) leads to

$$\begin{aligned} v_{b_i}^{1}({\hat{t}}_{b_i}^1(t_{-b_i})+\epsilon ,t_{-b_i})\ge \tau _{b_i}({\hat{t}}_{b_i}^1(t_{-b_i})+\epsilon ,t_{-b_i}). \end{aligned}$$

(20)

For $q=2,\ldots ,\pi _{b_i}(t)$, EPIC at $({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})$ with misreport ${\hat{t}}_{b_i}^{q-1}(t_{-b_i})+\epsilon$ implies

$$\begin{aligned} \begin{aligned} \sum _{q'=1}^{\pi _{b_i}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})}&v_{b_i}^{q'}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})-\tau _{b_i}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})\\&\ge \sum _{q'=1}^{\pi _{b_i}({\hat{t}}_{b_i}^{q-1}(t_{-b_i})+\epsilon ,t_{-b_i})}v_{b_i}^{q'}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})-\tau _{b_i}({\hat{t}}_{b_i}^{q-1}(t_{-b_i})+\epsilon ,t_{-b_i}). \end{aligned} \end{aligned}$$

(21)

Plugging (18) into (21) leads to

$$\begin{aligned} v_{b_i}^{q}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})\ge \tau _{b_i}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})-\tau _{b_i}({\hat{t}}_{b_i}^{q-1}(t_{-b_i})+\epsilon ,t_{-b_i}). \end{aligned}$$

(22)

If we add (22) for $q=2,\ldots ,\pi _{b_i}(t)$, and (20), we have

$$\begin{aligned} \sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^{q}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})\ge \tau _{b_i}({\hat{t}}_{b_i}^{\pi _{b_i}(t)}(t_{-b_i})+\epsilon ,t_{-b_i}). \end{aligned}$$

(23)

Because $\pi _{b_i}({\hat{t}}_{b_i}^{\pi _{b_i}(t)}(t_{-b_i})+\epsilon ,t_{-b_i})=\pi _{b_i}(t)$ from (18), Lemma 2 implies

$$\begin{aligned} \tau _{b_i}({\hat{t}}_{b_i}^{\pi _{b_i}(t)}(t_{-b_i})+\epsilon ,t_{-b_i})=\tau _{b_i}(t). \end{aligned}$$

(24)

Thus, from (23) and (24), we have

$$\begin{aligned} \sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^{q}({\hat{t}}_{b_i}^{q}(t_{-b_i})+\epsilon ,t_{-b_i})\ge \tau _{b_i}(t). \end{aligned}$$

This inequality holds for any sufficiently small $\epsilon >0$. Therefore, by the continuity of $v_{b_i}^q$,

$$\begin{aligned} \sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^{q}({\hat{t}}_{b_i}^{q}(t_{-b_i}),t_{-b_i})\ge \tau _{b_i}(t). \end{aligned}$$

(25)

(ii) $\pi _{b_i}(t)=0$: EPIR implies

$$\begin{aligned} 0\ge \tau _{b_i}(t). \end{aligned}$$

(26)

By adding up inequalities (25) and (26) for all $b_i\in B$, we have

$$\begin{aligned} \sum _{b_i\in \{b_i\in B|\pi _{b_i}(t)>0\}}\sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^{q}({\hat{t}}_{b_i}^{q}(t_{-b_i}),t_{-b_i})\ge \sum _{b_i\in B}\tau _{b_i}(t). \end{aligned}$$

(27)

If we repeat the same argument for the sellers, we have

$$\begin{aligned} \sum _{s_j\in S}\tau _{s_j}(t)\ge \sum _{s_j\in \{s_j\in S|\pi _{s_j}(t)>0\}}\sum _{q=1}^{\pi _{s_j}(t)}v_{s_j}^{q}({\hat{t}}_{s_j}^{q}(t_{-s_j}),t_{-s_j}). \end{aligned}$$

(28)

Thus, (27), (28), and EPBB imply

$$\begin{aligned} \sum _{b_i\in \{b_i\in B|\pi _{b_i}(t)>0\}}\sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^{q}({\hat{t}}_{b_i}^{q}(t_{-b_i}),t_{-b_i})\ge \sum _{s_j\in \{s_j\in S|\pi _{s_j}(t)>0\}}\sum _{q=1}^{\pi _{s_j}(t)}v_{s_j}^{q}({\hat{t}}_{s_j}^{q}(t_{-s_j}),t_{-s_j}). \end{aligned}$$

$\Leftarrow :$ Suppose that (14) holds for each $t\in T$ for some EF allocation rule $\pi$. Define the transfer rule $\tau$ as follows: for $b_i\in B$, $s_j\in S$, and $t\in T$,

$$\begin{aligned} \begin{aligned} \tau _{b_i}(t)&= {\left\{ \begin{array}{ll} \displaystyle \sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^q\big ({\hat{t}}_{b_i}^q(t_{-b_i}), t_{-b_i}\big ) &{}\quad \text {if } \pi _{b_i}(t)>0,\\ 0 &{}\quad \text {if} \pi _{b_i}(t)=0,\\ \end{array}\right. } \\\tau _{s_j}(t)&= {\left\{ \begin{array}{ll} \displaystyle \sum _{q=1}^{\pi _{s_j}(t)}v_{s_j}^q\big ({\hat{t}}_{s_j}^q(t_{-s_j}), t_{-s_j}\big ) &{}\quad \text {if } \pi _{s_j}(t)>0,\\ 0 &{}\quad \text {if } \pi _{s_j}(t)=0.\\ \end{array}\right. } \end{aligned} \end{aligned}$$

If t is a reported type profile, according to the mechanism, buyer $b_i$ purchases $\pi _{b_i}(t)$ units of the item and pays $v_{b_i}^q({\hat{t}}_{b_i}^q(t_{-b_i}), t_{-b_i})$ for the q-th unit. Similarly, seller $s_j$ sells $\pi _{s_j}(t)$ units and receives $v_{s_j}^q({\hat{t}}_{s_j}^q(t_{-s_j}), t_{-s_j})$ for the q-th unit. Note that the price of the q-th unit for agent $k\in B\cup S$, $v_k^q({\hat{t}}_k^q(t_{-k}),t_{-k})$, does not depend on her report $t_k$.

The mechanism $(\pi , \tau )$ defined above is the multi-price mechanism for the multiple units environment. It satisfies EPIC and EPIR. Consider buyer $b_i\in B$. The definition of ${\hat{t}}_{b_i}^q(t_{-b_i})$, EF of $\pi$, and (A1$^{\prime }$)–(A4$^{\prime }$) imply

$$\begin{aligned} \begin{aligned}&v_{b_i}^q(t) \ge v_{b_i}^q\big ({\hat{t}}_{b_i}^q(t_{-b_i}),t_{-b_i}\big )\quad \text { for }q=1,\ldots ,\pi _{b_i}(t),\text { and}\\&v_{b_i}^q(t) \le v_{b_i}^q\big ({\hat{t}}_{b_i}^q(t_{-b_i}),t_{-b_i}\big )\quad \text { for }q=\pi _{b_i}(t)+1,\ldots ,\pi _{b_i}({\overline{t}}_{b_i},t_{-b_i}). \end{aligned} \end{aligned}$$

(29)

Suppose that t is a profile of agent types, $b_i$ reports $t_{b_i}'$, and the other agents report $t_{-b_i}$. Then, $b_i$’s ex-post payoff is

$$\begin{aligned} \sum _{q=1}^{\pi _{b_i}(t'_{b_i},t_{-b_i})}v_{b_i}^q(t)-\tau _{b_i}(t'_{b_i},t_{-b_i}) = \sum _{q=1}^{\pi _{b_i}(t'_{b_i},t_{-b_i})}\big (v_{b_i}^q(t)-v_{b_i}^q({\hat{t}}_{b_i}^q(t_{-b_i}),t_{-b_i})\big ), \end{aligned}$$

(30)

which is the sum of the payoffs from trading $\pi _{b_i}(t'_{b_i},t_{-b_i})$ units. We note the following from (29): Up to $\pi _{b_i}(t)$ units, $b_i$ obtains a non-negative ex-post payoff from trading each unit. However, starting from the $(\pi _{b_i}(t)+1)$-th unit, $b_i$ obtains a non-positive payoff from trading each unit. Thus, $b_i$ maximizes her ex-post payoff, (30), by trading $\pi _{b_i}(t)$ units, which occurs when she reports her type truthfully—that is, $t'_{b_i}=t_{b_i}$. I omit the proof of EPIC and EPIR for the sellers, as it is similar to the one for the buyers. $(\pi , \tau )$ also satisfies EPBB because

$$\begin{aligned} \sum _{b_i\in B}\tau _{b_i}(t)&=\sum _{b_i\in \{b_i\in B|\pi _{b_i}(t)> 0\}}\sum _{q=1}^{\pi _{b_i}(t)}v_{b_i}^q\big ({\hat{t}}_{b_i}^q(t_{-b_i}),t_{-b_i}\big )\\&\ge \sum _{s_j\in \{s_j\in S|\pi _{s_j}(t)> 0\}}\sum _{q=1}^{\pi _{s_j}(t)}v_{s_j}^q\big ({\hat{t}}_{s_j}^q(t_{-s_j}),t_{-s_j}\big )=\sum _{s_j\in S}\tau _{s_j}(t). \end{aligned}$$

$\square$