The most ordinally egalitarian of random voting rules

• Published date: 01 April 2018
• DOI: http://doi.org/10.1111/jpet.12258
• Date: 01 April 2018
• Author: Anna Bogomolnaia

Received: 15 July 2016 Accepted: 14 May2017

DOI: 10.1111/jpet.12258

ARTICLE

The most ordinally egalitarian of random

voting rules

Anna Bogomolnaia

Universityof Glasgow and National Research

UniversityHigher School of Economics

Supportby the Basic Research Program of the

NationalResearch University Higher School of

Economicsis gratefully acknowledged.

AnnaBogomolnaia, University of Glasgow,

UK,and National Research University Higher

Schoolof Economics, St. Petersburg, Russia

(anna.bogomolnaia@glasgow.ac.uk).

Aziz and Stursberg propose an “Egalitarian Simultaneous Reserva-

tion” rule (ESR), a generalization of Serial rule, one of the most

discussed mechanisms in the random assignment problem, to the

more general random social choice domain. This article provides an

alternative definition, or characterization, of ESR as the unique most

ordinally egalitarian one. Specifically, given a lottery pover alterna-

tives,for each agent ithe author considers the total probability share

in pof objects from her first kindifference classes. ESR is shown to be

the unique one which leximinmaximizes the vector of all such shares

(calculated for all i,k). Serial rule is known to be characterized by

the same property. Thus, the author provides an alternative wayto

show that ESR, indeed, coincides with Serial rule on the assignment

domain. Moreover, since both rules are defined as the unique most

ordinallyegalitarian ones, the result shows that ESR is “the right way”

to think about generalizing Serial rule.

1INTRODUCTION

We consider the classical “voting”problem, when nagents have to jointly choose one common alternative from a given

set A={a1,…,a

w}. Our goal is to investigate plausible systematic preference aggregating mechanisms (rules) for this

problem, which do not use monetary transfers.

When preferences over Adiffer substantially, it might be difficult to choose an alternative that an agent would

consider a good compromise. One way to overcome this problem is to allow for an outcome to be a lottery over

A, or a vector of “shares” of alternatives, rather than a unique alternative. Potentially, any probability distribution

p=(p1,…,p

n)∈ΔAcan be jointly chosen as an outcome. We may interpret pas a real lottery to be performed. Hence,

the final ex post outcome still would be a single “pure” alternative. However, agents might regard the process (ifnot the

outcome)as more fair. Alternatively,pmay be interpreted as a vector of “time-shares,” fractions of total time each alter-

native is in place. Which interpretation is more appropriate depends on the particular economic situation. Weabstract

from it and concentrate on the formal model, which encompasses both.

A prominent paper by Gibbard (1977) on this random social choice model restricts attention to the “ordinal”mech-

anisms. Agents are assumed to have strict preferences over A, and are only askedabout their orderings of pure alter-

natives. It is implicitly assumed, however,that they have cardinal utilities over the alternatives, and compare lotteries

based on the expected utility. This assumption gives rise to a strong requirement of “strategy-proofness” : a rule is

Journal of Public Economic Theory.2018;20:271–276. wileyonlinelibrary.com/journal/jpet c

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