Directional equilibria
| Published date | 01 July 2018 |
| DOI | 10.1177/0951629818775515 |
| Author | Hun Chung,John Duggan |
| Date | 01 July 2018 |
| Subject Matter | Articles |
Article
Directional equilibria
Journal of Theoretical Politics
2018, Vol. 30(3) 272–305
©The Author(s) 2018
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DOI:10.1177/0951629818775515
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Hun Chung
School of Political Science and Economics, Waseda University, Tokyo, Japan
John Duggan
Department of Political Science and Department of Economics, University of Rochester, Rochester, NY, USA
Abstract
We propose the solution concept of directional equilibrium for the multidimensional model of
voting with general spatial preferences. This concept isolates alternatives that are stable with
respect to forces applied by all voters in the directions of their gradients, and it extends a known
concept from statistics for Euclidean preferences. We establish connections to the majority core,
Pareto optimality, and existence and closed graph, and we provide non-cooperative foundations
in terms of a local contest game played by voters.
Keywords
Equilibrium; geometric median; majority core; mediancentre; spatial median; spatial model;
structure-induced equilibrium; utilitarianism
1. Introduction
The multidimensional spatial model of politics provides an abstract framework for col-
lective choice, where points in a Euclidean space can represent vectors of positions on
different policy issues. Given an odd number of voters with single-peaked preferences
over a single policy issue, the median voter theorem dictates that the median voter’s ideal
point is the unique element of the majority core. An impediment to the analysis of the
general model is the instability of majority rule in multiple dimensions, formalized in
the symmetry conditions of Plott (1967) and a result of Schofield (1983) demonstrating
the generic emptiness of the majority core. In reaction to the indeterminacy of majority
Corresponding author:
John Duggan, Department of Political Science and Department of Economics, Wallis Institute of Political
Economy, University of Rochester, Rochester, NY 14627, USA.
Email: dugg@ur.rochester.edu
Chung and Duggan 273
rule, several solution concepts have discerned alternatives (or sets of alternatives) with
special properties as having positive or normative significance. We propose a concept of
directional equilibrium that isolates alternatives based on their stability with respect to
‘forces’ applied by voters in the directions of their gradients. This solution generalizes
a widely (but not well-)known concept from statistics for Euclidean preferences, and it
possesses desirable core consistency, existence, efficiency, and stability properties.
To convey the idea of directional equilibrium, we first consider the problem of max-
imizing the sum of strictly concave voter utilities over alternatives x∈Rdto obtain the
utilitarian equilibrium, which is the unique solution to the first-order condition
n
X
i=1
∇ui(x)=0. (1)
One interpretation of the above first-order condition is that the utilitarian equilibrium is
stable: if each voter applies to it a force equal to her gradient, then those forces cancel
each other out. However, this definition of stability is sensitive to scalings of voter utilities
and implicitly relies on an interpersonal comparison of utilities. We seek a criterion that
is free of interpersonal utility comparisons: in terms of the tug-of-war analogy, we want
to assume that all voters pull with equal force, and we therefore consider the normalized
gradients of the voters. Thus, for an alternative xthat is not an ideal point of any voter,
we say xis a directional equilibrium if it solves the system
n
X
i=1
1
k∇ui(x)k∇ui(x)=0 (2)
of equations of dequations in dunknowns. If the alternative is the ideal point of a voter,
then we assume that all such voters can resist a net pull in any direction, up to a force of
one unit.
This formulation has several desirable properties. We first establish that the concept
of directional equilibrium extends the majority core, in the sense that if the majority core
is non-empty, then every core alternative is a directional equilibrium. When the number
of voters is odd and no two voters share the same ideal point, the proof of this result is an
immediate corollary of Plott’s theorem: under these conditions, a core alternative must
satisfy radial symmetry, which means that each voter is balanced by another voter whose
gradient pulls in the opposite direction. This result extends easily to an even number of
voters when there is no voter whose ideal point is at the core, for then radial symmetry
again holds. When there is one voter for whom the core is ideal, then the result is non-
trivial and provides a new necessary condition for the majority core: if each voter acts to
displace the core alternative by the amount of her normalized gradient, then the sum of
these actions leads to an outcome in the unit ball around the core alternative. In general,
this core extension result allows for the possibility of directional equilibria in addition to
the core, but we prove that if the set of alternatives is one-dimensional, then the equiva-
lence is exact: if the majority core is non-empty, then the core alternative is the unique
directional equilibrium.
Wethen verify that directional equilibria are Pareto optimal and exist in great general-
ity, and that when voter preferences are continuously parameterized, the directional equi-
librium correspondence has closed graph. These attributes are appealing for any solution
274 Journal of Theoretical Politics 30(3)
to the spatial model, and existence in particular provides the potential for applicability of
the concept.
Finally, we explore the tug-of-war analogy in greater detail, expressing it in terms
of a ‘contest game’ at status quo alternative x, where voters’ strategies consist of the
application of an amount of force to move the outcome from x. We prove that under weak
background conditions, an alternative xis a directional equilibrium if and only if it is
the equilibrium outcome of a contest game at x, i.e., the status quo of xis maintained in
equilibrium, providing non-cooperative foundations for the solution concept.
The directional equilibria generalize a notion of multidimensional median from
the statistics literature that has been applied to the spatial model when voter prefer-
ences are Euclidean and utilities are linear in distance from a voter’s ideal point, i.e.,
ui(x)= −kˆxi−xk. In this setting, because the norm of a voter’s gradient is independent
of the alternative x6= ˆxiat which it is evaluated, the distinction between Equations (1)
and (2) disappears: an alternative maximizes the sum of voter utilities if and only if the
sum of normalized gradients is equal to zero.1Put differently, in this special case, the
concepts of utilitarian equilibrium and directional equilibrium coincide. Moreover, if the
number of voters is odd or the voters’ ideal points are not collinear, then it is known that
there is a unique utilitarian equilibrium, and thus a unique directional equilibrium.2Thus,
when utilities are linear functions of distance to the voters’ ideal points, the directional
equilibrium is well-understood and consistent with the utilitarian welfare criterion. In the
statistical context, the minimizer of total distance to a given number of points has a long
history as a notion of centrality. Weber (1909) introduced the idea in the context of locat-
ing a warehouse to minimize transportation costs, and it was imported to the statistics
literature by Gini and Galvani (1929) and rediscovered by Haldane (1948). The concept
has received various names: mediancentre (Gower, 1974), geometric median (Haldane,
1948), L1-median (Small, 1990), and spatial median (Brown, 1983).3
More recently, a literature has developed this concept in the context of social choice
theory, but under the assumption of Euclidean preferences. Baranchuk and Dybvig(2009)
assumed Euclidean preferences and used the term ‘consensus,’ and they applied the con-
cept to analyze decision making by a board of directors. Cervone et al. (2012) used the
terminology of ‘mediancentre’ and ‘Fermat–Weber point,’ and they discussed computa-
tional issues and cited earlier work on the topic. Brady and Chambers (2015) used the
term ‘geometric median,’ and assuming Euclidean preferences and a variable population,
they showed that whenindividual preferences are Euclidean, the geometric median is the
smallest rule that is Maskin monotonic and satisfies a number of background axioms; and
Brady and Chambers (2016) assumed three individuals with Euclidean preferences, and
they showedthat the geometric median is the unique r ule satisfying Maskin monotonicity,
anonymity, and neutrality.
The concept of directional equilibrium has not received explicit attention in the set-
ting of the general spatial model of politics, however. In general, the systems of equations
in (1) and (2) become distinct, and the utilitarian and directional equilibria diverge. The
existence of directional equilibria becomes a more subtle issue, as it is no longer clear
how to pose an optimization problem that generates directional equilibria as its solutions;
rather, we use a fixed-point argument to establish existence. Despite the relative techni-
cal complexity of directional equilibrium, an important benefit of this route is that our
concept is invariant with respect to smooth transformations of voter utilities with positive
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