Nonconvergent Electoral Equilibria under Scoring Rules: Beyond Plurality

• Published date: 01 April 2017
• Author: ARKADII SLINKO,DODGE CAHAN
• DOI: http://doi.org/10.1111/jpet.12207
• Date: 01 April 2017

NONCONVERGENT ELECTORAL EQUILIBRIA UNDER SCORING RULES:

BEYOND PLURALITY

DODGE CAHAN

University of California, San Diego

ARKADII SLINKO

University of Auckland

Abstract

We use Hotelling’s spatial model of competition to investigate the

position-taking behavior of political candidates under a class of elec-

toral systems known as scoring rules, though the model also has a

natural interpretation in the firm location context. Candidates choose

ideological positions so as to maximize their support in society.

Convergent Nash equilibria in which all candidates adopt the same

policy were characterized by Cox (1987). Here, we investigate noncon-

vergent equilibria, where candidates adopt divergent policies. We iden-

tify a number of classes of scoring rules exhibiting a range of different

equilibrium properties. For some of these, nonconvergent equilibria

do not exist. For others, nonconvergent equilibria in which candidates

cluster at positions spread across the issue space are observed. In partic-

ular, we prove that the class of convex rules does not have Nash equilib-

ria (convergent or nonconvergent) with the exception of some deriva-

tives of Borda rule. We also look at “two-party” equilibria. Implications

for the firm location model are discussed.

1. Introduction

Hotelling’s (1929) spatial model of competition has had a large and varied influence on

a number of fields. It has been applied not only in the original context of firms selecting

geographic locations along “Main Street” so as to maximize their share of the market,

but also to that of producers deciding on how much variety to incorporate into their

products (see Chamberlin 1933). Downs (1957) adapted it with minor modifications

Dodge Cahan, Department of Economics, University of California, San Diego, CA (dcahan@ucsd.edu).

Arkadii Slinko, Department of Mathematics, University of Auckland, Auckland, New Zealand

(a.slinko@auckland.ac.nz).

This paper develops the idea which Arkadii Slinko discussed with Murat Sertel during his visit to

Bogazici University in 1999. We thank the participants of the 7th Spain-Italy-Netherlands Meeting on

Game Theory (SING7), held in Paris on July 18–20, 2011, especially Hans Peters and Herve Moulin

for helpful discussions and suggestions and also participants of the 1st ATE Symposium held on 12–13

December,2013, at Massey University, Auckland. Wethank John McCabe-Dansted for help with compu-

tational experiments. Last but not least we thank the anonymous referees for many useful suggestions.

C2016 Wiley Periodicals, Inc.

Journal of Public Economic Theory, 19 (2), 2017, pp. 445–460.

445

446 Journal of Public Economic Theory

to model an election: in particular, the ideological position-taking behavior of political

candidates in their effort to win votes.1

The model in its simplest form features a number of candidates (firms) adopting

positions on a one-dimensional manifold, usually taken to be the interval [0,1], along

which voters’ ideal positions (consumers) are distributed. Voters cast their votes for the

candidate that is closest ideologically to them and candidates adopt positions so as to

maximize their share of the vote (market)2and cannot opt out. One can imagine parties

competing for votes in an election under a system of proportional representation. In the

economic interpretation of this model, firms compete for market share, competition on

price is excluded, and customers buy from the closest firm to minimize transportation

costs.

In most of the literature, voters have only one vote, which they allocate to their

favorite candidate. That is, the electoral system is taken to be plurality rule. The vot-

ers’ second, third, and other preferences do not come into play. Economically, this is

akin to saying customers only patronize the nearest firm, with more distant firms being

irrelevant to them.

In many situations, however, preferences other than first do indeed matter. Far from

all elections are held under plurality rule: electoral systems, both in use and theoreti-

cal, are diverse. So too are the incentives that candidates are faced with under differ-

ent electoral systems (Cox 1987). A thorough analysis of these incentives is important

for the design of voting institutions with desirable properties. In particular, a designer

should know whether candidates pursuing rational policies will cluster together, advo-

cating identical or similar policy positions and displaying what Hotelling called “exces-

sive sameness,” or will they adopt diverse positions that appeal to different groups of

voters? It is also important to know whether equilibrium situations exist since their ab-

sence may lead to permanent instability. This was the theme of Myerson’s Schumpeter

Lecture (1998, Berlin meetings of the European Economic Association), later published

in Myerson (1999).

One very general class of electoral systems that Myerson advocates are the scoring

rules. In an election held under a scoring rule, each voter ranks the candidates from

favorite to least favorite. The candidates are then assigned points according to where

they appear on the voter’s ballot. The total number of points across all voters is a mea-

sure of support for a candidate in society. Well-known examples of scoring rules include

plurality, Borda’s rule, k-approval, and antiplurality.

In an election—according to Stigler’s thesis—the candidates would then adopt po-

sitions with the aim of maximizing the total number of points received from all the

electorate, and hence, the equilibrium strategies will depend on the particular scoring

rule in use. This dependence is the focus of our investigation.

This is not the only possible interpretation of scoring rules in the context of the

Hotelling–Downs model; another plausible view is due to Cox (1990). For this interpre-

tation, rather than points for being ranked ith in a voter’s ballots, we think about the

probability with which a voter votes for the candidate that is ith closest ideologically.3

Indeed, ideological proximity is not the only factor at the ballot box: a single issue out of

many may put a voter off a candidate who, on the whole, is of a similar ideological bent;

or, it could simply be down to personal charisma, experience, prejudice, or any number

1Useful surveys include Shepsle (2001), Duggan (2005), and Osborne (1995).

2See Stigler’s (1972) argumentation for this assumption and a discussion of it in Denzau, Kats, and

Slutsky (1985).

3Of course, with a standard normalization.