Information concentration in common value environments

Abstract

We consider how information concentration affects a seller’s revenue in common value auctions. The common value is a function of $n$ random variables partitioned among $m \le n$ bidders. For each partition, the seller devises an optimal mechanism. We show that whenever the value function allows scalar sufficient statistics for each player’s signals, the mechanism design problem is well-defined. Additionally, whenever a common regularity condition is satisfied, a coarser partition always reduces revenues. In particular, any merger or collusion among bidders reduces revenue.

Appendix

1.1 Proof of Theorem  1

The proof requires three lemmas. Fix the player for whom the value functions admit sufficient statistics to be player 1.

Lemma 1

Under any incentive compatible mechanism, $\eta ,$ the expected surplus of equivalent types is equal.

Proof

For every mechanism $\eta =(p_{i}(\cdot ),\xi _{i}(\cdot ))$ define

$$\begin{aligned} \widetilde{V}_{j}(\mathbf{s}_{j};\mathbf{t}_{j})=\int V_{j}(\mathbf{s}_{j}, \mathbf{s}_{-j})p_{j}(\mathbf{t}_{j},\mathbf{s}_{-j})f_{-j}(\mathbf{s}_{-j})d \mathbf{s}_{-j} \end{aligned}$$

and

$$\begin{aligned} \widetilde{\xi }_{j}(\mathbf{t}_{j})=\int \xi _{j}(\mathbf{t}_{j},\mathbf{s} _{-j})f_{-j}(\mathbf{s}_{-j})d\mathbf{s}_{-j} \end{aligned}$$

Under mechanism $\eta$, the expected payoff for player $j$ who has information $\mathbf{s}_{j}$ and reports $\mathbf{t}_{j}$ is

$$\begin{aligned} \widetilde{V}_{j}(\mathbf{s}_{j};\mathbf{t}_{j})-\widetilde{\xi }_{j}( \mathbf{t}_{j}). \end{aligned}$$

The interim incentive compatibility constraint for player $1$ is

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{\xi }_{1}( \mathbf{s}_{1})\ge \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{t}_{1})- \widetilde{\xi }_{1}(\mathbf{t}_{1}) \end{aligned}$$

(5)

for all $\mathbf{s}_{1}$ and $\mathbf{t}_{1}.$ For two equivalent types $\mathbf{s}_{1}$ and $\mathbf{s}_{1}^{\prime }$ and any $\mathbf{t}_{1}$, we have by definition

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{t}_{1})=\widetilde{V}_{1}(\mathbf{s} _{1}^{\prime };\mathbf{t}_{1}) \end{aligned}$$

and in particular

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})&= \widetilde{V}_{1}(\mathbf s _{1}^{\prime };\mathbf{s}_{1}), \end{aligned}$$

(6)

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1}^{\prime };\mathbf{s}_{1}^{\prime })&= \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1}^{\prime }). \end{aligned}$$

(7)

Combining expressions, we have

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{V}_{1}(\mathbf{s} _{1}^{\prime };\mathbf{s}_{1}^{\prime })=\widetilde{V}_{1}(\mathbf{s}_{1}; \mathbf{s}_{1})-\widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1}^{\prime })\ge \widetilde{\xi }_{1}(\mathbf{s}_{1})-\widetilde{\xi }_{1}(\mathbf{s}_{1}^{\prime }) \end{aligned}$$

(8)

where the equality holds by (7) and the inequality by (5), substituting $\mathbf{s}_{1}^{\prime }$ for $\mathbf{t}_{1}.$ Also, by substituting $\mathbf{s}_{1}$ for $\mathbf{t}_{1}$ and $\mathbf{s} _{1}^{\prime }$ for $\mathbf{s}_{1}$ in (5), we have

$$\begin{aligned} \widetilde{\xi }_{1}(\mathbf{s}_{1})-\widetilde{\xi }_{1}(\mathbf{s} _{1}^{\prime })\ge \widetilde{V}_{1}(\mathbf{s}_{1}^{\prime };\mathbf{s} _{1})-\widetilde{V}_{1}(\mathbf{s}_{1}^{\prime };\mathbf{s}_{1}^{\prime })= \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{V}_{1}(\mathbf{s} _{1}^{\prime };\mathbf{s}_{1}^{\prime }), \end{aligned}$$

(9)

where the equality follows from (6). Combining (8) and (9), we obtain

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{V }_{1}(\mathbf{s}_{1}^{\prime };\mathbf{s}_{1}^{\prime }) = \widetilde{\xi }_{1}(\mathbf{s}_{1})-\widetilde{\xi }_{1}(\mathbf{s} _{1}^{\prime }) \end{aligned}$$

$\square$

Consider the mechanism $\eta ^{\prime }$ as defined by Eqs. (1) and (2).

Lemma 2

If mechanism $\eta$ is incentive compatible, then $\eta ^{\prime }$ is an incentive compatible mechanism.

Proof

Define $f_{\phi ^{A_1}}$ as the density of $\phi ^{A_1}(\mathbf{S}_{1}).$ Also, for every $\mathbf{s}_{1}$, such that $\phi ^{A_1}(\mathbf{s}_{1})=\alpha$, denote by

$$\begin{aligned} V_{i}^{A_{1}}(\alpha ;\mathbf{s}_{-1})=V_{i}^{{}}(\mathbf{s}_{1},\mathbf{s}_{-1}) \end{aligned}$$

the parametrization of $i$’s value function based on the sufficient statistic’s value. Since $\eta$ is a mechanism, we have $p_{i}(\mathbf{s} _{1},\mathbf{s}_{-1})\ge 0$ and $\sum p_{i}(\mathbf{s}_{1},\mathbf{s} _{-1})\le 1$ for every $i$ and every $(\mathbf{s}_{1},\mathbf s _{-1})$. This implies by integration over the set of equivalent types that for $i$ and $\mathbf{s}_{1},$

$$\begin{aligned} \int p_{i}(\mathbf{t}_{1},\mathbf{s}_{-1})f_{1}(\mathbf{t}_{1}|\phi ^{A_1}( \mathbf{t}_{1})=\phi ^{A_1}(\mathbf{s}_{1}))d\mathbf{t}_{1}\ge 0\Leftrightarrow p_{i}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1}),\mathbf{s} _{-1})\ge 0 \end{aligned}$$

and similarly

$$\begin{aligned} \sum p_{i}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1}),\mathbf{s}_{-1})\le 1. \end{aligned}$$

The interim incentive compatibility condition for player $j$ under mechanism $\eta$ is

$$\begin{aligned} \widetilde{V}_{j}(\mathbf{s}_{j};\mathbf{s}_{j})-\widetilde{\xi }_{j}( \mathbf{s}_{j})\ge \widetilde{V}_{j}(\mathbf{s}_{j};\mathbf{t}_{j})- \widetilde{\xi }_{j}(\mathbf{t}_{j}) \end{aligned}$$

for all $\mathbf{s}_{j}$ and $\mathbf{t}_{j}.$ For bidder $j\ne 1$,

$$\begin{aligned} \widetilde{\xi }_{j}(\mathbf{t}_{j})&\!=\!&\int \xi _{j}(\mathbf{t}_{j},\mathbf s _{1},\mathbf{s}_{-1j})f_{1}(\mathbf{s}_{1})f_{-1j}(\mathbf{s}_{-1j})d \mathbf{s}_{-j} \\&\!=\!&\int f_{-1j}(\mathbf{s}_{-1j})\int f_{\phi ^{A_1}}(\alpha )\int \xi _{j}(\mathbf{t}_{j},\mathbf{s}_{1},\mathbf{s}_{-1j})f_{1}(\mathbf{s}_{1}|\phi ^{A_1}(\mathbf{s}_{1})\!=\!\alpha )d\mathbf{s}_{1}d\alpha d\mathbf{s}_{-1j} \\&\!=\!&\int \int \xi _{j}^{\prime }(\alpha ,\mathbf{t}_{j},\mathbf{s} _{-1j})f_{-1j}(\mathbf{s}_{-1j})f_{\phi ^{A_1}}(\alpha )d\mathbf{s} _{-1j}d\alpha \\&\!=\!&\widetilde{\xi }_{j}^{\prime }(\mathbf{t}_{j}). \end{aligned}$$

Similarly, for all $\mathbf{s}_{j}$ and $\mathbf{t}_{j},$

$$\begin{aligned} \widetilde{V}_{j}(\mathbf{s}_{j};\mathbf{t}_{j})&= \int V_{j}(\mathbf{s}_{j}, \mathbf{s}_{1},\mathbf{s}_{-1j})p_{j}(\mathbf{t}_{j},\mathbf{s}_{1},\mathbf s _{-1j})f_{1}(\mathbf{s}_{1})f_{-1j}(\mathbf{s}_{-1j})d\mathbf{s}_{-j} \\&= \int \int V_{j}^{A_{1}}(\alpha ;\mathbf{s}_{j},\mathbf{s} _{-1j})p_{j}^{\prime }(\alpha ,\mathbf{t}_{j},\mathbf{s}_{-1j})f_{-1j}( \mathbf{s}_{-1j})f_{\phi ^{A_1}}(\alpha )d\mathbf{s}_{-1j}d\alpha \\&= \widetilde{V}_{j}^{\prime }(\mathbf{s}_{j};\mathbf{t}_{j}). \end{aligned}$$

Substituting the identical terms into the incentive compatibility constraint we get

$$\begin{aligned} \widetilde{V}_{j}^{\prime }(\mathbf{s}_{j};\mathbf{s}_{j})-\widetilde{\xi } _{j}^{\prime }(\mathbf{s}_{j})\ge \widetilde{V}_{j}^{\prime }(\mathbf{s} _{j};\mathbf{t}_{j})-\widetilde{\xi }_{j}^{\prime }(\mathbf{t}_{j}) \end{aligned}$$

for all $\mathbf{s}_{j}$ and $\mathbf{t}_{j}$ which indicates that mechanism $\eta ^{\prime }$ is incentive compatible for player $j$.

Finally, we need to show that player $1$’s incentive compatibility constraint is satisfied under mechanism $\eta ^{\prime }.$ Define

$$\begin{aligned} \widetilde{V}_{1}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1});\phi ^{A_1}(\mathbf{t} _{1}))=\int \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf y _{1})f_{1}(\mathbf y _{1}^{{}}|\phi ^{A_1}(\mathbf y _{1})=\phi ^{A_1}(\mathbf{t}_{1}))d\mathbf y _1. \end{aligned}$$

and

$$\begin{aligned} \widetilde{\xi }_{1}^{\prime }(\phi ^{A_1}(\mathbf{t}_{1}))=\int \widetilde{ \xi }_{1}(\mathbf y _{1})f_{1}(\mathbf y _{1}^{{}}|\phi ^{A_1}(\mathbf y _{1}^{{}})=\phi ^{A_1}(\mathbf{t}_{1}))d\mathbf y _1 \end{aligned}$$

which are player 1’s expected asset value and expected payment when his type is equivalent to $\mathbf{s}_1$ and he reports a type equivalent to $\mathbf{t}_1$.

The previous lemma establishes that for equivalent types $\mathbf{s}_{1}$ and $\mathbf{s}_{1}^{\prime }$,

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{\xi }_{1}( \mathbf{s}_{1})=\widetilde{V}_{1}(\mathbf{s}_{1}^{\prime };\mathbf{s}_{1}^{\prime })- \widetilde{\xi }_{1}(\mathbf{s}_{1}^{\prime }). \end{aligned}$$

and therefore, by the definition of equivalent types,

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{\xi }_{1}( \mathbf{s}_{1})=\widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1}^{\prime })- \widetilde{\xi }_{1}(\mathbf{s}_{1}^{\prime }). \end{aligned}$$

Integrating these relationships along the set of equivalent types with respect to $f_{1}(\mathbf{s}_{1}^{\prime }|\phi ^{A_1}(\mathbf{s}_{1}^{\prime })=\phi ^{A_1}(\mathbf{s}_{1}))$ results in

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{\xi }_{1}( \mathbf{s}_{1}) = \widetilde{V}_{1}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1});\phi ^{A_1}(\mathbf{s}_{1}))-\widetilde{\xi }_{1}^{\prime }(\phi ^{A_1}(\mathbf{s} _{1})). \end{aligned}$$

(10)

Player 1’s incentive compatibility constraint under $\eta$ for nonequivalent types $\mathbf{s}_{1}$ and $\mathbf{t}_{1}$ is

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{\xi }_{1}( \mathbf{s}_{1})\ge \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{t}_{1}) - \widetilde{\xi }_{1}(\mathbf{t}_{1}). \end{aligned}$$

Integrating these relationships along the set of equivalent types to $\mathbf{t}_1$ yields

$$\begin{aligned} \widetilde{V}_{1}(\mathbf{s}_{1};\mathbf{s}_{1})-\widetilde{\xi }_{1}( \mathbf{s}_{1})\ge \widetilde{V}_{1}^{\prime }(\phi ^{A_1} (\mathbf{s} _{1});\phi ^{A_1} (\mathbf{t}_{1})) -\widetilde{\xi }_{1}^{\prime }(\phi ^{A_1}(\mathbf t _{1})). \end{aligned}$$

(11)

Combining (10) and (11) yields

$$\begin{aligned} \widetilde{V}_{1}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1});\phi ^{A_1}(\mathbf{s} _{1}))-\widetilde{\xi }_{1}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1}))\ge \widetilde{V}_{1}^{\prime }(\phi ^{A_1}(\mathbf{s}_{1});\phi ^{A_1}(\mathbf{t} _{1}))-\widetilde{\xi }_{1}^{\prime }(\phi ^{A_1}(\mathbf{t}_{1})). \end{aligned}$$

which is the incentive compatibility constraint for player $1$ under mechanism $\eta ^{\prime }.$ $\square$

Lemma 3

The mechanisms $\eta$ and $\eta ^{\prime }$ are revenue-equivalent.

Proof

The expected revenue for the seller under mechanism $\eta$ is

$$\begin{aligned} ER(\eta )&= \int \sum _{k}\xi _{k}(\mathbf{s})f(\mathbf{s})d\mathbf{s} \\&= \sum _{k}\int \xi _{k}(\mathbf{s})\prod f_{i}(\mathbf{s}_{i})d\mathbf{s}_{i} \\&= \sum _{k}\int _{{}}f_{-1}(\mathbf{s}_{-1})\int f_{\phi ^{A_1}}(\alpha )\int \xi _{k}(\mathbf{s}_{1},\mathbf{s}_{-1})f_{1}(\mathbf{s}_{1}|\phi ^{A_1} (\mathbf{s}_{1})=\alpha )d\mathbf{s}_{1}d\alpha d\mathbf{s}_{-1}. \\&= \sum _{k} \int \int \xi _{k}^{\prime }(\alpha ,\mathbf{s}_{-i})f_{-1}(\mathbf{s}_{-1})f_{\phi ^{A_1}}(\alpha )d\alpha d\mathbf{s}_{-1} \\&= ER(\eta ^{\prime }) \end{aligned}$$

$\square$

Proof of Theorem 1

Follows from the above three lemmas. $\square$

1.2 Proof of Theorem  2

1.2.1 Notation

For notational convenience, we consider the case where buyer $1$ and $2$’s information is centralized under the control of a new buyer $c$. Let

$$\begin{aligned} \mathbf{A}=\left\{ A_{1},A_{2},A_{3},\ldots ,A_{m}\right\} \end{aligned}$$

and

$$\begin{aligned} \mathbf{A}^{\prime }=\left\{ A_{c},A_{3}^{\prime },\ldots ,A_{m}^{\prime }\right\} \end{aligned}$$

where $A_{c}=A_{1}\cup A_{2}$ and $A_{i}=A_{i}^{\prime }$ for all $i\ge 3$. All the other cases can be derived from this analysis. For simplicity, we drop the superscript in $\phi ^{A_c}(X_1,X_2)$ and denote the sufficient statistic by $\phi (X_{1},X_{2})$.

For every $t$ in the support of $\phi (X_{1},X_{2})$ define:

$$\begin{aligned} \underline{x}_{1}(t)=\inf \left\{ \left. x_{1}\right| \exists x_{2}\in [\underline{z}_{2},\overline{z}_{2}],\phi (x_{1},x_{2})=t\right\} \end{aligned}$$

(12)

and

$$\begin{aligned} \overline{x}_{1}(t)=\sup \left\{ \left. x_{1}\right| \exists x_{2}\in [\underline{z}_{2},\overline{z}_{2}],\phi (x_{1},x_{2})=t\right\} , \end{aligned}$$

(13)

analogously define $\underline{x}_{2}(t)$ and $\overline{x}_{2}(t).$ Note that these objects are well-defined and increasing when $\phi (\cdot ,\cdot )$ is increasing.

Define implicitly for every $t, \psi _{2}(\cdot ,t):[\underline{x}_{1}(t), \overline{x}_{1}(t)]\rightarrow [\underline{x}_{2}(t),\overline{x} _{2}(t)]$

$$\begin{aligned} \phi (x_{1},\psi _{2}(x_{1},t))=t \end{aligned}$$

and $\psi _{1}(\cdot ,t):[\underline{x}_{2}(t),\overline{x} _{2}(t)]\rightarrow [\underline{x}_{1}(t),\overline{x}_{1}(t)]$

$$\begin{aligned} \phi (\psi _{1}(x_{2},t),x_{2})=t. \end{aligned}$$

Note that $\psi _{1}(\cdot ,t)$ and $\psi _{2}(\cdot ,t)$ are well-defined decreasing functions and $\psi _{1}^{-1}(\cdot ,t)=\psi _{2}(\cdot ,t)$. Further, when $\phi$ is differentiable then so are $\psi$ and $\psi _{2},$ and

$$\begin{aligned} \frac{d}{dx_1}\phi (x_1,\psi _{2}(x_1,t))=0\Leftrightarrow \partial _{1}\phi (x_1,\psi _{2}(x_1,t))=-\partial _{2}\phi (x_1,\psi _{2}(x_1,t))\partial _{x_1}\psi _{2}(x_1,t) \end{aligned}$$

or

$$\begin{aligned} \partial _{x_1}\psi _{2}(x_1,t)=-\frac{\partial _{1}\phi (x_1,\psi _{2}(x_1,t))}{\partial _{2}\phi (x_1,\psi _{2}(x_1,t))} \end{aligned}$$

and analogously

$$\begin{aligned} \partial _{x_2}\psi _{1}(x_2,t)=-\frac{\partial _{2}\phi (\psi _{1}(x_2,t),x_2)}{\partial _{1}\phi (\psi _{1}(x_2,t),x_2)}. \end{aligned}$$

Note that if $\phi (x_{1},x_{2})=t$

$$\begin{aligned} V^{\mathbf{A}^{\prime }}(t,t_{-12})=V^{\mathbf{A}}(x_{1},\psi _{2}(x_{1},t),t_{-12}) \end{aligned}$$

(14)

and therefore

$$\begin{aligned} \partial _{t}V^{\mathbf{A}^{\prime }}(t,t_{-12})&= \partial _{2}V^\mathbf{A }(x_{1},\psi _{2}(x_{1},t),t_{-12})\partial _{t}\psi _{2}(x_{1},t)\nonumber \\&= \frac{\partial _{2}V^{\mathbf{A}}(x_{1},\psi _{2}(x_{1},t),t_{-12})}{\partial _{2}\phi \left( x_{1},\psi _{2}(x_{1},t)\right) }. \end{aligned}$$

(15)

By a similar argument one can show that

$$\begin{aligned} \partial _{t}V^{\mathbf{A}^{\prime }}(t,t_{-12})=\frac{\partial _{1}V^{\mathbf{A}}(\psi _{1}(x_{2},t),x_{2},t_{-12})}{\partial _{1}\phi \left( \psi _{1}(x_{2},t),x_{2}\right) }. \end{aligned}$$

and for all $j\ge 3$

$$\begin{aligned} \partial _{tj}V^{\mathbf{A}^{\prime }}(t,t_{-1})=\frac{\partial _{1j}V^{ \mathbf{A}}(\psi _{1}(x_{2},t),x_{2},t_{-12})}{\partial _{1}\phi \left( \psi _{1}(x_{2},t),x_{2}\right) }, \end{aligned}$$

which means that $\partial _{tj}V^{\mathbf{A}^{\prime }}$ and $\partial _{1j}V^{\mathbf{A}}$ will have the same sign.

Lemma 4

The survival function and density function of $\phi (X_{1},X_{2})$ are given by

$$\begin{aligned} \overline{F}_{\phi }(t)&= \int \limits _{-\infty }^{\infty }f_{1}(x)\overline{F} _{2}(\psi _{2}(x,t))dx =\int \limits _{-\infty }^{\infty }f_{2}(x)\overline{F} _{1}(\psi _{1}(x,t))dx , \text{ and }\end{aligned}$$

(16)

$$\begin{aligned} f_{\phi }(t)&= \int \limits _{-\infty }^{\infty }\frac{f_{1}(x)f_{2}(\psi _{2}(x,t))}{\partial _{2}\phi (x,\psi _{2}(x,t))}dx =\int \limits _{-\infty }^{\infty }\frac{ f_{2}(x)f_{1}(\psi _{1}(x,t))}{\partial _{1}\phi (\psi _{1}(x,t),x)}dx. \end{aligned}$$

(17)

Proof

Note that

$$\begin{aligned} \Pr \left[ \phi (X_{1},X_{2})\ge t\right] =\overline{F}_{\phi }(t)=\int \Pr \left[ \phi (X_{1},X_{2})\ge t|X_{1}=x\right] f_{1}(x)dx. \end{aligned}$$

Since $X_{1}$ and $X_{2}$ are independent, we have

$$\begin{aligned} \Pr \left[ \phi (X_{1},X_{2})\ge t|X_{1}=x\right] =\Pr \left[ X_{2}\ge \psi _{2}(x,t)|X_{1}=x\right] =\overline{F}_{2}(\psi _{2}(x,t)). \end{aligned}$$

Substituting back into the integral yields the first expression. Note that

$$\begin{aligned} \frac{d}{dt}\phi (x,\psi _{2}(x,t))=1\Leftrightarrow \partial _{2}\phi (x,\psi _{2}(x,t))\partial _{t}\psi _{2}(x,t)=1 \end{aligned}$$

or

$$\begin{aligned} \partial _{t}\psi _{2}(x,t)=\frac{1}{\partial _{2}\phi (x,\psi _{2}(x,t))}. \end{aligned}$$

(18)

Then,

$$\begin{aligned} f_{\phi }(t)&= -\partial _t \overline{F}_{\phi }(t) \\&= \int \limits _{-\infty }^{\infty }f_{1}(x)f_{2}(\psi _{2}(x,t))\partial _{t}\psi _{2}(x,t)dx \end{aligned}$$

Substituting Eq. (18) yields:

$$\begin{aligned} =\int \limits _{-\infty }^{\infty }\frac{f_{1}(x)f_{2}(\psi _{2}(x,t))}{\partial _{2}\phi (x,\psi _{2}(x,t))}dx \end{aligned}$$

$\square$

Proof of Theorem 2

Under our assumptions, every buyer’s information can be summarized by $T_{i} =\phi ^{A_{i}}(\mathbf{S}_{i})$ and in any scalar mechanism for information profile $\mathbf{A}$ only reports $t_{i}$ are required from buyers. In the concentrated environment $\mathbf{A}^{\prime }$, a scalar mechanism will require reports $t_{i} =T_{i}$ from buyers $3$ through $m$, while it will ask buyer $c$ to submit a report $t_{c}=\phi (t_{1},t_{2}).$ Denote by $f_{i}(t_{i}),F_{i}(t_{i})$ and $[\underline{a}_{i},\overline{a}_{i}]$ the density, distribution function and, respectively, support of the random variables $T_{i}.$ Further, denote by $f_{\phi }$ and $F_{\phi }$ the density and distribution function of the random variable $\phi (T_{1},T_{2}).$ Denote by $t_{-i}$ and $t_{-ij}$ the vector of reports excluding buyer $i$ or buyers $i$ and $j$.

Consider the virtual valuation of player $c$,

$$\begin{aligned} H_{c}^{\mathbf{A}^{\prime }}(t_{c},t_{-12})=V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})-\frac{\overline{F}_{\phi }(t_{c})}{f_{\phi }(t_{c})} \partial _{t_{c}}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12}). \end{aligned}$$

Under the regularity assumption, $H_{c}^{\mathbf{A}^{\prime }}(\cdot ,t_{-12})$ is non-decreasing.

Following (Bulow and Klemperer 1996), for any mechanism $\eta ^{\prime }\!=\!\left( p_{i}^{\prime },\xi _{i}^{\prime }\right) _{i\in \{c,3,\ldots ,m\}}$ in environment $\mathbf{A}^{\prime },$ the seller’s revenue is

$$\begin{aligned} R(\eta ^{\prime })&= \int \left( \sum _{i\ge 3}p_{i}^{\prime }(t_{i},t_{-i})H_{i}^{\mathbf{A}^{\prime }}(t_{i},t_{-i})+p_{c}^{\prime }(t_{c},t_{-12})H_{c}^{\mathbf{A}^{\prime }}(t_{c},t_{-12})+p_{0}^{\prime }v_{0}\right) \nonumber \\&\times f_{\phi }(t_{c})f(t_{-12})dt_{c}dt_{-12}, \end{aligned}$$

where $v_{0}$ is the seller’s reservation value. Point-by-point maximization of the integrand yields the optimal solution $\mu ^{\mathbf{A}^{\prime }}$ where

$$\begin{aligned} p_{i}^{\prime }(t_{i},t_{-i})=1\Leftrightarrow H_{i}^{\mathbf{A}^{\prime }}(t_{i},t_{-i})\ge \underset{j\ne i}{\max }(v_{0},H_{c}^{\mathbf{A} ^{\prime }}(t_{c},t_{-12}),H_{j}^{\mathbf{A}^{\prime }}(t_{j},t_{-j})) \end{aligned}$$

and zero otherwise. Since the functions $H_{i}^{\mathbf{A}^{\prime }}$ are non-decreasing, $H_{i}^{\mathbf{A}^{\prime }}(t_{i},t_{-i})\ge v_{0}$ implies $H_{i}^{\mathbf{A}^{\prime }}(t_{i}^{\prime },t_{-i})\ge v_{0}$ for all $t_{i}^{\prime }\ge t_{i}.$ Further, for all $j$, and for all $t_{i}^{\prime }\ge t_{i}$, by Assumption $1$ we have $H_{i}^{\mathbf{A}^{\prime }}(t_{i}^{\prime },t_{-i})\ge H_{j}^{\mathbf{A}^{\prime }}(t_{i}^{\prime },t_{-i}).$ In particular, for bidder $c$ we have

$$\begin{aligned} p_{c}^{\prime }(t_{c},t_{-12})=1\Leftrightarrow H_{c}^{\mathbf{A}^{\prime }}(t_{c},t_{-12})\ge \underset{i}{\max }(v_{0},H_{i}^{\mathbf{A}^{\prime }}(t_{i},t_{-i})) \end{aligned}$$

and zero otherwise. We conclude thus that for any $t_{-12}$ and $v_{0}$, the set of types for which bidder $c$ gets the object is given by

$$\begin{aligned} M_{c}=\left\{ (t_{1},t_{2})|\phi (t_{1},t_{2})=t_{c}\ge \tau (t_{-12},v_{0})\right\} . \end{aligned}$$

for some function, $\tau$. The expected payment received by the auctioneer from bidder $c$ is therefore,

$$\begin{aligned} \xi _{c}^{\mathbf{A}^{^{\prime }}}=\int \left( \,\,\, \int \limits _{t_{c}>\tau (t_{-12},v_{0})}H_{c}^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{\phi }(t_{c})dt_{c}\right) f(t_{-12})dt_{-12}. \end{aligned}$$

Define

$$\begin{aligned} Q^{\prime }(t)=\int \limits _{t_{c}>t}H_{c}^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{\phi }(t_{c})dt_{c}, \end{aligned}$$

then

$$\begin{aligned} \xi _{c}^{\mathbf{A}^{^{\prime }}}=\int Q^{\prime }(\tau (t_{-12},v_{0}))f(t_{-12})dt_{-12}. \end{aligned}$$

Consider a mechanism $\mu ^{\mathbf{A}}=\left( p_{i},\xi _{i}\right) _{i\in \{1,2,\ldots ,m\}}$ for the environment $\mathbf{A}$, with the following properties $p_{1}(t_{1},t_{-1})\equiv 0,p_{2}(t_{1},t_{2},t_{-12})\equiv p_{c}^{\prime }(\phi \left( t_{1},t_{2}\right) ,t_{-12})$ and $p_{i}(t_{i},t_{-i})\equiv p_{i}^{\prime }(t_{i},t_{-i})$ for all $i\ge 3.$ The mechanism $\mu ^{\mathbf{A}}$ is incentive compatible since all the functions are non-decreasing in own type.⁷ Furthermore, the expected payment received from bidders $3$ through $n$ ,under $\mu ^{\mathbf{A}}$ and $\mu ^{\mathbf{A} ^{\prime }}$are the same$.$

The expected payment from bidder $2$ in this case is therefore

$$\begin{aligned} \xi _{2}^{\mathbf{A}}=\int \left( \int \int \limits _{M_{c}}H_{2}^{\mathbf{A} }(t_{1},t_{2},t_{-12})f_{1}(t_{1})f_{2}(t_{2})dt_{1}dt_{2}\right) f(t_{-12})dt_{-12}. \end{aligned}$$

Define

$$\begin{aligned} Q(t)=\underset{\phi (t_{1},t_{2})\ge t}{\int \int }H_{2}^{\mathbf{A} }(t_{1},t_{2},t_{-12})f_{1}(t_{1})f_{2}(t_{2})dt_{1}dt_{2}, \end{aligned}$$

then

$$\begin{aligned} \xi _{2}^{\mathbf{A}}=\int \left( Q(\tau (t_{-12},v_{0}))\right) f(t_{-12})dt_{-12}. \end{aligned}$$

In particular, we will show that for all $t$

$$\begin{aligned} Q^{\prime }(t)=Q(t) \end{aligned}$$

the expected payments of bidder $c$ under $\mu ^{\mathbf{A}^{\prime }}$ and those of bidder $2$ under $\mu ^{\mathbf{A}}$ coincide, which makes the two mechanisms revenue-equivalent. Define

$$\begin{aligned} \underline{t}_{1}(t)=\inf \left\{ \left. t_{1}\right| \exists t_{2},\phi (t_{1},t_{2})=t\right\} \text{ and } \overline{t}_{1}(t)=\sup \left\{ \left. t_{1}\right| \exists t_{2},\phi (t_{1},t_{2})=t\right\} , \end{aligned}$$

(19)

and note that $Q(t)$ may be expressed as

$$\begin{aligned} Q(t)=\underset{\phi (t_{1},t_{2})\ge t}{\int \int }\left( V^{\mathbf{A}}(t_{1},t_{2},t_{-12})-\frac{\overline{F}_{2}(t_{2})}{f_{2}(t_{2})}\partial _{2}V^{\mathbf{A}}(t_{1},t_{2},t_{-12})\right) f_{1}(t_{1})f_{2}(t_{2})dt_{1}dt_{2} \nonumber \\ \end{aligned}$$

(20)

We consider each component separately. First,

$$\begin{aligned}&\underset{\phi (t_{1},t_{2})\ge t}{\int \int }V^{\mathbf{A} }(t_{1},t_{2},t_{-12})f_{1}(t_{1})f_{2}(t_{2})dt_{1}dt_{2}\nonumber \\&\qquad =\int \limits _{\underline{t }_{1}(t)}^{\overline{t}_{1}(t)}\int \limits _{t_{2}\ge \psi _{2}(t_{1},t)}V^\mathbf{ A }(t_{1},t_{2},t_{-12})f_{1}(t_{1})f_{2}(t_{2})dt_{2}dt_{1} \end{aligned}$$

Fix $t_{1}$ and introduce the change in variable $t_{2}=\psi _{2}(t_{1},t_{c}),$observing that $dt_{2}=\partial _{t_{c}}\psi _{2}(t_{1},t_{c})dt_{c}.$

The integral becomes

$$\begin{aligned}&\int \limits _{\underline{t}_{1}(t)}^{\overline{t}_{1}(t)}\int \limits _{t_{c}\ge t}V^{\mathbf{A}}(t_{1},\psi _{2}(t_{1},t_{c}),t_{-12})f_{1}(t_{1})f_{2}(\psi _{2}(t_{1},t_{c}))\partial _{t_{c}}\psi _{2}(t_{1},t_{c})dt_{c}dt_{1} \\&= \int \limits _{\underline{t}_{1}(t)}^{\overline{t}_{1}(t)}\int \limits _{t_{c}\ge t}\frac{ V^{\mathbf{A}}(t_{1},\psi _{2}(t_{1},t_{c}),t_{-12})f_{1}(t_{1})f_{2}(\psi _{2}(t_{1},t_{c}))}{\partial _{2}\phi (t_{1},\psi _{2}(t_{1},t_{c}))} dt_{c}dt_{1} \\&= \int \limits _{t_{c}\ge t}\int \limits _{\underline{t}_{1}(t)}^{\overline{t}_{1}(t)}\frac{ V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{1}(t_{1})f_{2}(\psi _{2}(t_{1},t_{c}))}{\partial _{2}\phi (t_{1},\psi _{2}(t_{1},t_{c}))} dt_{1}dt_{c} \\&= \int \limits _{t_{c}\ge t}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})\int \limits _{\underline{ t}_{1}(t)}^{\overline{t}_{1}(t)}\frac{f_{1}(t_{1})f_{2}(\psi _{2}(t_{1},t_{c}))}{\partial _{2}\phi (t_{1},\psi _{2}(t_{1},t_{c}))} dt_{1}dt_{c} \\&= \int \limits _{t_{c}\ge t}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{\phi }(t)dt_{c}. \end{aligned}$$

Where the first equality follows from Eq. (18), the second from (14) and Fubini’s theorem, and the last by Lemma 4.

The second part of the integral in (20) is

$$\begin{aligned}&-\underset{\phi (t_{1},t_{2})\ge t}{\int \int }\left( \frac{ \overline{F}_{2}(t_{2})}{f_{2}(t_{2})}\partial _{2}V^{\mathbf{A} }(t_{1},t_{2},t_{-12})\right) f_{1}(t_{1})f_{2}(t_{2})dt_{1}dt_{2} \\&= -\underset{\phi (t_{1},t_{2})\ge t}{\int \int }f_{1}(t_{1})\partial _{2}V^{\mathbf{A}}(t_{1},t_{2},t_{-12})\overline{F}_{2}(t_{2})dt_{1}dt_{2} \\&= -\int \limits _{\underline{t}_{1}(t)}^{\overline{t}_{1}(t)}\int \limits _{t_{2}\ge \psi _{2}(t_{1},t)}f_{1}(t_{1})\partial _{2}V^{\mathbf{A}}(t_{1},t_{2},t_{-12}) \overline{F}_{2}(t_{2})dt_{2}dt_{1} \\&= -\int \limits _{\underline{t}_{1}(t)}^{\overline{t}_{1}(t)}\int \limits _{t_{c}\ge t}f_{1}(t_{1})\partial _{2}V^{\mathbf{A}}(t_{1},\psi _{2}(t_{1},t_{c}),t_{-12})\overline{F}_{2}(\psi _{2}(t_{1},t_{c}))\partial _{t_{c}}\psi _{2}(t_{1},t_{c})dt_{c}dt_{1} \\&= -\int \limits _{\underline{t}_{1}(t)}^{\overline{t}_{1}(t)}\int \limits _{t_{c}\ge t}\frac{ \partial _{2}V^{\mathbf{A}}(t_{1},\psi _{2}(t_{1},t_{c}),t_{-12})}{\partial _{2}\phi (t_{1},\psi _{2}(t_{1},t_{c}))}f_{1}(t_{1})\overline{F}_{2}(\psi _{2}(t_{1},t_{c}))dt_{c}dt_{1} \\&= -\int \limits _{t_{c}\ge t}\int \limits _{\underline{t}_{1}(t)}^{\overline{t} _{1}(t)}\partial _{t_{c}}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{1}(t_{1}) \overline{F}_{2}(\psi _{2}(t_{1},t_{c}))dt_{1}dt_{c} \\&= -\int \limits _{t_{c}\ge t}\partial _{t_{c}}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})\int \limits _{\underline{t}_{1}(t)}^{\overline{t} _{1}(t)}f_{1}(t_{1})\overline{F}_{2}(\psi _{2}(t_{1},t_{c}))dt_{1}dt_{c} \\&= -\int \limits _{t_{c}\ge t}\partial _{t_{c}}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})\overline{F}_{\phi }(t_{c})dt_{c} \end{aligned}$$

Combining the two results we have

$$\begin{aligned} Q(t)&= \int \limits _{t_{c}\ge t}\left( V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{\phi }(t_c)-\partial _{t_{c}}V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})\overline{F}_{\phi }(t_{c})\right) dt_{c} \\&= \int \limits _{t_{c}\ge t}\left( V^{\mathbf{A}^{\prime }}(t_{c},t_{-12})-\frac{ \overline{F}_{\phi }(t_{c})}{f_{\phi }(t_c)}\partial _{t_{c}}V^{\mathbf{A} ^{\prime }}(t_{c},t_{-12})\right) f_{\phi }(t_{c})dt_{c} \\&= \int \limits _{t_{c}>t}H_{c}^{\mathbf{A}^{\prime }}(t_{c},t_{-12})f_{\phi }(t_{c})dt_{c} \\&= Q^{\prime }(t) \end{aligned}$$

which means that $\mu ^{\mathbf{A}}$ and $\mu ^{\mathbf{A}^{\prime }}$ generate the same expected revenue. However, under $\mu ^{\mathbf{A}}$, buyer $1$ receives the good with probability zero. If the optimal mechanism in environment $\mathbf{A}$ allocates to buyer $1$ with strictly positive probability, then it is, by definition, revenue superior to $\mu ^{\mathbf{A} }$ and hence to $\mu ^{\mathbf{A}^{\prime }}$. $\square$