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.     

Enlargement and Institutional Reform in the European Union: Separate or Connected Issues?

In this paper, I analyze how the enlargement process is connected to the currently proposed institutional reforms in the European Union. Although the official claim is that enlargement requires reform, the relationship between both processes needs clarification. The use of unanimity voting for policy issues leads to substantial deadlock in the current Union, a problem that could be solved by the introduction of (qualified) majority voting. Moreover, in view of its current inflexibility enlargement would only marginally decrease the Union's capacity to make decisions. Under qualified majority rule, enlargement is likely to have little or no effect on flexibility given the distribution of preferences of the member-states. Also, if qualified majority voting is used, enlargement would decrease the power of the current member-states in a future Union, especially the larger ones. This, and the possibility for current member-states to link enlargement with specific policy demands, could be the crucial factors in the timing and shaping of the enlargement process.     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Repeated Elections with Asymmetric Information

An infinite sequence of elections with no term limits is modelled. In each period a challenger with privately known preferences is randomly drawn from the electorate to run against the incumbent, and the winner chooses a policy outcome in a one-dimensional issue space. One theorem is that there exists an equilibrium in which the median voter is decisive: an incumbent wins re-election if and only if his most recent policy choice gives the median voter a payoff at least as high as he would expect from a challenger. The equilibrium is symmetric, stationary, and the behavior of voters is consistent with both retrospective and prospective voting. A second theorem is that, in fact, it is the only equilibrium possessing the latter four conditions — decisiveness of the median voter is implied by them.     

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     

Anonymous yes–no voting with abstention and multiple levels of approval

Symmetric (3,2) simple games serve as models for anonymous voting systems in which each voter may vote “yes,” abstain, or vote “no,” the outcome is “yes” or “no,” and all voters play interchangeable roles. The extension to symmetric (j,2) simple games, in which each voter chooses from among j ordered levels of approval, also models some natural decision rules, such as pass–fail grading systems. Each such game is determined by the set of (anonymous) minimal winning profiles. This makes it possible to count the possible systems, and the counts suggest some interesting patterns. In the (3,2) case, the approach yields a version of May's Theorem, classifying all possible anonymous voting rules with abstention in terms of quota functions. In contrast to the situation for ordinary simple games these results reveal that the class of simple games with 3 or more levels of approval remains large and varied, even after the imposition of symmetry.     

Coordinated voting in sequential and simultaneous elections: some experimental evidence

This paper studies a situation wherein a set of voters choose between two alternatives in the presence of a payoff externality. Specifically, regardless of her intrinsic preference, a voter’s payoff is maximized should she vote for the alternative that garners a majority of the votes cast. Are votes coordinated on a single alternative? Using laboratory experiments, we examine voting patterns in sequential voting and simultaneous voting elections. Across both election types, we also vary the amount of information that an individual voter has regarding the intrinsic preferences of the other voters. Our main findings are as follows. In the “low” information treatment, sequential voting elections facilitate coordinated voting. However, in the “high” information treatment, voting patterns are not dependent on how the election is structured.     

The swing voter's curse with adversarial preferences

We analyze voting behavior in a large electorate in which voters have adversarial state-contingent preferences with incomplete information about the state of the world. We show that one type of voter can suffer from the swing voter's curse à la Feddersen and Pesendorfer [The swing voter's curse, Amer. Econ. Rev. 86 (1996) 408-424], and go on to characterize the symmetric Nash equilibria of this model under different parameter values. We prove that unlike settings with nonadversarial preferences, there are equilibria in which in one state of the world, a minority-preferred candidate almost surely wins the election and thus the election may fail to correctly aggregate information. Indeed, we show that the fraction of the electorate dissatisfied with the result can be as large as .     

Electoral fraud and voter turnout: An experimental study

In this paper we experimentally investigate the consequences of electoral fraud on voter turnout. The experiment is based on a strategic binary voting model where voters decide whether to cast a costly vote in favour of their preferred candidate or to abstain. The electoral process is illicitly influenced by applying ballot-box stuffing. In the experiment we implement two different framings: we compare voter turnout in a neutral environment and with framed instructions to explicitly replicate elections. This approach enables us to both test the model's predictions and to estimate the framing effects of voting and fraud. Comparison of experimental results with theoretical predictions reveals over-voting, which is exacerbated when fraud occurs. Turnout increases as predicted with a moderate level of fraud while, with higher levels of electoral fraud, voters fail to recognize that the existence of a relatively larger number of “agents” voting with certainty considerably decreases the benefits of voting. Importantly, framing matters, as revealed by the higher turnout of those in the majority group, against which the fraud is applied.     

Voting Equilibria in Multi-candidate Elections

We consider a general plurality voting game with multiple candidates, where voter preferences over candidates are exogenously given. In particular, we allow for arbitrary voter indifferences, as may arise in voting subgames of citizen-candidate or locational models of elections. We prove that the voting game admits pure strategy equilibria in undominated strategies. The proof is constructive: we exhibit an algorithm, the "best winning deviation" algorithm, that produces such an equilibrium in finite time. A byproduct of the algorithm is a simple story for how voters might learn to coordinate on such an equilibrium.     

Power,luck and the right index

Summary We have pointed out the theoretical drawbacks of the traditional indices for measuring a priori voting power inasmuch as they are implied in considering the coalition value a private good. This criticism caused us to view the coalition outcome as a public good. From this aspect and additional considerations with respect to power, luck, and decisiveness, we obtained a story describing the characteristics of an adequate measure of a priori voting power. These characteristics were found to be fulfilled by an index presented by Holler (1978). Through the above analysis this index has received its theoretical justification. An independent view of this index was then provided by means of an axiomatic characterization. This characterization makes possible abstract comparison of the index with previously established private good indices.While we have restricted our attention to simple games, the index presented can be generalized to provide a value on games in characteristic function form. We leave this topic for future conideration.     

An institutional analysis of voter turnout: the role of primary type and the expressive and instrumental voting hypotheses

Recent events highlight primary type as an institutional variable that merits further examination in the economics literature on voter turnout. Using panel data for U.S. gubernatorial elections and treating primary type as a proxy for candidate deviation from the median voter, we test whether primary type changes voter turnout and whether that change is dominated by instrumental or expressive voting. The results show that states with more open primaries tend to have greater voter turnout in general elections, and that this increase reflects the effect of open primaries on expressive voting.

An IMPOSSIBILITY THEOREM IN PURE PUBLIC GOODS ECONOMIES WITH FEASIBILITY CONSTRAINTS: VOTING BY COMMITTEES IN NON-RECTANGULAR FEASIBLE SETS*

We consider the problem of choosing an alternative in a pure public goods economy with feasibility constraints when voters have "additively-separable and single-peaked"preferences.Our purpose is to identify tops-only voting procedures satisfying "nonmanipulability" and "voter sovereignty". First, we show that such procedures are generalizations of the schemes of "voting by committees"¹)introduced by Barbera, Sonnenschein and Zhou (1991) in the sense that these procedures are defined similarly for the generalized feasible set. Second, we establish that when no two goods can be simultaneously produced at their maximal feasible levels, the procedures are characterized by the existence of a very powerful voter.     

Approval voting with endogenous candidates

We present a formal model of political competition under approval voting which allows for endogenous candidate entry. Our analysis yields a number of novel insights. First, we develop a notion of sincere voting behavior under approval voting, called relative sincerity. We then show that the relatively sincere voting behavior is consistent with the strategic calculus of voting. Second, we show that in a one-dimensional model with distance preferences, equilibria in relatively sincere strategies and without spoiler candidates always generate outcomes close to the median voter. Moreover, approval voting satisfies Duverger's Law in the sense that there are at most two winning positions! Finally, we extend our analysis to arbitrary policy spaces. In the general setting, approval voting is shown to be susceptible to the same kinds of problems as the plurality rule, such as the possibility of non-majoritarian outcomes, failure to elect the Condorcet winner and existence of spoiler candidates.     

Is the median voter decisive? Evidence from referenda voting patterns

This paper examines whether the voter with the median income is decisive in local spending decisions. Previous tests have relied on cross-sectional data while we make use of a pair of California referenda to estimate a first difference specification. The referenda proposed to lower the required vote share for passing local educational bonding initiatives from 67 to 50% and 67 to 55%, respectively. We find that voters rationally consider future public service decisions when deciding how to vote on voting rules. However, the empirical evidence strongly suggests that an income percentile below the median is decisive for majority voting rules, especially in communities that have a large share of high-income voters with attributes that suggest low demand for public services. Based on a model that explicitly recognizes that each community contains voters with both high and low demand for public school spending, we also find that an increase in the share of low demand voters is associated with a lower decisive voter income percentile for the high demand group. This two type model implies that our low demand types (individuals over age 45 with no children) have demands that are 45% lower than other voters. Collectively, these findings are consistent with high-income voters with weak preferences for public educational services voting with the poor against increases in public spending on education.     

Fully sincere voting

In a general social choice framework where the requirement of strategy-proofness may not be sensible, we call a social choice rule fully sincere if it never gives any individual an incentive to vote for a less-preferred alternative over a more-preferred one and provides an incentive to vote for an alternative if and only if it is preferred to the default option that would result from abstaining. If the social choice rule can depend only on the number of votes that each alternative receives, those rules satisfying full sincerity are convex combinations of the rule that chooses each alternative with probability equal to the proportion of the vote it receives and an arbitrary rule that ignores voters' preferences. We note a sense in which the natural probabilistic analog of approval voting is the fully sincere rule that allows voters maximal flexibility in expressing their preferences and gives these preferences maximal weight.     

Direct democracy in Australia: Voter behavior in the choice between constitutional monarchy and a republic

Opinion polls conducted throughout the 1990s indicate most Australians favor a republic. A referendum making that constitutional change, however, was defeated. This paper investigates whether voters employ a loss-minimization rule, as opposed to a value-maximization rule, when making political decisions. Based on the predictions of each rule, political strategies are devised and compared to the official arguments employed by republicans and monarchists during the period preceding the vote. Empirical research relates voting outcomes at the individual voter level to influences that are likely to be correlated with political risk aversion. The results are consistent with the conclusion that voters do not employ value-maximization rules.     

A passion for voting

We model expressive voting as a dynamic game with informed and ignorant voters. A voter has selective memory for actions and he is aware of it. We find a unique symmetric equilibrium with ignorant voting. Public signal in favor of one particular alternative creates the bandwagon and underdog effects. When the signal is sufficiently strong, the majority outcome is biased. This is a possible reason for persistence of public policies.     

A live experiment on approval voting

This paper presents a large-scale experiment on the Approval Voting rule that took place during the 2002 French presidential election. We describe the experiment and its main results. The findings are as follows: (i) Such an experiment is feasible, and very well accepted by voters. (ii) The principle of approval voting is easily understood and accepted. (iii) Within the observed political context, compared to the official first-round vote, approval voting modifies the overall ranking of candidates. (iv) The candidates Le Pen and Chirac, more than the others, were able to convert approval votes into official first-round votes. JEL Classification C93, D70, D72     

Winning probabilities in a pairwise lottery system with three alternatives

Summary. The pairwise lottery system is a multiple round voting procedure which chooses by lot a winner from a pair of alternatives to advance to the next round where in each round the odds of selection are based on each alternatives majority rule votes. We develop a framework for determining the asymptotic relative likelihood of the lottery selecting in the final round the Borda winner, Condorcet winner, and Condorcet loser for the three alternative case. We also show the procedure is equivalent to a Borda lottery when only a single round of voting is conducted. Finally, we present an alternative voting rule which yields the same winning probabilities as the pairwise lottery in the limiting case as the number of rounds of the pairwise lottery becomes large.Received: 5 June 2003, Revised: 17 June 2004, JEL Classification Numbers: D71. Correspondence to: Jac C. HeckelmanWe thank Keith Dougherty and Andrew Yates for their comments.