Linear and quadratic utility loss functions in voting behavior research
| Published date | 01 January 2014 |
| Date | 01 January 2014 |
| DOI | http://doi.org/10.1177/0951629813488985 |
| Subject Matter | Articles |
Article
Linear and quadratic utility
loss functions in voting
behavior research
Journal of Theoretical Politics
2014, Vol 26(1)35–58
©The Author(s) 2013
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DOI:10.1177/0951629813488985
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Shane Singh
University of Georgia, Athens, USA
Abstract
Numerous studies examine voting behavior based on the formal theoretical predictions of the
spatial utility model. These studies model individual utility from the election of a preferred party
or candidate as decreasing as the alternative deviates from one’s ideal point, but differ as to
whether this loss should be modeled linearly or quadratically. After advancing a theoretical argu-
ment for linear loss, this paper uses a wealth of data across 20 countries to empirically examine
the predictive power of these two loss functions in terms of both voter choice and voter turnout.
Results indicate that the linear loss function outperforms the quadratic loss function. The findings
have important implications for theoretical scholars seeking to model voter behavior, for obser-
vational scholars, who must assign utility values across parties to individuals under study, and for
experimental researchers, who must entice individuals with particular utility loss functions.
Keywords
Loss functions; party behavior; proximity voting; utility; voter behavior
The spatial proximity model of voter utility, most famously articulated by Anthony
Downs (1957), has weathered numerous theoretical and empirical tests (see, for example,
Blais et al., 2001; Gilljam, 1997; Krämer and Rattinger, 1997; Pierce, 1997; Westholm,
1997) and is the cornerstone of voting behavior research (Adams et al., 2005). In short,
the model predicts that voters derive the most utility from the candidate or party closest
to them on some ideological or policy continuum. As such, one is most likely to vote for
the option nearest one’s ideal point, and one is more likely to vote at all when the party
system provides one with an ideologically close option.
The decrease in utility associated with an alternative’s deviation from one’s ideal point
is called loss, and the function that maps this loss to one’s utility value is called a loss
function. Linear loss imposes a constant ‘penalty’ on deviations from one’s ideal point,
Corresponding author:
Shane Singh, Department of International Affairs, University of Georgia, 303 Candler Hall, Athens, GA 30602,
USA.
Email: singh@uga.edu
36 Journal of Theoretical Politics 26(1)
while quadratic loss imposes a harsher penalty as outcomes become more and more dis-
tant. While the linear and quadratic loss functions dominate voting behavior research,
decisions about which formulation to employ are generally neither made a priori, with
theoretical justification, nor a posteriori, with empirical guidance, but are instead regu-
larly made ad hoc, with little or no justification. The purpose of this paper is twofold:
first, to provide theoretical guidance for which loss function voters are most likely to
employ, and second, to empirically examine which functional form has more predictive
power.
Determining the theoretical and empirical applicability of these functions is impor-
tant for two principal reasons. First, a substantial portion of voter behavior research,
whether quantitative or formal theoretical, conceives of voter utility in terms of ideal
point distances. Future studies will benefit from guidance on how to best model decreas-
ing utility as ideal points deviate. Second, much experimentalresearch attempts to under-
stand voter turnout and choice in the laboratory, often generating voter preferences with a
particular loss function. Participants are rewarded with prizes, monetary or otherwise, for
selecting the ‘best’ option. This research provides guidance for experimental researchers
who seek to simulate real-world voter behavior inside the lab.
I start with a theoretical argument against quadratic loss based on its overly harsh
punishment of distant alternatives. I then present an argument for linear loss based on
findings from psychological research indicating that humans tend to conceive of the
world in linear terms. After reviewing previous work on spatial voting, I conduct a
series of examinations of voter choice and turnout using data over 30 elections and
20 countries from Modules I and II of the Comparative Study of Electoral Systems
(CSES). This broad analysis allows me to test the relativeperfor mance of each loss func-
tion across a broad range of countries, providing variation in cultural and institutional
constructs.
From the tests, I reach three main conclusions. First, the prevalence of proximity
behavior varies widely across countries, and both observational researchers and experi-
mentalists, regardless of particular subfield, should be awarethat contextual factors have
the potential to influence the usefulness of the proximity model. Second, the linear loss
function is shown to better model voter utility in regard to vote choice than the quadratic
loss function. Third, linear loss also outperforms quadratic loss in regard to voter turnout,
though the difference is less pronounced. As assigning utility values based on linear loss
better captures intuition (Ordeshook, 1986), it is not advisable to deviate from the model
without a proven alternative. I suggest that both experimental and empirical researchers
formulate utility based on linear loss functions in the absence of compelling evidence to
do otherwise.
1. Linear and quadratic loss
Voting behavior is often modeled in terms of the physical distance between the individ-
ual and the objects in the choice set. As noted by Olson and Gale (1968: 231), ‘The
minimization of physical distance is only a theoretical notion intended to simplify the
formulation of abstract concepts.’ Thus, the dimension or dimensions along which such
physical distances exist are not strictly observable.
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