Models of inter-election change in partisan vote share
| Author | Mark C. Wilson,Bernard N. Grofman |
| DOI | http://doi.org/10.1177/09516298221123263 |
| Published date | 01 October 2022 |
| Date | 01 October 2022 |
| Subject Matter | Articles |
Models of inter-election change
in partisan vote share
Mark C. Wilson
University of Massachusetts Amherst, Amherst, MA, USA
Bernard N. Grofman
University of California Irvine, Irvine, CA, USA
Abstract
For a two-party electoral competition in a districted legislature, the change in mean vote share for
party A from one election to the next is commonly referred to as swing. A key question, highly
relevant to election forecasting and the measurement of partisan gerrymandering, is: “How do we
expect the swing to be distributed across the districts as a function of previous vote share?”.
The literature gives two main answers: uniform swing and proportional swing. Which is better
has been unresolved for decades. Here we (a) provide an axiomatic foundation for desirable prop-
erties of a model of swing; (b) show axiomatically that using uniform swing or proportional swing
is a bad idea, (c) provide a simple swing model that does satisfy the axioms, and (d) show how to
integrate a reversion to the mean effect into models swing.
We show that all the above models can be expected to work well when (a) elections are close,
or (b) when we restrict to data where swing is low, or (c) when we eliminate the cases where the
model is most likely to go wrong. We show on a large US Congressional dataset that in addition to
its superior axiomatic properties, our new model provides an overall equal or better fitonfive
indicators: mistakes about directionality of change, mistakes in winner, estimates that are outside
the [0..1] bounds, mean-square error, and correlation between actual and predicted values. We
recommend replacing the uniform and proportional swing models with the new model.
Keywords
Uniform swing, proportional swing, vote share prediction
Corresponding author:
Mark C. Wilson, Department of Mathematics & Statistics, University of Massachusetts Amherst, Amherst, MA
01002 USA.
Email: markwilson@umass.edu
Article
Journal of Theoretical Politics
2022, Vol. 34(4) 481–498
© The Author(s) 2022
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DOI: 10.1177/09516298221123263
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1. Introduction
Therearemanysituationswherewehavedataattwo(orpossiblymorethantwo)pointsin
time, and we are interested in comparing values between and among the data points for each
subject or case. One such situation is an experiment in which there is a treatment effect which
results in an overall mean change of sunits in some variable of interest after the intervention.
The situation in which we are chiefly interested here is a two-party plurality vote contest
(parties Aand B) in the set of electoral units (districts, states, etc), for which we have data
at two distinct points in time. For simplicity, we assume an idealized situation in which
there are Kdistricts each of equal size, and turnout is equal in each district.
Consider two elections, one at time t=1 and one at time t=2. Let xidenote the vote
share of a given party at time 1 in district i, and x′
ithe vote share at time 2 in that district.
We use bars to denote mean values over districts, so that xdenotes the mean over all dis-
tricts of xi, namely the overall vote fraction for that party.
The aggregrate inter-election swing
1
is simply x′−x, and we denote this by s.Bysym-
metry, in a two party contest, swing for party A is swing against party B, and conversely.
At the district level we denote by sithe district-level inter-election swing x′
i−xi.
A key problem in election forecasting and the study of partisan gerrymandering,
among other areas, is simply: given sand the previous election result, estimate sifor
each i. The literature gives two main answers to the above question: uniform swing
and proportional swing.
Butler, in a chapter in McCallum and Readman (1999: pp. 263–265) is credited with
first using uniform swing to model British elections. While its limitations for multi-party
contexts were noted by Butler and by later authors such as Curtice and Steed (1982); Rose
(1991); Dorling et al. (1993), the concept has nonetheless subsequently become a work-
horse model in the U.K. Multiparty competition in the U.K. is often dealt with by focus-
ing on competition between the two largest parties. The use of the uniform swing model was
promoted by Butler and Van Beek (1990) and is now common in the U.S., with a similar way
of treating multi-party competition. Following Gelman and King (1994), swing is sometimes
taken to be uniformly distributed with a stochastic error term with mean zero and some fixed
variance (often estimated from data on the standard deviation of past inter-election vote share
shifts in the polity). Some stochastic versions of swing include covariates, such as incum-
bency, in the model (see e.g. Katz et al. (2020)). When we study congressional elections
in the U.S. we also restrict ourselves to two-party competition.
Proportional swing, on the other hand, positsthat each unit experiencesthe same percent-
age change from one election to the next, and thus units that have higher starting values
experience greater absolute vote changes. We believe that this model was first proposed
by Berrington (1965), where it was offered as an alternative to uniform swing. In subsequent
literature, proportional swing has become the main alternative to uniform swing.
Even in the context of two-party politics the use of these models has always been prob-
lematic, since each had properties that made a good fit to empirical data implausible. And
yet, when applied to British elections, by Butler and then by other authors in every
post-WWII British National Election Study, or in U.S. congressional and other elections,
the fit of these models, especially uniform swing, has been shown to be amazingly good –
see e.g. Butler and Stokes (1969: pp. 140–151) and Katz et al. (2020). The remarkable
482 Journal of Theoretical Politics 34(4)
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