Consider the problem of deciding a winner among three alternatives when voters have common values, but private information regarding the values of the alternatives. We compare approval voting with other scoring rules. For any finite electorate, the best equilibrium under approval voting is more efficient than either plurality rule or negative voting. If any scoring rule yields a sequence of equilibria that aggregates information in large elections, then approval voting must do so as well

Ahn, DS

Oliveros, S

University of Essex Research Repository

ISSN 1755-5361                   Discussion Paper Series          Approval Voting and Scoring Rules with Common Values  David S. Ahny and Santiago Oliveros  Note : The Discussion Papers in this series are prepared by members of the Department of Economics, University of Essex, for private circulation to interested readers. They often represent preliminary reports on work in progress and should therefore be neither quoted nor referred to in published work without the written consent of the author.                                  University of Essex           Department of Economics    No. 732 October 2013 Approval Voting and Scoring Ruleswith Common Values∗David S. Ahn†University of California, BerkeleySantiago Oliveros‡Haas School of Business, UC BerkeleySeptember 2010AbstractConsider the problem of deciding a winner among three alternatives when voters have com-mon values, but private information regarding the values of the alternatives. We compareapproval voting with other scoring rules. For any finite electorate, the best equilibrium underapproval voting is more efficient than either plurality rule or negative voting. If any scoringrule yields a sequence of equilibria that aggregates information in large elections, then approvalvoting must do so as well.1 ModelThere are three candidates K = {1, 2, 3}. There is a finite set of I voters. Each voter i can submita ballot, or vote vector, (ci(1), ci(2), ci(3)), where ci(k) denotes the score allocated by voter i tocandidate k. The scoring rule defines the permitted ballots. As in Myerson (2002), we consider(A,B)-scoring rules, that are defined by two parameters 0 ≤ A ≤ B ≤ 1. For a fixed (A,B)-scoringrule, each voter can submit either a permutation of (1, B, 0) or of (1, A, 0). Let C denote the spaceof all possible ballots. Given a profile of ballots (c1, . . . , cI), the winner W (c1, . . . , cI) of the electionis the candidates whose total score∑I ci(k) is maximal. In the case of a tie, uniform randomizationis used to select among the winners.Some specific (A,B)-scoring rules are well known. The case (A,B) = (0, 0) is plurality voting,where each voter can support a single candidate. The case (A,B) = (1, 1) is negative voting, whereeach voter can oppose a single candidate. The case (A,B) = (0.5, 0.5) is the Borda count, wherecandidates are totally ranked and receive scores proportional to their ranks. Of particular interestis the case (A,B) = (0, 1). This case is approval voting, where each voter decides a set of one ortwo candidates to support.∗We acknowledge the National Science Foundation for financial support under Grant SES-0851704.†Department of Economics, University of California, 508-1 Evans Hall #3880, Berkeley, CA 94720-3880. Email:dahn@econ.berkeley.edu.‡545 Student Services Building #1900, Haas School of Business, UC Berkeley, Berkeley, CA 94720-1900. Email:soliveros@haas.berkeley.edu.1Most of the literature on approval voting focuses on private values. We focus on the case ofpure common values. Let Ω be a finite set of states of the world. The prior probability of state ωis P (ω). All voters share a common utility U(k|ω) for candidate k = 1, 2, 3 in state ω. We assumethat there is a unique best candidate kω that maximizes U(k|ω) for each state of the world.The finite set S is a set of possible signals. The conditional probability of signal s ∈ S given ω isdenoted F (s|ω). Given the state ω, each voter receives a conditionally independent signal followingthe distribution F (s|ω).A strategy σ : S → C is a function assigning a vote vector to each signal. When σi(s) is adegenerate lottery with probability one of c, we slightly abuse notation and write σi(s) = c. Aprofile (σ1, . . . , σI) of strategies is symmetric if σi = σj for all i, j. When referring to symmetricstrategy profiles, we drop the subscript. The common expected utility for the strategy profileσ(s) = σ1(s1), . . . , σI(sI) isEU(σ) =∫Ω∫S∫X IU (W (c1, . . . , cI) |ω) dσ(s) dF (s|ω) dP (ω).Consider a sequence of symmetric strategies (σI). We say that the probability of error goes to zeroif, for every ω, ∫S∫X IU (W (c1, . . . , cI) |ω |ω) dσI(sI) dF (sI |ω)→ U(kω|ω),as I goes to infinity. The probability of error goes to zero if and only if the probability of the bestcandidate kω winning the election goes to one for every state of the world.A smaller and newer literature studies information aggregation for multiple candidates withcommon or interdependent values. A theme in these studies is that approval voting outperformsother institutions. However, this comparison is currently understood only for restricted classesof environments. Goertz and Maniquet (2009) consider a class of environments where voters areindifferent between the two inferior candidates. Within this class, approval voting is the only simplescoring rule that admits an informationally efficient limit equilibrium.1 Bouton and Castanheira(2010) consider a class of environments where a majority of voters prefer two candidates to a thirdminority candidate, but have incomplete information about which of the two candidates is better.Within this class, approval voting yields a unique limit equilibrium that efficiently aggregatesinformation, while plurality rule can have multiple equilibria. Both papers work with Poissonpopulation uncertainty, which allows for a cleaner construction of limit equilibrium.Our analysis, which makes no assumptions either on the information or on the utility, appliesto arbitrary environments. However, we do not attempt to explicitly construct limit equilibria.Instead, we argue indirectly by adapting an insight due to McLennan (1998): in a common valueelection, any strategy that maximizes utility is an equilibrium. While our hypotheses are moregeneral than those in the literature, our conclusions are also less sharp. We cannot speak to the1A scoring rule is simple if B = 0. For example, negative voting is not a simple scoring rule.2uniqueness of equilibrium nor to its characterization.2 ResultsOur first result asserts that the maximal equilibrium utility under approval voting is greater thanor equal to the maximal equilibrium utility under approval voting or under negative voting.Proposition 1. If σ∗ is a symmetric equilibrium for plurality rule or for negative voting, thenthere exists a symmetric equilibrium ρ∗ such that U(ρ∗) ≥ U(σ∗).Proof. First take the case where σ∗ is a symmetric equilibrium of plurality rule. Then define thestrategy profile ρ for approval voting by ρ(s) = σ∗(s).2 Then the expected utility of ρ is identical tothe expected utility of σ∗. Since U is continuous and the space of symmetric strategies is compact,there exists a symmetric strategy profile ρ∗ that maximizes the common expected utility U forapproval voting among all symmetric strategy profiles. By Theorem 2 of McLennan (1998), ρ∗ isan equilibrium for approval voting. By its construction, U(ρ∗) ≥ U(ρ) = U(σ).The argument for σ∗ as a symmetric equilibrium of negative voting is identical.While the reasoning for Proposition 1 is mathematically straightforward, to our knowledge ithas not been previously observed. Specifically, any ballot that can be submitted under pluralityrule can also be submitted under approval voting. So the expected utility under plurality rule canbe replicated under approval voting, by having each voter approve the singleton set correspondingto the candidate that she would support under plurality rule. Of course, approval voting also allowsother ballots that support two candidates. However, the observation of McLennan (1998) is that thebest equilibrium in a game of common values must be as good as any strategy profile. So the bestequilibrium under approval voting must be at least as efficient as any equilibrium under pluralityrule. The argument makes transparent the connection between approval voting and plurality rule:the fundamental reason that approval voting outperforms plurality rule (or negative voting) isthat approval voting allows a larger set of feasible ballots. While similar arguments for approvalvoting are suggested in private value environments, its flexibility has direct force in common valueenvironments.Our second finding is that the best asymptotic equilibrium under approval voting is at least asgood as the best asymptotic equilibrium under any interior (A,B)-scoring rule where 0 < A ≤ B <1. Unlike plurality rule or negative voting, the strategies under general (A,B)-scoring rules cannotbe exactly replicated by approval voting. When A 6= B, a general (A,B)-scoring rule allows fortwelve distinct vote vectors, while approval voting allows only for six. However, the outcome of the(A,B)-scoring rule can be approximated through appropriate randomization of the ballots underapproval voting. Specifically, the expected vote counts can be maintained under approval voting.This approximation becomes asymptotically precise, so the limit outcome under a general scoringrule can be replicated by approval voting.2To be pedantic, we identify σ∗(s) within the strictly large strategy simplex for approval voting.3Proposition 2. Suppose A,B ∈ (0, 1). If there exists a sequence (σI) of symmetric strategies forthe (A,B) scoring rule that takes the error probability to zero, then there exists a sequence (ρ∗I) ofsymmetric equilibria for approval voting that takes the error probability to zero.Proof. For any symmetric strategy profile σ of the (A,B) scoring rule, define the symmetric strategyprofile ρ of approval voting with permutations of the following:[ρ(s)](1, 1, 0) =∑X=A,BX {[σ(s)](1, X, 0) + [σ(s)](X, 1, 0)} (1)[ρ(s)](1, 0, 0) =∑X=A,B(1−X) {[σ(s)](1, X, 0) + [σ(s)](1, 0, X)} (2)Without loss of generality, fix a state ω such that kω = 1. We now prove that the limit proba-bility that candidate 1 wins the election goes to one, conditional on the state ω. All probabilitiesand expectations hereon are conditional on ω. Define the random variableδIi = σIi · (1, 0, 0)− σIi · (0, 1, 0),where the “·” operation denotes the dot product. The random variable δIi is the difference inthe scores given to candidate 1 and candidate 2 by voter i when playing the strategy σI in the(A,B)-scoring rule. Similarly, let∆Ii = ρIi · (1, 0, 0)− ρIi · (0, 1, 0),i.e. the difference in the scores of candidate 1 and candidate 2 when playing the strategy ρI underapproval voting.We first show that the expectation of the scores differences are identical under σ and ρ.Lemma 1. E(δIi) = E(∆Ii).Proof. We will show that the expectations conditional on a fixed signal are equal: E(δIi|si = s) =E(∆Ii|si = s) for all signals si. Then the unconditional expectations are also equal.First, computing the expectation for the (A,B)-scoring rule:E(δIi|si = s) =∑X=A,B{[σI(s)](1, 0, X)− [σI(s)](0, 1, X)}+∑X=A,BX {[σI(s)](X, 0, 1)− [σI(s)](0, X, 1)}+∑X=A,B(1−X) {[σI(s)](1, X, 0)− [σI(s)](X, 1, 0)}4Next, using the construction of ρI in (1) and (2) for approval voting:E(∆Ii|si = s) = [ρI(s)](1, 0, 0) + [ρI(s)](1, 0, 1)− [ρI(s)](0, 1, 0)− [ρI(s)](0, 1, 1)]=∑X=A,B(1−X) {[σ(s)](1, X, 0) + [σ(s)](1, 0, X)}+∑X=A,BX {[σ(s)](1, 0, X) + [σ(s)](X, 0, 1)}−∑X=A,B(1−X) {[σ(s)](X, 1, 0) + [σ(s)](0, 1, X)}−∑X=A,BX {[σ(s)](0, 1, X) + [σ(s)](0, X, 1)}The right hand sides are equal to each other.Consider the case where limV(δIi) → 0 for sufficiently large I. Then δIi converges to a pointmass on some constant E for large I. However, because (A,B) are interior, 0 is not part of thesupport of δIi because it is impossible for a ballot to provide equal scores to candidates 1 and 2.Therefore, since the probability that∑Ii δIi > 0 goes to one, it must be the case that the pointmass is located at a strictly positive point. This point defines the expectation limE(δIi) = E > 0for large enough I. By Lemma 1, it is also the case that limE(∆Ii) = E > 0 for large enough I.By the weak law of large numbers for triangular arrays, the probability that∑Ii=1 ∆Ii/I > 0 goesto one. Hence the probability∑Ii=1 ∆Ii > 0 also goes to one.So, without loss of generality, suppose limV(δIi) > 0 for all I, i. First, note that:Pr(I∑i=1δIi ≤ 0)= Pr(∑Ii=1 δIi − IE(δIi)√IV(δIi)≤ − IE(δIi)√IV(δIi))By the Central Limit Theorem for triangular arrays:∣∣∣∣∣Pr(∑Ii=1 δIi − IE(δIi)√IV(δIi)≤ − IE(δIi)√IV(δIi))− Φ(− IE(δIi)√IV(δIi))∣∣∣∣∣→ 0Since Pr(∑Ii=1 δIi ≤ 0)must go to zero by the assumed efficiency of σI , the triangle inequalityimplies that Φ(− IE(δIi)√IV(δIi))must go to zero. Hence:√IE(δIi)√V(δIi)→∞.5Then√IE(∆Ii)√V(∆Ii)=√IE(δIi)√V(∆Ii), by Lemma 1=√IE(δIi)√V(δIi)×√V(δIi)V(∆Ii).The second factor√V(δIi)V(∆Ii)is strictly positive. It is uniformly bounded away from zero, sincelimV(δIi) > 0 and V(∆Ii) is uniformly bounded ∆Ii takes values in a bounded set [−1, 1]. Sincethe first factor E(δIi)√V(δIi)goes to infinity, so does the product:√IE(∆Ii)√V(∆Ii)→∞.Hence Φ(−√I E(∆Ii)√V(∆Ii))→ 0. Applying the Central Limit Theorem for triangular arrays and thetriangle inequality as before, we have that Pr(∑Ii=1 ∆Ii ≤ 0)goes to zero.Since the probability that Pr(∑Ii=1 ∆Ii ≤ 0)goes to zero, the probability that the total scorefor candidate 1 is strictly larger than the total score for candidate 2 must go to one. Similarly, theprobability that the total score for candidate 1 is strictly larger than the total score for candidate3 also goes to one. The intersection event is that candidate 1 wins the election, and the probabilityof this event goes to one.By Theorem 2 of McLennan (1998), there must exist a sequence of symmetric equilibrium ρ∗Iithat provides weakly more common utility than ρIi for each fixed population I. But since ρIi takesthe probability of error to zero, then ρ∗Ii must also take the probability of error to zero.ReferencesBouton, L., and M. Castanheira (2010): “One Person, Many Votes: Divided Majority and Information Aggre-gation,” Working paper, Boston University and Universite´ Libre de Bruxelles.Goertz, J. M., and F. Maniquet (2009): “On the Informational Efficiency of Simple Scoring rules,” Workingpaper, University of Guelph and Universite´ Catholique de Louvain.McLennan, A. (1998): “Consequences of the Condorcet Jury Theorem for Beneficial Information Aggregation byRational Agents,” American Political Science Review, 92, 413–418.Myerson, R. B. (2002): “Comparison of Scoring Rules in Poisson Voting Games,” Journal of Economic Theory,103, 219–251.6

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