The downsides of information transmission and voting

Abstract

I analyze a model of information transmission in collective choice environments. An Expert possesses private information about the consequences of passing an exogenous proposal and engages in cheap talk to persuade voters to pass or reject the proposal. The Expert may successfully persuade the voters to take her preferred action even when all or most voters would receive a better ex ante payoff with no information transmission. I consider several remedies that an institutional designer may consider in order to avoid this problem while allowing information transmission that benefits the voters. I evaluate the effects of (1) limiting Expert communication to binary endorsements, (2) encouraging competition between Experts, and (3) restricting the agenda in order to consider only one dimension at a time. None of these proposals completely eliminate negative persuasion outcomes, but limiting the Expert to binary endorsements avoids the worst manipulation while preserving beneficial information transmission.

Proofs of propositions

The verbal arguments in the body of the paper are sufficient to prove Propositions 1 and 4. The proofs of the remaining results are below.

Proof of Proposition 2

Proposition 1 implies that any equilibrium with influential cheap talk results in passing the proposal any time $\omega _{E}>0$ and rejecting the proposal any time $\omega _{E}<0$. Thus, an influential equilibrium strategy profile $(\sigma ^{*},v^{*})$ partitions $\Omega$ into two sets

$$\begin{aligned}&S_{P}\supseteq \{x:\omega _{E}>0\}, \text{ and } \\&S_{F}\supseteq \{x:\omega _{E}<0\} \end{aligned}$$

such that the outcome is $x=1$ when $\omega \in S_{P}$ and $x=0$ when $\omega \in S_{F}$. Let $x_{0}\in \{0,1\}$ be the voting outcome if all players vote according to their prior. For any $D\in {\mathscr {D}}$, the strategy profile $(\sigma ^{*},v^{*})$ is strictly preferred to a babbling equilibrium by the members of D if and only if

$$\begin{aligned} \Pr [\omega \in S_{P}]{\mathbb {E}}_{\mu _{0}}[\omega _{i}|\omega \in S_{P}]>x_{0}{\mathbb {E}}_{\mu _{0}}[\omega _{i}] \end{aligned}$$

(2)

for all $i\in D$. Equation 2 is also the condition under which all members of D would strictly prefer to delegate to the Expert.

Furthermore, if (2) holds, then we can support an equilibrium with two messages $s\ne s^{\prime }$ where $\sigma (s|\omega )=1$ if $\omega \in S_{P}$ and $\sigma (s^{\prime }|\omega )=1$ if $\omega \in S_{F}$. To prove this statement, note that

$$\begin{aligned} {\mathbb {E}}_{\mu _{0}}[\omega _{i}]=\Pr [\omega \in S_{P}]{\mathbb {E}}_{\mu _{0}}[\omega _{i}|\omega \in S_{P}]+\Pr [\omega \in S_{F}]{\mathbb {E}}_{\mu _{0}}[\omega _{i}|\omega \in S_{F}] \end{aligned}$$

by the law of total expectation. Thus, (2) implies that

$$\begin{aligned} \Pr [\omega \in S_{F}]{\mathbb {E}}_{\mu _{0}}[\omega _{i}|\omega \in S_{F}]<{\mathbb {E}}_{\mu _{0}}[\omega _{i}]. \end{aligned}$$

(3)

Thus, all members of D will vote in favor of passage when it is revealed that $\omega \in S_{P}$ (by 2) and against when it is revealed that $\omega \in S_{F}$ (by 3). Furthermore, given these outcomes, E has no incentive to deviate from the messaging strategy, so this describes an equilibrium. $\square$

Proof of Proposition 3

The law of total probability and Proposition 1 imply that an influential equilibrium does not exist if $\mu _{0}(\omega |\omega _{E}>0)\not \in \text{ co }(W_{{\mathscr {D}}})$ or $\mu _{0}(\omega |\omega _{E}<0)\not \in \text{ co }(L_{{\mathscr {D}}}),$ since no feasible distribution of posteriors can lead to outcomes consistent with the preferences of the Expert. Thus, I show that $\mu _{0}(\omega |\omega _{E}>0)\in \text{ co }(W_{{\mathscr {D}}})$ and $\mu _{0}(\omega |\omega _{E}<0)\in \text{ co }(L_{{\mathscr {D}}})$ implies the existence of an influential equilibrium. Suppose that $\mu _{0}(\omega |\omega _{E}>0)\in \text{ co }(W_{{\mathscr {D}}})$ and $\mu _{0}(\omega |\omega _{E}<0)\in \text{ co }(L_{{\mathscr {D}}}).$. By Caratheodory’s theorem, there are finite sets of priors (with size $K_{P}$ and $K_{F}$) $\{\mu _{1},\dots ,\mu _{K_{P}}\}\subset W_{{\mathscr {D}}}$ and $\{\mu _{K_{P}+1},\dots ,\mu _{K_{P}+K_{F}}^{F}\}\subset L_{{\mathscr {D}}}$ and real vectors $(\rho _{1},\dots ,\rho _{K_{P}})$ and $(\rho _{K_{P}+1},\dots ,\rho _{K_{P}+K_{F}})$ such that:

• $\rho _{j}\ge 0$ for all $j\in \{1,\dots ,K_{P}+K_{F}\}$

• $\sum _{j=1}^{K_{P}}\rho _{j}=1$ and $\sum _{j=K_{P}+1}^{K_{P}+K_{F}}\rho _{j}=1$;

• $\mu _{0}(\omega |\omega _{E}>0)=\sum _{j=1}^{K_{P}}\rho _{j}\mu _{j}(\omega )$ and $\mu _{0}(\omega |\omega _{E}<0)=\sum _{j=K_{P}+1}^{K_{P}+K_{F}}\rho _{j}\mu _{j}(\omega )$.

• $\mu _{j}(\omega )=0$ if for $j\in \{1,\dots ,K_{P}\}$ if $\omega _{E}<0$ and $\mu _{j}(\omega )=0$ for $j\in \{K_{P}+1,\dots ,K_{P}+K_{F}\}$ if $\omega _{E}>0$.

Consider the following mixed strategy for E with support on a finite set of messages $\{s_{1},\dots ,s_{K_{P}},s_{K_{P}+1},\dots ,s_{K_{P}+K_{F}}\}$:

$$\begin{aligned} \sigma ^{*}(s_{j}|\omega )=\frac{\mu _{j}(\omega )\rho _{j}}{\mu _{0}(\omega )}. \end{aligned}$$

(4)

Note that this strategy has the property that $\sigma ^{*}(s_{j}|\omega )=0$ if $\omega _{E}<0$ and $j\le K_{P}$ and $\sigma ^{*}(s_{j}|\omega )=0$ if $\omega _{E}>0$ and $j\ge K_{P}$. The voters’ posterior beliefs following $s=s_{j}$ are then

$$\begin{aligned} \mu _{s_{j}}(\omega )=\frac{\mu _{0}(\omega )\frac{\mu _{j}(\omega )\rho _{j}}{\mu _{0}(\omega )}}{\int _{\Omega }\mu _{0}(\omega ^{\prime })\frac{\mu _{j}(\omega ^{\prime })\rho _{j}}{\mu _{0}(\omega )}d\omega ^{\prime }}=\frac{\mu _{j}(\omega )\rho _{j}}{\int _{\Omega }\mu _{j}(\omega ^{\prime })\rho _{j}d\omega ^{\prime }}=\frac{\mu _{j}(\omega )\rho _{j}}{\rho _{j}}=\mu _{j}(\omega ). \end{aligned}$$

(5)

Thus, $\sigma ^{*}$ induces posterior beliefs in $\{\mu _{1},\dots ,\mu _{K_{P}}\}\subset W_{{\mathscr {D}}}$ for all $\omega _{E}>0$ and induces posterior beliefs in $\{\mu _{K_{P}+1},\dots ,\mu _{K_{P}+K_{F}}\}\subset L_{{\mathscr {D}}}$ for all $\omega _{E}<0$. Thus, given sequentially rational voting strategies we have $x=1$ when $\omega _{E}>0$ and $x=0$ when $\omega _{E}<0$, and E has no incentive to deviate from this strategy. $\square$

Proof of Proposition 5

Let $\overline{z}$ be the median of $(z_{1},\dots ,z_{n})$ and let $\overline{i}$ be the index of the voter with $z_{i}=\overline{z}$. Since the number of voters is odd, $\overline{z}$ is preferred to all points in $\Theta (q)$. Banks and Duggan (2006) show that, if voters’ utility functions are quadratic and there exists a core point at a voter’s ideal point, then the core voter is decisive over lotteries. That is, for all $\mu \in \Delta (\Omega )$, $\{i\in N:{\mathbb {E}}_{\mu }[\omega _{i}]\ge 0\}\in {\mathscr {D}}^{m}$ if and only if ${\mathbb {E}}_{\mu }[\omega _{\overline{i}}]\ge 0\}$. Thus, since proposals only pass if ${\mathbb {E}}_{\mu _{s}}[\omega _{\overline{i}}]>0$, it must be the case that $U_{\overline{i}}^{*}(\sigma ,v^{*})\ge x_{0}{\mathbb {E}}_{\mu _{0}}[\omega _{\overline{i}}]$ if $(\sigma ,v^{*})$ is a persuasive equilibrium. Furthermore, by the result of Banks and Duggan (2006), this implies that $\{i\in N:U_{i}^{*}(\sigma ,v^{*})\ge x_{0}{\mathbb {E}}_{\mu _{0}}[\omega _{i}]\}\in {\mathscr {D}}^{m}$.