Majority voting on orders

Social Choice and Welfare, 1996

This paper considers the construction of sets of preferences that give consistent outcomes under majority voting. Fishburn [7] shows that by combining the concepts of single-peaked and single-troughed preferences (which are themselves examples of value restriction) it is possible to provide a simple description of the extent of agreement between individuals that allows the construction of sets that are as large as those previously known (for fewer than 7 alternatives) and larger than those previously known (for 7 or more alternatives). This paper gives a characterisation of the preferences generated through these agreements and makes observations on the relation between the sizes of such sets as the number of alternatives increases.

Mathematical Social Sciences, 2002

We develop a general concept of majority rule for finitely many choice alternatives that is consistent with arbitrary binary preference relations, real-valued utility functions, probability distributions over binary preference relations, and random utility representations. The underlying framework is applicable to virtually any type of choice, rating, or ranking data, not just the linear orders or paired comparisons assumed by classic majority rule social welfare functions. Our general definition of majority rule for arbitrary binary relations contains the standard definition for linear orders as a special case.

2010

If a decision maker prefers x to y to z, would he choose orderd set [x;z] or [y;x]? This article studies extension of preferences over individual alternatives to an ordered set which is prevalent in closed ballot elections with proportional representation and other real life problems where the decision maker is to choose from groups with an associated hierarchy inside. I introduce ve ordinal decision rules: highest-position, top-q, lexicographic order, max-best, highest-of-best rules and provide axiomatic characterization of them. I also investigate the relationship between ordinal decision rules and the expected utility rule. In particular, whether some ordinal rules induce the same (weak) ranking of ordered sets as the expected utility rule.

TOP, 2013

A society has to choose within a set X of programs, each defining a decision regqrding a finite number D of yes-no issues. An X-profile associates with every program x in X a finite number of voters who support x. We prove that the outcome of the issue-wise simple majority rule Maj is an element of X at any X-profile where Maj is well-defined if and only if this is true when Maj is applied to any profile involving only 3 elements of X, each being supported by exactly one voter. We call this property triple-consistency. Moreover, we characterize the class of anonymous issue-wise choice functions that are triple-consistent. We discuss three applications of the results. First, interpreting X as a domain of preference relations over a finite set of alternatives, we argue that they generalize a well-known consequence of the value-restriction propery . Second, we can characterize the sets of approval ballots for which the strong version of paradox of multiple elections never occurs. Third,we can provide some new insights to the dynamics of club formation.

Journal of Mathematical Economics, 2011

Collective rationality of voting rules, requiring transitivity of social preferences (or quasi-transitivity, acyclicity for weaker notions), has been known to be incompatible with other standard conditions for voting rules when there is no prior information, thus no restriction, on individual preferences Sen, 1970). proposes two restricted domains of individual preferences where majority voting generates transitive social preferences; they are the domain consisting of preferences that have at most two indifference classes, and the domain where any set of three alternatives is partitioned into two non-empty subsets and alternatives in one set are strictly preferred to alternatives in the other set. On these two domains, we investigate whether majority voting is the unique way of generating transitive, quasi-transitive, or acyclic social preferences. First of all, we rule out non-standard voting rules by imposing monotonicity, anonymity, and neutrality. Our main results show that majority rule is the unique voting rule satisfying transitivity, yet all other voting rules satisfy acyclicity (also quasi-transitivity on the second domain). Thus we find a very thin border dividing majority and other voting rules, namely, the gap between transitivity and acyclicity.

Public Choice, 2012

Suppose an organization has a committee with multiple seats, and the committee members are to be elected by a group of voters. For the organization, the possible alternatives are the possible sets of individuals who could serve together. A common approach is to choose from among these alternatives by having each voter cast separate votes on the candidates for each seat. When this type of ballot is used, important characteristics of the set of individuals on the committee (such as what percentage of the members will be female) might not be explicitly considered by the voters. Another approach that has been used is to have each voter cast a ballot which ranks all possible sets of members. However, this approach can require the voters to weigh a relatively large number of alternatives. This paper considers group decisions where it is desirable to: (1) explicitly consider characteristics of alternatives and (2) have a relatively small number of options upon which a voter has to express his preferences. The approach that we propose has two steps: First voters vote directly on pertinent characteristics of alternatives; Then these votes are used to indirectly specify preferences on alternatives. The indirectly specified preferences are ones that are naturally modeled using

2018

A collective choice rule selects a set of alternatives for each collective choice problem. Suppose that the alternative ’x’, is in the set selected by a collective choice rule for some collective choice problem. Now suppose that ‘x’ rises above another selected alternative ‘y’ in some individual’s preferences. If the collective choice rule is “positively responsive”, ‘x’ remains selected but ‘y’ is no longer selected. If the set of alternatives contains two members, an anonymous and neutral collective choice rule is positively responsive if and only if it is majority rule (May 1952). If the set of alternatives contains three or more members, a large set of collective choice rules satisfy these three conditions. We show, however, that in this case only the rule that assigns to every problem its strict Condorcet winner satisfies the three conditions plus Nash’s version of “independence of irrelevant alternatives” for the domain of problems that have strict Condorcet winners. Further, ...

International Economic Review

May's theorem (1952) shows that if the set of alternatives contains two members, an anonymous and neutral collective choice rule is positively responsive if and only if it is majority rule. We show that if the set of alternatives contains three or more alternatives only the rule that assigns to every problem its strict Condorcet winner satisfies the three conditions plus Nash's version of "independence of irrelevant alternatives" for the domain of problems that have strict Condorcet winners. We show also that no rule satisfies the four conditions for domains that are more than slightly larger.

American Political Science Review, 2002

In pairwise voting, when a simple majority rule produces a winner, that winner is robust to the minority's preferences. The typical means of protecting the minority from the decisiveness of the majority is by increasing the required majority or by augmenting the simple majority rule with constitutional constraints. In the former case the required majority q becomes larger than one-half, and this implies that the q-majority rule becomes biased in favor of one of the alternatives, usually the status quo. In the latter case the augmented rule becomes biased in favor of the minority. The main issue examined in this paper is whether the amelioration of majority decisiveness can be attained by unbiased voting rules that allow some restricted expression of preference intensities. Our results clarify that the use of scoring rules provides a positive answer to the above question when voters resort to variable degrees of coordinated strategic voting. The results are illustrated in the spe...

Mathematical Social Sciences, 1981