A Bayesian Model of Voting in Juries

We model voting in juries as a game of incomplete information, allowing jurors to receive a continuum of signals. We characterize the unique symmetric equilibrium of the game, and give a condition under which no asymmetric equilibria exist under unanimity rule. We offer a condition under which unanimity rule exhibits a bias toward convicting the innocent, regardless of the size of the jury, and give an example showing that this bias can be reversed. We prove a “jury theorem” for our general model: As the size of the jury increases, the probability of a mistaken judgment goes to zero for every voting rule except unanimity rule. For unanimity rule, the probability of making a mistake is bounded strictly above zero if and only if there do not exist arbitrarily strong signals of innocence. Our results explain the asymptotic inefficiency of unanimity rule in finite models and establishes the possibility of asymptotic efficiency, a property that could emerge only in a continuous model. Journal of Economic Literature Classification Numbers: C72, D72.     

Deliberative voting

We analyze a model of jury decision making in which jurors deliberate before casting their votes. We consider a wide range of voting institutions and show that deliberations render these equivalent with respect to the sequential equilibrium outcomes they generate. In particular, in the context of a jury setup, all voting rules excluding the two types of unanimity rules (one requiring a unanimous consensus to acquit, one requiring a unanimous consensus to convict) induce the same set of equilibria outcomes. We show the robustness of our results with respect to several restrictions on communication protocols and jurors’ strategies. Furthermore, we demonstrate that our observations extend to practically all of the voting structures commonly studied in the voting literature. The paper suggests the importance of accounting for communication in models of collective choice.     

The structure of bias in peer voting systems: lessons from the Eurovision Song Contest

This paper assesses whether and how common characteristics of jury members or peer voters affect the outcomes of voting systems. In particular, we analyze to what extent these common features result in voting bias. We take as a case study the Eurovision Song Contest for which an extensive amount of historical data is available. In contrast to earlier studies we analyze the impact of common factors on the bias individually for each country, which is necessary to substantiate the publicly debated accusations of regional block voting by certain groups of countries. We establish strong evidence for voting bias in the song contest on the basis of geography, even after correction for culture, language, religion and ethnicity. However, these effects do generally not correspond to the usual accusations. We believe that our findings extend to all instances where groups of jury members or peer voters share certain common factors, which may cause voting bias. It is important to identify such structures explicitly, as it can help avoiding bias in the first place. The authors are grateful to Marieke van Dijk for excellent research assistance and to Laurens Swinkels, Ieva Pudane, Gijsbert van Lomwel, Jelena Stefanovic, and Bas van den Heuvel for useful comments. The usual disclaimer applies.     

The Costs of Implementing the Majority Principle: The Golden Voting Rule

In a context of constitutional choice of a voting rule, this paper presents an economic analysis of scoring rules that identifies the golden voting rule under the impartial culture assumption. This golden rule depends on the weights β and (1−β) assigned to two types of costs: the cost of majority decisiveness (‘tyranny’) and the cost of the ‘erosion’ in the majority principle. Our first main result establishes that in voting contexts where the number of voters n is typically considerably larger than the number of candidates k, the golden voting rule is the inverse plurality rule for almost any positive β. Irrespective of n and k, the golden voting rule is the inverse plurality rule if β ≥ 1/2 .. This hitherto almost unnoticed rule outperforms any other scoring rule in eliminating majority decisiveness. The golden voting rule is, however, the plurality rule, the most widely used voting rule that does not allow even the slightest ‘erosion’ in the majority principle, when β=0. Our second main result establishes that for sufficiently “small size” voting bodies, the set of potential golden rules consists at most of just three rules: the plurality rule, the Borda rule and the inverse plurality rule. On the one hand, this finding provides a new rationalization to the central role the former two rules play in practice and in the voting theory literature. On the other hand, it provides further support to the inverse plurality rule; not only that it is the golden rule in voting contexts, it also belongs, together with the plurality rule and the Borda method of counts, to the “exclusive” set of potential golden voting rules in small committees. We are indebted to Jim Buchanan, Amichai Glazer, Noa Nitzan, Ken Shepsle, and an anonymous referee for their useful comments.     

A theory of hung juries and informative voting

This paper investigates a jury decision when hung juries and retrials are possible. When jurors in subsequent trials know that previous trials resulted in hung juries, informative voting cannot be an equilibrium regardless of voting rules unless the probability that each juror receives the correct signal when the defendant is guilty is identical to the one when he is innocent. Thus, while Coughlan (2000) claims that mistrials facilitate informative voting, our result shows that such an assertion holds only in limited circumstances.     

Bargaining in committees as an extension of Nash's bargaining theory

This paper addresses the following issue: if a set of agents bargain on a set of feasible alternatives ‘in the shadow’ of a voting rule, that is, any agreement can be enforced if a ‘winning coalition’ supports it, what general agreements are likely to arise? In other words: what influence can the voting rule used to settle (possibly nonunanimous) agreements have on the outcome of consensus? We model the situation as an extension of the Nash bargaining problem in which an arbitrary voting rule replaces unanimity. In this setting a natural extension of Nash's solution is characterized.     

Minority voting and long-term decisions

In this paper we propose minority voting as a scheme that can partially protect individuals from the risk of repeated exploitation. We consider a committee that meets twice to decide about projects including a first-period project that may have long-lasting impact. In the first period, a simple open majority voting scheme takes place. Voting splits the committee into three groups: voting winners, voting losers, and absentees. Under minority voting, only voting losers retain their voting rights in the second period. We show that as soon as absolute risk aversion exceeds a threshold value, minority voting is superior to repeated application of the simple majority rule.     

The Disunity of Unanimity

Unanimity is the optimal voting rule in a world of zero transactions costs, when side payments are impossible. When side payments are available and transactions costs are zero, the voting rule is irrelevant to the ultimate outcome. In the more realistic situation where side payments are allowed but transactions costs are positive, a unanimity voting rule creates situations where the collective choice may fail a proposed measure even if all members favor the measure in principle. This evidences a disunity between unanimity rules and unanimous outcomes. Constitutional design should focus on rules leading to unanimous outcomes, as opposed to unanimity rules.     

Noncooperative foundations of bargaining power in committees and the Shapley–Shubik index

In this paper we explore the noncooperative foundations of the bargaining power that a voting rule gives to each member of a committee that bargains in search of consensus over a set of feasible agreements under a voting rule. Assuming complete information, we model a variety of bargaining protocols and investigate their stationary subgame perfect equilibria. We show how the Shapley–Shubik index and other power indices can be interpreted as measures of ‘bargaining power’ that appear in this light as limit cases.     

Jury Size and the Free Rider Problem

In recent times, judges in the United States have said that6-person juries are inferior to 12-person juries. But by whatreasoning is a smaller jury inferior? One argument is the Condorcetjury theorem, which says that a larger jury will reach a moreaccurate decision. This, however, assumes that the informationof each juror is independent of the size of the jury. I showthat a juror's information does depend on the size of the jury.In a larger jury panel each juror has less incentive to payattention in the court, even though they are all pledged tohear and deliver a verdict on a trial. Because of the free-riderproblem, a larger jury may actually make poorer decisions. Theresults apply to many environments in which decisions are madeby committees and work teams.     

A Coalition-Form Analysis Of The “One Country - One Vote” Rule In The Governing Council Of The European Central Bank *

This paper analyses the “one country—one vote” rule for monetary policy decision

making of the Governing Council of the European Central Bank in a framework of cooperative game theory. The Shapley value is used as a solution concept. In contrast to former papers analysing the allocation of abstract "voting power" in committees of international organisations, preferences for monetary policy are modelled to obtain a prediction about potential transfers implied by an equal allocation of voting rights when countries are of different size. It is shown that if the number of countries participating to a currency union grows and the weight of the largest country within the currency union becomes small, the allocation of voting rights becomes irrelevant in the sense that transfers per country tend in any case to zero. [C71, E5, F02]     

Cooperative bargaining foundations of the Shapley–Shubik index

In a recent paper, we provide an interpretation of the Shapley–Shubik index as a measure of the ‘bargaining power’ that a voting rule gives to each member of a committee negotiating in search of consensus. It is assumed that negotiation takes place under the condition that every winning coalition can impose any agreement. In this paper we further investigate the axiomatic foundations of this interpretation of the Shapley–Shubik index. To this end a wider framework admitting random voting rules is considered.     

Protecting Minorities through the Average Voting Rule

In the average voting rule, the outcome is some weighted average of votes. The unique average voting outcome is characterized by a median formula, which depends on voters' preferred allocations and some parameters constructed from voters' weights. A minority is said to be protected by a switch in voting rule if the outcome becomes closer to the median bliss point of the minority. Sufficient conditions for minority protection are that, either the minority's weight is sufficiently large or the majority outcome is too unfavorable to the minority. Applications to the composition of public goods and to public expenditures level are considered. We explore the combined use of average and majority voting in a two‐stage procedure for determining the level and the composition of public expenditures.     

An experimental test of a committee search model

The objective of this paper is to design a laboratory experiment for an infinite-horizon sequential committee search model in order to test some of the implications obtained by the model in Albrecht et al. (2010) (AAV). We find that, compared with single-agent search, the search duration is longer for committee search under the unanimity rule, but is shorter for committee search in which at least one vote is required to stop searching. In addition, according to estimates from round-based search decisions, subjects are more likely to vote to stop searching in committee search than in single-agent search. This confirms that agents are less picky in committee search. Overall, the experimental outcomes are consistent with the implications suggested by the AAV model. However, despite the prediction from the AAV model, we could not obtain a significant outcome in relation to the size order of the probabilities of voting to stop searching in committee search for the various plurality voting rules.     

On the informational efficiency of simple scoring rules

We study information aggregation in large elections. With two candidates, efficient information aggregation is possible (e.g., Feddersen and Pesendorfer [5], [6] and [7]). We show that this result does not extend to elections with more than two candidates. We study a class of simple scoring rules in voting games with Poisson population uncertainty and three candidates. No simple scoring rule aggregates information efficiently, even if preferences are dichotomous and a Condorcet winner always exists. We introduce a weaker criterion of informational efficiency that requires a voting rule to have at least one efficient equilibrium. Only approval voting satisfies this criterion.     

Approval voting reconsidered

Summary. The merit of approval voting has been widely discussed in the past 25 years. The distinct property of this rule is the extent of flexibility it allows; any voter can approve as many alternatives/candidates as he wishes. Nevertheless, this advantage is the very reason for two drawbacks of approval voting: its extreme vulnerability to majority decisiveness (Theorem 1) and its extreme vulnerability to erosion of the majority principle (Theorem 2). On the one hand, under some feasible voting strategies any majority of more than 1/2 of the voters can guarantee the selection of its most favorable candidate, regardless of the preferences of the other voters. On the other hand, under alternative voting strategies even the largest majority cannot impose its common most preferred candidate. A simultaneous resolution of the two problems is possible by restricted approval voting (RAV), a voting rule that allows partial voter flexibility by restricting the minimal and maximal number of candidates that can be approved. Our main result (Theorem 3) clarifies how the foregone flexibility in voters sovereignty mitigates the above mentioned drawbacks under sincere and insincere coordinated voting. Our findings suggest a new possible justification of a particular voting rule which is based on the significance assigned to three considerations: the advantages of voters flexibility, immunity to majority decisiveness and immunity to erosion of the majority principle. Such justification can provide a possible explanation to the prevalent use of some special cases of RAV, notably, of the plurality rule and of approval voting.Received: 8 October 2003, Revised: 17 June 2004, JEL Classification Numbers: D71, D72. Correspondence to: Shmuel NitzanWe are indebted to Steve Brams, Remzi Sanver and an anonymous referee for their very useful comments and suggestions.     

PUBLIC GOOD PROVISION WITH CONSTITUTIONAL CONSTRAINT

This article studies the public good provision problem, in which the principal faces a constitutional constraint in the sense that in order for a public good provision mechanism to be implemented, it must first be approved by agents under a prespecified voting rule. I find that as long as the voting rule is not the unanimity rule, the principal can propose a mechanism such that first‐best efficiency of provision of the public good is achieved. I also consider various constraints, such as prohibition of discriminatory mechanisms and the existence of vote buying, and discuss optimal voting rules in these situations.     

An axiomatic theory of political representation

We discuss the theory of gerrymandering-proof voting rules. Our approach is axiomatic. We show that, for votes over a binary set of alternatives, any rule that is unanimous, anonymous, and gerrymandering-proof must decide a social outcome as a function of the proportions of agents voting for each alternative, and must either be independent of this proportion, or be in one-to-one correspondence with the proportions. In an extended model in which the outcome of a vote at the district level can be a composition of a governing body (with two possible parties), we discuss the quasi-proportional rules (characterized by unanimity, anonymity, gerrymandering-proofness, strict monotonicity, and continuity). We show that we can always (pointwise) approximate a single-member district quota rule with a quasi-proportional rule. We also discuss a more general environment, where there may be more than two parties.     

Committee Design with Endogenous Information

Identical agents gather costly information, and then aggregate it through voting. Because information is a public good, information is underprovided relative to the social optimum. A "good" voting rule must give incentives to acquire information, as well as aggregate information efficiently. A voting rule that requires a large plurality (in the extreme, unanimity) to upset the status quo can be optimal only if the information available to each agent is sufficiently accurate. This result is independent of the preferences of voters and of the cost of information.     

The Borda rule,Condorcet consistency and Condorcet stability

Summary. The Borda rule is known to be the least vulnerable scoring rule to Condorcet inconsistency, Saari (2000). Such inconsistency occurs when the Condorcet winner (the alternative which is preferred to any other alternative by a simple majority) is not selected by the Borda rule. This note exposes the relationship between the Borda rule and the Condorcet q-majority principle as well as the Condorcet q-majority voting rule. The main result establishes that the Borda rule is Condorcet q-majority consistent when where k is the number of alternatives. The second result establishes that is the minimal degree of majority decisiveness corresponding to the Borda rule under sincere voting. The same majority is required to ensure decisiveness under the Borda rule and to ensure that a q-rule (the generalized q-majority Condorcet rule) is a voting rule. Received: April 8, 2002; revised version: July 17, 2002 Correspondence to:S. Nitzan