Information aggregation with a continuum of types
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 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 . In that case, a specific quota rule, depending on , must be used in order to make voting by strategies with cut-off 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 voters, and two possible states of the world and . The prior probability of state is equal to , with .
Voter ’s type is denoted by , and represents ’s private information about the true state. Each is distributed according to the density or , depending on whether the state is or . We assume that and are piecewise continuous positive functions, and that is weakly decreasing on . 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 and . The density functions are In this case, (1) is satisfied for , and 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 are derived as: (a) Suppose efficient information aggregation is feasible. Hence, a voting rule makes some cut-off strategy-profile efficient. We show that satisfies (1) for every .
First, consider a type profile and suppose that a decision is efficient for . This means
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