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Journal ArticleDOI

Optimal voting schemes with costly information acquisition

01 Jan 2009-Journal of Economic Theory (Academic Press)-Vol. 144, Iss: 1, pp 36-68

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%)

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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 conictofinterestamongthe
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 ecient
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 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 inecient.
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, Je 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
conict of interest among the group mem bers, an individual may not want to acquire information
if the probability that she will inuence the nal decision is too small compared with her cost.
That is, she compares the cost of information only to her own benet, 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
suciently 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 ts a vacant job. (For example, the committee may want to hire somebody who
can teach and research a narrowly dened eld.) 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 nal decision. Thus
the members will not exert eort 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 rm, 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 nal 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 benets, 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 nance the project. New taxes have both
political and economic consequences. In addition, a new highway may generate heavier trac,
which may increase air and soil pollution. Even understanding the complicated calculation of the
2

project’s net benet requires the committee members to incur signicant 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 inecient 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 ecient decision making in the presence of conicts 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 inuence the nal decision, which can lead to
inecient outcomes. We believe that another source of ineciency is the diculty of aggregating
information. Hence, in this paper, we completely abstract from potential conicts 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 conict 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 nal 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 ecient mechanisms. Committing to ex-post inecient
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 qualied than the successful applicant.) Therefore,wecharacterizeoptimalvotingschemes
among the ex-post ecient ones. The optimal scheme in this class has a particularly simple form.
Essen tially it can be characterized by a cut-o 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-o 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-o 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
inecient. That is, ex-post inecient 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 specic 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 nd equivalence between dierent voting
schemes in terms of equilibrium outcome. In particular, they nd 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 dierent classical mechanisms.
Persico (2004) considers two-stage voting games of the following form. At the rst 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 ecient. Mukhopadhaya (2003) analyzes the eect of committee size on the accuracy of the
nal 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 aect the accuracy of the nal 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-veriable 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 nal 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 nal deci-
sion. Li (2001) considers a committee with a xed 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 aects 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-o between the total cost of information
acquisition and the accuracy of the nal 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
rst-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,
5

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References
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Book
01 Jun 1970
Abstract: 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. 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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. Sequential Sampling.12.1 Gains from Sequential Sampling.12.2 Sequential Decision Procedures.12.3 The Risk of a Sequential Decision Procedure.12.4 Backward Induction.12.5 Optimal Bounded Sequential Decision procedures.12.6 Illustrative Examples.12.7 Unbounded Sequential Decision Procedures.12.8 Regular Sequential Decision Procedures.12.9 Existence of an Optimal Procedure.12.10 Approximating an Optimal Procedure by Bounded Procedures.12.11 Regions for Continuing or Terminating Sampling.12.12 The Functional Equation.12.13 Approximations and Bounds for the Bayes Risk.12.14 The Sequential Probability-ratio Test.12.15 Characteristics of Sequential Probability-ratio Tests.12.16 Approximating the Expected Number of Observations.Exercises.Chapter 13. Optimal Stopping.13.1 Introduction.13.2 The Statistician's Reward.13.3 Choice of the Utility Function.13.4 Sampling Without Recall.13.5 Further Problems of Sampling with Recall and Sampling without Recall.13.6 Sampling without Recall from a Normal Distribution with Unknown Mean.13.7 Sampling with Recall from a Normal Distribution with Unknown Mean.13.8 Existence of Optimal Stopping Rules.13.9 Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall.13.10 Martingales.13.11 Stopping Rules for Martingales.13.12 Uniformly Integrable Sequences of Random Variables.13.13 Martingales Formed from Sums and Products of Random Variables.13.14 Regular Supermartingales.13.15 Supermartingales and General Problems of Optimal Stopping.13.16 Markov Processes.13.17 Stationary Stopping Rules for Markov Processes.13.18 Entrance-fee Problems.13.19 The Functional Equation for a Markov Process.Exercises.Chapter 14. Sequential Choice of Experiments.14.1 Introduction.14.2 Markovian Decision Processes with a Finite Number of Stages.14.3 Markovian Decision Processes with an Infinite Number of Stages.14.4 Some Betting Problems.14.5 Two-armed-bandit Problems.14.6 Two-armed-bandit Problems When the Value of One Parameter Is Known.14.7 Two-armed-bandit Problems When the Parameters Are Dependent.14.8 Inventory Problems.14.9 Inventory Problems with an Infinite Number of Stages.14.10 Control Problems.14.11 Optimal Control When the Process Cannot Be Observed without Error.14.12 Multidimensional Control Problems.14.13 Control Problems with Actuation Errors.14.14 Search Problems.14.15 Search Problems with Equal Costs.14.16 Uncertainty Functions and Statistical Decision Problems.14.17 Sufficient Experiments.14.18 Examples of Sufficient Experiments.Exercises.References.Supplementary Bibliography.Name Index.Subject Index.

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Roger B. Myerson1Institutions (1)
Abstract: The general principal–agent problem is formulated, in which agents have both private information and private decisions, unobservable to the principal. It is shown that the principal can restrict himself to incentive-compatible direct coordination mechanisms, in which agents report their information to the principal, who then recommends to them decisions forming a correlated equilibrium. In the finite case, optimal coordination mechanisms can be found by linear programming. Some basic issues relating to systems with many principals are also discussed. Non-cooperative equilibria between interacting principals do not necessarily exist, so quasi-equilibria are defined and shown to exist.

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Abstract: We consider auctions for a single indivisible object, in the case where the bidders have information about each other which is not available to the seller. We show that the seller can use this information to his own benefit, and we completely characterize the environ- ments in which a well chosen auction gives him the same expected payoff as that obtainable were he able to sell the object with full information about each bidder's willingness to pay. We provide this characterization for auctions in which the bidders have dominant strate- gies, and for those where the relevant equilibrium concept is Bayesian Nash. In both set-ups, the existence of these auctions hinges on the possibility of constructing lotteries with the correct properties. WE CONSIDER the situation in which an agent, the seller, possesses one indivisible unit of a good to which he attaches no value. But the good has value to a number of potential buyers, and its transfer to one of them would increase social welfare. In particular, the transfer to the buyer with the highest valuation maximizes social welfare. In this paper, we completely characterize environments in which the seller can design an auction that will enable him to capture for himself the full increase in social welfare induced by the transfer of the good to the bidder with the highest willingness to pay. If the seller had full information about the reservation prices of potential buyers, his optimal selling strategy would be very simple. He would announce a price equal or very close to the highest reservation value. The optimal strategy for the bidder with the highest evaluation would be to accept the offer. (Note that we are treating a situation in which the seller can commit himself to a price.) As a result of the exchange, the utility of the seller increases by the full amount of the increase in social welfare, and he has been able to fully extract the surplus. In many circumstances, however, a seller has only imperfect knowledge of the buyers' willingnesses to pay. In this case, he must find some mechanism, or auction, which will enable him to maximize his benefit from the sale of the object. The auction literature starts with this observation and shows how the seller can, by an astute choice of auction, extract the largest possible fraction of the surplus. In general, the literature has shown that this proportion is strictly less than one. In some circumstances, the bidders will have information about each other which is not available to the seller. For instance, in auctions for petroleum drilling rights, bidders know the results of geological tests which they have

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