POSITIVELY RESPONSIVE COLLECTIVE CHOICE RULES AND MAJORITY RULE: A GENERALIZATION OF MAY'S THEOREM TO MANY ALTERNATIVES

• Date: 01 November 2019
• DOI: http://doi.org/10.1111/iere.12394
• Author: Sean Horan,M. Remzi Sanver,Martin J. Osborne
• Published date: 01 November 2019

INTERNATIONAL ECONOMIC REVIEW

Vol. 60, No. 4, November 2019 DOI: 10.1111/iere.12394

POSITIVELY RESPONSIVE COLLECTIVE CHOICE RULES AND MAJORITY

RULE: A GENERALIZATION OF MAY’S THEOREM TO MANY ALTERNATIVES∗

BYSEAN HORAN,MARTIN J. OSBORNE,AND M. REMZI SANVER1

Universit´

e de Montr´

eal and CIREQ, Canada; University of Toronto, Canada; Universit´

e

Paris-Dauphine,Universit ´

e PSL, CNRS, LAMSADE, France

May’s theorem 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.

1. INTRODUCTION

May’s theorem (1952, p. 682) says that for collective choice problems with two alternatives,

majority rule is the only anonymous, neutral, and positively responsive collective choice rule.2

These three conditions are attractive. Anonymity says that the alternative chosen does not

depend on the names of the individuals and neutrality says that it does not depend on the

names of the alternatives. Positive responsiveness says that the rule responds sensibly when

any individual’s favorite alternative changes: If for some profile of preference relations the

rule selects alternative a(possibly together with b, in a tie) and some individual switches from

preferring bto preferring a, then the rule selects aalone.

What happens if there are more than two alternatives? A standard answer points to

Arrow’s “general possibility theorem” (1963, Theorem 2, p. 97), which says that no preference

aggregation rule3is Pareto-efficient,4independent of irrelevant alternatives,5and nondictato-

rial.6This answer is unsatisfying because Arrow’s conditions are disjoint from May’s, which

leaves open some basic questions. When there are three or more alternatives, which collective

choice rules satisfy May’s conditions? For such an environment, does May’s theorem have a

natural generalization?

Arrow himself writes that a complete characterization of the collective choice rules satisfying

May’s conditions when there are three or more alternatives “does not appear to be easy to

∗Manuscript received April 2018; revised December 2018.

1We thank Salvador Barber`

a, Jean-Pierre Benoˆ

ıt, Felix Brandt, Markus Brill, Donald E. Campbell, Christopher

Dobronyi, John Duggan, Justin Kruger, Herv´

e Moulin, Mat´

ıas N ´

u˜

nez, and Maurice Salles for discussions and comments.

We are grateful also to a referee and the editor for suggestions that led us to improve the article. In particular, they

pushed us to significantly tighten the results. Horan’s work was partly supported by FRQSC. Sanver’s work was partly

supported by the projects ANR-14-CE24-0007-01 “CoCoRICo-CoDec” and IDEX ANR-10-IDEX-0001-02 PSL*

“MIFID.” Please address correspondence to: Martin J. Osborne, Department of Economics, University of Toronto,

150 St. George Street, Toronto, ON M5S 3G7, Canada (CA). E-mail: martin.osborne@utoronto.ca.

2A collective choice rule is a function that associates with every profile of preference relations a set of alternatives.

3A preference aggregation rule is a function that associates with every profile of preference relations a (“social”)

preference relation. Preference aggregation rules are sometimes called social welfare functions.

4If all individuals prefer ato b,thenais socially preferred to b.

5If every individual’s preference between aand bis the same in two profiles of preference relations, the social

preference between aand bis the same for both profiles.

6For no individual is it the case that the social preference between any pair of alternatives is the same as the

individual’s preferences regardless of the other individuals’ preferences.

1489

C

(2019) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1490 HORAN,OSBORNE,AND SANVER

achieve” (1963, footnote 26, p. 101). We agree. Furthermore, Arrow’s theorem and the subse-

quent vast literature suggest that when preferences are unrestricted, no collective choice rule

for problems with many alternatives satisfies conditions like May’s plus a condition requiring

consistency across problems with different numbers of alternatives. But what if we restrict

to a limited set of preference profiles? Is a natural generalization of majority rule character-

ized by May’s conditions plus a consistency condition across problems with different numbers

of alternatives?

We show that the answer is affirmative. The consistency condition we add is a set-valued

version of the independence condition used by Nash (1950, condition 7 on p. 159) in the context

of his bargaining model, which we call Nash independence.7This condition says that removing

unchosen alternatives does not affect the set of alternatives selected. We show that for the

domain of collective choice problems that have a strict Condorcet winner,8an adaptation of

May’s conditions plus Nash independence characterize the collective choice rule that selects

the strict Condorcet winner (Theorem 1). We also show that when the preference profile is one

step or more away from having a strict Condorcet winner, no collective choice rule satisfies a

slight variant of these conditions if there are at least three individuals and three alternatives

(Theorem 2). When preferences are strict, a similar result holds if there are either at least three

individuals and four alternatives or at least four individuals and three alternatives (Theorem 3).

A strict Condorcet winner is an appealing outcome if it exists, and the conditions of anonymity,

neutrality, positive responsiveness, and Nash independence also are appealing. We interpret

Theorem 1 to increase the appeal of both the collective choice rule that selects the strict

Condorcet winner and the four conditions. It shows that the combination of the conditions

is “just right” for collective choice problems with a strict Condorcet winner: it implies the

collective choice rule that selects the strict Condorcet winner. If no collective choice rule were

to satisfy the conditions on this domain, the combination of conditions would be too strong, and

our subsequent results that no collective choice rule satisfies the conditions on any domain that

is more than slightly larger would be less significant. If collective choice rules other than the

one that selects the strict Condorcet winner were to satisfy the combination of conditions on

the domain of collective choice problems with a strict Condorcet winner, then the combination

of conditions would be too weak.

The key condition in our results is an adaptation of May’s positive responsiveness to an

environment with many alternatives. Suppose that the alternatives aand bare both selected, in

a tie, for some problem, and some individual ranks babove a. Now suppose that the individual’s

preferences change to rank aabove b. Our condition requires that aremains one of the selected

alternatives, bis no longer selected, and no alternative that was not selected originally is now

selected.9This condition captures the spirit of May’s condition: A change in the relative ranking

of two alternatives by a single individual breaks a tie between the alternatives. More loosely,

the condition ensures that every individual’s preferences matter.

2. MODEL

Throughout we fix a finite set Nof individuals and a finite set Aof all possible alternatives, and

assume that both sets contain at least two elements. In any given instance, the set of individuals

has to choose an alternative from the set of available alternatives, which is a subset of A.

DEFINITION 1(COLLECTIVE CHOICE PROBLEM). A collective choice problem is a pair (X,),

where Xis a subset of Awith at least two members and is a profile (i)i∈Nof complete and

transitive binary relations (preference relations)onA.

7The condition is known by several other names, including the strong superset property. See Brandt and Harrenstein

(2011) (who call the condition α) for an analysis of the condition and an account of its previous use. It neither implies

nor is implied by the related Chernoff condition (see Subsection 4.2).

8An alternative asuch that for every other alternative ba strict majority of individuals prefer ato b.

9A stronger condition is sometimes given the same name. See the discussion after Definition 8.