A stability property in social choice theory

• Published date: 01 March 2018
• Date: 01 March 2018
• DOI: http://doi.org/10.1111/ijet.12141
• Author: Shaofang Qi,Jerry S. Kelly,Donald E. Campbell

doi: 10.1111/ijet.12141

A stability property in social choice theory

Donald E. Campbell,∗Jerry S. Kelly†and Shaofang Qi‡

A social choice function gis stable if whenever an alternative xis selected at profile uand then

profile u∗is constructed from uby only switching xwith yfor one individual iwho has yjust

below xat u, then rule gselects either xor yat u∗and not some third alternative. Stability is

strictly weaker than strategy-proofness but strong enough to establish two kinds of impossibility

theorems.

Key wor ds stability, Pareto, dictatorship, tops-only, monotonicity, plurality, Condorcet,

maximin, Borda

JEL classification D70, D71

Accepted 20 July2017

1 Introduction

To introduce stability, we start with a domain D, a subset of the collection L(X)Nof all profiles of

strong preferences. HereXis the set of all alternatives and Nis the set of all individuals. A social choice

function on Dmaps Dto X. We say that a social choice function on Dsatisfies stability if whenever

an alternative xis selected at a profile uand then profile u∗is constructed from ubyonly sw itching

xwith yfor one individual iwho has yjust below xat u, and if u∗∈D, then g(u∗) is either xor y

and not some third alternative.1Stability is strictly weaker than strategy-proofness, but, as we show

in this paper, it is strong enough to establish several interesting impossibility results. We will focus

on two classes of such results.

rConsider plurality rule. At profile u, alternative xis a plurality winner if xis top-ranked for at

least as many individuals as any other alternative. Let Pbe the domain consisting of all profiles at

which there is a unique plurality winner and then define gPto be the social choice function on P

such that gP(u) is the plurality winner at ufor u∈P. It is easy to check that gPsatisfies stability

on P. Next consider an extension gof gPto the collection of all possible profiles,so g(u)=gP(u)

for all u∈P. We will prove that any such extension gmust violate stability. In part, this implies

that introducing tie-breaks into the plurality correspondence will necessarily cause a violation of

stability. Similar results are obtained for the Condorcet,maximin, and Borda correspondences.

rGibbard (1973) and Satterthwaite (1975) showed that the social choice function gon L(X)N

is dictatorial if gis strategy-proof and the range of gis X. Since we do not want to give up

∗Department of Economics and The Program in Public Policy, The College of William and Mary, Williamsburg,

VA,USA.

†Department of Economics, Syracuse University,Syracuse, NY, USA. Email: jskelly@maxwell.syr.edu

‡School of Business and Economics Humboldt UniversityBerlin, Berlin, Germany.

1Richard Bellman (1984, p. 181) writes in his autobiography: “Change one small feature, and the structure of the solution

was strongly altered. There was no stability!”

International Journal of Economic Theory 14 (2018) 85–95 © IAET 85

International Journal of Economic Theory