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Journal Article•DOI•

Optimal voting schemes with costly information acquisition

Alex Gershkov1, Balázs Szentes2•
University of Bonn1, University of Chicago2
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.
About: This article is published in Journal of Economic Theory.The article was published on 2009-01-01 and is currently open access. It has received 96 citations till now. The article focuses on the topics: Optimal decision.

Summary (3 min read)

Jump to: [1 Introduction] – [2 The Model] – [3 The First-Best Mechanism] – [4 Preliminaries Canonical Mechanisms] – [Incentive Compatibility] – [Continuation Mechanisms] – [5 Optimal Mechanisms] – [5.1 Optimal Ex-post Efficient Mechanism] – [Theorem 2] – [5.2 Discussion of the Ex-post Optimal Mechanism Infinitely Many Voters] – [Robustness] – [5.3 Ex-ante Optimal Mechanism] and [6 Discussion]

1 Introduction

  • In many environments, groups rather than individuals make decisions.
  • The committee must analyze the proposed road's costs and benefits, which are complex.
  • 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 information themselves.
  • The total amount of information aggregated in the optimal mechanism is an endogenous object of their interest.

2 The Model

  • There are two possible states of the world: H and L. Each of these states occurs with probability one-half.
  • (We also implicitly assume that information acquisition takes no time.)the authors.the authors.
  • There is a Social Planner (SP) who wants to maximize expected sum of the individuals' utilities, net of the expected total cost of information acquisition: EQUATION where L is the expected number of voters who collect information.
  • The utilities are not transferable; that is, the SP cannot use a transfer scheme to induce the voters to acquire information.
  • The lemma above states that the posteriors about the state of the world and about the next signal after observing a sequence of signals depend only on the difference between the numbers of signals H and L observed so far.

3 The First-Best Mechanism

  • This section characterizes the first-best voting scheme.
  • The following proposition characterizes the first-best mechanism.
  • Notice that not all of the states can be reached.
  • Hence, given a certain d, the more voters have been already asked, the more likely it is that the SP makes a decision.

4 Preliminaries Canonical Mechanisms

  • A voting mechanism is an extensive-form game with imperfect information where the players are the voters.
  • Chance may randomize when taking an action.
  • When a voter is asked to collect information, she does not know her position in the sequence, and she does not know what other voters reported to the SP.
  • (b) If, at some information set, a certain voter does not acquire information but takes an action in e, then modify the game such that chance moves at that information set and takes the same action as the voter took in (G, e). 2 Myerson (1986) claims a similar result for a more general class of multistage games with communication.
  • Let us assume that the optimal mechanism is asymmetric.

Incentive Compatibility

  • The goal of this subsection is to explicitly characterize the incentive compatibility constraint, that is, the constraint that guarantees the voters indeed have incentive to collect information when asked instead of just reporting something.
  • Hence, when she computes the probability of a sequence, she takes away a signal H and computes the likelihood of the remainder of the sequence.
  • For each mechanism, there exists another one that operates as follows:.
  • The probability mixture of incentive compatible mechanisms is also incentive compatible.

Continuation Mechanisms

  • The arguments of most proofs regarding optimality of mechanisms involve modifying the mechanism at some states.
  • When the scheme must be incentive compatible, there is an interaction between different continuation mechanisms.
  • Furthermore, the continuation mechanism M never specifies that the SP asks more voters than the number available after reaching V (l, d), that is N − l.
  • The right side of the inequality is the SP's posterior at V (l, d) about the true state of the world.
  • This lemma is essential for characterizing the optimal ex-post efficient mechanism.

5 Optimal Mechanisms

  • The authors first characterize the optimal ex-post efficient mechanism.
  • The authors show that it has very similar properties to the first-best voting scheme.
  • Then the authors discuss some attributes of this mechanism.
  • Finally, the authors show that the ex-ante optimal mechanism sometimes involves ex-post inefficient decisions.

5.1 Optimal Ex-post Efficient Mechanism

  • The authors are ready to characterize the optimal mechanism in the class of ex-post efficient mechanisms, that is, the class of mechanisms where the SP always makes a majority decision.
  • The SP keeps asking the voters sequentially to collect information and report it to him.
  • Since the function f is decreasing, the more voters the SP has already asked, the less precise a posterior induces the SP to stop asking voters and take an action.
  • Furthermore, the function f never jumps down by more than one.
  • As the authors pointed out earlier, from Lemma 3, it follows that the optimal mechanism generically involves randomization.

Theorem 2

  • The optimal ex-post efficient mechanism described in Theorem 1 is generically unique and involves randomization only at a single state.
  • First, the authors show that if, for a certain pair (p, c), there are at least two different optimal mechanisms, then there exists an optimal mechanism that involves randomizations in at least two different states.
  • By Lemma 6, it follows that there are at least two continuation mechanisms that have the same efficiency.
  • Since there are only finitely many continuation mechanisms, if the optimal mechanism were not generically unique, there would exist two continuation mechanisms with the same efficiency for a positive measure of (p, c).

5.2 Discussion of the Ex-post Optimal Mechanism Infinitely Many Voters

  • Recall that the function g was decreasing because, as the number of voters who have not yet collected information decreases, the value of asking more voters also decreases.
  • The way to guarantee a large probability of being pivotal to the voters is to make decisions after signal sequences where the difference between the numbers of different signals is small.
  • (Otherwise the model is the same as before.).
  • This is because as k 0 goes to infinity, the probability of being pivotal, and hence also the benefit from collecting information, goes to zero.
  • Suppose that there are infinitely many voters, and the first-best mechanism is not incentive compatible.

Robustness

  • A common critique of Bayesian mechanism design is that to design the optimal mechanism, the SP has to have perfect knowledge about the information structure of the environment.
  • If the decreasing step function is too large, then this equilibrium is one in which the voters do not collect any information.
  • Hence, the value of the objective function of the SP is also close to the value corresponding to the incentive compatible mechanism.
  • Then there exists a unique symmetric mixed-strategy equilibrium.

5.3 Ex-ante Optimal Mechanism

  • The authors show that the optimal mechanism sometimes involves ex-post inefficient decisions.
  • That is, the mechanism characterized in Theorem 1 is not always optimal.
  • It will be shown that if the cost of information acquisition is small enough, then the optimal ex-post efficient mechanism can be improved upon by replacing a continuation mechanism with an ex-post inefficient continuation mechanism.
  • At V (K, 1), the SP asks an additional voter, and if she confirms his posterior, he makes the majority decision.
  • If their reports are the same, he again makes the majority decision.

6 Discussion

  • This paper analyzed optimal voting schemes in environments where information acquisition is costly and unobservable.
  • The Social Planner stops asking voters if and only if his posterior is more precise than the value corresponding to the number of voters already asked.
  • This intuition does not involve any assumption about the distribution of the states of the world and the signals.
  • Recall that having the explicit form of the incentive compatibility constraint made it possible to compute the efficiency of continuation mechanisms.
  • From ( 26) and ( 27) it follows that the absolute value of the ratio of the coefficients corresponding to the terms with the largest power is 1/l (s).

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Citations
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Book Chapter•DOI•
Advances in Economics and Econometrics: Giving and Receiving Advice

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Joel Sobel1•
University of California, San Diego1
01 May 2013
TL;DR: In this paper, the authors classify the literature on communication between informed experts and uninformed decision makers along four dimensions: strategic, technological, institutional, and cultural, and provide some insight into what constitutes a persuasive statement and under what conditions a decision maker will benefit from consulting an expert.
Abstract: This paper reviews literature on communication between informed experts and uninformed decision makers. The research provides some insight into what constitutes a persuasive statement and under what conditions a decision maker will benefit from consulting an expert. I classify the literature along four dimensions: strategic, technological, institutional, and cultural. To the extent that decision makers and experts have different preferences, communication creates strategic problems. Technological considerations describe the domain of uncertainty, the cost of acquiring information, and the cost of manipulating information. The institution determines who has responsibility for making decisions and the rules that govern communication. Cultural factors describe the way in which agents interpret language.

111 citations

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Abstract: This article studies how a recommender system may incentivize users to learn about a product collaboratively. To improve the incentives for early exploration, the optimal design trades off fully transparent disclosure by selectively overrecommending the product (or “spamming”) to a fraction of users. Under the optimal scheme, the designer spams very little on a product immediately after its release but gradually increases its frequency; she stops it altogether when she becomes sufficiently pessimistic about the product. The recommender’s product research and intrinsic/naive users “seed” incentives for user exploration and determine the speed and trajectory of social learning. Potential applications for various Internet recommendation platforms and implications for review/ratings inflation are discussed.

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Optimal auctions with information acquisition

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01 Mar 2012-Games and Economic Behavior
TL;DR: In this article, the optimal auction design in a private value setting with endogenous information gathering was studied and it was shown that the optimal monopoly price is always lower than the standard monopoly price.

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Progressive Screening: Long-Term Contracting with a Privately Known Stochastic Process

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Raphael Boleslavsky1, Maher Said2•
University of Miami1, Washington University in St. Louis2
01 Jan 2013-The Review of Economic Studies
TL;DR: In this article, a model of long-term contracting in which the buyer is privately informed about the stochastic process by which her value for a good evolves is examined. And the realized values are also private information.
Abstract: We examine a model of long-term contracting in which the buyer is privately informed about the stochastic process by which her value for a good evolves. In addition, the realized values are also private information. We characterize a class of environments in which the profit-maximizing long-term contract offered by a monopolist takes an especially simple structure: we derive sufficient conditions on primitives under which the optimal contract consists of a menu of deterministic sequences of static contracts. Within each sequence, higher realized values lead to greater quantity provision; however, an increasing proportion of buyer types are excluded over time, eventually leading to inefficiently early termination of the relationship. Moreover, the menu choices differ by future generosity, with more costly (up front) plans guaranteeing greater quantity provision in the future. Thus, the seller screens process information in the initial period and then progressively screens across realized values so as to reduce the information rents paid in future periods. Copyright , Oxford University Press.

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A resurrection of the Condorcet Jury Theorem

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Yukio Koriyama1, Balázs Szentes2•
École Polytechnique1, University College London2
09 Jun 2009-Theoretical Economics
TL;DR: In this article, the optimal size of a deliberating committee where there is no conflict of interest among individuals and information acquisition is costly is analyzed, and it is shown that any arbitrarily large committee aggregates the decentralized information more efficiently than the committee of size k*-2.
Abstract: This paper analyzes the optimal size of a deliberating committee where (i) there is no conflict of interest among individuals and (ii) information acquisition is costly. The committee members simultaneously decide whether to acquire information, and then make the ex-post efficient decision. The optimal committee size, k*, is shown to be bounded. The main result of this paper is that any arbitrarily large committee aggregates the decentralized information more efficiently than the committee of size k*-2. This result implies that oversized committees generate only small inefficiencies.

64 citations

References
More filters
Book•
Optimal Statistical Decisions

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Morris H. DeGroot
01 Jun 1970
TL;DR: In this article, the authors present a survey of probability theory in the context of sample spaces and decision problems, including the following: 1.1 Experiments and Sample Spaces, and Probability 2.2.3 Random Variables, Random Vectors and Distributions Functions.
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. 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. 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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.

4,287 citations

Journal Article•DOI•
Information Aggregation, Rationality, and the Condorcet Jury Theorem

[...]

David Austen-Smith, Jeffrey S. Banks1•
University of Rochester1
01 Jan 1996-American Political Science Review
TL;DR: The Condorcet Jury Theorem states that majorities are more likely than any single individual to select the "better" of two alternatives when there exists uncertainty about which of the two alternatives is in fact preferred as discussed by the authors.
Abstract: The Condorcet Jury Theorem states that majorities are more likely than any single individual to select the "better" of two alternatives when there exists uncertainty about which of the two alternatives is in fact preferred Most extant proofs of this theorem implicitly make the behavioral assumption that individuals vote "sincerely" in the collective decision making, a seemingly innocuous assumption, given that individuals are taken to possess a common preference for selecting the better alternative However, in the model analyzed here we find that sincere behavior by all individuals is not rational even when individuals have such a common preference In particular, sincere voting does not constitute a Nash equilibrium A satisfactory rational choice foundation for the claim that majorities invariably "do better" than individuals, therefore, has yet to be derived

948 citations

Journal Article•DOI•
Optimal coordination mechanisms in generalized principal–agent problems

[...]

Roger B. Myerson1•
Northwestern University1
01 Jun 1982-Journal of Mathematical Economics
TL;DR: In this article, 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.

934 citations

"Optimal voting schemes with costly ..." refers result in this paper

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Journal Article•DOI•
Full extraction of the surplus in bayesian and dominant strategy auctions

[...]

Jacques Cremer, Richard P. McLean
01 Nov 1988-Econometrica
TL;DR: In this article, the authors consider auctions for a single indivisible object, in which the bidders have information about each other which is not available to the seller and show that the seller can use this information to his own benefit, and 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 the bidder's willingness to pay.
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

925 citations

Posted Content•
The Swing Voter's Curse

[...]

Timothy J. Feddersen, Wolfgang Pesendorfer1•
Northwestern University1
01 Jun 1996-The American Economic Review
TL;DR: In this article, the existence of a swing voter's curse is demonstrated: less informed indifferent voters strictly prefer to abstain rather than vote for either candidate even when voting is costless, and the equilibrium result that a substantial fraction of the electorate will abstain even though all abstainers strictly prefer voting for one candidate over voting for another.
Abstract: The authors analyze two-candidate elections in which some voters are uncertain about the realization of a state variable that affects the utility of all voters. They demonstrate the existence of a swing voter's curse: less informed indifferent voters strictly prefer to abstain rather than vote for either candidate even when voting is costless. The swing voter's curse leads to the equilibrium result that a substantial fraction of the electorate will abstain even though all abstainers strictly prefer voting for one candidate over voting for another. Copyright 1996 by American Economic Association.

793 citations

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Frequently Asked Questions (5)
Q1. What are the contributions in "Optimal voting schemes with costly information acquisition∗" ?

This paper analyzes a voting model where ( i ) there is no conflict of interest among the voters, and ( ii ) information acquisition is costly and unobservable. The social 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. Obeying the planner by investing and reporting truthfully is optimal for voters. In this scheme, the social planner stops aggregating information and makes a decision when the precision of his posterior exceeds a cut-off which decreases with each additional report. 

The authors also show that, if the cost of information acquisition is small, then, surprisingly, the ex-ante optimal mechanism is often ex-post inefficient. 

It will be shown that if the cost of information acquisition is small enough, then the optimal ex-post efficient mechanism can be improved upon by replacing a continuation mechanism with an ex-post inefficient continuation mechanism. 

Since the SP orders the voters independently of the realizations of the signals,p (A ∩B) = p (s) 2i+ d N ,where (2i+ d) /N is the probability that the deviator is asked to report a signal if a decision is made after a sequence with length 2i+d. 

The authors argue that for given p and c, there always exists a k0 ∈ N such that the mechanism is not incentive compatible if the SP stops asking voters only if |d| ≥ k0.