May’s theorem in one dimension
| Date | 01 January 2017 |
| Published date | 01 January 2017 |
| DOI | 10.1177/0951629815603694 |
| Subject Matter | Articles |
Article
Journal of Theoretical Politics
2017, Vol.29(1) 3–21
ÓThe Author(s) 2015
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DOI: 10.1177/0951629815603694
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May’s theorem in one
dimension
John Duggan
Department of PoliticalScience and Department of Economics, University of Rochester,
Rochester,USA
Abstract
This paper provides three versions of May’s theorem on majority rule, adapted to the one-
dimensional model common in formal political modeling applications. The key contribution is that
single peakedness of voter preferences allows us to drop May’s restrictive positiveresponsiveness
axiom. The simplest statement of the result holds when voter preferences are single peaked and
linear (no indifferences), in which case a voting rule satisfies anonymity, neutrality, Pareto, and
transitivity of weak social preference if and only if the number of individuals is odd and the rule is
majority rule.
Keywords
Majority rule; May’s theorem; single peaked;transitivity
1. Introduction
Majority rule occupies a central role in democratic decision making, and it has
accordingly received close attention in formal political theory. A well-known axio-
matization due to May (1952) provides some theoretical justification for the status
of majority rule: when individual preferences are unrestricted, it is the only voting
rule that is unbiased toward individuals, unbiased toward alternatives, and posi-
tively responsive to changes individuals preferences. In the terminology of this
paper, majority rule is uniquely characterized by the axioms of anonymity, neutral-
ity, and strong tie break. The latter axiom is indeed strong: it requires that if social
indifference holds between two alternatives and if even just one individual increases
Corresponding author:
John Duggan, Department of PoliticalScience and Department of Economics, University of Rochester,
Rochester,NY 14627, USA.
Email: dugg@ur.rochester.edu
her support for one of these alternatives (e.g. indifference is replaced by strict pre-
ference for one alternative), then the social ‘tie’ between the alternatives is broken
in favor of the one with increased support. Although desirable in a world where
each vote counts and social ties represent an exact balancing of individual prefer-
ences, the strong tie break axiom is violated by most voting rules —for example, it
is violated by all quota rules other than majority rule—and its normative impor-
tance is less compelling than the other axioms.
In this paper, I provide an alternative to May’s axiomatization that is closer to
the subject matter of political science and eschews his strong tie break axiom. The
key structure added to the problem is the assumption of single peakedness: whereas
May assumes individual preferences are unrestricted, consistent with the social
choice literature, formal modeling in political science often assumes alternatives
are one-dimensional and individual preferences are single peaked. This preference
restriction is prevalent in work on electoral modeling in the tradition of Calvert
(1983); it was used in the analysis of agenda control by Romer and Rosenthal
(1978), introduced to legislative bargaining by Banks and Duggan (2000), and
Penn et al. (2011) recently explore the implications of single peakedness in an axio-
matic analysis of strategy-proof voting mechanisms. A compelling property of
single-peaked preferences, established by Arrow (1951) and Black (1948, 1958), is
that it precludes the Condorcet paradox and confers desirable transitivity proper-
ties on majority rule: majority voting generates transitive strict social preferences,
and when the number of individuals is odd, majority indifference relation is also
transitive. This full transitivity of majority rule with an odd number of individuals
is a highly specialized property—for example, it is violated by all quota rules other
than majority rule—and in this sense, transitivity with single-peaked individual
preferences seems to impose restrictions similar to strong tie break with unrest-
ricted preferences.
I maintain May’s other axioms and investigate the question, ‘Assuming single-
peaked preferences, does May’s characterization carry over if we replace the tie
break axiom with transitivity?’ A positive answer to this question would be of
interest because it would provide further justification for majority rule in many
applications considered in political science, and because, compared to the tie break
axiom, transitivity arguably has greater normative relevance: when a voting rule
generates an ordering of alternatives, we can view social choices as deriving from a
representative agent; this simplifies the process of choosing from a finite set of fea-
sible options, as we simply choose the top-ranked of the available options; and it
precludes inconsistencies when options are added or deleted. The answer to the
above question depends on the details—on the possibility of individual indiffer-
ences between alternatives, on the form of majority rule considered, and on the
nature of the transitivity condition imposed. Regarding the definition of majority
rule, a majority preference for xover ymay hold if more than half of all individu-
als strictly prefer xto y(simple majority rule), or it may hold if more individuals
strictly prefer xthan strictly prefer y(relative majority rule). One transitivity condi-
tion is that strict majority preferences are transitive, and a stronger one is that
weak majority preferences are transitive (i.e. both strict majority preference and
4Journal of Theoretical Politics 29(1)
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