Necessary and sufficient conditions for transitivity in voting theory

In this paper we provide necessary and sufficient conditions for transitivity of the social preferences given by two types of social decision processes. We consider transitivity of the dominance relation of a proper simple game and that of the social preference given by the simple majority decision rule.     

Unsettling aspects of voting theory

Summary. Voting procedures are known to be plagued with a variety of difficulties such as strategic voting, or where a voter is rewarded with a better election outcome by not voting, or where a winning candidate can lose by receiving more support. Once we know that these problems can occur, the next objective should be to completely understand how, why, and where they arise. Namely, for each election procedure, the new goal is to determine when such problems can occur, all voter types who can cause these difficulties, and the actions they must take. This paper develops an easily used approach to handle all of these issues for standard voting methods. New intuitive explanations for these various oddities follow from this approach. Received: August 14, 2002 ; revised version: September 4, 2002 RID="*" ID="*"My thanks to Richard Barrett and, in particular, to Hannu Nurmi for corrections, several useful comments, and suggestions they made about earlier versions. Also, my thanks to a referee for calling some excellent references to my attention. An earlier draft was presented at the 2002 Public Choice Society meeting. This research was supported by an NSF grant.     

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.     

Necessity of transversality conditions for stochastic problems

This paper shows stochastic versions of (i) Michel's (Econometrica 58 (1990) 705, Theorem 1) necessity result, (ii) a generalization of the TVC results of Weitzman (Management Sci. 19 (1973) 783) and Ekeland and Scheinkman (Math. Oper. Res. 11 (1986) 216), and (iii) Kamihigashi's (Econometrica 69 (2001) 995, Theorem 3.4) result, which is useful particularly in the case of homogeneous returns. These stochastic extensions are established for an extremely general stochastic reduced-form model that assumes neither differentiability nor continuity.     

Sophisticated voting with information for two voting functions

The necessary and sufficient conditions for a single rational voter to reveal a sophisticated preference ordering under the Borda and Kemeny voting functions are presented. Algorithms for determining the dominant strategic ballot are given for a key voter having perfect information regarding the true preference orderings of the other voters. An example illustrating the stated concepts is provided.     

Non-consequentialist voting

Standard theory assumes that voters’ preferences over actions (voting) are induced by their preferences over electoral outcomes (policies, candidates). But voters may also have non-consequentialist (NC) motivations: they may care about how they vote even if it does not affect the outcome. When the likelihood of being pivotal is small, NC motivations can dominate voting behavior. To examine the prevalence of NC motivations, we design an experiment that exogenously varies the probability of being pivotal yet holds constant other features of the decision environment. We find a significant effect, consistent with at least 12.5 percent of subjects being motivated by NC concerns.     

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.     

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.     

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.     

Reciprocity and voting

In this paper we present a two-period model where a left-wing and a right-wing political party are solely interested in the policies they pursue. We assume that voters display reciprocal behavior. By contrast, political parties are not motivated by reciprocity. We show that reciprocity may have dramatic consequences for models of voting behavior. The incentive to be kind to the median voter may ensure that a position closer to the median voter's position is adopted even if political parties are not directly interested in being elected and cannot commit to a political stance during an election campaign. Moreover, reciprocity increases incumbency advantages.     

Approval voting in subset elections

Summary. Approval voting is designed to be “insensitive to numbers” of voters, and likely to elect a Condorcet candidate. However, the result of an election among one group of candidates gives no information about the results of elections among any other groups, even if every voter follows the recommended utility-maximizing strategy, which places strong restrictions on the individual voter's subset ballots. Thus the addition of a single candidate could completely reverse the outcome of an election, or a Condorcet candidate could finish last. Received: November 5, 1998; revised version: November 30, 1998     

Weighted approval voting

To allow society to treat unequal alternatives distinctly we propose a natural extension of Approval Voting by relaxing the assumption of neutrality. According to this extension, every alternative receives ex-ante a strictly positive and finite weight. These weights may differ across alternatives. Given the voting decisions of every individual (individuals are allowed to vote for, or approve of, as many alternatives as they wish to), society elects the alternative for which the product of total number of votes times exogenous weight is maximal. If the product is maximal for more than one alternative, a pre-specified tie-breaking rule is applied. Our main result is an axiomatic characterization of this family of voting procedures.      

Size approval voting

We propose a new class of voting procedures, called Size Approval Voting, according to which, the effective weight of a vote from a given individual depends on how many other candidates that individual votes for. In particular, weights are assumed to be non-negative and weakly decreasing in the number of approved candidates. Then, for a given profile of individual votes, all candidates with the maximal sum of weighted votes are elected. We show in our axiomatic analysis that the family of all Size Approval Voting procedures is characterized by a set of natural properties.     

The geometry of voting power: Weighted voting and hyper-ellipsoids

Suppose legislators represent districts of varying population, and their assembly's voting rule is intended to implement the principle of one person, one vote. How should legislators' voting weights appropriately reflect these population differences? An analysis requires an understanding of the relationship between voting weight and some measure of the influence that each legislator has over collective decisions. We provide three new characterizations of weighted voting that embody this relationship. Each is based on the intuition that winning coalitions should be close to one another. The locally minimal and tightly packed characterizations use a weighted Hamming metric. Ellipsoidal separability employs the Euclidean metric: a separating hyper-ellipsoid contains all winning coalitions, and omits losing ones. The ellipsoid's proportions, and the Hamming weights, reflect the ratio of voting weight to influence, measured as Penrose–Banzhaf voting power. In particular, the spherically separable rules are those for which voting powers can serve as voting weights.     

Macroeconomic conditions in the U.S. and congressional voting on environmental policy: 1970-2008

Using the Environmental Scorecard ratings of Congressmen and Senators published annually by the League of Conservation Voters, we explore empirically whether political support for pro-environment legislation, aggregated across each legislative body, is sensitive over time to changing economic conditions — that is, whether there is a political trade-off between economic conditions and the environment. Using LCV scorecard ratings from 1970 to 2008, we find evidence, consistent across both the House and Senate, that political support for the environment is related to per capita income, but this general tendency can be decomposed into sharp differences by party.     

Reciprocity and voting

In this paper we present a two-period model where a left-wing and a right-wing political party are solely interested in the policies they pursue. We assume that voters display reciprocal behavior. By contrast, political parties are not motivated by reciprocity. We show that reciprocity may have dramatic consequences for models of voting behavior. The incentive to be kind to the median voter may ensure that a position closer to the median voter's position is adopted even if political parties are not directly interested in being elected and cannot commit to a political stance during an election campaign. Moreover, reciprocity increases incumbency advantages.     

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.     

Implementing alternative voting in kingmaker trees

The alternative vote procedure (the single transferable vote system) can be implemented as the subgame perfect equilibrium of a kingmaker tree. We call the implementation procedure “the method of descending chains.” A number of examples are given.