# Optimal voting schemes with costly information acquisition

TL;DR: It is shown that, of all mechanisms, a sequential one is optimal and works as follows: one agent at a time is selected to acquire information and report the resulting signal and the restriction to ex-post efficiency is shown to be without loss when the available signals are sufficiently imprecise.

Abstract: A group of individuals with identical preferences must make a decision under uncertainty about which decision is best. Before the decision is made, each agent can privately acquire a costly and imperfect signal. We discuss how to design a mechanism for eliciting and aggregating the collected information so as to maximize ex-ante social welfare.
We first show that, of all mechanisms, a sequential one is optimal and works as follows. At random, one agent at a time is selected to acquire information and report the resulting signal. Agents are informed of neither their position in the sequence nor of other reports. Acquiring information when called upon and reporting truthfully is an equilibrium.
We next characterize the ex-ante optimal scheme among all ex-post efficient mechanisms. In this mechanism, a decision is made when the precision of the posterior exceeds a cut-off that decreases with each additional report. The restriction to ex-post efficiency is shown to be without loss when the available signals are sufficiently imprecise. On the other hand, ex-post efficient mechanisms are shown to be suboptimal when the cost of information acquisition is sufficiently small.

Topics: Optimal decision (60%)

Op tima l Vot in g S ch eme s w ith Co st ly In fo rma tio n

A c q u isitio n

∗

Alex Gershkov

†

Department of Economics, Hebrew Universit y of Jerusalem

Balázs Szentes

‡

Departm ent of Econom ics, Univ ersity of Chicago

Abstract

This paper analyzes a voting model where (i) there is no conﬂictofinterestamongthe

voters, and (ii) information ac quisition is costly and uno b servable. T h e optimal mechanism is

shown to be s equential. The so cial planner asks, at random, one voter at a time to invest in

information and to report the resulting signal. Voters are informed of neither their position in

the sequence nor the reports of previous voters. Ob eying the planner by investing and reporting

truthfully is optimal for voters. The ex-ante optimal voting scheme among the ex-post eﬃcient

ones is characterized. In this scheme, the so cial planner stops aggregating information and

makes a dec ision w hen th e precision of his p osterior exce eds a cut-o ﬀ which decrease s with

each additional report. It is also shown that if the cost of information acquisition is small,

then the ex-ante optimal mechanism is sometimes necessarily ex-post ineﬃcient.

1Introduction

In many environments, groups rather than individuals make decisions. Information about the

desirability of the possible decisions is often dispersed: Individual group members must separately

∗

We ar e grate fu l for helpfu l discu ss ion s to E dd ie D ekel, Jeﬀ Ely, Tim Federsen, A li Hortacsu, M otty Perry, Phil

Reny, Rob Shimer, and Hugo Sonnenschein. We thank seminar participants at the U niversity of Chicago, Iowa

State U n iversity, Tel A viv U n iversity, He brew Un iversity, Boston Un iversity, No rthwestern, and Qu een ’s U niversity

for comm e nts.

†

M o u nt Sc opus , Je ru sa le m, 91 9 0 5, Is ra e l. E ma il: ma r @po b .huji. ac .il.

‡

1126 E. 59th Street, C hicago, IL 606 57, USA . E m ail: szentes@uchicago.edu.

1

acquire information about the alternatives. The group is more likely to make the right decision

if it aggregates individuals’ information. However, it often is costly for individuals to acquire

information, and whether they have done so is often unobservable. Hence, even when there is no

conﬂict of interest among the group mem bers, an individual may not want to acquire information

if the probability that she will inﬂuence the ﬁnal decision is too small compared with her cost.

That is, she compares the cost of information only to her own beneﬁt, not to society’s. The

naturally arising mechanism design problem is to design an information aggregation scheme that

maximizes the group’s expected gain from making the right decision, net of the total expected

cost of acquiring information. The scheme must make the probability that an individual is pivotal

suﬃciently large, compared with the cost of information, that individuals have adequate incentive

to collect information. This paper analyzes optimal voting schemes in such environments.

Small groups make decisions in many economic and political situations. Consider, for instance,

a recruiting committee in an academic department. The committee’s objective is to identify the

candidate who best ﬁts a vacant job. (For example, the committee may want to hire somebody who

can teach and research a narrowly deﬁned ﬁeld.) The committee members review applicants’ CVs,

teaching evaluations, and research papers. The opportunity cost of this work may be substantial

because time spent reading applications cannot be devoted to a member’s own research. The

chair of such a committee faces the following dilemma: On the one hand, to mak e an accurate

assessment, he wants several committee members to review the same candidate. On the other hand,

if several members review the same candidate, the members may lack the incentive to work hard.

Each member understands that, ev en if she does not spend much time reviewing the candidate and

hence forms an inaccurate opinion, the others’ reports may induce a correct ﬁnal decision. Thus

the members will not exert eﬀort if their reports are not likely to be decisive.

Consider, also, the board of a company that must evaluate a potential merger. The board mem-

bers’ goal should be to maximize the value of their company. Properly forecasting the consequences

of a merger requires detailed analysis of data about the target ﬁrm, such as its past performance,

the value of its capital stock, its market share, and so on. The board may hire experts to help

with this analysis, but the board members ultimately must reach their own conclusions. If a board

member trusts other members’ assessments, she has no incentive to invest energy in forming her

ow n opinion; she will prefer to let the other members decide. B ut if all members think this way,

the ﬁnal decision may be very poor.

Another example is a legislative committee considering whether to build a highway. The com-

mittee must analyze the proposed road’s costs and beneﬁts, which are complex. A new highway

may reduce not only the time residents spend traveling but also the cost of local trade, which in

turn can reduce the prices of some commodities. A new highway may also lead to more tourism.

On the other hand, the government must levy taxes to ﬁnance the project. New taxes have both

political and economic consequences. In addition, a new highway may generate heavier traﬃc,

which may increase air and soil pollution. Even understanding the complicated calculation of the

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project’s net beneﬁt requires the committee members to incur signiﬁcant costs. If they do not

live near the site of the proposed highway, they may simply leave the decision to other committee

members.

Finally, suppose that a patient is diagnosed with lung cancer. Two treatments are available:

surgery and chemotherapy. The probability of each treatment’s success depends on the type and

size of the tumor, the patient’s other medical problems, her age, etc. A group of doctors, each an

expert on some of these factors, must agree on the best treatment to recommend to the patient.

But, as in the previous examples, making a responsible recommendation is costly; eac h doctor may

w ant to save that cost if she trusts her colleagues’ judgment.

In all of these cases, even though the committee members have the same objective, decisions

may be ineﬃcient because members may prefer to rely on others’ opinions instead of acquiring in-

formation themselves. We view this problem as a social choice theoretical one. Most of social choice

theory, however, focuses on eﬃcient decision making in the presence of conﬂicts of interest among

the members of a society. If individuals disagree about the merits of possible social alternatives,

they may want to misrepresent their preferences to inﬂuence the ﬁnal decision, which can lead to

ineﬃcient outcomes. We believe that another source of ineﬃciency is the diﬃculty of aggregating

information. Hence, in this paper, we completely abstract from potential conﬂicts of interest and

focus solely on the costly and unobservable nature of information acquisition. Surprisingly, the

modern Bayesian mechanism design literature has not yet fully explored this important and basic

line of research.

We set up probably the simplest possible model where (i) there is no conﬂict of interest among

voters, and (ii) information acquisition is costly and unobservable. A theoretical interest of our

model is that the Rev elation Principle cannot be applied. The voters have no information, and

hence no type, to start with. Therefore, the optimal mechanism cannot be based simply on asking

the voters to report their information. The total amount of information aggregated in the optimal

mechanism is an endogenous object of our interest.

We show that for an y equilibrium in any voting scheme, there exists another outcome-equivalent

mechanism and equilibrium with the following properties. First, the voters are randomly ordered.

Next, the voters are asked sequentially to acquire and report information. A voter receives no

information about either her position in the sequence or the actions of previous voters. After eac h

voter’s report, either a ﬁnal decision is made or an additional voter is asked. A voter’s set of

actions is the set of signals she can possibly draw and report about the state of the world. (We call

suc h a mechanism a canonical mechanism.) In equilibrium, whenever a voter is asked to acquire

information, she does so and reports the information truthfully.

Most of this paper focuses on ex-post eﬃcient mechanisms. Committing to ex-post ineﬃcient

decisions is often impossible or absurd in the en vironments of our interest. (It is unlikely that

a group of doctors can commit to recommending a treatment that will almost surely kill the

patient; a rejected job candidate may sue the academic department if she can prove that she is

3

better qualiﬁed than the successful applicant.) Therefore,wecharacterizeoptimalvotingschemes

among the ex-post eﬃcient ones. The optimal scheme in this class has a particularly simple form.

Essen tially it can be characterized by a cut-oﬀ function, f, mapping the number of solicited voters

to the precision of the posterior distribution over states of the world. If, after receiving reports

from n voters, the posterior has precision above the cut-oﬀ f (n), information acquisition stops and

a decision is made. Otherwise, an additional voter is asked to acquire information and to report

her resulting signal. Our main result is that f is nonincreasing. Consequently, with each additional

report the cut-oﬀ precision of the posterior decreases. That is, less precise information is required

to induce a decision as the number of solicited voters increases We also show that, if the cost of

information acquisition is small, then, surprisingly, the ex-ante optimal mechanism is often ex-post

ineﬃcient. That is, ex-post ineﬃcient decisions can be used as a threat to induce voters to acquire

information.

We view this mechanism design problem as a particularly important and applied one. This

line of research may not only deepen understanding of voting theory and existing information

aggregation procedures but also lead to speciﬁc recommendations for improving voting schemes

now in use.

Most of the existing literature has focused on voting models with exogenously given information

structure. See Austen-Smith and Banks (1996), Chwe (1999), Federsen and Pesendorfer (1996,

1997, 1998), Li, Rosen, and Suen (2001), and McLennan (1998). For example, Dekel and Piccione

(2000) compare sequential and simultaneous voting and ﬁnd equivalence between diﬀerent voting

schemes in terms of equilibrium outcome. In particular, they ﬁnd that for any choice rule, there is

a Pareto best equilibrium outcome that is the same whether voting is sequential or simultaneous.

In contrast, this paper shows that sequential schemes always dominate simultaneous ones when

information acquisition is costly.

There is a growing literature on information acquisition in mechanism design. Most of this

literature deals with auction and public good models where utilities are transferable, unlike in

voting models. See Bergemann and Valimaki (2002), Persico (2000), Milgrom (1981), and the

references therein. These papers focus on simultaneous information acquisition. Their goal is to

analyze the incentives to acquire information in diﬀerent classical mechanisms.

Persico (2004) considers two-stage voting games of the following form. At the ﬁrst stage, all

committee members simu ltaneously decide whether to acquire a noisy signal about the state of the

w o rld or to remain uninformed. At the second stage, they vote. The author analyzes the optimal

voting scheme among threshold voting rules and the optimal size of the committee. Gerardi and

Yariv (2003) analyze environments similar to the ones in Persico (2004). They enrich the model by

in troducing a communication stage before voting. This cheap talk stage can make the mechanism

more eﬃcient. Mukhopadhaya (2003) analyzes the eﬀect of committee size on the accuracy of the

ﬁnal decision. He shows that in symmetric mixed-strategy equilibria, increasing the committee size

may lead to a less accurate decision. In asymmetric pure-strategy equilibria, however, changing

4

the committee size does not aﬀect the accuracy of the ﬁnal decision.

All of the papers described in the previous paragraph analyze settings with homogeneous pref-

erences. Cai (2003) allows for heterogeneous preferences with both non-veriﬁable information

acquisition and costly participation in a committee. In this model, mem bers are supposed to re-

port their information to a principal. The principal makes the ﬁnal decision based on the reports.

Cai (2003) characterizes the optimal committee size when the signals as w ell as the decision are

con tinuous variables and the principal uses the mean decision rule to determine the ﬁnal deci-

sion. Li (2001) considers a committee with a ﬁxed number of members. Each member acquires

a noisy signal. The signals become publicly observable by all the other committee members after

acquisition. He analyzes the properties of the optimaldecisionruleforthecaseofsimultaneous

information acquisition. To provide incentives for acquiring information, it is optimal to distort

the decision rule away from the ex-post optimal one. Cai (2003) treats the decision rule as giv en

and focuses on the committee size; Li (2001) does just the opposite. Austen-Smith and Federsen

(2002) also consider a model where the voters have heterogeneous preferences over two alternatives.

Their model allows the voters to communicate prior to voting. They show that unanimous voting

makes it impossible to reveal all the private information.

Smorodinsky and Tennenholtz (2003) analyze the problem of free riding in multi-party com-

putations. A group of people has to compute the value of a certain function. Each person must

incur a cost to retrieve a piece of input. The agents face a free-rider problem similar to ours: If the

probability that a single piece of input aﬀects the value of the function is too small, an individual

prefers not to incur the cost and provides an input at random. Smorodinsky and Tennenholtz

(2003) analyze mechanisms in whic h the true value of the function is computed with probability

one. Therefore, unlike in our paper, there is no trade-oﬀ between the total cost of information

acquisition and the accuracy of the ﬁnal decision. On the other hand, Smorodinsky and Tennen-

holtz (2003) derive a canonical mechanism, independently from us, that has similar attributes to

our canonical mechanism. In particular, they show that the mechanism can be assumed to be

sequential and that each player is told only whether she should or should not retrieve the input.

The paper is organized as follows. Section 2 describes the model. Section 3 characterizes the

ﬁrst-best voting scheme. Section 4 characterizes the canonical mechanisms, explicitly derives the

incentive compatibility constraint, and prov es some basic properties of the optimal mec hanisms.

ThemainresultsareinSection5. Section6concludes. Most of the proofs are relegated to the

appendixes.

2 The Model

There is a population consisting of N(∈ N) individuals. There are two possible states of the world:

H and L. Each of these states occurs with probability one-half. The society must take an action,

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##### References

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01 Jun 1970Abstract: Foreword.Preface.PART ONE. SURVEY OF PROBABILITY THEORY.Chapter 1. Introduction.Chapter 2. Experiments, Sample Spaces, and Probability.2.1 Experiments and Sample Spaces.2.2 Set Theory.2.3 Events and Probability.2.4 Conditional Probability.2.5 Binomial Coefficients.Exercises.Chapter 3. Random Variables, Random Vectors, and Distributions Functions.3.1 Random Variables and Their Distributions.3.2 Multivariate Distributions.3.3 Sums and Integrals.3.4 Marginal Distributions and Independence.3.5 Vectors and Matrices.3.6 Expectations, Moments, and Characteristic Functions.3.7 Transformations of Random Variables.3.8 Conditional Distributions.Exercises.Chapter 4. Some Special Univariate Distributions.4.1 Introduction.4.2 The Bernoulli Distributions.4.3 The Binomial Distribution.4.4 The Poisson Distribution.4.5 The Negative Binomial Distribution.4.6 The Hypergeometric Distribution.4.7 The Normal Distribution.4.8 The Gamma Distribution.4.9 The Beta Distribution.4.10 The Uniform Distribution.4.11 The Pareto Distribution.4.12 The t Distribution.4.13 The F Distribution.Exercises.Chapter 5. Some Special Multivariate Distributions.5.1 Introduction.5.2 The Multinomial Distribution.5.3 The Dirichlet Distribution.5.4 The Multivariate Normal Distribution.5.5 The Wishart Distribution.5.6 The Multivariate t Distribution.5.7 The Bilateral Bivariate Pareto Distribution.Exercises.PART TWO. SUBJECTIVE PROBABILITY AND UTILITY.Chapter 6. Subjective Probability.6.1 Introduction.6.2 Relative Likelihood.6.3 The Auxiliary Experiment.6.4 Construction of the Probability Distribution.6.5 Verification of the Properties of a Probability Distribution.6.6 Conditional Likelihoods.Exercises.Chapter 7. Utility.7.1 Preferences Among Rewards.7.2 Preferences Among Probability Distributions.7.3 The Definitions of a Utility Function.7.4 Some Properties of Utility Functions.7.5 The Utility of Monetary Rewards.7.6 Convex and Concave Utility Functions.7.7 The Anxiomatic Development of Utility.7.8 Construction of the Utility Function.7.9 Verification of the Properties of a Utility Function.7.10 Extension of the Properties of a Utility Function to the Class ?E.Exercises.PART THREE. STATISTICAL DECISION PROBLEMS.Chapter 8. Decision Problems.8.1 Elements of a Decision Problem.8.2 Bayes Risk and Bayes Decisions.8.3 Nonnegative Loss Functions.8.4 Concavity of the Bayes Risk.8.5 Randomization and Mixed Decisions.8.6 Convex Sets.8.7 Decision Problems in Which ~2 and D Are Finite.8.8 Decision Problems with Observations.8.9 Construction of Bayes Decision Functions.8.10 The Cost of Observation.8.11 Statistical Decision Problems in Which Both ? and D contains Two Points.8.12 Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage.Exercises.Chapter 9. Conjugate Prior Distributions.9.1 Sufficient Statistics.9.2 Conjugate Families of Distributions.9.3 Construction of the Conjugate Family.9.4 Conjugate Families for Samples from Various Standard Distributions.9.5 Conjugate Families for Samples from a Normal Distribution.9.6 Sampling from a Normal Distribution with Unknown Mean and Unknown Precision.9.7 Sampling from a Uniform Distribution.9.8 A Conjugate Family for Multinomial Observations.9.9 Conjugate Families for Samples from a Multivariate Normal Distribution.9.10 Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision matrix.9.11 The Marginal Distribution of the Mean Vector.9.12 The Distribution of a Correlation.9.13 Precision Matrices Having an Unknown Factor.Exercises.Chapter 10. Limiting Posterior Distributions.10.1 Improper Prior Distributions.10.2 Improper Prior Distributions for Samples from a Normal Distribution.10.3 Improper Prior Distributions for Samples from a Multivariate Normal Distribution.10.4 Precise Measurement.10.5 Convergence of Posterior Distributions.10.6 Supercontinuity.10.7 Solutions of the Likelihood Equation.10.8 Convergence of Supercontinuous Functions.10.9 Limiting Properties of the Likelihood Function.10.10 Normal Approximation to the Posterior Distribution.10.11 Approximation for Vector Parameters.10.12 Posterior Ratios.Exercises.Chapter 11. Estimation, Testing Hypotheses, and linear Statistical Models.11.1 Estimation.11.2 Quadratic Loss.11.3 Loss Proportional to the Absolute Value of the Error.11.4 Estimation of a Vector.11.5 Problems of Testing Hypotheses.11.6 Testing a Simple Hypothesis About the Mean of a Normal Distribution.11.7 Testing Hypotheses about the Mean of a Normal Distribution.11.8 Deciding Whether a Parameter Is Smaller or larger Than a Specific Value.11.9 Deciding Whether the Mean of a Normal Distribution Is Smaller or larger Than a Specific Value.11.10 Linear Models.11.11 Testing Hypotheses in Linear Models.11.12 Investigating the Hypothesis That Certain Regression Coefficients Vanish.11.13 One-Way Analysis of Variance.Exercises.PART FOUR. SEQUENTIAL DECISIONS.Chapter 12. 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