Elsevier

Economics Letters

Volume 180, July 2019, Pages 46-49
Economics Letters

Information aggregation with a continuum of types

https://doi.org/10.1016/j.econlet.2019.03.035Get rights and content

Highlights

  • Voters receive continuous signals but can only vote yes or no.

  • All voters wish the most likely decision to be taken.

  • Symmetric cut-off strategy profiles are considered.

  • Efficient voting can only be reached under a restrictive condition and using quota rules, essentially reproducing the case of binary signals.

Abstract

We study the problem of designing a voting rule which makes voting by cut-off strategies efficient for settings where voters have state-dependent common preferences over and vote on accepting or rejecting an issue but hold private information in the form of continuous types about the true state. We show that such rules only exist under a restrictive condition on the model parameters.

Introduction

Consider decision making situations where a group needs to accept or reject an issue. A policy is approved or rejected in a referendum, a defendant is convicted or acquitted by the jury in a court trial, or a job candidate is hired or not by a hiring committee. We model such situations as voting problems where individuals have state-dependent common preferences and private information in the form of types about the true state. We assume that types are distributed from a state-dependent continuous density. Our model is similar to that of Duggan and Martinelli (2001) and Meirowitz (2002), who derive a Condorcet jury type theorem for a fixed mechanism. We instead focus on the problem of designing a voting rule which makes voting by cut-off strategies efficient for any fixed number of voters. A cut-off strategy means that a voter votes ‘yes’ if and only if the type of the voter (a number between zero and one) exceeds a certain threshold, the cut-off. Efficiency means that the most likely correct outcome is chosen given the available information, i.e., types. Our paper can also be seen as an extension of Austen-Smith and Banks (1996) to a continuum of types.

A cut-off strategy reflects what is often called ‘informative voting’. In models where the set of types and the set of possible votes are equal or have the same size, defining informative voting is straightforward. Cut-off strategies constitute the most obvious and natural form of informative voting in our model. We consider cut-off strategy profiles where each voter uses the same cut-off. This is a natural assumption, given that in our model all voters are ex ante completely symmetric.

We show that a voting rule which makes voting according to such cut-off strategy profiles efficient, exists under a specific and restrictive condition on the model parameters (see Theorem 1). This condition says that there exists a number τ(0,1) such that whenever an issue is more likely to be true than false given the types, it should be more likely to be true than false based on the number of types exceeding τ. In that case, a specific quota rule, depending on τ, must be used in order to make voting by strategies with cut-off τ efficient. The theorem supports the intuition that when private information reflects a set of possibilities richer than indicating ‘yes’ or ‘no’ only, a binary voting rule usually cannot aggregate the whole available information efficiently.

Other related contributions focusing on information aggregation include Barelli et al. (2017), who again study asymptotic efficiency. Azrieli and Kim (2014) and Schmitz and Tröger (2012) focus on mechanism design for collective choice problems with two alternatives and private values. In some sense our paper can be seen as a much more detailed version of the model in the latter paper, and therefore we are able to express our results on the basis of these details, namely the objective probabilities of the states and the type functions, as well as the binary voting method.

Section snippets

The model

There are n2 voters, and two possible states of the world 0 and 1. The prior probability of state 1 is equal to π, with 0<π<1.

Voter i’s type is denoted by ti[0,1], and represents i’s private information about the true state. Each ti is distributed according to the density f1 or f0, depending on whether the state is 1 or 0. We assume that f1 and f0 are piecewise continuous positive functions, and that f0f1 is weakly decreasing on [0,1]. The latter is the familiar monotone likelihood ratio

When is voting by cut-off strategies efficient?

Our goal is to design a voting rule such that there is an efficient symmetric cut-off strategy profile. Obviously, any efficient strategy profile is a (Bayesian Nash) equilibrium (see also McLennan, 1998), but not conversely. If such a voting rule exists, we say that efficient information aggregation is feasible.

In the case where private information is binary, there always exist voting rules which efficiently aggregate private information (Austen-Smith and Banks, 1996, Bozbay et al., 2014), so

Examples and discussion

In the first example condition (1) is satisfied, and in the second example it is not.3

Example 1

This example is by Duggan and Martinelli (2001). Let n=2 and π=0.6. The density functions are f0(t)=32if 0<t<1212if 12t1, and f1(t)=12if 0<t<1232if 12t1.

In this case, (1) is satisfied for τ=12, and m=1 in Theorem 1. In fact, it is not hard to see that this generates the same efficient decisions as the binary information model

Proof of Theorem 1

The probabilities of the states conditional on the full information t[0,1]n are derived as: Pr(1|t)=πi{1,,n}f1(ti)i{1,,n}(πf1(ti)+(1π)f0(ti))Pr(0|t)=(1π)i{1,,n}f0(ti)i{1,,n}(πf1(ti)+(1π)f0(ti)). (a) Suppose efficient information aggregation is feasible. Hence, a voting rule g makes some cut-off strategy-profile σtˆ efficient. We show that τ=tˆ satisfies (1) for every t.

First, consider a type profile t and suppose that a decision d is efficient for t. This means Pr(d|t)>Pr(d|t)

References (9)

There are more references available in the full text version of this article.

Cited by (0)

View full text