Equilibrium in the Spatial ‘Valence’ Model of Politics
| Author | Norman Schofield |
| Published date | 01 October 2004 |
| Date | 01 October 2004 |
| DOI | http://doi.org/10.1177/0951629804046150 |
| Subject Matter | Articles |
EQUILIBRIUM IN THE SPATIAL ‘VALENCE’
MODEL OF POLITICS
Norman Schofield
ABSTRACT
It has been a standard result of the stochastic, or probabilistic, spatial model
of voting that vote maximizing candidates, or parties, will converge to the
electoral mean (the origin). This conclusion has appeared to be contradicted
by empirical studies.
Here, a more general stochastic model, incorporating ‘exogeneous’ valence,
is constructed. Contrary to the standard result, it is shown in Theorem 1 of this
paper that a potentially severe domain constraint (determined by the electoral
and stochastic variance, valence as well as the dimension of the space) is neces-
sary for the existence of equilibrium at the electoral mean. A more stringent
condition, independent of the dimension of the space, is shown to be sufficient.
An empirical study of Israel for 1992 shows that the necessary condition failed.
This suggests that, in proportional electoral systems, a pure strategy equili-
brium will almost always fail to exist at the electoral mean. Instead, in both
the formal and empirical models, each party positions itself along a major
electoral axis in a way which is determined by the valence terms.
A second empirical analysis for Britain for the elections of 1992 and 1997
shows that, in fact, the necessary and sufficient condition for the validity of
the ‘mean voter theorem’ was satisfied, under the assumption of uni-
dimensionality of the policy space. Indeed the low valence party, the Liberal
Democrat Party, did appear to locate at the electoral center. However, the
high valence parties, Labour and the Conservatives, did not. This suggests
that, in polities based on plurality rule, valence is a function of activist support
rather than a purely exogenous factor. Theorem 2 shows, as in Britain, that
exogeneous and activist valence produce opposite effects.
KEY WORDS .elections .stochastic model .valence
Journal of Theoretical Politics 16(4): 447–481 Copyright &2004 Sage Publications
DOI: 10.1177/0951629804046150 London, Thousand Oaks, CA and New Delhi
Earlier versions of this paper were presented at the European Public Choice Meeting, Aarhus,
Denmark, April 2003, and the World Congress of the International Economic Association,
Lisbon July, 2002. I thank my colleagues Randy Calvert, John Carey, Gary Miller and Olga
Shvetsova, as well as seminar participants at CERGE (Prague), School of Economics
(Warsaw), Trinity College (Dublin) and Nuffield College (Oxford). I would like to express my
appreciation for support from the Fulbright Foundation, from Humboldt University and from
Washington University in St Louis, as well as from the NSF under grant SES 0241732. Joint
work with Andrew Martin, David Nixon, Kevin Quinn and Itai Sened formed the basis for the
empirical analyses discussed in Section 3 of this paper. In addition, Dganit Ofek and Martin
Battle provided invaluable research assistance and Lexi Shankster kindly typed the manuscript.
John Duggan and an anonymous referee of the journal made a number of very helpful comments.
1. Introduction
Formal spatial models of voting have been available for many decades and
have been, to some degree, the inspiration for empirical analyses of elections
in the US (Poole and Rosenthal, 1984) and, more recently, in the context of
multiparty competition under proportional electoral rules (Merrill and Grof-
man, 1999; Adams and Merrill, 1999; Adams, 2001). It has become apparent,
however, that conclusions derived from at least one class of formal models
are contradicted by some of the empirical evidence.
This paper will focus on the ‘stochastic’ formal voting model and will offer
a variation of the model which appears compatible with empirical analyses
of Israel and Britain. The conclusion of the standard stochastic model is
the ‘mean voter theorem’ – that ‘vote-maximizing’ parties or candidates
will converge to a ‘pure strategy Nash equilibrium’(PSNE) located at the
center of the electoral distribution (Hinich, 1977; Lin et al., 1999). The
empirical evidence offered here is that the standard stochastic model has to
be modified to include non-policy preferences of the electorate. In particular,
it is necessary to add to each voter’s evaluation of candidate, j, say, a valence
term (Stokes, 1992). This term, labeled
ij
, can be interpreted as the weight
that voter igives to the competence of candidate j. In empirical estimations,
it is usual to assume that the term
ij
comprises an expected ‘exogeneous’
term,
j,
together with a ‘stochastic’ error term, "
j
. The error vector, ",is
drawn from a multivariate normal or log-Weibull extreme value distribution
(Poole and Rosenthal, 1984).
Recent formal analyses of the non-stochastic voting model with valence
(but limited to the case with two candidates) suggest that the candidates
will not converge to the electoral center (Ansolabehere and Snyder, 2000;
Groseclose, 2001).The purpose of this paper is to extend the work of Grose-
close (2001) on the stochastic model with valence to the case with an arbitrary
number of candidates and dimensions. We shall obtain necessary and suffi-
cient conditions for convergence to the electoral mean in this general
model. The first theorem of this paper asserts that the formal stochastic
model is classified by a number called the ‘convergence coefficient’, c, defined
in terms of all the parameters of the model. The parameters include: the
vector of valences, {
p
,...,
1
}, the variance of the stochastic errors,
2
,
the ‘variance’ of the voter ideal points, v
2
, the spatial coefficient, , the
number, p, of political parties, and the dimension, w, of the ‘policy’ space, X.
Suppose the exogeneous valences are ranked p...1. Define
avð1Þ¼½1=ðp1Þ X
j¼2
j
where this is the average valence of the parties other than the lowest ranked.
Define the valence difference for the lowest valence party to be ¼
448 JOURNAL OF THEORETICAL POLITICS 16(4)
avð1Þ1. This valence difference is the key to the analysis. Obviously if all
valences are equal then ¼0.
Define the ‘convergence coefficient’, c, of the model to be
c¼2ðp1Þv2=½p2
The main theorem shows that if cis bounded above by 1, then a centrist equi-
librium will exist. If the policy space has dimension wand if cexceeds w, then
such a centrist equilibrium is impossible. It is inferred, therefore, that the
vector of vote-maximizing party positions will depend in a subtle fashion
on the actual distribution of voter-preferred positions.
An empirical analysis of Israel (developing earlier work by Schofield et al.,
1998) for 1992 shows that the estimated ‘non-convergent’ party positions are
indeed close to the vote-maximizing positions derived from a multinomial
logit (MNL) model. To interpret this analysis, the notion of a ‘local Nash
equilibrium’ (LNE) is introduced. This concept depends on the idea that a
vector of party positions is in ‘local’ equilibrium if no party may effect a uni-
lateral, but small, change in position so as to increase its expected vote share.
The formal analysis is then used to obtain conditions for existence of a LNE
at the mean voter position.
In the case of Israel for the election of 1992, the empirical MNL analysis in
two policy dimensions obtained high estimates for the valence terms (
Likud
and
Labor
) for the two major parties, Likud and Labor, and a very low
valence term for one of the small parties, Shas. This gave a maximum valence
difference of ¼5:08 and a value for the convergence coefficient of 9.72,
well above the critical value of 2.0. Theorem 1 asserts that the electoral
origin cannot be an equilibrium for the low valence parties such as Shas.
Instead, this party should, in equilibrium, adopt a policy position far from
the electoral center. This is confirmed by the empirical analysis. Moreover,
parties with large valence (such as Likud and Labor) will not adopt identical
positions at the electoral center. Instead, their equilibrium positions will
depend on the differences in their valences and on the distribution of voter
ideal points.
Whether or not a vector of party positions is a LNE can be determined by
examining first- and second-order conditions on the differentials of the
expected vote functions. Typically, these conditions can be met by more
than one vector of party positions. However, once the empirical model has
been developed, it is possible to determine the full set of LNE by simulation.
These simulations have been carried out for a number of polities (Schofield
et al., 1998; Schofield and Sened, 2004a,b) and have found no evidence of
convergence. Since the empirical analysis mandates the inclusion of valence
terms, the formal anlysis presented here gives an explanation as to why con-
vergence is generally not observed in polities using proportional electoral
methods.
SCHOFIELD: EQUILIBRIUM IN THE SPATIAL ‘VALENCE’ MODEL 449
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