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.
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Notes
Previous work establishes that public information may harm voters under majority rule and ties this phenomenon to collective preference cycles and distributional uncertainty (Gersbach 1991, 2000a, b). The more recent work of Schnakenberg (2015, 2016) and Alonso and Câmara (2016) incorporates communication or signal design by a strategic sender and shows how manipulation may occur in these settings.
For instance, Krehbiel (1991) argues that legislative organization reflects a need to facilitate information acquisition and transmission.
Though mixed strategies are not always necessary to support the equilibria of interest in this paper, they usually are needed for the discrete examples I present here and are convenient for proving existence of persuasive equilibrium strategies in the general case. One objection to my use of mixed strategies may be that experts do not appear to randomize in practice. Fortunately, we need not rely on such a literal interpretation of mixed strategies. Following the purification approach of Harsanyi (1973), we may imagine that the expert has a small bias in favor of one message or another that is unknown and therefore seemingly random to the voters. Thus, choices are random from the perspective of the voters even though the expert does not experience the choice as random. Thus, mixed strategy equilibiria in these games may be seen as the limit of pure strategy equilibria in games where the expert has a small, unknown bias. Though I stop short of directly demonstrating the validity of this interpretation here, previous work has demonstrated the soundness of this interpretation in similar communication games (Ambrus et al. 2013, for example).
Recall that we assume no player is indifferent under \(\mu _{0}\) so \(x_{0}\) is unique.
As in any cheap talk model, there always exist a babbling equilibria with no information transmission. The argument for such an equilibrium is as follows: Suppose E randomizes uniformly over all messages following any state of the world. Then every message is uninformative to the voters and the voters choose according to their priors. Since voting strategies do not depend on messages, E has no strict incentive to deviate from this strategy, which shows that this is an equilibrium. In this paper I sidestep issues of equilibrium selection, focusing instead on whether or not influence is possible and whether or not it is desirable.
Proposition 1 does not rule out equilibria that are bad for the Expert. For instance, the equilibrium in Example 2 need not change if we assumed that the Expert always wants the proposal to fail: if we assume the voters would also vote to approve for any messages off the path of play, there would still be no profitable deviation for the Expert. Note, however, that such situations are excluded from our definition of influential equilibrium. This avoids further discussion of unreasonable equilibria that are supported only because of assumptions about offpath beliefs.
In previous models, delegation may be preferred to cheap talk if preference divergence between the informed and uninformed agent is not too large (Dessein 2002). That result does not hold in the discrete game considered here: if the Expert’s advice influences the outcome then it is outcomeequivalent to delegation, and when all cheap talk equilibria are noninformative the voters generally would have preferred not to delegate.
Since N is finite and each voter can take one of two actions, any equilibrium outcome can be supported using a finite number of messages. Thus, to simplify notation, I focus on signaling strategies that place positive probability on a finite number of messages.
The convex hull of \(\{\mu _{1},\mu _{2},\dots ,\mu _{K}\}\subset \Delta (\Omega )\) is the set of all \(\mu \in \Delta (\Omega )\) such that, for some \((\rho _{1},\rho _{2},\dots ,\rho _{K})\) with \(\rho _{j}\ge 0\) for all j and \(\sum _{j=1}^{K}\rho _{j}=1\), we have \(\mu =\sum _{j=1}^{K}\rho _{j}\mu _{j}\).
The remedies considered here maintain the assumption that the voters are are facing an exogenous binary agenda. Under some circumstances, allowing amendments to the agenda by the voters may remedy some of the problems associated with information transmission and voting. However, since we are most interested in cases where there is no representative voter, the effect of such a protocol is unpredictable and in most cases the welfare effects are unclear.
In a setting where barring the Expert to speak is feasible, Proposition 4 can be interpreted as a prediction about when voters would allow the Expert to speak: voters would take advice from the Expert when they would be equally willing to delegate their decision to that Expert.
Limiting communication to endorsements essentially imposes a requirement that the maximum number of unique messages used by the expert is 2. A related question is whether any other limitations on the number of unique messages would achieve the same result. The answer to this question depends on the voting rule. Constructing a unanimously harmful manipulative equilibrium requires using (a) one unique message for each winning coalition that is the target of persuasion along the equilibrium path and (b) ensuring that no voter is a member of every targeted coalition. Thus, the minimum number of messages required to support a unanimously harmful manipulative equilibrium is equal to the cardinality of the smallest set T of winning coalitions under the voting rule such that the intersection of all the coalitions of T is empty. This number is called the Nakamura number of the voting rule (Nakamura 1979). The Nakamura number for majority rule is 3 so allowing any number of messages greater than 2 could still allow unanimously harmful manipulative persuasion. However, some other voting rules (for instance, some stricter supermajority rules) have larger Nakamura numbers and for those rules limiting the expert to binary endorsements is more restrictive than necessary. I focus here on binary endorsements because it is the only solution that works for all proper voting rules and because Proposition 2 implies that any beneficial equilibrium is outcomeequivalent to one with endorsements.
This example also demonstrates that obfuscation is not necessary for manipulative persuasion. Instead, the result simply shows that some ambiguity is helpful from an ex ante perspective when it avoids pitfalls of majority voting. This argument resembles the philosophical concept of a veil of ignorance: voter behavior can be made more consistent with (ex ante) social welfare when the voters are uncertain about who the identities of the winners and losers.
The results may differ somewhat from games with simultaneous communication which are not analyzed here. Sequential communication is a more realistic model of political communication since advocates rarely are asked to speak without responding to one another.
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Proofs of propositions
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
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
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
by the law of total expectation. Thus, (2) implies that
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 \(\{s_{1},\dots ,s_{K_{P}},s_{K_{P}+1},\dots ,s_{K_{P}+K_{F}}\}\):
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
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 \(\{i\in N:U_{i}^{*}(\sigma ,v^{*})\ge x_{0}{\mathbb {E}}_{\mu _{0}}[\omega _{i}]\}\in {\mathscr {D}}^{m}\).
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Schnakenberg, K.E. The downsides of information transmission and voting. Public Choice 173, 43–59 (2017). https://doi.org/10.1007/s1112701704628
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DOI: https://doi.org/10.1007/s1112701704628