Optimal monotone signals in Bayesian persuasion mechanisms

Abstract

This paper develops a new approach—based on the majorization theory—to the information design problem in Bayesian persuasion mechanisms, i.e., models in which the sender selects the signal structure of the agent(s) who then reports it to the non-strategic receiver. We consider a class of mechanisms in which the posterior payoff of the sender depends on the value of a realized posterior mean of the state, its order in the sequence of possible means, and the marginal distribution of signals. We provide a simple characterization of mechanisms in which optimal signal structures are monotone partitional. Our approach has two economic implications: it is invariant to monotone transformations of the state and allows to decompose setups with multiple agents into independent Bayesian persuasion mechanisms with a single agent. As the main application of our characterization, we show the optimality of monotone partitional signal structures in all selling mechanisms with independent private values and quasi-linear preferences. We also provide sharp characterization of optimal signal structures for second-price auctions and posted-price sales.

Appendix

Proof of Lemma 1

Consider a distribution of posterior means $\left\{ g,\omega \right\} \in {\mathcal {G}}_{n}$, such that $\omega _{s}=E\left[ \upsilon \left( \theta \right) |s\right] ,s\in \left[ n\right]$, where $\upsilon \left( \theta \right)$ is right-continuous and strictly increasing (that is, integrable). Let $\sigma \in \Pi _{n}$ be a signal structure that generates $\left\{ g,\omega \right\}$ and $H\left( s,\theta \right)$ be the joint distribution of $\left( s,\theta \right)$ induced by $\sigma$ according to (2).

Because $f\left( \theta \right) >0$ almost everywhere on $\left[ 0,1\right]$, $F\left( \theta \right)$ is strictly increasing and differentiable in $\theta$ for $\theta \in \left( 0,1\right)$. As a result, $F^{-1}\left( x\right)$ exists, uniquely determined, and strictly increasing in x on $\left( 0,1\right)$. Consider the monotone partitional $\sigma ^{o}\in \Pi _{n}$ with the cutoffs $\left\{ \theta _{s}^{o}\right\} _{s=1}^{n-1}$, such that $\theta _{s}^{o}=F^{-1}\left( G_{s}\right)$, where $G_{s} =\sum \nolimits _{i=1}^{s}g_{i}$ is the cdf of signals generated by $\sigma$. Because $g>0$ and $F\left( \theta \right)$ is strictly increasing, the sequence $\left\{ \theta _{s}^{o}\right\} _{s=1}^{n}$ is unique, strictly increasing, and $\theta _{s}^{o}\in \left( 0,1\right) ,s\in \left[ n-1\right]$. By construction, we have

$$\begin{aligned} G_{s}^{o}=F\left( \theta _{s}^{o}\right) =G_{s}\text { for all }s\in \left[ n\right] , \end{aligned}$$

(48)

or, equivalently, the marginal distribution of signals generated by $\sigma ^{o}$ is g. Hence, $H\left( s,\theta \right) \in {\mathcal {M}}(G,F)$ and $H^{o}\left( s,\theta \right) \in {\mathcal {M}}(G,F)$, where ${\mathcal {M}}(G,F)$ is the class of all bivariate distributions with fixed marginal cdf $F\left( \theta \right)$ and $G_{s}$. Also, there is only one monotone partitional $\left\{ g,\omega ^{o}\right\}$ $\in {\mathcal {G}}_{n}^{+}$, since any changes in $\left\{ \theta _{s}^{o}\right\} _{s=1}^{n-1}$ result in the violation of (48) due to the strict monotonicity of $F\left( \theta \right)$.

Second, since the step function $\varkappa \left( \theta \right) =\sum_{k-1}^n k{\mathbf {1}}_{(\theta^{o} _{k-1},\theta^{o} _{k}]}\left( \theta \right)$ is increasing in $\theta$ on $\left[ 0,1\right]$, then the joint distribution $H^{o}\left( s,\theta \right)$ generated by $\sigma ^{o}$ is given by

$$\begin{aligned} H^{o}\left( s,\theta \right) =\Pr \left\{ s^{\prime }\le s,\theta ^{\prime }\le \theta \right\} =\Pr \left\{ \varkappa \left( \theta ^{\prime }\right) \le s,\theta ^{\prime }\le \theta \right\} =\min \left\{ G_{s},F\left( \theta \right) \right\} . \end{aligned}$$

As shown by Joe (1997), if $H\left( s,\theta \right) \in {\mathcal {M}}(G,F)$, then

$$\begin{aligned} H^{o}\left( s,\theta \right) =\min \left\{ G_{s},F\left( \theta \right) \right\} \ge H\left( s,\theta \right) \text { for all }s\in \left[ n\right] \text { and}\ \theta \in \left[ 0,1\right] . \end{aligned}$$

(49)

Finally, Tchen (1980) shows that if (49) holds for $H^{o}\left( s,\theta \right) \in {\mathcal {M}}(G,F)$ and $H\left( s,\theta \right) \in {\mathcal {M}}(G,F)$, and $\ \upsilon \left( \theta \right)$ are right continuous, increasing, and integrable, then

$$\begin{aligned} E\left[ E^{o}\left[ \upsilon \left( \theta \right) |s\right] |s\le k\right] \le E\left[ E\left[ \upsilon \left( \theta \right) |s\right] |s\le k\right] \text { for all }k\in \left[ n\right] , \end{aligned}$$

(50)

which implies (33). Finally, $E\left[ \omega _{s}^{o}\right] =E\left[ \omega _{s}\right] =E\left[ \upsilon \left( \theta \right) \right]$ and (48) result in $\omega ^{o}\succ _{g}\omega$. $\square$

Proof of Theorem 1

Consider a distribution of posterior means $\left\{ g,\omega \right\} \in {\mathcal {G}}_{n}$, which provides the ex-ante payoff $EV\left( g,\omega \right)$ according to (13), where $V_{s}\left( g,\omega _{s}\right)$ satisfies Condition 1 for g. We show below that $EV\left( g,\omega \right) \le EV\left( g,\omega ^{o}\right)$, where $\left\{ g,\omega ^{o}\right\} \in {\mathcal {G}}_{n}^{+}$ is generated by the monotone partitional $\sigma ^{o} \in \Pi _{n}$ with cutoffs $\left\{ \theta _{s}\right\} _{s=1}^{n-1}=\left\{ F^{-1}\left( G_{s}\right) \right\} _{s=1}^{n-1}$.

By Lemma 1, it follows that $\omega ^{o}$ is unique and $\omega ^{o} \succ _{g}\omega$. Then, Theorem 2.4 in Cheng (1977) implies that there is a finite collection of weakly increasing sequences $\left\{ \omega ^{t}\right\} _{t=1}^{T}$, such that $\omega ^{o}=\omega ^{T+1}\succ _{g}\omega ^{T}\succ _{g}\cdots \succ _{g}\omega ^{1}\succ _{g}\omega ^{0}=\omega$, and all pairs $\omega ^{t},\omega ^{t+1},t=0,\ldots ,T$ differ in only two elements. Hence, it is sufficient to consider $\omega ^{t}$ and $\omega ^{t+1}$, which differ in $\left\{ \omega _{k}^{t},\omega _{s}^{t}\right\}$ and $\left\{ \omega _{k}^{t+1},\omega _{s}^{t+1}\right\}$, where $k<s$. Since $\omega ^{t+1} \succ _{g}\omega ^{t}$, this means

$$\begin{aligned} \omega _{k}^{t+1}&<\omega _{k}^{t}\le \omega _{s}^{t}<\omega _{s}^{t+1}\text {, and}\\ g_{k}\omega _{k}^{t}+g_{s}\omega _{s}^{t}&=g_{k}\omega _{k}^{t+1}+g_{s} \omega _{s}^{t+1}, \end{aligned}$$

where the last equality can be written as

$$\begin{aligned} g_{k}\left( \omega _{k}^{t+1}-\omega _{k}^{t}\right) =-g_{s}\left( \omega _{s}^{t+1}-\omega _{s}^{t}\right) . \end{aligned}$$

(51)

Also, because $\omega ^{o}\succ _{g}\omega ^{t},t=1,\ldots ,T$, and $\omega ^{o} \in \Omega _{n}^{+}$, we have $\upsilon \left( 0\right) <\omega _{1}^{o} \le \omega _{1}^{t}$ and $\omega _{n}^{t}\le \omega _{n}^{o}\le \upsilon \left( 1\right)$. Since $\omega ^{t}$ is also weakly increasing, it follows that $\omega ^{t}\in \Omega _{n}$, that is, $\left\{ g,\omega ^{t}\right\} \in \Delta _{n}\times \Omega _{n}$.³⁷ Therefore, $EV\left( g,\omega ^{t}\right)$ is defined for all $t=1,\ldots ,T$.

Suppose first that $\omega ^{t}\in \Omega _{n}^{+}$ and $\omega ^{t+1}\in \Omega _{n}^{+}$, i.e., $\omega ^{t}$ and $\omega ^{t+1}$ are strictly increasing. Since $\Omega _{n}^{+}$ is convex, then $\omega ^{\theta }=\left( 1-\theta \right) \omega ^{t}+\theta \omega ^{t+1}\in \Omega _{n}^{+}$ for any $\theta \in \left( 0,1\right)$. Also, because $\Omega _{n}^{+}$ is open and $EV\left( g,\omega \right)$ is differentiable in $\omega$ on $\Omega _{n} ^{+}$ for all $g\in \Delta _{n}$, then $\frac{\partial }{\partial \omega _{i} }EV\left( g,\omega ^{\theta }\right)$ exists for all $i\in \left[ n\right]$ and $\theta \in \left( 0,1\right)$ and is given by

$$\begin{aligned} \frac{\partial EV\left( g,\omega ^{\theta }\right) }{\partial \omega _{i}} =g_{i}\frac{\partial V_{i}\left( g,\omega _{i}^{\theta }\right) }{\partial \omega _{i}}. \end{aligned}$$

(52)

Then, Lagrange’s mean value theorem implies

$$\begin{aligned} \Delta EV= & {} EV\left( g,\omega ^{t+1}\right) -EV\left( g,\omega ^{t}\right) =\left( \omega _{k}^{t+1}-\omega _{k}^{t}\right) \frac{\partial EV\left( g,\omega ^{\theta }\right) }{\partial \omega _{k}}\\&+\left( \omega _{s} ^{t+1} -\omega _{s}^{t}\right) \frac{\partial EV\left( g,\omega ^{\theta }\right) }{\partial \omega _{s}}, \end{aligned}$$

for some $\theta \in \left( 0,1\right)$. This gives

$$\begin{aligned} \Delta EV&=g_{k}\left( \omega _{k}^{t+1}-\omega _{k}^{t}\right) \frac{\partial V_{k}\left( g,\omega _{k}^{\theta }\right) }{\partial \omega _{k}}+g_{s}\left( \omega _{s}^{t+1}-\omega _{s}^{t}\right) \frac{\partial V_{s}\left( g,\omega _{s}^{\theta }\right) }{\partial \omega _{s}}\nonumber \\&=g_{s}\left( \omega _{s}^{t+1}-\omega _{s}^{t}\right) \left( \frac{\partial V_{s}\left( g,\omega _{s}^{\theta }\right) }{\partial \omega _{s}}-\frac{\partial V_{k}\left( g,\omega _{k}^{\theta }\right) }{\partial \omega _{k}}\right) \ge 0, \end{aligned}$$

(53)

where first and second equalities follow from (52) and (51), respectively, and the inequality follows from Condition 1, $s>k$, and $\omega _{s}^{\theta }>\omega _{k}^{\theta }$.

If $\omega ^{t}\in \Omega _{n}$ and/or $\omega ^{t+1}\in \Omega _{n}$ are not strictly increasing, then consider $\omega ^{t,\varepsilon }=\left( 1-\varepsilon \right) \omega ^{t}+\varepsilon \omega ^{o}$ and $\omega ^{t+1,\varepsilon }=\left( 1-\varepsilon \right) \omega ^{t+1}+\varepsilon \omega ^{o}$, where $\varepsilon \in \left( 0,1\right)$. Because $\omega ^{o}\in \Omega _{n}^{+}$, then $\omega ^{t,\varepsilon }\in \Omega _{n}^{+}$ and $\omega ^{t+1,\varepsilon }\in \Omega _{n}^{+}$ for any $\varepsilon \in \left( 0,1\right)$. Then, $\omega _{k}^{t+1}<\omega _{k}^{t}\le \omega _{s}^{t} <\omega _{s}^{t+1}$ and $\omega _{k}^{o}<\omega _{s}^{o}$ result in

$$\begin{aligned} \left( 1-\varepsilon \right) \omega _{k}^{t+1}+\varepsilon \omega _{k} ^{o}<\left( 1-\varepsilon \right) \omega _{k}^{t}+\varepsilon \omega _{k} ^{o}<\left( 1-\varepsilon \right) \omega _{s}^{t}+\varepsilon \omega _{s} ^{o}<\left( 1-\varepsilon \right) \omega _{s}^{t+1}+\varepsilon \omega _{s}^{o} \end{aligned}$$

for all $\varepsilon \in \left( 0,1\right)$. Also, $\omega _{i}^{t}=\omega _{i}^{t+1},i\notin \left\{ k,s\right\}$ implies

$$\begin{aligned} \omega _{i}^{t,\varepsilon }=\left( 1-\varepsilon \right) \omega _{i} ^{t}+\varepsilon \omega _{i}^{o}=\left( 1-\varepsilon \right) \omega _{i} ^{t+1}+\varepsilon \omega _{i}^{o}=\omega _{i}^{t+1,\varepsilon },i\notin \left\{ k,s\right\} . \end{aligned}$$

Thus, $\omega ^{t,\varepsilon }$ and $\omega ^{t+1,\varepsilon }$ differ only in $\left\{ \omega _{k}^{t,\varepsilon },\omega _{s}^{t,\varepsilon }\right\}$ and $\left\{ \omega _{k}^{t+1,\varepsilon },\omega _{s}^{t+1,\varepsilon }\right\}$, and $\omega ^{t+1,\varepsilon }\succ _{g}\omega ^{t,\varepsilon }$. As a result, (53) holds for $\omega ^{t,\varepsilon }$ and $\omega ^{t+1,\varepsilon }$:

$$\begin{aligned} EV\left( g,\omega ^{t+1,\varepsilon }\right) -EV\left( g,\omega ^{t,\varepsilon }\right) \ge 0\text { for all }\varepsilon \in \left( 0,1\right) . \end{aligned}$$

(54)

Finally, (54) and the continuity of $EV\left( g,\omega \right)$ in $\omega$ on $\Omega _{n}$ for all $g\in \Delta _{n}$ imply

$$\begin{aligned} EV\left( g,\omega ^{t+1}\right) -EV\left( g,\omega ^{t}\right) =\lim _{\varepsilon \downarrow 0}\left( EV\left( g,\omega ^{t+1,\varepsilon }\right) -EV\left( g,\omega ^{t,\varepsilon }\right) \right) \ge 0\text {.} \end{aligned}$$

$\square$

Proof of Corollary 1

If $\frac{\partial }{\partial x}V_{s}\left( g,x\right) |_{x=\omega _{s}}\le \frac{\partial }{\partial x}V_{k}\left( g,x\right) |_{x=\omega _{k}}$ for all $s>k$ and $\omega _{s}\ge \omega _{k}$, then the result follows from the same lines as those in the proof of Theorem 1 by taking $\omega _{s}^{o}=E\left[ \upsilon \left( \theta \right) \right] ,s\in \left[ n\right]$ and noting that $\omega \succ _{g}\omega ^{o}$. Finally, $\left\{ g,\omega ^{o}\right\} \in {\mathcal {G}}_{n}$, since it can be generated by $\sigma ^{o}\in \Pi _{n}$, such that $\sigma _{s}^{o}\left( \theta \right) =g_{s},s\in \left[ n\right]$. $\square$

Proof of Theorem 2

Consider a Bayesian persuasion mechanism with the ex-ante payoff to the sender of the form

$$\begin{aligned} EV\left( g,\omega \right) =\sum \limits _{s=1}^{n}g_{s}\left( \alpha _{s}\left( g\right) \omega _{s}+\beta _{s}\left( g\right) \right) , \end{aligned}$$

and suppose that $\alpha _{s}\$ is not weakly increasing in s. Given any signal structure $\sigma \in \Pi _{n}$, we will construct a redundant monotone partitional $\sigma ^{o}\in \Pi _{n}$, which generates the same marginal distribution of signals, i.e., $g^{o}=g$, and provides a higher ex-ante payoff to the sender, $EV\left( g,\omega ^{o}\right) \ge EV\left( g,\omega \right)$. A new signal structure $\sigma ^{o}$ is derived from $\sigma$ via a finite iterated sequence $\left\{ \sigma ^{t}\right\} _{t=1}^{T}$, where $T\le n$, such that each $\sigma ^{t}$ improves the ex-ante payoff of the sender relative to $\sigma ^{t-1}$.

We start the proof by noting first that $EV\left( g,\omega \right)$ represents the sender’s ex-ante payoff in the hypothetical Bayesian persuasion mechanism such that the receiver’s action rule is $a=\left\{ \alpha _{s}\left( g\right) ,\beta _{s}\left( g\right) \right\}$; the sender’s payoff function is $U\left( a,\theta \right) =\alpha _{s}\left( g\right) \upsilon \left( \theta \right) +\beta _{s}\left( g\right)$; and the agent’s payoff function is $U^{M}\left( a,\theta \right) =0$ for all $\left( a,\theta \right)$.

Next, consider the following finite iterative sequence of hypothetical mechanisms. Each iteration $t=1,\ldots ,T$ consists of two hypothetical mechanisms: one with an initial signal structure $\sigma$ and the modified $\alpha _{s}^{t}\left( g\right)$; and the other one with an initial $\alpha _{s}\left( g\right)$ and a modified $\sigma ^{t}$. Along this sequence, $\beta _{s}\left( .\right)$ and the marginal distribution of signals $g^{t}$ remain unchanged, i.e., $g^{t}=g$ for all t. Because of that, without loss of generality we put $\beta _{s}\left( g\right) =0$ for all $s\in \left[ n\right]$. Then, we call $\left\{ \alpha _{s}\left( g\right) \right\} _{s=1}^{n}$ the action rule and denote it $\alpha _{s}$ for the notational simplicity.

Put $\sigma ^{0}=\sigma$ and $\omega ^{0}=\omega$. Also, denote $\alpha _{s} ^{0}=\alpha _{s}$ the initial action rule, and $\alpha _{s}^{t},t>0$ the auxiliary “ironing” action rule, which is defined below. Denote ${\mathcal {S}}_{\alpha }=\left\{ s|\alpha _{s} =\alpha \right\} \subset \left[ n\right]$ the subset of signals which induce an action $\alpha$ (initial or ironing, depending on the mechanism). Thus, the probability of inducing an action $\alpha$ is $g_{\alpha }=\sum \nolimits _{s\in {\mathcal {S}}_{\alpha }}g_{s}$. In the t-mechanism with a modified action rule $\alpha ^{t}$, where $t=0,\ldots ,n-1$, let $k_{t}\in \left[ n\right] \backslash \left\{ 1\right\}$ be such that $\alpha _{k_{t}}^{t}<\alpha _{k_{t}-1}^{t}$. Consider a $\left( t+1\right)$-mechanism with the signal structure $\sigma ^{t+1}$, which is derived from $\sigma ^{t}$ and $\alpha ^{t}\$ as follows:

$$\begin{aligned} \sigma _{s}^{t+1}\left( \theta \right) =\left\{ \begin{array} [c]{ll} \sigma _{s}^{t}\left( \theta \right) &{} \quad \text { if }s\notin {\mathcal {S}} _{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}\text {, and}\\ \frac{g_{s}}{g_{\alpha _{k_{t}-1}^{t}}+g_{\alpha _{k_{t}}^{t}}}\sum \limits _{j\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t} }^{t}}}\sigma _{j}^{t}\left( \theta \right) &{} \quad \text { if }s\in {\mathcal {S}} _{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}. \end{array} \right. \end{aligned}$$

That is, $\sigma ^{t+1}=\left\{ \sigma _{s}^{t+1}\left( \theta \right) \right\} _{s=1}^{n}$ is identical to $\sigma ^{t}$ for signals $s\notin {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}$, but randomizes among signals $s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}} \cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}$ with probabilities $g_{s}^{t+1} =g_{s}^{t}=g_{s}$. Thus, $\sigma ^{t+1}$ generates a sequence of n posterior means $\left\{ \omega _{s}^{t+1}\right\} _{s=1}^{n}$, such that $\omega _{s}^{t+1}=\omega _{s}^{t}\$ if $s\notin {\mathcal {S}}_{\alpha _{k-1}^{t}} \cup {\mathcal {S}}_{\alpha _{k}^{t}}$. If $s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t} }\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}$, then

$$\begin{aligned} \omega _{s}^{t+1}&=E\left[ \omega _{s}|s\in {\mathcal {S}}_{\alpha _{k_{t} -1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}\right] =E\left[ \omega _{s} ^{t}|s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}} ^{t}}\right] \\&=\frac{1}{\sum \limits _{j\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}} \cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{j}}\sum \limits _{j\in {\mathcal {S}} _{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{j}\omega _{j}=\frac{g_{\alpha _{k_{t}-1}^{t}}}{g_{\alpha _{k_{t}-1}^{t}}+g_{\alpha _{k_{t}}^{t}}}\omega _{k_{t}-1}^{t}+\frac{g_{\alpha _{k_{t}}^{t}}}{g_{\alpha _{k_{t}-1}^{t}}+g_{\alpha _{k_{t}}^{t}}}\omega _{k_{t}}^{t}. \end{aligned}$$

By construction, $\left\{ g,\omega ^{t+1}\right\} \in {\mathcal {G}}_{n}^{+}$. Next, consider the $\left( t+1\right)$-mechanism with a modified “ironing” action rule $\left\{ \alpha _{s}^{t+1}\right\} _{s=1}^{n}\$ defined as follows. First, $\alpha _{s} ^{t+1}=\alpha _{s}^{t}\$ if $s\notin {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}} \cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}$. If $s\in {\mathcal {S}}_{\alpha _{k_{t} -1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}$, then

$$\begin{aligned} \alpha _{s}^{t+1}&=E\left[ \alpha _{s}|s\in {\mathcal {S}}_{\alpha _{k_{t} -1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}\right] =E\left[ \alpha _{s} ^{t}|s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}} ^{t}}\right] \\&=\frac{1}{\sum \limits _{j\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}} \cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{j}}\sum \limits _{j\in {\mathcal {S}} _{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{j}\alpha _{j}=\frac{g_{\alpha _{k_{t}-1}^{t}}}{g_{\alpha _{k_{t}-1}^{t}}+g_{\alpha _{k_{t}}^{t}}}\alpha _{k_{t}-1}^{t}+\frac{g_{\alpha _{k_{t}}^{t}}}{g_{\alpha _{k_{t}-1}^{t}}+g_{\alpha _{k_{t}}^{t}}}\alpha _{k_{t}}^{t}. \end{aligned}$$

Denote $EV\left( g,\omega ^{t}\right)$ the sender’s ex-ante payoff in the t-mechanism with the signal structure $\sigma ^{t}$ and the initial action rule $\left\{ \alpha _{s}\right\} _{s=1}^{n}$. Then, the marginal ex-ante payoff to the sender in $\left( t+1\right)$-mechanism, $\Delta EV^{t}=EV\left( g,\omega ^{t+1}\right) -EV\left( g,\omega ^{t}\right)$, is determined as

$$\begin{aligned} \Delta EV^{t}= & {} \sum \limits _{s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}} \cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{s}\alpha _{s}\left( \omega _{s} ^{t+1}-\omega _{s}^{t}\right) =\sum \limits _{s\in {\mathcal {S}}_{\alpha _{k_{t} -1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{s}\alpha _{s}E\left[ \omega _{s}|s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}} _{\alpha _{k_{t}}^{t}}\right] \\&-\left( \sum \limits _{s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}}g_{s}\alpha _{s}E\left[ \omega _{s}|s\in {\mathcal {S}}_{\alpha _{k_{t}-1}^{t}}\right] +\sum \limits _{s\in {\mathcal {S}}_{\alpha _{k_{t}}^{t}}}g_{s}\alpha _{s}E\left[ \omega _{s}|s\in {\mathcal {S}}_{\alpha _{k_{t}}^{t}}\right] \right) \\= & {} \left( \alpha _{k_{t}-1}^{t}g_{\alpha _{k_{t}-1}^{t}}+\alpha _{k_{t}} ^{t}g_{\alpha _{k_{t}-1}^{t}}\right) E\left[ \omega _{s}|s\in {\mathcal {S}} _{\alpha _{k_{t}-1}^{t}}\cup {\mathcal {S}}_{\alpha _{k_{t}}^{t}}\right] \\&-\left( \alpha _{k_{t}-1}^{t}g_{\alpha _{k_{t}-1}^{t}}\omega _{k_{t}-1} ^{t}+\alpha _{k_{t}}^{t}g_{\alpha _{k_{t}}^{t} }\omega _{k_{t} }^{t}\right) \\= & {} \sum \limits _{l\in \left\{ k_{t}-1,k_{t}\right\} }g_{\alpha _{l}^{t}} V^{t}\left( E\left[ \omega _{l}^{t}|l\in \left\{ k_{t}-1,k_{t}\right\} \right] ,l\right) -\sum \limits _{l\in \left\{ k_{t}-1,k_{t}\right\} }g_{\alpha _{l}^{t}}V^{t}\left( \omega _{l}^{t},l\right) , \end{aligned}$$

where $V^{t}\left( x,l\right) =\alpha _{l}^{t}x$. By construction, $\omega ^{t}\succ _{g}\omega ^{t+1}$. Also, since $\alpha _{k_{t}}^{t} <\alpha _{k_{t}-1}^{t}$, then Condition 1 is reversed for $k=k_{t}-1,s=k_{t}$, and $\omega _{k_{t}}\ge \omega _{k_{t}-1}$. Therefore, $\Delta EV^{t}\ge 0$ by Corollary 1.

Because $n\mathcal {\ }$ is finite, after $T\le n-1$ iterations of the original mechanism we obtain the signal structure $\sigma ^{T}$ and the weakly increasing ironing action rule $\left\{ \alpha _{s}^{T}\right\} _{s=1}^{n}$, such that $EV\left( g,\omega ^{T}\right) \ge EV\left( g,\omega \right)$, where $EV\left( g,\omega ^{T}\right)$ is the sender’s ex-ante payoff in the mechanism with the signal structure $\sigma ^{T}$ and the initial action rule $\left\{ \alpha _{s}\right\} _{s=1}^{n}$.

Let $\left\{ \alpha _{s_{k}}^{T}\right\} _{k=1}^{n^{*}}$ be a strictly increasing subsequence of $\left\{ \alpha _{s}^{T}\right\} _{s=1}^{n}$, which includes all values of $\left\{ \alpha _{s}^{T}\right\} _{s=1}^{n}$.³⁸ Since $\omega _{s}^{T}=\omega _{s_{k}}^{T}$ for all $s\in {\mathcal {S}}_{\alpha _{s_{k}}^{T}}$, $\left\{ \omega _{s_{k}}^{T}\right\} _{k=1}^{n^{*}}$ is a strictly increasing subsequence of $\left\{ \omega _{s}^{T}\right\} _{s=1}^{n}$ which includes all values of $\left\{ \omega _{s}^{T}\right\} _{s=1}^{n}$. Hence, the probability of inducing an ironing action $\alpha _{s_{k}}^{T}$ or, equivalently, generating a mean $\omega _{s_{k}}^{T}$ is $g_{k}^{*} =g_{\alpha _{s_{k}}^{T}}=\sum \limits _{s\in {\mathcal {S}}_{\alpha _{s_{k}}^{T}} }g_{s},k\in \left[ {n^{*}}\right]$. Then, the sender’s ex-ante payoff in the T-mechanism is given by

$$\begin{aligned} EV\left( g,\omega ^{T}\right) =\sum \limits _{s=1}^{n}g_{s}\alpha _{s}\omega _{s}^{T}=\sum \limits _{k=1}^{n^{*}}g_{k}^{*}\sum \limits _{s\in {\mathcal {S}}_{\alpha _{s_{k}}^{T}}}\frac{g_{s}}{g_{k}^{*}}\alpha _{s} \omega _{s_{k}}^{T}=\sum \limits _{k=1}^{n^{*}}g_{k}^{*}\alpha _{s_{k}} ^{T}\omega _{s_{k}}^{T}. \end{aligned}$$

Now, consider a redundant monotone partitional $\sigma ^{*}\in \Pi _{n}$ defined as follows. First, partition $\Theta$ into $n^{*}$ intervals $\left\{ \Theta _{k}\right\} _{k=1}^{n^{*}}$, such that $g_{k}^{*} =\Pr \left\{ \theta \in \Theta _{k}\right\}$. Then, for each $k\in \left[ n^{*}\right]$, $\sigma ^{*}$ generates $s\in {\mathcal {S}}_{\alpha _{s_{k} }^{T}}$ with the probability $\frac{g_{s}}{g_{k}^{*}}$. Thus, each $s\in {\mathcal {S}}_{\alpha _{s_{k}}^{T}}$ induces $\omega _{s}^{*} =\omega _{s_{k}}^{*}$ with probability $g_{s}$. Then, the sender’s ex-ante payoff in the mechanism with the signal structure $\sigma ^{*}$ and the initial action rule $\alpha _{s}$ is

$$\begin{aligned} EV\left( g,\omega ^{*}\right)&=\sum \limits _{s=1}^{n}g_{s}\alpha _{s}\omega _{s}^{*}=\sum \limits _{k=1}^{n^{*}}g_{k}^{*}\sum \limits _{s\in {\mathcal {S}}_{\alpha _{s_{k}}^{T}}}\frac{g_{s}}{g_{k}^{*}} \alpha _{s}\omega _{s_{k}}^{*}\\&=\sum \limits _{k=1}^{n^{*}}g_{k}^{*}\omega _{s_{k}}^{*} \sum \limits _{s\in {\mathcal {S}}_{\alpha _{s_{k}}^{T}}}\frac{g_{s}}{g_{k}^{*} }\alpha _{s}=\sum \limits _{k=1}^{n^{*}}g_{k}^{*}\alpha _{s_{k}}^{T} \omega _{s_{k}}^{*}. \end{aligned}$$

Since $\left\{ g_{k}^{*},\omega _{k}^{*}\right\} _{k=1}^{n^{*}}$ is monotone partitional, we have $\left\{ \omega _{s_{k}}^{*}\right\} _{k=1}^{n^{*}}\succ _{g^{*}}\left\{ \omega _{s_{k}}^{T}\right\} _{k=1}^{n^{*}}$ by Lemma 1. Also, since $\left\{ \alpha _{s_{k}} ^{T}\right\} _{k=1}^{n^{*}}$ is strictly increasing, then Condition 1 holds. Therefore, Theorem 1 implies that $EV\left( g,\omega ^{*}\right) \ge EV\left( g,\omega ^{T}\right)$. $\square$

Proof of Theorem 3

Consider a mechanism ${\mathcal {M}}$ with an interim incentive-compatible and individually rational receiver ${\mathcal {R}} =\left\{ {\vec {q}}\left( {\vec {\omega }}_{{\vec {s}}}\right) ,{\vec {t}}\left( {\vec {\omega }}_{{\vec {s}}}\right) \right\} =\left\{ q_{i}\left( {\vec {\omega }}_{{\vec {s}}}\right) ,t_{i}\left( {\vec {\omega }}_{{\vec {s}}}\right) \right\} _{i\in {\mathcal {N}}}$ and a profile of distributions $\left\{ {\vec {g}} ,\vec {\omega }^{*}\right\} \in {\mathcal {G}}_{\left[ {\vec {n}}\right] } =\times _{i\in {\mathcal {N}}}{\mathcal {G}}_{n_{i}}$. Then, the interim payoff of buyer i upon observing signal $s_{i}$ and reporting $z_{i}$ in the mechanism ${\mathcal {M}}$ is

$$\begin{aligned} U_{s_{i},z_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) =Q_{z_{i}} ^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) \omega _{s_{i}}^{*i}-T_{z_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) . \end{aligned}$$

By Bergemann and Pesendorfer (2007), the interim payoff of buyer i with a signal $s_{i}$ in a mechanism ${\mathcal {M}}$ with an incentive-compatible and individually rational receiver ${\mathcal {R}}$ and a profile of distributions $\left\{ {\vec {g}},{\vec {\omega }}\right\} \in {\mathcal {G}}_{\left[ {\vec {n}}\right] }$ can be expressed as

$$\begin{aligned} U_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}\right)&=Q_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}\right) \omega _{s_{i}}^{i}-T_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}\right) \nonumber \\&=U_{s_{i}-1}^{i}\left( g^{-i},\omega _{i}\right) +{\tilde{Q}}_{s_{i}} ^{i}\left( g^{-i},{\vec {\omega }}\right) \left( \omega _{s_{i}}-\omega _{s_{i}-1}\right) , \end{aligned}$$

(55)

where

$$\begin{aligned} {\tilde{Q}}_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}\right) \in \left[ Q_{s_{i}-1}^{i}\left( g^{-i},{\vec {\omega }}\right) ,Q_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}\right) \right] . \end{aligned}$$

(56)

Also, the expected payment to the seller from buyer i with a signal $s_{i}$ in a mechanism ${\mathcal {M}}$ with a profile of distributions $\left\{ {\vec {g}},{\vec {\omega }}^{*}\right\}$ is

$$\begin{aligned} T_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right)&=T_{s_{i}-1} ^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) +Q_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) \omega _{s_{i}}^{*i}\nonumber \\&\quad -Q_{s_{i}-1}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) \omega _{s_{i} -1}^{*i}-{\tilde{Q}}_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) \left( \omega _{s_{i}}^{*i}-\omega _{s_{i}-1}^{*i}\right) . \end{aligned}$$

(57)

Given an initial mechanism ${\mathcal {M}}$ with a profile of sequences ${\vec {\omega }}^{*}\in \Omega _{{\vec {n}}}$, consider a reduced receiver ${\mathcal {R}}^{*}=\left\{ {\vec {q}}^{*}\left( {\vec {s}}\right) ,{\vec {t}}^{*}\left( {\vec {s}},\omega ^{i}\right) \right\}$ such that

$$\begin{aligned} {\vec {q}}^{*}\left( {\vec {s}}\right) ={\vec {q}}\left( {\vec {\omega }}_{{\vec {s}} }^{*}\right) \text { for all }{\vec {s}}\in \left[ {\vec {n}}\right] \text {.} \end{aligned}$$

(58)

That is, ${\vec {q}}^{*}\left( {\vec {s}}\right)$ replicates the allocation rule ${\vec {q}}\left( {\vec {\omega }}_{{\vec {s}}}\right)$ for the profile of sequences ${\vec {\omega }}^{*}$. This implies

$$\begin{aligned} Q_{s_{i}}^{*,i}\left( g^{-i}\right) & =E_{s_{-i}}\left[ q_{i}^{*}\left( {\vec {s}}\right) \right] =E_{s_{-i}}\left[ q_{i}\left( {\vec {\omega }}_{{\vec {s}}}^{*}\right) \right] \\ & =Q_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) \text { for all }s_{i}\in \left[ n_{i}\right] \text { and }g^{-i}\in \Delta ^{-i}. \end{aligned}$$

For an arbitrary $\left\{ {\vec {g}},{\vec {\omega }}\right\} \in {\mathcal {G}} _{\left[ {\vec {n}}\right] }$, consider a payment rule ${\vec {t}}^{*}\left( {\vec {s}},\omega ^{i}\right)$, such that the expected payment $T_{s_{i}} ^{*,i}\left( g^{-i},\omega ^{i}\right) =E_{s_{-i}}\left[ t_{i}^{*}\left( {\vec {s}},\omega ^{i}\right) \right]$ takes the form³⁹

$$\begin{aligned} T_{s_{i}}^{*,i}\left( g^{-i},\omega ^{i}\right)&=T_{s_{i}-1}^{*,i}\left( g^{-i},\omega ^{i}\right) +Q_{s_{i}}^{*,i}\left( g^{-i}\right) \omega _{s_{i}}^{i}\nonumber \\&\quad -Q_{s_{i}-1}^{*,i}\left( g^{-i}\right) \omega _{s_{i}-1}^{i}-{\tilde{Q}}_{s_{i}}^{*,i}\left( g^{-i}\right) \left( \omega _{s_{i}}^{i} -\omega _{s_{i}-1}^{i}\right) , \end{aligned}$$

(59)

where

$$\begin{aligned} {\tilde{Q}}_{s_{i}}^{*,i}\left( g^{-i}\right) ={\tilde{Q}}_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) . \end{aligned}$$

(60)

Then, (59) is equivalent to

$$\begin{aligned}&T_{s_{i}}^{*,i}\left( g^{-i},\omega ^{i}\right) =\left( {\tilde{Q}} _{s_{i}+1}^{*,i}\left( g^{-i}\right) -Q_{s_{i}}^{*,i}\left( g^{-i}\right) \right) \omega _{s_{i}}\\&\quad +\sum \limits _{k_{i}=1}^{s_{i}}\left( {\tilde{Q}}_{k_{i}+1}^{*,i}\left( g^{-i}\right) -{\tilde{Q}}_{k_{i}}^{*,i}\left( g^{-i}\right) \right) \omega _{k_{i}}. \end{aligned}$$

This results in the buyer’s interim payoff in a mechanism ${\mathcal {M}}^{*}=\left\{ {\vec {g}},{\vec {\omega }},{\mathcal {R}}^{*}\right\}$ with a reduced receiver ${\mathcal {R}}^{*}$ and a profile of distributions $\left\{ {\vec {g}},{\vec {\omega }}\right\} \in {\mathcal {G}}_{\left[ {\vec {n}}\right] }$:

$$\begin{aligned} U_{s_{i}}^{*,i}\left( g^{-i},\omega ^{i}\right)&=Q_{s_{i}}^{*,i}\left( g^{-i}\right) \omega _{s_{i}}^{i}-T_{s_{i}}^{*,i}\left( g^{-i},\omega ^{i}\right) \nonumber \\&=U_{s_{i}-1}^{*,i}\left( g^{-i},\omega ^{i}\right) +{\tilde{Q}}_{s_{i} }^{*,i}\left( g^{-i}\right) \left( \omega _{s_{i}}^{i}-\omega _{s_{i} -1}^{i}\right) , \end{aligned}$$

(61)

where ${\tilde{Q}}_{s_{i}}^{*,i}\left( g^{-i}\right) \in \left[ Q_{s_{i} -1}^{*,i}\left( g^{-i}\right) ,Q_{s_{i}}^{*,i}\left( g^{-i}\right) \right]$ by (56) and (60). Since (61) is a special case of (55) under the condition (60), the reduced receiver ${\mathcal {R}}^{*}$ is interim incentive compatible and individually rational. Moreover, (57) and (59) imply

$$\begin{aligned} T_{s_{i}}^{*,i}\left( g^{-i},\omega ^{*,i}\right) =T_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) \text { for all }i\in {\mathcal {N}}\text {,} \end{aligned}$$

that is, the ex-ante payments of a single buyer in the mechanisms ${\mathcal {M}}$ and ${\mathcal {M}}^{*}$ with the profile of distributions $\left\{ {\vec {g}},{\vec {\omega }}^{*}\right\}$ are equal:

$$\begin{aligned} EV_{i}^{{\mathcal {M}}^{*}}=\sum \limits _{s_{i}=1}^{n_{i}}g_{s_{i}}^{i} T_{s_{i}}^{*,i}\left( g^{-i},\omega ^{*,i}\right) =\sum \limits _{s_{i} =1}^{n_{i}}g_{s_{i}}^{i}T_{s_{i}}^{i}\left( g^{-i},{\vec {\omega }}^{*}\right) =EV_{i}^{{\mathcal {M}}}. \end{aligned}$$

By (26) and (27), the ex-ante payment $EV_{i}$ of buyer i in a mechanism with the profile of distributions $\left\{ {\vec {g}},{\vec {\omega }}\right\} \in {\mathcal {G}}_{\left[ {\vec {n}}\right] }$ and the reduced receiver ${\mathcal {R}}^{*}$ can be expressed as

$$\begin{aligned} EV_{i}=\sum \limits _{s_{i}=1}^{n_{i}}g_{s_{i}}^{i}V_{s_{i}}^{i}\left( g^{i},\omega _{s_{i}}^{i}\right) =\sum \limits _{s_{i}=1}^{n_{i}}g_{s_{i}} ^{i}\alpha _{s_{i}}^{i}\left( {\vec {g}}\right) \omega _{s_{i}}^{i}. \end{aligned}$$

(62)

Here, $V_{s_{i}}^{i}\left( \omega _{s_{i}}^{i},{\vec {g}}\right)$ is the posterior payoff function:

$$\begin{aligned} V_{s_{i}}^{i}\left( {\vec {g}},\omega _{s_{i}}\right) =\alpha _{s_{i}}^{i}\left( {\vec {g}}\right) \omega _{s_{i}}^{i}, \end{aligned}$$

such that

$$\begin{aligned} \alpha _{s_{i}}^{i}\left( {\vec {g}}\right) ={\tilde{Q}}_{s_{i}+1}^{*,i}\left( g^{-i}\right) -Q_{s_{i}}^{*,i}\left( g^{-i}\right) +\Delta \tilde{Q}_{s_{i}}^{*,i}\left( g^{-i}\right) \left( 1+\frac{1}{\lambda _{s_{i} }\left( g^{i}\right) }\right) , \end{aligned}$$

where $\Delta {\tilde{Q}}_{s_{i}}^{*,i}\left( g^{-i}\right) =\tilde{Q}_{s_{i}+1}^{*,i}\left( g^{-i}\right) -{\tilde{Q}}_{s_{i}}^{*,i}\left( g^{-i}\right) ,G_{s_{i}}^{i}=\sum \limits _{k_{i}=1}^{s_{i}} g_{k_{i}}^{i}$, and $\lambda _{s_{i}}\left( g^{i}\right) =\frac{g_{s_{i}} ^{i}}{1-G_{s_{i}}^{i}}$.

Since $V_{s_{i}}^{i}\left( {\vec {g}},\omega _{s_{i}}\right)$ takes the form (38) for all $i\in {\mathcal {N}}$, and ex-ante payments $EV_{i}$ of form (62) are independent across $i\in {\mathcal {N}}$ for all profiles of distributions $\left\{ {\vec {g}},\vec {\omega }\right\} \in {\mathcal {G}}_{\left[ {\vec {n}}\right] }$ with a fixed ${\vec {g}}$, Theorem 2 implies that there exists a profile of redundant monotone partitional distributions $\left\{ {\vec {g}},\vec {\omega }^{o}\right\} \in {\mathcal {G}}_{\left[ {\vec {n}}\right] }$, such that the ex-ante payment of each buyer $i\in {\mathcal {N}}$ in the mechanism ${\mathcal {M}}^{o}=\left\{ {\vec {g}},{\vec {\omega }}^{o},{\mathcal {R}}^{*}\right\}$ is higher than that in the mechanism ${\mathcal {M}}^{*}=\left\{ {\vec {g}},{\vec {\omega }}^{*},{\mathcal {R}}^{*}\right\}$:

$$\begin{aligned} EV_{i}^{{\mathcal {M}}^{o}}=\sum \limits _{s_{i}=1}^{n_{i}}g_{s_{i}}^{i} \alpha _{s_{i}}^{i}\left( {\vec {g}}\right) \omega _{s_{i}}^{o,i}\ge \sum \limits _{s_{i}=1}^{n_{i}}g_{s_{i}}^{i}\alpha _{s_{i}}^{i}\left( {\vec {g}}\right) \omega _{s_{i}}^{*,i}=EV_{i}^{{\mathcal {M}}^{*}}\text { for all }i\in {\mathcal {N}}\text {.} \end{aligned}$$

We complete the proof by showing the equivalence of distributions over interim allocation profiles in all three mechanisms, ${\mathcal {D}}^{{\mathcal {M}} }={\mathcal {D}}^{{\mathcal {M}}^{*}}={\mathcal {D}}^{{\mathcal {M}}^{o}}$. First, note that the distributions of signals ${\vec {g}}$ in mechanisms ${\mathcal {M}}$, ${\mathcal {M}}^{*}$, and ${\mathcal {M}}^{o}$ are identical. In addition, mechanism ${\mathcal {M}}^{*}$ replicates the interim allocation of mechanism ${\mathcal {M}}$ for the distribution $\left\{ {\vec {g}},{\vec {\omega }}^{*}\right\}$ by (58). Finally, the distributions of interim allocations in mechanisms ${\mathcal {M}}^{*}$ and ${\mathcal {M}}^{o}$ are identical, since these mechanisms include the same receiver ${\mathcal {R}} ^{*}$, which allocates the product on the basis of signals only. Together, these arguments imply ${\mathcal {D}}^{{\mathcal {M}}^{o}}={\mathcal {D}} ^{{\mathcal {M}}^{*}}={\mathcal {D}}^{{\mathcal {M}}}$. Formally, the probability $g_{q}^{{\mathcal {M}}}$ of inducing an allocation profile $q\in {\mathcal {Q}} ^{{\mathcal {M}}}$ in the mechanism ${\mathcal {M}}$ is given by

$$\begin{aligned} g_{q}^{{\mathcal {M}}}=\sum \limits _{{\vec {s}}\in \left[ n\right] :{\vec {q}}\left( {\vec {\omega }}_{{\vec {s}}}^{*}\right) =q}\prod \limits _{i\in {\mathcal {N}} }g_{s_{i}}^{i}=\sum \limits _{{\vec {s}}\in \left[ n\right] :{\vec {q}}^{*}\left( {\vec {s}}\right) =q}\prod \limits _{i\in {\mathcal {N}}}g_{s_{i}}^{i}=g_{q} ^{{\mathcal {M}}^{*}}=g_{q}^{{\mathcal {M}}^{o}}. \end{aligned}$$

$\square$

Proof of Theorem 4

We start the proof by showing the optimality of monotone partitional signal structures. First, it follows from (42) and (44) that $V_{s}^{1}\left( \omega _{s},g\right)$ and $V_{s}^{2}\left( \omega _{s},g\right)$ are of the form (38). Thus, given the maximum number of signals $n<\infty$, Theorem 2 implies that there are optimal redundant monotone partitional distributions $\left\{ {\bar{g}}^{r},{\bar{\omega }}^{r}\right\} \in {\mathcal {G}}_{n},r=1,2$. Denote ${\bar{\sigma }}^{r}\in \Pi _{n}$ the signal structure that generates $\left\{ {\bar{g}}^{r},{\bar{\omega }}^{r}\right\}$.

Second, the allocation and payments in the auction depend only on the values of posterior means rather than on signals, which induce them. Thus, if different signals in a distribution $\left\{ {\bar{g}}^{r},{\bar{\omega }} ^{r}\right\} \in {\mathcal {G}}_{n}$ induce identical posterior means, then ${\mathcal {V}}_{r:N}\left( {\bar{g}}^{r},{\bar{\omega }}^{r}\right) ={\mathcal {V}} _{r:N}\left( g^{r},\omega ^{r}\right)$, where $\left\{ g^{r},\omega ^{r}\right\} \in {\mathcal {G}}_{m_{r}}^{+}$ is derived from $\left\{ {\bar{g}}^{r},{\bar{\omega }}^{r}\right\}$ by collapsing all signals that induce identical posterior means and relabeling signals in such a way that the new signal space is $\left[ m\right]$. This implies that $\left\{ g^{r} ,\omega ^{r}\right\}$ is also optimal.

• (i) Consider any $n>1$ and $\left\{ g,\omega \right\} \in {\mathcal {G}}_{n}$. Since $\upsilon \left( 0\right) \ge 0$, $\upsilon \left( \theta \right)$ is strictly increasing, and $g_{s}>0,s\in \left[ n\right]$, then $\omega _{s}>0$ for all $s\in \left[ n\right]$. From (43) and $g_{s}=G_{s} -G_{s-1}$, we get

  $$\begin{aligned} g_{s}\alpha _{1,s}\left( g,N\right)&=G_{s}^{N}-G_{s-1}^{N}=\left( G_{s}-G_{s-1}\right) \left( \sum \limits _{j=0}^{N-1}G_{s}^{N-1-j}G_{s-1} ^{j}\right) \\&=g_{s}\sum \limits _{j=0}^{N-1}G_{s}^{N-1-j}G_{s-1}^{j}\text {, and}\\ \alpha _{1,s}\left( g,N\right)&=\sum \limits _{j=0}^{N-1}G_{s}^{N-1-j} G_{s-1}^{j}, \end{aligned}$$

  which is strictly increasing in s. Thus, given a fixed number of signals m, there is an optimal monotone partitional $\left\{ g^{*},\omega ^{*}\right\} \in {\mathcal {G}}_{m}^{+}$ by Theorem 1.

  In order to show that any $m<n$ is suboptimal, consider the monotone partitional $\left\{ g^{o},\omega ^{o}\right\} \in {\mathcal {G}}_{m+1}^{+}$, which is derived from $\ \left\{ g^{*},\omega ^{*}\right\} \in {\mathcal {G}}_{m}^{+}$ by splitting the interval (partition element) $\Theta _{1}^{*}=[0, \theta_1^{*}]$ into two subintervals of a positive length, $\Theta _{1}^{o}$ and $\Theta _{2}^{o}$, and relabeling the signals in $\left\{ g^{o},\omega ^{o}\right\}$ so that $\omega _{s}^{o}=\omega _{s-1}^{*}$ for $s=3,\ldots ,m+1$. This transformation decomposes the lowest posterior mean $\omega _{1}^{*}$ into two means, $\omega _{1}^{o}\$ and $\omega _{2}^{o}$, where $\omega _{1}^{o}<\omega _{1}^{*}<\omega _{2}^{o}<\omega _{3}^{o} =\omega _{2}^{*}$ by construction. Also, $\omega _{1}^{o}\$ and $\omega _{2}^{o}$ are induced with probabilities $g_{1}^{o}=\Pr \left\{ \theta \in \Theta _{1}^{o}\right\} >0$ and $g_{2}^{o}=\Pr \left\{ \theta \in \Theta _{2}^{o}\right\} >0$, where $g_{1}^{o}+g_{2}^{o}=g_{1}^{*}$ and $g_{1} ^{o}\omega _{1}^{o}+g_{2}^{o}\omega _{2}^{o}=g_{1}^{*}\omega _{1}^{*}$. Then, the difference $\Delta {\mathcal {V}}_{1}={\mathcal {V}}_{1:N}\left( g^{o},\omega ^{o}\right) -{\mathcal {V}}_{1:N}\left( g^{*},\omega ^{*}\right)$ between the expected valuations of the winning bidder under $\left\{ g^{o},\omega ^{o}\right\}$ and $\left\{ g^{*},\omega ^{*}\right\}$ is equal to

  $$\begin{aligned} \Delta {\mathcal {V}}_{1}&=\sum \limits _{s=1}^{m+1}g_{s}^{o}\alpha _{1,s}\left( g^{o}\right) \omega _{s}^{o}-\sum \limits _{s=1}^{m}g_{s}^{*}\alpha _{1,s}\left( g^{*}\right) \omega _{s}^{*}\\&=g_{1}^{o}\alpha _{1,1}\left( g^{o}\right) \omega _{1}^{o}+g_{2}^{o} \alpha _{1,2}\left( g^{o}\right) \omega _{2}^{o}-g_{1}^{*}\alpha _{1,1}\left( g^{*}\right) \omega _{1}^{*}\\&=\left( G_{1}^{o}\right) ^{N}\omega _{1}^{o}+\left( \left( G_{2} ^{o}\right) ^{N}-\left( G_{1}^{o}\right) ^{N}\right) \omega _{2} ^{o}-\left( G_{1}^{*}\right) ^{N}\omega _{1}^{*}. \end{aligned}$$

  Because $G_{1}^{o}=g_{1}^{o},G_{2}^{o}=G_{1}^{*}=g_{1}^{*}=g_{1} ^{o}+g_{2}^{o}$, and $G_{0}^{o}=G_{0}^{*}=0$, we get

  $$\begin{aligned} \Delta {\mathcal {V}}_{1}&=\left( g_{1}^{o}\right) ^{N}\omega _{1} ^{o}+\left( \left( g_{1}^{o}+g_{2}^{o}\right) ^{N}-\left( g_{1} ^{o}\right) ^{N}\right) \omega _{2}^{o}-\left( g_{1}^{*}\right) ^{N-1}g_{1}^{*}\omega _{1}^{*}\\&=\left( g_{1}^{o}\right) ^{N}\omega _{1}^{o}+\left( \left( g_{1} ^{o}+g_{2}^{o}\right) ^{N}-\left( g_{1}^{o}\right) ^{N}\right) \omega _{2}^{o}-\left( g_{1}^{o}+g_{2}^{o}\right) ^{N-1}\left( g_{1}^{o}\omega _{1}^{o}+g_{2}^{o}\omega _{2}^{o}\right) . \end{aligned}$$

  Consider the function

  $$\begin{aligned} \varphi \left( x,y\right) =c^{N}y+\left( \left( c+d\right) ^{N} -c^{N}\right) x-\left( c+d\right) ^{N-1}\left( cy+dx\right) , \end{aligned}$$

  where $c>0,d>0$, and $N>1$. Then, we have

  $$\begin{aligned} \varphi \left( y,y\right)&=c^{N}y+\left( \left( c+d\right) ^{N} -c^{N}\right) y-\left( c+d\right) ^{N-1}\left( cy+dy\right) \\&=c^{N}y+\left( c+d\right) ^{N}y-c^{N}y-\left( c+d\right) ^{N}y\equiv 0, \end{aligned}$$

  and

  $$\begin{aligned} \frac{\partial \varphi \left( x,y\right) }{\partial x}=\left( c+d\right) ^{N}-c^{N}-d\left( c+d\right) ^{N-1}=\left( c+d\right) ^{N-1}c-c^{N}. \end{aligned}$$

  Thus, $\frac{\partial \varphi \left( x,y\right) }{\partial x}>\left( c+0\right) ^{N-1}c-c^{N}=0$ for $c\ge 0,d>0$, and $N\ge 1$. This property and $\varphi \left( y,y\right) \equiv 0$ imply $\varphi \left( x,y\right) >0$ for $x>y,c>0,d>0$, and $N>1$. Finally, note that

  $$\begin{aligned} \Delta {\mathcal {V}}_{1}=\varphi \left( \omega _{2}^{o},\omega _{1}^{o}\right) \text { for }c=g_{1}^{o}\text { and }d=g_{2}^{o}. \end{aligned}$$

  Because $g_{1}^{o}>0,g_{2}^{o}>0,N>1$, and $\omega _{2}^{o}>\omega _{1}^{o}$, we have $\varphi \left( \omega _{2}^{o},\omega _{1}^{o}\right) >0$; that is, ${\mathcal {V}}_{1:N}\left( g^{o},\omega ^{o}\right) >{\mathcal {V}}_{1:N}\left( g^{*},\omega ^{*}\right)$ for all N.

• (ii) Consider any $\left\{ g,\omega \right\} \in {\mathcal {G}}_{n}$, where $n>1$. For $N=2$, it follows from (45) that

  $$\begin{aligned} \alpha _{2,s}\left( g\right) =\frac{2G_{s}-2G_{s}^{2}-2G_{s-1}+2G_{s-1} ^{2}+G_{s}^{2}-G_{s-1}^{2}}{g_{s}}=2-G_{s}-G_{s-1}, \end{aligned}$$

  which is decreasing in s, i.e., Condition 1 is reversed. Then, Corollary 1 results in ${\mathcal {V}}_{2:N}\left( g,\omega \right) \le {\mathcal {V}}_{2:N}\left( g,\omega ^{0}\right)$, where $\left\{ g,\omega ^{0}\right\} \in {\mathcal {G}}_{n}$ is such that each signal $s\in \left[ n\right]$ induces the same posterior mean $\omega _{s}^{0}=$ $E\left[ \upsilon \left( \theta \right) \right]$ with probability $g_{s}$. Finally, $\left\{ g,\omega ^{0}\right\}$ is ex-ante payoff equivalent to the degenerated $\left\{ g_{1}^{0},\omega _{1}^{0}\right\} =\left\{ 1,E\left[ \upsilon \left( \theta \right) \right] \right\}$, which is induced by the uninformative signal structure $\sigma _{1}\left( \theta \right) =1,\theta \in \Theta$.

• (iii) For a given maximum number of signals $n<\infty$, there is a solution to (46) in the set of monotone partitional signal structures of size $m\le n$ by the first part of the proof. In order to show that any $m<n$ is suboptimal if N is large enough, we follow the lines similar to those in part (i). In particular, given any $m<n$ and a monotone partitional $\left\{ g^{*},\omega ^{*}\right\} \in {\mathcal {G}}_{m}^{+}$, consider the monotone partitional $\left\{ g^{o},\omega ^{o}\right\} \in {\mathcal {G}}_{m+1}^{+}$, which is derived from $\ \left\{ g^{*},\omega ^{*}\right\} \in {\mathcal {G}}_{m}^{+}$ by decomposing the highest posterior mean $\omega _{m}^{*}$ into two means, $\omega _{m}^{o}\$ and $\omega _{m+1}^{o}$, where $\omega _{m-1}^{o}=\omega _{m-1}^{*}<\omega _{m}^{o}<\omega _{m}^{*} <\omega _{m+1}^{o}$ by construction. Posterior means $\omega _{m}^{o}\$ and $\omega _{m+1}^{o}$ are induced with probabilities $g_{m}^{o}\$and $g_{m+1}^{o}$, respectively, such that $g_{m}^{o}+g_{m+1}^{o}=g_{m}^{*}$ and $g_{m} ^{o}\omega _{m}^{o}+g_{m+1}^{o}\omega _{m+1}^{o}=g_{m}^{*}\omega _{m}^{*}$. Then, the difference $\Delta {\mathcal {V}}_{2}\left( N\right) ={\mathcal {V}} _{2:N}\left( g^{o},\omega ^{o}\right) -{\mathcal {V}}_{2:N}\left( g^{*},\omega ^{*}\right)$ in the seller’s ex-ante revenues under $\left\{ g^{o},\omega ^{o}\right\}$ and $\left\{ g^{*},\omega ^{*}\right\}$ is equal to

  $$\begin{aligned} \Delta {\mathcal {V}}_{2}\left( N\right)&=\sum \limits _{s=1}^{m+1}g_{s} ^{o}\alpha _{2,s}\left( g^{o},N\right) \omega _{s}^{o}-\sum \limits _{s=1} ^{m}g_{s}^{*}\alpha _{2,s}\left( g^{*},N\right) \omega _{s}^{*}\\&=g_{m}^{o}\alpha _{2,m}\left( g^{o},N\right) \omega _{m}^{o}+g_{m+1} ^{o}\alpha _{2,m+1}\left( g^{o},N\right) \omega _{m+1}^{o}-g_{m}^{*} \alpha _{2,m}\left( g^{*},N\right) \omega _{m}^{*}. \end{aligned}$$

  From (45), we have

  $$\begin{aligned}&g_{m}^{o}\alpha _{2,m}\left( g^{o},N\right) \\&\quad =N\left( \left( G_{m} ^{o}\right) ^{N-1}\left( 1-G_{m}^{o}\right) -\left( G_{m-1}^{o}\right) ^{N-1}\left( 1-G_{m-1}^{o}\right) \right) +\left( G_{m}^{o}\right) ^{N}-\left( G_{m-1}^{o}\right) ^{N}\\&\quad =N\left( \left( 1-g_{m+1}^{o}\right) ^{N-1}g_{m+1}^{o}-\left( 1-g_{m}^{o}-g_{m+1}^{o}\right) ^{N-1}\left( g_{m}^{o}+g_{m+1}^{o}\right) \right) \\&\qquad +\left( 1-g_{m+1}^{o}\right) ^{N}-\left( 1-g_{m}^{o}-g_{m+1}^{o}\right) ^{N}, \\&g_{m+1}^{o}\alpha _{2,m+1}\left( g^{o},N\right) =N\left( \left( G_{m+1}^{o}\right) ^{N-1}\left( 1-G_{m+1}^{o}\right) -\left( G_{m} ^{o}\right) ^{N-1}\left( 1-G_{m}^{o}\right) \right) \\&\qquad +\left( G_{m+1} ^{o}\right) ^{N}-\left( G_{m}^{o}\right) ^{N}\\&\quad =-N\left( 1-g_{m+1}^{o}\right) ^{N-1}g_{m+1}^{o}+1-\left( 1-g_{m+1} ^{o}\right) ^{N}\text {, and} \\&g_{m}^{*}\alpha _{2,m}\left( g^{*},N\right) =N\left( \left( G_{m}^{*}\right) ^{N-1}\left( 1-G_{m}^{*}\right) \right. \\&\qquad \left. -\left( G_{m-1}^{*}\right) ^{N-1}\left( 1-G_{m-1}^{*}\right) \right) +\left( G_{m}^{*}\right) ^{N}-\left( G_{m-1}^{*}\right) ^{N}\\&\quad =-N\left( 1-g_{m}^{*}\right) ^{N-1}g_{m}^{*}+1-\left( 1-g_{m}^{*}\right) ^{N}, \end{aligned}$$

  where $G_{m}^{o}=1-g_{m+1}^{o},G_{m-1}^{*}=G_{m-1}^{o}=1-g_{m}^{*}$, and $G_{m}^{*}=G_{m+1}^{o}=1$.

  Since $g_{m}^{o}>0,g_{m+1}^{o}>0$, and $g_{m}^{o}+g_{m+1}^{o}=g_{m}^{*} \le 1$, it follows that

  $$\begin{aligned} \lim _{N\rightarrow \infty }g_{m}^{o}\alpha _{2,m}\left( g^{o},N\right) & =0\text {, }\lim _{N\rightarrow \infty }g_{m+1}^{o}\alpha _{2,m+1}\left( g^{o},N\right) =1, \\ &\quad \text { and }\lim _{N\rightarrow \infty }g_{m}^{*} \alpha _{2,m}\left( g^{*},N\right) =1. \end{aligned}$$

  This results in $\lim _{N\rightarrow \infty }\Delta {\mathcal {V}}_{2}\left( N\right) =\omega _{m+1}^{o}-\omega _{m}^{*}>0$.