Mechanism design for land acquisition

Abstract

We consider the issue of designing Bayesian incentive-compatible, efficient, individually rational and balanced mechanisms for Land Acquisition. This is a problem of great practical importance in developing countries. Several sellers, each with one unit of land, are located at the nodes of a graph. Two sellers are contiguous if they are connected by an edge in the graph. The buyer realizes a positive value only if he can purchase plots that constitute a path of given length. Our main result is that there is a robust set of priors for which successful mechanisms exist when there are at least two distinct feasible sets of contiguous sellers. The analysis also identifies the role of critical sellers who lie on all such feasible sets.

Appendices

Appendix

A The VCG mechanism

Here we present the well-known VCG mechanism and mention a related result derived by Williams (1999) and Krishna and Perry (2000) that will be useful for our analysis. We will be closely following the notation of the textbook by Krishna (2002) who also presents these results.

The VCG mechanism is a generalization of the second-price auction scheme conceived by Vickrey (1961). The VCG payment for each agent is interpreted as the externality she imposes on other agents.

Let $SW(v)=\sum _{j=0}^{n}v_jP_{j}^{*}(v)$ and $SW_{-j}(v)=\sum _{i\ne j}v_{i}P_{i}^{*}(v)$.

Definition 8

(VCG mechanism) The VCG mechanism is the pair $(P^{*}, t^{V})$ where $P^{*}$ is an efficient allocation rule and $t_{j}^{V}$, $j=0,\ldots , n$, is defined for all profiles v as follows:

$$\begin{aligned} t_{0}^{V}(v)= & {} SW(\underline{v}_{0}, v_{-0})-SW_{-0}(v) \end{aligned}$$

(14)

$$\begin{aligned} t_{i}^{V}(v)= & {} SW(\bar{v}, v_{-i})-SW_{-i}(v) \quad \hbox {for } i=1,\ldots , n. \end{aligned}$$

(15)

It is well-known that the VCG mechanism is efficient, BIC¹¹, interim individually rational but not budget-balanced.

Example 6

In Example 1 let us suppose that the valuation for both agents lie in [0, 1], so that, $\underline{v}_{0}=0$ and $\bar{v}=1$. The efficient rule is to transfer the good to the buyer if $v_{0}>v_{1}$, and not to transfer if $v_{0}\le v_{1}$. Then $SW(v)=v_{0}-v_{1}$ if $v_{0}>v_{1}$ and 0 otherwise. Using this we get, $t_{0}^{V}(v)=SW(0,v_{1})-SW_{1}(v_{0},v_{1})$ which is $v_{1}$ if $v_{0}>v_{1}$ and 0 otherwise. Similarly, $t_{1}^{V}(v)=SW(v_{0},1)-SW_{0}(v_{0},v_{1})$ which is $-v_{0}$ if $v_{0}>v_{1}$ and 0 otherwise. Thus the sum of payments is $v_{1}-v_{0}$ if $v_{0}>v_{1}$ and 0 otherwise. For all profiles where $v_{0}>v_{1}$, budget balance is violated.

The following Proposition is central to the proof of our main result. See Krishna (2002) for a proof of this result that explicitly constructs a successful mechanism when the given condition is true.

Proposition 1

(WKP) There exists a successful mechanism if and only if the VCG mechanism generates a non-negative expected surplus, i.e.,

$$\begin{aligned} E \left( \sum _{j=0}^{n} t_{j}^{V} (v)\right) \ge 0. \end{aligned}$$

The Proposition can be used to prove the Myerson–Satterthwaite impossibility result. To see this, observe that in Example 6, the sum of VCG payments is negative if $v_{0}>v_{1}$ and 0 otherwise. Therefore, the expectation of this sum is negative. An application of the Proposition yields the result immediately.

B Proof of Theorem 1

Part A. We rely on Proposition 1 and compute the expected value of the sum of VCG payments.

In Part I, we will show that for all v,

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}&\ge \underline{v}_{0}-\sum _{i\in \mathcal {P}_{[1]}(v)}S_{[1]}(\bar{v},v_{-i})+(k-1)S_{[1]}(v). \end{aligned}$$

If (2) holds, then the expected sum of payments is non-negative and the result follows by an application of Proposition 1.

The VCG payment of the buyer is given by Lemma 1.

Lemma 1

The VCG payment of the buyer is given by

$$\begin{aligned} t_{0}^{V}(v)&= \left\{ \begin{array}{ll} \underline{v}_{0} &{}\quad \hbox {if } v_{0}\ge \underline{v}_{0}>S_{[1]}(v),\\ S_{[1]}(v)&{}\quad \hbox {if } v_{0}> S_{[1]}(v)\ge \underline{v}_{0},\\ 0 &{}\quad \hbox {if } \underline{v}_{0}\le v_{0}\le S_{[1]}(v). \end{array} \right. \end{aligned}$$

Proof

If $v_{0}\ge \underline{v}_{0}>S_{[1]}(v)$,

$$\begin{aligned} SW(\underline{v}_{0}, v_{-0})&=\underline{v}_{0}-S_{[1]}(v),\\ \hbox {and } SW_{-0}({v}_{0}, v_{-0})&=-S_{[1]}(v).\\ \hbox {Hence } t_{0}^{V}(v)&=\underline{v}_{0}. \end{aligned}$$

If $v_{0}> S_{[1]}(v)\ge \underline{v}_{0}$,

$$\begin{aligned} SW(\underline{v}_{0}, v_{-0})&=0, \\ \hbox {and }SW_{-0}({v}_{0}, v_{-0})&=-S_{[1]}(v). \\ \hbox {Hence } t_{0}^{V}(v)&=S_{[1]}(v). \end{aligned}$$

If $v_{0}\le S_{[1]}(v)$,

$$\begin{aligned} SW(\underline{v}_{0}, v_{-0})&=0,\\ \hbox {and }SW_{-0}({v}_{0}, v_{-0})&=0.\\ \hbox {Hence }t_{0}^{V}(v)&=0. \end{aligned}$$

$\square$

For the VCG payment of the sellers, we refer to the next two Lemmas.

Lemma 2

The VCG payment of any seller $i\in \mathcal {P}_{[j]}(v)\backslash \mathcal {P}_{[1]}(v), j>1$, is 0.

Proof

Let $v_{0}> S_{[1]}(v)$. For any seller $i\in \mathcal {P}_{[j]}(v)\backslash \mathcal {P}_{[1]}(v), j>1$,

$$\begin{aligned} SW(\bar{v},v_{-i})&=v_{0}-S_{[1]}(v),\\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=v_{0}-S_{[1]}(v).\\ \hbox {Hence } t_{i}^{V}(v)&=0. \end{aligned}$$

If $v_{0}\le S_{[1]}(v)$,

$$\begin{aligned} SW(\bar{v},v_{-i})&=0, \\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=0. \\ \hbox {Hence } t_{i}^{V}(v)&=0. \end{aligned}$$

$\square$

Lemma 3

The VCG payment of any seller $i\in \mathcal {P}_{[1]}(v)$, is given by

$$\begin{aligned} t_{i}^{V}(v)&= \left\{ \begin{array}{ll} -(v_{0}-S_{[1]}(v)+v_{i} ) &{}\quad \hbox {if } S_{[1]}(v)< v_{0}\le S_{[1]}(\bar{v},v_{-i}),\\ -(S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}) &{}\quad \hbox {if } S_{[1]}(v) \le S_{[1]}(\bar{v},v_{-i})< v_{0},\\ 0 &{}\quad \hbox {if } v_{0}\le S_{[1]}(v) \le S_{[1]}(\bar{v},v_{-i}).\end{array} \right. \end{aligned}$$

Proof

If $S_{[1]}(v)< v_{0}\le S_{[1]}(\bar{v},v_{-i})$,

$$\begin{aligned} SW(\bar{v},v_{-i})&=0,\\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=v_{0}-S_{[1]}(v)+v_{i} .\\ \hbox {Hence } t_{i}^{V}(v)&=-(v_{0}-S_{[1]}(v)+v_{i}). \end{aligned}$$

If $S_{[1]}(v) \le S_{[1]}(\bar{v},v_{-i})< v_{0}$,

$$\begin{aligned} SW(\bar{v},v_{-i})&=v_{0}-S_{[1]}(\bar{v},v_{-i}),\\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=v_{0}-S_{[1]}(v)+v_{i}. \\ \hbox {Hence } t_{i}^{V}(v)&=-(S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}). \end{aligned}$$

If $v_{0}\le S_{[1]}(v) \le S_{[1]}(\bar{v},v_{-i})$,

$$\begin{aligned} SW(\bar{v},v_{-i})&=0,\\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=0. \\ \hbox {Hence } t_{i}^{V}(v)&=0. \end{aligned}$$

$\square$

When $q>1$, let $A(v)=\{h\in \mathcal {P}_{[1]}: S_{[1]}(v)< v_{0}\le S_{[1]}(\bar{v}, v_{-i})\}$. As before, A(v) represents the set of trade-pivotal sellers at profile v: they influence the possibility of trade by reporting their highest valuation.

Then we have the following cases for $\sum _{j=0}^{n}t_{j}^{V}(v)$.

Case I $v_{0}\ge \underline{v}_{0}>S_{[1]}(v)$ and $A(v)\ne \emptyset$

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&=\underline{v}_{0}-\sum _{i\in A(v)} \left( v_{0}-S_{[1]}(v)+v_{i}\right) -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}\right) \\&=\underline{v}_{0}-|A(v)|\left( v_{0}-S_{[1]}(v)\right) -\sum _{i\in A(v)}v_{i} \\&\quad -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} v_{i}\\&=\underline{v}_{0}-S_{[1]}(v) -|A(v)|\left( v_{0}-S_{[1]}(v)\right) -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) \\&\le 0. \end{aligned}$$

The first equality follows from Lemmas 1, 2 and 3. The second follows by taking the $v_{i}$ terms out from the parentheses. The next equality sums these $v_{i}$ terms up to $S_{[1]}(v)$. The inequality follows because $\underline{v}_{0}\le v_{0}$ and A(v) is nonempty.

Case II $v_{0}\ge \underline{v}_{0}>S_{[1]}(v)$ and $A(v)= \emptyset$

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)= & {} \underline{v}_{0}-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}\right) \\= & {} \underline{v}_{0}-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) -S_{[1]}(v)\\= & {} \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i}) \end{aligned}$$

The first equality follows from Lemmas 1, 2 and 3. The second follows by taking out the $v_{i}$ terms out of the parentheses. The third follows by taking out the $S_{[1]}(v)$ terms out of the parentheses and collecting them together. The sign of the resulting expression can be positive or negative, as examples will show.

Case III $v_{0}> S_{[1]}(v)\ge \underline{v}_{0}$ and $A(v)\ne \emptyset$

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&=S_{[1]}(v)-\sum _{i\in A(v)} \left( v_{0}-S_{[1]}(v)+v_{i}\right) \\&\quad -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}\right) \\&=S_{[1]}(v)-|A(v)|\left( v_{0}-S_{[1]}(v)\right) -\sum _{i\in A(v)}v_{i} \\&\quad -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} v_{i}\\&=-|A(v)|\left( v_{0}-S_{[1]}(v)\right) -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) \\&< 0. \end{aligned}$$

The first equality follows from Lemmas 1, 2 and 3. The second follows by taking out the $v_{i}$ terms out of the parentheses. The third follows after canceling out the $S_{[1]}(v)$’s. The resulting expression is nonpositive since $v_{0}> S_{[1]}(v)$ and $S_{[1]}(\bar{v},v_{-i})\ge S_{[1]}(v)$.

Case IV $v_{0}> S_{[1]}(v)\ge \underline{v}_{0}$ and $A(v)= \emptyset$

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&=S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) -S_{[1]}(v)\\&=-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) \\&\le 0. \end{aligned}$$

The first equality follows from Lemmas 1, 2 and 3. The second follows by taking out the $v_{i}$ terms out of the parentheses. The third follows after canceling out the $S_{[1]}(v)$’s. The resulting expression is nonpositive since $S_{[1]}(\bar{v},v_{-i})\ge S_{[1]}(v)$.

Case V $v_{0}\le S_{[1]}(v)$

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&=0. \end{aligned}$$

These cases are summarized in Table 1.

Lemma 4

For all v,

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&\ge \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i}). \end{aligned}$$

Proof

For $i\in A(v)$, $v_{0}\le S_{[1]}(\bar{v},v_{-i})$. Therefore, in Case I of Table 1,

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&= \underline{v}_{0}-S_{[1]}(v) -|A(v)|(v_{0}-S_{[1]}(v))-\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} (S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)) \\&\ge \underline{v}_{0}-S_{[1]}(v) -\sum _{i\in \mathcal {P}_{[1]}(v)} (S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v))\\&= \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i}). \end{aligned}$$

In Case II, $\sum _{j=0}^{n}t_{j}^{V}(v)= \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i})$.

In Case III,

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&= -|A(v)|(v_{0}-S_{[1]}(v))-\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} (S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v))\\&= S_{[1]}(v)-\sum _{i\in A(v)} \left( v_{0}-S_{[1]}(v)+v_{i}\right) \\&\quad -\sum _{\begin{array}{c} i\in \mathcal {P}_{[1]}(v)\\ i\notin A(v) \end{array}} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}\right) \\&\ge S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)+v_{i}\right) \\&\ge \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i}). \end{aligned}$$

In Case IV,

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&= -\sum _{i\in \mathcal {P}_{[1]}(v)} (S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v))\\&=S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})- S_{[1]}(v)\right) -S_{[1]}(v)\\&\ge \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i}). \end{aligned}$$

In Case V,

$$\begin{aligned} \sum _{j=0}^{n}t_{j}^{V}(v)&= 0 \ge \underline{v}_{0}-S_{[1]}(v)\\&\ge \underline{v}_{0}-S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} \left( S_{[1]}(\bar{v},v_{-i})-S_{[1]}(v)\right) \\&= \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i}). \end{aligned}$$

$\square$

If (2) holds, Lemma 4 implies

$$\begin{aligned}&E\left( \sum _{j=0}^{n}t_{j}^{V}(v)\right) \ge \underline{v}_{0}+(k-1)E\left( S_{[1]}(v)\right) -E\left( \sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i})\right) \ge 0. \end{aligned}$$

Part I now follows by Proposition 1.

For part II, note that the profiles in $\tilde{V}$ correspond to Case II of Table 1 and all other profiles correspond to the other Cases. If a successful mechanism exists then by Proposition 1,

$$\begin{aligned} 0&\le E\left( \sum _{j=0}^{n}t_{j}^{V}(v)\right) \\&= E\left( \sum _{j=0}^{n}t_{j}^{V}(v)\Bigg | v\in \tilde{V}\right) \times \Pr \left( v\in \tilde{V}\right) \\&\quad +E\left( \sum _{j=0}^{n}t_{j}^{V}(v)\Bigg |v\notin \tilde{V} \right) \times \Pr \left( v\notin \tilde{V} \right) \end{aligned}$$

The second component of this sum of products is negative since $\sum _{j=0}^{n}t_{j}^{V}(v)$ takes negative or zero value when $v\notin \tilde{V}$. Therefore,

$$\begin{aligned} E\left( \sum _{j=0}^{n} t_{j}^{V}(v)\Bigg | v\in \tilde{V}\right)&>0 \end{aligned}$$

Since $\sum _{j=0}^{n} t_{j}^{V}(v)= \underline{v}_{0}+(k-1) S_{[1]}(v)-\sum _{i\in \mathcal {P}_{[1]}(v)} S_{[1]}(\bar{v},v_{-i})$ when $v\in \tilde{V}$, the claim follows.

By Proposition 1, a successful mechanism exists if and only if the expected sum of VCG payments is non-negative. In Lemma 4, we showed that the sum of VCG payments is bounded below by $\underline{v}_{0}-\sum _{i\in \mathcal {P}_{[1]}(v)}S_{[1]}(\bar{v},v_{-i})+(k-1)S_{[1]}(v)$ at any profile v. Part I of Theorem 1 is a direct consequence of this Lemma. For Part II, note that the set $\tilde{V}$ corresponds to Case II in Table 1. For all $v\in \tilde{V}$, $\underline{v}_{0}>S_{[1]}(v)$, i.e., trade takes place at $(\underline{v}_{0}, v_{-0})$ and therefore, also at v. Furthermore, $v_{0}>S_{[1]}(\bar{v},v_{-i})$ for all $i\in \mathcal {P}_{[1]}(v)$: trade takes place if any seller i, who is successful at v, changes his valuation to $\bar{v}$. Therefore, no successful seller at v is trade-pivotal, i.e., the set A(v) is empty. According to Table 1, the sum of VCG payments, $\underline{v}_{0}-\sum _{i\in \mathcal {P}_{[1]}(v)}S_{[1]}(\bar{v},v_{-i})+(k-1)S_{[1]}(v)$, can be positive only at such profiles. For all other profiles the sum of VCG payments is non-positive. It follows that the conditional expectation in Condition 3 must be positive in order for the expected sum of VCG payments to be non-negative.

Part B. First, we omit all isolated paths of length less than k. Since the VCG mechanism is efficient, trade would have never taken place with sellers on such paths. Consequently, their VCG payments would have been zero in any case. The sellers on the unique feasible path are labelled $1,\ldots , k$. We will show that the sum of VCG payments $\sum _{j=0}^{k}t_{j}^{V}$ is nonpositive at all profiles and negative in some interval of profiles. Therefore, its expectation is negative. The result follows by applying Proposition 1.

We have

$$\begin{aligned} SW(v)=\left\{ \begin{array}{ll} v_0 - \sum \nolimits _{i=1}^{k}v_{i} &{}\quad \hbox {if } v_0 > \sum \nolimits _{i=1}^{k}v_{i},\\ 0&{}\quad \hbox {otherwise.} \end{array}\right. \end{aligned}$$

The next lemmas specify the VCG payments for the agents.

Lemma 5

The VCG payment of the buyer is given by

$$\begin{aligned} t_{0}^{V}(v)&= \left\{ \begin{array}{ll} \underline{v}_{0} &{}\quad \hbox {if } v_{0}\ge \underline{v}_{0}>\sum \nolimits _{i=1}^{k}v_{i},\\ \sum \nolimits _{i=1}^{k}v_{i} &{}\quad \hbox {if } v_{0}> \sum \nolimits _{i=1}^{k}v_{i}\ge \underline{v}_{0},\\ 0 &{}\quad \hbox {if } \underline{v}_{0}\le v_{0}\le \sum \nolimits _{i=1}^{k}v_{i}. \end{array} \right. \end{aligned}$$

Proof

If $v_{0}\ge \underline{v}_{0}>\sum _{i=1}^{k}v_{i}$,

$$\begin{aligned} SW(\underline{v}_{0}, v_{-0})&=\underline{v}_{0}-\sum _{i=1}^{k}v_{i},\\ \hbox {and } SW_{-0}({v}_{0}, v_{-0})&=-\sum _{i=1}^{k}v_{i}.\\ \hbox {Hence } t_{0}^{V}(v)&=\underline{v}_{0}. \end{aligned}$$

If $v_{0}> \sum _{i=1}^{n}v_{i}\ge \underline{v}_{0}$,

$$\begin{aligned} SW(\underline{v}_{0}, v_{-0})&=0, \\ \hbox {and }SW_{-0}({v}_{0}, v_{-0})&=-\sum _{i=1}^{k}v_{i}. \\ \hbox {Hence } t_{0}^{V}(v)&=\sum _{i=1}^{k}v_{i}. \end{aligned}$$

If $v_{0}\le \sum _{i=1}^{k}v_{i}$,

$$\begin{aligned} SW(\underline{v}_{0}, v_{-0})&=0,\\ \hbox {and }SW_{-0}({v}_{0}, v_{-0})&=0.\\ \hbox {Hence }t_{0}^{V}(v)&=0. \end{aligned}$$

$\square$

Lemma 6

When $q=1$, the VCG payment of the seller corresponding to $v_{i}$, $i=1, \ldots , n$, is given by

$$\begin{aligned} t_{i}^{V}(v)&=\left\{ \begin{array}{ll}-\left( v_{0}-\sum \nolimits _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j} \right) &{}\quad \hbox {if } \sum \nolimits _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}+\bar{v}\ge v_{0}>\sum \nolimits _{j=1}^{k }v_{j}, \\ -\bar{v} &{}\quad \hbox {if } v_{0}> \sum \nolimits _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}+\bar{v}\ge \sum \nolimits _{j=1}^{k }v_{j}, \\ 0 &{}\quad \hbox {if } v_{0}\le \sum \nolimits _{j=1}^{k }v_{j} \le \sum \nolimits _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}+\bar{v}. \end{array} \right. \end{aligned}$$

Proof

If $\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}+\bar{v}\ge v_{0} >\sum _{j=1}^{k }v_{j}$,

$$\begin{aligned} SW(\bar{v},v_{-i})&=0,\\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=v_{0}-\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}. \\ \hbox {Hence } t_{i}^{V}(v)&=-\left( v_{0}-\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j} \right) . \end{aligned}$$

If $v_{0}> \sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}+\bar{v}\ge \sum _{j=1}^{k }v_{j}$,

$$\begin{aligned} SW(\bar{v},v_{-i})&= v_{0}- \sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}-\bar{v},\\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=v_{0}-\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}.\\ \hbox {Hence } t_{i}^{V}(v)&=-\bar{v}. \end{aligned}$$

If $v_{0}\le \sum _{j=1}^{n }v_{j}\le \sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^k {v}_{j}+\bar{v}$,

$$\begin{aligned} SW(\bar{v},v_{-i})&= 0, \\ \hbox {and }SW_{-i}(v_{i},v_{-i})&=0.\\ \hbox {Hence } t_{i}^{V}(v)&=0. \end{aligned}$$

$\square$

For any profile v, let $A(v)=\{h\in \{1, \ldots , k\}: \sum _{i=1}^k {v}_{i}<v_{0}\le \sum _{\begin{array}{c} i=1\\ i\ne h \end{array}}^k {v}_i +\bar{v}\}$. In other words, A(v) is the set of sellers who influence the possibility of trade by reporting their highest valuation. Such sellers will be referred to as trade-pivotal.

Different mutually exclusive and exhaustive cases and corresponding sum of payments as obtained from Lemmas 5 and 6 are presented below.

Case I $v_{0}\ge \underline{v}_{0}>\sum _{i=1}^{k}v_{i}$ and $A(v)\ne \emptyset$

$$\begin{aligned} \sum _{j=0}^{k}t_{j}^{V}(v)= & {} \underline{v}_{0}-\sum _{h\in A(v)} \left( v_{0}-\sum _{\begin{array}{c} i=1\\ i\ne h \end{array}}^k {v}_{i} \right) -\left( k-|A(v)|\right) \bar{v}\\= & {} \underline{v}_{0}-\sum _{h\in A(v)} v_{0}+\sum _{h\in A(v)}\sum _{i=1}^k {v}_{i}-\sum _{h\in A(v)}v_{[h]} -\left( k-|A(v)|\right) \bar{v}\\= & {} \left( \underline{v}_{0}-\sum _{h\in A(v)}v_{[h]}-\left( k-|A(v)|\right) \bar{v}\right) -\sum _{h\in A(v)}\left( v_{0}-\sum _{i=1}^{k} v_{i}\right) \\\le & {} \left( \underline{v}_{0}-\sum _{i=1}^{k}v_{i} \right) -\sum _{h\in A(v)}\left( v_{0}-\sum _{i=1}^{k}v_{i}\right) \\\le & {} 0. \end{aligned}$$

Here, $k-|A(v)|$ is the number of sellers not in A(v) at profile v. The second equality is obtained by expanding $\sum _{\begin{array}{c} i=1\\ i\ne h \end{array}}^k {v}_{i}$; the third is obtained by re-arranging terms; the first inequality holds since $v_{i}\le \bar{v}$ for any $i=1, \ldots , k$ and the second inequality holds since $v_{0}\ge \underline{v}_{0}$.

Case II $v_{0}\ge \underline{v}_{0}>\sum _{i=1}^{k}v_{i}$ and $A(v)= \emptyset$

$$\begin{aligned} \sum _{j=0}^{k}t_{j}^{V}(v)= & {} \underline{v}_{0}-k\bar{v}\\< & {} 0. \end{aligned}$$

This holds by virtue of the assumption $\underline{v}_{0}<k\bar{v}$.

Case III $v_{0}> \sum _{i=1}^{k}v_{i}\ge \underline{v}_{0}$ and $A(v)\ne \emptyset$

$$\begin{aligned} \sum _{j=0}^{k}t_{j}^{V}(v)= & {} \sum _{i=1}^{k}v_{i}-\sum _{h\in A(v)}\left( v_{0}-\sum _{\begin{array}{c} i=1\\ i\ne h \end{array}}^k {v}_{i}\right) -\left( k-|A(v)|\right) \bar{v}\\= & {} \sum _{h\in A(v)}v_{h} +\sum _{\begin{array}{c} i=1\\ i\notin A(v) \end{array}}^{k} v_{i}-\sum _{h\in A(v)}\left( v_{0}-\sum _{\begin{array}{c} i=1\\ i\ne h \end{array}}^k {v}_{i}\right) -\left( k-|A(v)|\right) \bar{v}\\= & {} -\sum _{h \in A(v)}\left( v_{0}-\sum _{i=1}^{k} {v}_{i}\right) +\sum _{\begin{array}{c} i=1\\ i\notin A(v) \end{array}}^{k}\left( v_{i}-\bar{v}\right) \\< & {} 0. \end{aligned}$$

The second equality is obtained by expanding $\sum _{i=1}^{k}v_{i}$; the third adds the first and the third term of the left hand side to obtain the first term, and the other two terms constitute the second term. The inequality follows because $v_{0}> \sum _{i=1}^{k}v_{i}$ and $v_{i}\le \bar{v}$ for $i= 1, \ldots , k$.

Case IV $v_{0}> \sum _{i=1}^{k}v_{i}\ge \underline{v}_{0}$ and $A(v)= \emptyset$

$$\begin{aligned} \sum _{j=0}^{k}t_{j}^{V}(v)= & {} \sum _{i=1}^{k}v_{i}-k\bar{v}\\= & {} \sum _{i=1}^{k}v_{i}-k\bar{v}\\< & {} 0. \end{aligned}$$

The inequality follows because $v_{i}\le \bar{v}$ for $i= 1, \ldots , k$.

Case V $v_{0}\le \sum _{i=1}^{k}v_{i}$

$$\begin{aligned} \sum _{j=0}^{k}t_{j}^{V}(v)&=0. \end{aligned}$$

The calculations are summarized in Table 2. Observe that for any profile v, $\sum _{j=0}^{k}t_{j}^{V}(v)\le 0$ and in Cases II, III and IV, $\sum _{j=0}^{k}t_{j}^{V}(v)< 0$. Therefore, as required,

$$\begin{aligned} E\left( \sum _{j=0}^{k}t_{j}^{V}(v)\right)&<0. \end{aligned}$$

Table 2 Sum of payments when $q=1$