Spatial Models, Cognitive Metrics
And Majority Voting Equilibria
Macartan Humphreys (Columbia University)
Michael Laver (New York University)∗
Abstract
If policy choices are defined in spaces with more than one dimension, it has long been known that a majority rule equilibrium
fails to exist for a very general class of smooth preference profiles.
However we show here that, if agents perceive political similarity
and difference in “city-block” terms, then a majority rule equilibrium is possible in spaces with an arbitrarily large number of
dimensions; we provide necessary and sufficient conditions for the
existence of such an equilibrium. This result is important because
city block preferences accord more closely with empirical research
on human perception than do many smooth preference profiles.
The implication is that, if the already extensive empirical research
findings on human perceptions of similarity and difference extend
also to perceptions of political similarity and difference, then the
possibility of equilibrium under majority rule re-emerges.
January 2006
∗
Our thanks to David Austen Smith, Neal Beck, Steven Brams, John Huber, and
Kenneth Shepsle for valuable comments on an earlier draft of this paper.
1
1
Introduction
It is well known that for a very general class of preference profiles a
Majority Rule Equilibrium (MRE) does not exist in settings with a minimally complex set of political alternatives (Plott 1967, DeMeyer and
Plott 1970; Schofield 1983; McKelvey and Schofield 1986; McKelvey and
Schofield 1987). More specifically, when voters have smooth preferences
over outcomes in multidimensional choice spaces, for any given outcome
there will almost always exist some majority of agents that prefers some
other outcome. Normatively, these results suggest that there is no privileged outcome that democratic institutions “ought” to select (Riker
1982). Substantively, although these results do not predict cycling, they
render the prediction of voting outcomes difficult in complex environments.
Some theorists have responded to these concerns by using a form
of institutionally structured equilibrium (e.g. Shepsle 1979; Shepsle and
Weingast 1987; Laver and Shepsle 1996). Others have turned their attention to non-cooperative models that successfully generate majority-rule
equilibria in n-dimensional outcome spaces, but pay a price in terms of a
narrowing of empirical applicability. These costs arise because stronger
assumptions about the institutional form of the decision-making process
makes results difficult to generalize to settings where the precise model
assumptions are violated (e.g. Baron, 1991; Besley and Coate, 1997;
Jackson and Moselle, 2002). Many others effectively sidestep the theoretical problem by simply assuming that the set of feasible political
outcomes can be described as points in a one-dimensional space (e.g.
Osborne, 2000; Snyder and Ting, 2002; Morelli, 2004).
The different tradeoffs made by these different approaches are especially important in situations in which an MRE does not exist. In cases
where there is a well defined equilibrium however–for example in cases
where the median voter result applies–the MRE can be supported for
a wide class of games not only as a cooperative equilibrium but also as
a non-cooperative equilibrium (Moulin 1980, Austen-Smith and Banks,
1998). For this reason, knowing when there is indeed a well defined social choice under majority rule is of importance to researchers in both
traditions. In this paper we argue that, classic non-existence results
notwithstanding, there may be a well defined social choice even in an
outcome space of many dimensions if agents employ a “city block” metric
to measure similarities and differences between political outcomes. Cityblock preferences (defined formally below) fall within the class of convex
preferences. However, standard results do not apply, since these rely
fundamentally on analytical techniques that assume human preferences
to be smooth, while city block preferences are not smooth.
2
Our results are substantively significant because the city block metric
is widely used by cognitive scientists who analyze human perceptions of
similarity and difference in terms of distances in some cognitive space.
Standard empirical findings in this field, based on over 50 years of research, have led to a received wisdom that, when differences between
stimuli are constructed in terms of separable1 features or dimensions,
spatial representations using the city block metric best fit empirical data
on human perceptions of similarity and difference (Attneave 1950; Garner 1974; Shepard 1987; Shepard 1991). For an excellent overview of
this research paradigm, see Gärdenfors (2000).
Following this research tradition and assuming that human voters
may cognitively analyze political similarities and differences in terms of
city block distances, we retrieve, strengthen, and generalize, an “ancient”
and largely neglected analytical result due to Rae and Taylor (1971). The
Rae and Taylor result was subsequently constrained to two-dimensional
spaces by Kats and Nitzan (1977) and was generated, independently, for
two dimensional spaces by Wendell and Thorson (1974) and generalized
by McKelvey and Wendell (1976). McKelvey and Wendell (1976) further identified sufficient conditions for equilibrium in higher dimensional
settings.
The line of research initiated by Rae and Taylor did not however
identify necessary and sufficient conditions for the existence of an equilibrium in n-dimensional settings. In this paper we identify these conditions for the first time. This allows us to characterize the conditions
for equilibria and quantify the likelihood of equilibria. In doing so, we
show that it is quite possible, though not inevitable, for the dimensionby-dimension median (DDM), and only the DDM, of voters’ ideal points
to be the equilibrium in a majority voting game in a setting with an
arbitrarily large number of dimensions. Our finding contradicts received
wisdoms about the existence of an equilibrium under majority rule, but
does so conditional upon the assumption of a particular class of preferences. This indicates the need for a program of research designed to
assess the extent to which real voters typically do use a city block metric
when perceiving political differences, as opposed to using metrics that
are closer in form to those traditionally assumed in spatial models of
politics.
We proceed as follows. In the next section we first distinguish between classes of spatial model in terms of whether or not the analyst
requires cognitive assumptions about how the human agents being mod1
This uses the term “separable” in the standard sense that a set of separable
dimensions has the property that perceived differences between stimuli on one dimension are independent of perceived differences on other dimensions in the set.
3
eled perceive political similarity and difference. We then argue that,
for spatial models that do make such assumptions, there are a priori
reasons to expect the city block metric to describe how agents analyze
alternative political options. Having argued that the city block metric
is an inherently plausible assumption for use in such spatial models, we
demonstrate that, if the agents being modeled do indeed think this way,
standard results about the generic non-existence of an MRE do not apply. We also show the likelihood of an MRE can be high, at least for
low numbers of agents and in a low number of dimensions. We construct
our main existence argument analytically, and then use simulations to
show the probability of an MRE may be much higher if agent ideal
points are correlated across dimensions and/or if agents have different
voting weights. Taken together, these results imply that, if city-block
distances are a reasonable way to characterize how humans compare different possible political outcomes, this has a considerable impact on our
understanding of political equilibrium.
2
Spatial Models and Cognitive Metrics
The term “spatial model” is informally used to cover a wide class of
models in which political agents have rational preferences over sets of
outcomes that can be characterized as points in a space. However, it
is not always appreciated that there is an important epistemological
distinction between the analyst’s mathematical description of a set of
outcomes as points in a space, and a human agent’s cognitive perception
of these same outcomes in spatial terms. This important distinction
leads us to distinguish two types of spatial model.
In a weakly spatial model, the set of outcomes, W , can be mathematically represented as a space, for example a vector space, a smooth
manifold or, most commonly, a Euclidean space. Agents are assumed
to have a rational (complete, reflexive and transitive) binary preference
relation–%i –over all pairs of elements in this set. But it is not required
that agents cognitively perceive the relative utilities of outcomes in terms
of relative distances in the underlying mathematical space. Commonly,
agent preferences are characterized simply by an abstract agent-specific
utility function u : W → R1 with the property that, for any pair of
outcomes y, y 0 , we have y %i y 0 ⇔ ui (y) ≥ ui (y 0 ).
In a strongly spatial model, the set of outcomes is characterized as
being located in a space, but human agents’ evaluations of these same
outcomes are also assumed to be “spatial.” This implies that each agent
locates the set of outcomes, as well as his/her most-preferred outcome
(ideal point) xi , in a cognitive space and uses some distance measure
4
di (., .) defined over the entire space2 , to characterize the relative utility
of outcomes. This means that, given preference relation %i , we have
y %i y 0 ⇔ di (xi , y) ≤ di (xi , y 0 ). Commonly, these preferences are represented by an agent specific utility function u : W → R1 with the
property that for any two points y, y 0 we have y %i y 0 ⇔ ui (y) ≥ ui (y 0 )
where ui (.) is itself a composition of a loss function f and the distance
function di ; hence, ui (y) = fi (di (xi , y)). Political theorists often assume
the function f to be either a linear or quadratic function of distance
from the ideal point. Psychologists and cognitive scientists, on the other
hand, typically assume that the perceived similarity of two points is an
exponentially decaying function of the distance between them (Shepard
1987; Gärdenfors 2000; but see also Poole and Rosenthal 1997). For the
questions of concern here we do not need to impose any structure on f
other than that it be strictly decreasing in d(., .).
The models in McKelvey and Schofield (1986, 1987) are weakly spatial models that seek to characterize agent preferences in the most general and abstract form. Most spatial models of party competition in the
classic Downsian tradition are strongly spatial, in our sense, since they
assume each voter to evaluate the set of potential outcomes in terms of
each potential outcome’s relative “closeness” to the voter’s ideal point.
The class of strongly spatial models extends, for essentially the same
reason, to spatial models of probabilistic voting, with or without valence parameters (e.g. Groseclose 2001; Schofield, 2003, 2004). A useful
rule of thumb is that any model requiring the notion of an agent “ideal
point” is a strongly spatial model. This is because the distance between
the agent ideal point and each element in the set of potential outcomes
typically enters as an argument in the agent’s utility schedule.
Strongly spatial models of political choice require a very much stronger
cognitive assumption about real agents than do weakly spatial models
since they require information about the perceptual geometry used (consciously or not) by agents to measure the distance between two stimuli in
the assumed cognitive space. Any assumption about the cognitive metric
used to measure such distances contains crucial psychological assumptions about how agents trade off perceived differences on one dimension
of difference against perceived differences on another dimension. Using
2
A distance measure is a function d : W × W → R1 that has the properties that
for points a,b, and c in W :
1. d(a, b) = 0 ⇔ a = b
2. d(a, b) = d(b, a)
3. d(a, b) + d(b, c) ≥ d(a, c)
5
the Euclidean metric involves assuming that agents measure the psychological distance between two points in cognitive space using the same
Pythagorean Theorem they use to measure the distance between two
points in local physical space.3 Using a city block metric implies that
agents simply add up, or take an average of, the distances along each
dimension which, as we have seen, is what many cognitive scientists have
found that humans tend to do when dimensions of perceived difference
are separable.
It is important to emphasize that the assumption of a Euclidean cognitive metric to measure distance is quite distinct from the assumption
of a quadratic loss function, despite the use of quadratic terms in each.
The metric assumption is a cognitive assumption about how agents trade
off differences between stimuli on distinct dimensions of difference. The
loss function is a cognitive assumption about how these distances, calculated having made these trade-offs, map into agent utility. Thus it
is quite consistent to apply quadratic or exponential loss functions to
city-block distances.
Clearly, strongly spatial models are a special case of weakly spatial
models and hence results that can be obtained for weakly spatial models
also apply to strongly spatial models. However, strongly spatial models
merit separate examination if two conditions obtain: (i) if results from
the study of weakly spatial models are indeterminate, or if they are good
only for a subset of weakly spatial models that does not itself contain
the class of strongly spatial models, and (ii) if strongly spatial models
do in fact approximate how real humans think about political choices.
Since most results generated for weakly spatial models, such as the
non-existence results described above, are good only for a subset of models (those in which agents have smooth preferences) that does not contain the entire class of strongly spatial models, condition (i) is satisfied.
In determining the plausibility of (ii), much depends on the choice of
metrics. Mathematically, the number of possible cognitive metrics is infinite. Indeed, restricting our attention to the class of Minkowski metrics,
there remains an infinite number of possibilities.4 There are thus many
3
We say that a player, i ∈ N , has “Euclidean preferences” over points in Rm
if there exists an xi ∈ Rm and a semipositive definite matrix Ai such that for all
y, y 0 ∈ Rm :
y ºi y 0 ⇔ (xi − y)0 A(xi − y) ≤ (xi − y 0 )0 A(xi − y 0 )
4
The Euclidean metric is a Minkowski metric or order 2 and the city block metric
a Minkowski metric of order 1. But the order of a Minkowski metric can be any positive real number, up to and including ∞.Indeed the ∞−metric is a not-implausible
candidate for a cognitive metric, assuming as it does that the perceived distance
between two stimuli is simply the distance between them on the dimension on which
6
metrics to choose from, yet rarely is the choice of a particular metric
well-defended in the spatial modeling literature.
The problem becomes acute when we set out to estimate the ideal
points of real “wet life” political agents in some assumed space. In such
situations we must employ a measurement device. But any measurement
device contains deeply embedded and sometimes obscure assumptions
about the metric of the space in which measurements are being made.
For example, a common approach to estimating the ideal points of
real citizens is to use opinion surveys. These typically involve batteries
of attitude questions on matters considered a priori to be substantively
related — attitudes on a series of “moral” issues, for example, or on the
economy. Citizens’ responses to these questions are then analyzed to investigate the extent to which these can be combined into a single reliable
additive (e.g. Likert) scale. Such a scale might be used to measure the
“conservatism” of respondents on moral issues, for example, or their leftright positions on economic policy, and thereby to map a set of points
measured in a high dimensional response space into a set of points on
a single synthetic analytical dimension assumed to underlie these. The
additive combination of survey responses into a single Likert-like scale,
common when survey data are used to estimate the ideal points of respondents, makes the cognitive scaling assumption about respondents
that they use the city block metric when judging similarity and difference between conceivable political outcomes within the domain of the
scaling exercise. An alternative approach would be to use a data reduction technique such as factor analysis to search for latent dimensions,
with which responses to batteries of attitude questions are correlated.
Note that most reported factor analyses are based, “under the hood”,
on algorithms that minimize Euclidean rather than City Block distances
between latent dimensions and the observed indicators from which they
are constructed. Using factor analysis to derive respondents’ positions
on a single dimension from a high dimensional response space thus makes
different cognitive assumptions about how agents perceive political similarity and difference.
Another striking example of the intrinsic embedding of a metric into
any real measurement instrument can be found in the Nominate procedure used to estimate legislators’ ideal points from their roll call voting
behavior, for which the technique of dimensional analysis is a version
of multidimensional scaling. Before any actual ideal points can be estimated from roll call voting behavior, the Nominate technique assumes
that these ideal points are embedded in a Euclidean real space (Poole
and Rosenthal 1997). Thus a Euclidean metric is used by Nominate to
they are farthest apart.
7
measure distances, while Nominate also assumes that legislators have
exponential (as opposed to linear or quadratic) loss functions with respect to these distances (Poole and Rosenthal 1997). The cognitive basis
of these assumptions could of course be debated and different assumptions could be made, but our point here is that some specific assumption
about the cognitive metric of the space in which distances are to be measured is fundamental in this, as in any other, measurement device.
The choice of distance metric is thus crucial, both for analysts developing strongly spatial models and for those estimating agent ideal
points from empirical data, since such models makes cognitive assumptions about how human agents think about politics. Which metric best
models the perceptions of “real” humans is of course an entirely empirical matter. So what do we know empirically about which metrics best
describe how humans think about political similarity and difference?
As we noted above cognitive scientists, on the basis of over 50 years
of empirical research into human perceptions of similarity and difference
between different (non-political) stimuli, typically see the choice as being
between a Euclidean or a city block metric, with the latter a more appropriate assumption when dimensions of perceived difference are separable
(Attneave 1950; Garner 1974; Shepard 1987; Shepard 1991; Gärdenfors
2000).5 For a rigorous formal review of the conceptual foundations of
this cognitive paradigm, and the role of metrics in this, see Aisbett and
Gibbon (2001). To this mainstream psychological evidence we can add
a range of research findings on various aspects of artificial intelligence
(Lee, 1997) and human-machine interaction (Jacob et al.,1994).
In relation to human-machine interaction, for example, Jacob et al
(1994: 24-25) find that:
“choosing an input device for a task requires looking at
the deeper perceptual structure of the task, the device, and
the interrelationship between task and device. [...] Tasks in
which the perceptual structure is integral operate by similarity and follow the Euclidean metric. The attributes of an
integral object do not dominate and are viewed as a unitary
whole. They are manipulated as a unit. Tasks set in a separable space operate by dimensional structure and obey the
city-block metric. The attributes of a separable object cannot be ignored and are manipulated along each attribute in
turn.”
The conclusion is that, when tasks are more efficiently performed
5
In recognition of the extent to which this work had become part of the psychological mainsteam, Shepard was awarded the National Medal of Science in 1995.
8
by humans using separable input devices (for example a joystick and
a button; a steering wheel and a brake pedal), the implied conceptual
structure is a cognitive space with separable dimensions, in which similarity and distance is best measured using the city-block metric. This
suggests that what we should think about is whether humans typically
view two dimensions of politics–economic policy and foreign policy, for
example–as one integral object to be evaluated as a whole or as two
separate objects whose attributes can be manipulated independently. In
the absence of systematic research, our intuition suggests the latter.
An interesting element of the general class of city-block metrics is
the Hamming (or signal) distance. For any set of dimensions, the Hamming distance between two points is the number of dimensions on which
the points differ. This is widely used by computer scientists in efficient
algorithms designed to search high-dimensional spaces for cognitively
“similar” objects. The Levenshtein (or edit) distance, an important notion in information theory, is a generalization of the Hamming distance
for vectors of unequal length.6 This city-block metric is used in computer
spell-checkers and error-tolerant computer search engines, that work by
minimizing the distance between an unrecognized word and some list
of potential “targets”, as well as in computerized plagiarism detectors.
As many readers will appreciate, these essentially feral research projects
show that algorithms using city-block metrics are empirically quite impressive in retrieving unobservable human perceptions of similarity and
difference.
Our purpose here is not to review the vast, growing and multidisciplinary literature on cognitive metrics. Rather it is to demonstrate
that two important matters should be very much on the intellectual
agenda of those who build spatial models of political choice. The first
concerns the importance of choosing a suitable cognitive metric for a
strongly spatial model. The second, motivating the core argument of
this paper, draws attention to the empirical suitability of the city-block
metric in particular, demonstrated in a range of different contexts, as a
way to model human perceptions of similarity when dimensions of difference are separable. Having argued this, we now move on to demonstrate
that equilibrium outcomes for majority voting games in which agents
have city-block preferences differ significantly from those derived from
weakly spatial models that assume strictly convex preferences.
6
See www.levenshtein.net
9
3
Equilibrium with Uniform City Block Preference
Profiles
3.1
The Model
We consider a situation with a set of agents given by N = {1, 2, ..., n},
n odd, an outcome space given by Rm and player specific rational preference relations over elements of Rm , (%i )i∈N . In this context we define
city block preferences as follows.
Definition 1 We say that an agent, i ∈ N, has “city block preferences” if there exists a yi ∈ Rm and positive scalars (αi,j )j=1,2,...,m such
that for any x, x0 ∈ Rm :
Xm
Xm
x %i x0 ⇔
αi,j |yi,j − xj | ≤
αi,j |yi,j − x0j |
j=1
j=1
The weighting parameters, (αi,j )j=1,2,...,m depend on how the space
itself is defined. However, further regularity conditions can be imposed
upon these weights if agents are seen to view the space in similar ways.
In what follows we generally employ such a regularity condition in order
to define a “uniform city block preference profile.”
Definition 2 We say that a collection of agents, N = {1, 2, ...n}, has
a“uniform city block preference profile” if there exists for each i ∈
N a vector yi ∈ Rm and positive scalars (αi,j )j=1,2,...,m such that for any
x, x0 ∈ Rm :
Xm
Xm
x %i x0 ⇔
αi,j |yi,j − xj | ≤
αi,j |yi,j − x0j |
j=1
αi,j
αi,k
j=1
and for any two agents i and h and any two dimensions j and k,
αh,j
= αh,k
.
In other words there is a uniform city block preference profile when
all agents not only have city block preferences but they all also attach
the same relative weights to the different dimensions. Note that many
strongly spatial models, whatever distance metric they employ, also make
the assumption that agents share a common perception of the space in
the sense of using the same set of relative dimension weights. In particular all models in which all agents have circular, or spherical, indifference
curves make this assumption.
Now, consider a strongly spatial model in which a set of agents N =
{1, 2, ..., n}, n odd, has a uniform city block preference profile.
10
We are interested in conditions under which a point x ∈ Rm is a
“Majority Rule Equilibrium” (MRE) point in the sense that there
is no other point x0 ∈ Rm that at least n+1
agents strictly prefer to x.
2
For simplicity we choose labels for the axes such that the dimension
by dimension median is located at the origin. We assume furthermore
that agents have ideals in “general position,” and in particular that only
one agent is median on each dimension. Finally when we study uniform
city block preference profiles we normalize such that αi,j = 1 for all
agents i, and dimensions, j.
The discussion that follows focuses on the origin as a candidate MRE.
We do this because, given our normalization, it turns out that under
city block preferences no point other than the origin can be an MRE.
To see this assume that x is an MRE but that for some dimension j,
xj 6= 0. Consider now a rival policy, x0 that differs from x only in that
x0j = 0. Clearly we then have x %i x0 ⇔ |yi,j − xj | ≤ |yi,j − x0j | and
hence x0 Âi x ⇔ |yi,j − x0j | < |yi,j − xj |. Since x0j is the median on
dimension j there is a majority, including the dimension j median, for
whom |yi,j − x0j | < |yi,j − xj | and hence a majority that prefers x0 to x.
Hence a necessary condition for a point x to be an equilibrium, given
our normalization, is that x = 0.
For a point to be the DDM is not, however, a sufficient condition
for it to be an MRE. Standard sufficiency conditions for the existence of
an MRE use information on the gradient vectors of agents’ utility functions around the equilibrium point to construct relations of opposition.
Essentially agent i can be termed opposed to agent k in relation to an
outcome at the origin if the gradient vectors, ∇i (0) and ∇k (0) are well
defined and for some scalar β < 0, ∇i (0) = β∇k (0). But in cases with
city block preferences, the fact that the utility functions cannot be differentiated everywhere prevents us from using conditions such as those
found in Plott (1967) or McKelvey and Schofield (1987), at the candidate
equilibrium point. The difficulty arises for agents with yi,j = 0 for some
but not all dimensions. In such cases agents have utility functions that
are not differentiable at the dimension by dimension median. However
symmetry conditions that are analogous to Plott’s can be used.
In the next section we provide such symmetry conditions. The necessary and sufficient condition for the existence of an MRE that we identify
is akin to the result for games with Euclidean preferences that a point,
x, is an MRE if and only if for every supporting hyperplane, H of x,
each closed half space of H contains a majority of voter ideals (see for
example Cox and McKelvey 1984), but is, we shall see, considerably less
restrictive.
11
3.2
Results
Our main result relies on the fact that, with a uniform city block preference profile, all agents with ideal points in a given (open) orthant have
similar preferences over small movements away from the origin. Given
this feature, in order to locate an equilibrium, we require a language to
talk about collections of orthants in which agent ideal points may lie.
We do this as follows.
Let O denote the collection of 2m open orthants of Rm and define
X ≡ {−1, 1}m . Clearly there is a one-to-one correspondence between
the set of orthants and the elements of X: the set X contains 2m points
with each point lying in one of the 2m orthants of Rm . Specifically, let
the function f : X → O, choose the (open) orthant in which x ∈ X
lies: f (x) = {z ∈ Rm : sgn(zj ) = xj for j = 1, 2, ..., m}. Since f is
invertible, the inverse mapping is denoted f −1 : O → X and is given
by f −1 (O) = {x ∈ X : xj =sgn(oj ) for all o ∈ O and j = 1, 2, ..., m}.
With this inverse mapping, for a collection of orthants C we can without
ambiguity denote the corresponding subset of points in X by XC .
We use the inverse mapping f −1 to define the following family of
subsets of Rm that will prove central to our analysis.
Definition 3 We say that a set of points is a “rugged halfspace” if
it is the closure of a collection, C, of 2m−1 (open) orthants that has the
property that there exists a vector v ∈ Rm such that for every orthant
O ∈ C, v.f −1 (O) > 0.
It is easy to see that the set of rugged halfspaces is non-empty (any v
for which v.x 6= 0 for all x ∈ X induces a rugged halfspace, including all
vectors that have a zero in all but one entry); further for every rugged
halfspace with “normal” vector v, there exists an opposite rugged halfspace with normal −v; finally rugged halfspaces are half spaces in the
sense that the union of two opposite rugged halfspaces is the set Rm . A
number of other features of rugged halfspaces are however worth noting.
First, since a rugged halfspace is the closure of a collection of (open)
orthants, it includes all the boundary points of the orthants; in particular, since 0 is in the boundary of each orthant it is contained in every
rugged halfspace. More generally if a point x is in a rugged halfspace,
so is any point x0 that differs from x only in that for some dimension j,
x0j = 0 but xj 6= 0.
Second, rugged halfspaces differ from the standard notion of a halfspace generated by a hyperplane through the origin. Conventionally,
such a halfspace is given by a set of points Z ⊂ Rm for which there exists a normal vector v ∈ Rm with v.z ≥ 0 for all z ∈ Z. However whereas
12
in R1 and R2 , all rugged halfspaces are also halfspaces in this standard
sense, this is not generally the case. In R3 the vectors XC = {{1, 1, 1},
{−1, 1, 1}, {1, −1, 1}, {1, 1, −1}} induce a rugged halfspace (with a normal given by v = {1, 1, 1}) but the rugged halfspace, being blocky, can
not be described as the half space lying above a hyperplane. This rugged
half space is illustrated in Figure 1.
Figure 1: Illustration of two rugged half spaces in R3 . With v = {1, 1, 1}
the upper right rugged halfspace is generated by the set of elements of
X with x.v > 0; the lower left rugged half space is generated by the
vectors in X for which x.(−v) > 0.
We now state and prove our main result:
Proposition 1 In a majority rule game with a uniform city block
preference profile and an odd number of voters, an MRE exists (and
corresponds to the dimension by dimension median) if and only if a
majority of ideal points lies in every rugged halfspace.
Proof. See Appendix.
Proposition 1 tells us that we can fully characterize the conditions for
MRE existence using only information about the distribution of agent
ideal points across orthants. Any perturbation of ideal points that does
not alter their distribution across orthants does not affect the equilibrium. Such perturbations may be quite large. Thus the importance of
this result is that it establishes that an MRE, if it exists under a uniform
city block preference profile, will be robust to significant perturbations
of agent ideal points. In this sense, the non-existence of an MRE is not
“generic” under uniform city block preference profiles (we discuss this
notion of “genericity” in more detail in the next section).
13
3.3
Qualitative Implications
Proposition 1 has five implications of interest.
A first implication of the proposition is that with uniform city block
preferences an equilibrium always exists in games with only three agents.
This follows from the fact that with only three agents, a majority (two)
necessarily lie in every rugged halfspace.7
A second implication is that with uniform city block preferences an
equilibrium always exists in games with only two dimensions. This result,
given also in Rae and Taylor (1971) and Wendell and Thorson (1974)
follows directly from the proposition since with only two dimensions a
majority necessarily lies in each rugged halfspace. In two dimensions the
only rugged halfspaces are composed of the pairs of adjacent orthants
either above or below the horizontal axis or to the left or the right of the
vertical axis. The origin is defined however such that there is a majority
on or to one side of each axis.8
A third implication is that Proposition 1 generalizes Plott’s results
(conditional upon our restriction to city block preferences) on radial
symmetry since the conditions of the proposition are implied by Plott’s
proposition whenever this can be applied (hence whenever Plott’s sufficient conditions for an equilibrium are satisfied for agents with city
block preferences, so too are the conditions of our proposition). To see
this, consider a situation in which one agent has an ideal at the dimension by dimension median, 0, and all other agents can be divided into
pairs that are opposed in the sense described above. Assuming a smooth
loss function, a utility function that represents city-block preferences is
differentiable at a point 0 for an agent i with ideal point yi if there is
no dimension j such that yi,j = 0. Hence all agents other than the
7
To see this let x, y and z denote the ideals of three players and assume that x and
only x lies in a rugged halfspace, H, with normal, v.½ Without loss of generality assume
sgn(yj ) if yj 6= 0
that vj ≥ 0 for j = 1, 2, ...m. Define y 0 by yj0 =
. Since y lies in
1 if yj = 0 P
P
clos(f (y 0 )) but not in H we have y 0 .v < 0, or {j:sgn(yj )=−1} vj > {j:sgn(yj )≥0} vj .
P
P
Defining z 0 in the same way we have
{j:sgn(zj )=−1} vj >
{j:sgn(zj )≥0} vj .
But with
P only three players
P we have sgn(yj ) = −1 implies sgn(zj ) ≥ 0 and
hence
v
≥
Together these inequalities imply
j
{j:sgn(z
)≥0}
{j:sgn(yj )=−1} vj .
j
P
P
v
>
v
.
But
by
the same logic, if sgn(zj ) = −1
j
j
{j:sgn(zj )=−1}
{j:sgn(y
P j )≥0}
P
then sgn(yj ) ≥ 0 and so {j:sgn(yj )≥0} vj ≥ {j:sgn(zj )=−1} vj , contradicting the
last inequality
8
But as shown in Wendell and Thorson (1974) and Kats and Nitzan (1977) this
existence result in two dimensions does not extend to higher dimensions. In terms of
our proposition, the reason is that the set of rugged halfspaces in higher dimensions
is not constrained to the set of halfspaces that have a normal vector given by a basis
vector of the space.
14
one with yi = 0 have well defined gradients that are proportional to
[sgn(yi,1 ), sgn(yi,2 ), ..., sgn(yi,m )]. Hence two agents are opposed if their
ideals lie in opposite orthants. With all remaining agents divided into
pairs with ideals in opposite orthants we necessarily have a majority in
every rugged halfspace.
A fourth implication, following from the last, is that the conditions
for symmetry are considerably less onerous than in other spatial models.
Say that a set of preference profiles is “generic” if it is open and dense in
the set of all admissible preference profiles. Essentially this means that,
whether or not a profile is in a generic set, A, some small perturbation
of the profile will be in A. In the case of strictly convex preferences with
the DDM fixed at the origin, the set of preference profiles for which there
exists no scalar β < 0, with ∇i (0) = β∇k (0) is generic. In other words
generically, two (non-median) agents, i and k are not opposed. In the
case of Euclidean preferences, opposition of agents i and k (relative to the
origin) implies that there is a scalar β < 0, with yi = βyk and so we can
apply the notion of genericity to the distribution of ideal points rather
than to the gradient vectors. When we do this, we find that generically
a distribution of ideal points that induces opposition between pairs does
not obtain. This is not the case with a uniform city block preference profile. With gradient vectors lying in a finite set, {−1, 1}m , rather than
in Rm , a small change in the location of ideal points does not imply a
change in gradient vectors. Thus the set of profiles of ideal points that do
not produce an MRE is not dense. Hence, robust equilibria that involve
non-pathological distributions of ideal points are possible for any m and
n. This highlights an important intuition behind our results, since we
should not forget that the existing non-existence results depend upon
setting aside a class of exceptions, which for the case of Euclidean preferences requires “radial symmetry” of ideal points, deemed pathological.
In a sense, our results show that the radial symmetry exception expands
explosively under city-block preferences, rendering the non-existence results non-generic.
A fifth implication is that even if Plott conditions do not hold and
there is no agent ideal point at the dimension-by-dimension median,
there may still exist classes of cases where an MRE exists at the dimensionby-dimension median, in any number of dimensions. For an example,
consider a case with 3 voters in three dimensions with ideals with a
corresponding sign ordering, s, (with si,j = sgn(yi,j )) given by:
⎡
⎤
0 −1 1
s = ⎣ 1 0 −1 ⎦
−1 1 0
15
This class of sign orderings can be readily extended to any odd number of dimensions, m, with m voters such that, for each i, yi,i = 0, and,
for k ∈ 1, 2, ...m − 1, yi,(i+k)(mod(m)) < 0 if k is odd and yi,(i+k)(mod(m)) > 0
if k is even. To see that this set of sign patterns satisfies the condition of
the Proposition note that for any “normal” v, for each pair of neighboring points, yi and y(i+1)(mod(m)) , one belongs in one rugged half space and
the other belongs in the opposite rugged half space.9 This guarantees
that there is a majority in every rugged half space.10 For n > m voters
such classes can obtain if the remaining n − m voters can be divided into
pairs with ideals in opposite orthants.
The intuition behind Proposition 1 is illustrated in Figures 2 and 3.
Figure 2 shows the positions of three voters with ideal points at A, B,
and C, all elements of R2 . The dimension-by-dimension median of the
voter positions is marked as DDM — re-scaled without loss of generality
to the origin of the space. The dashed axes thus contain the median
voter on each dimension.
The dashed circles in Figure 2 show Euclidean indifference curves for
each voter with respect to the DDM. The three lens-shaped intersections
of these indifference curves show the “winset” of the DDM–the set of
points in the policy space that are majority preferred to the DDM. The
DDM will lose in a majority vote to any point in this set and thus is
not an equilibrium outcome. The non-existence results show that the
winset of any arbitrary point in this space, and the DDM in particular, is
“generically” non-empty under majority rule in multidimensional policy
spaces.
The rotated squares in the figure show the city-block indifference
curves for each voter with respect to the DDM. These curves do not
intersect and thus the DDM has an empty winset. It is therefore a
majority-voting equilibrium. In terms of Proposition 1, it can be seen
that each rugged halfspace contains the ideal points of two of the three
voters; a movement in any direction will be opposed by some such pair
of voters.
9
To see this let ½
k be given by k = (i + 1)(mod(m)).
Define ỹi and ỹk for j =
½
sgn(yi,j ) if yi,j 6= 0
sgn(yk,j ) if yk,j 6= 0
1, 2, ..., m by ỹi,j =
and ỹk,j =
.
−1 if yi,j = 0
1 if yk,j = 0
Note that yi is in the closure of f (ỹi ) and yk is in the closure of f (ỹk ) but that
ỹi = −ỹk . It follows that for any v such that ỹi .v > (<)0 we must have ỹk .v < (>)0,
and hence yi and yk lie in opposite halfspaces.
10
Assume to the contrary that a rugged halfspace H(v) contained only a minority of
ideal points, then a majority of ideals would lie in the interior of the opposite rugged
halfspace, H(−v). But such a majority would necessarily contain two neighboring
points that do not belong to opposite halfspaces.
16
Figure 2: Majority voting equilibrium at the dimension-by-dimension
median with city block preferences
Intuitively the difference between the two cases arises because of two
features of uniform city block preference profiles. The first is that the
indifference curves of agents that are median on some dimension (but not
all dimensions) are kinked at the origin. To generate an improvement in
policy for an agent with such a kink is as difficult as it is to generate an
improvement for two non-median agents with ideals in distinct (but not
opposite) orthants–in Figure 2 for example agents A and B are willing
to accept only one quarter of all possible directions of movement from
the origin whereas agent C will accept one half. The movements that
are acceptable to Agent B correspond to those that are simultaneously
acceptable to agents in the upper left and upper right orthants. In
games with strictly convex preferences, median agents do not hold such
privileged positions and are not more difficult than non-median agents
to satisfy.
The second feature of city block preferences is that all agents with
ideal points in the interior of a given orthant have indifference curves of
the same slope (a) as each other and (b) as agents in opposite orthants.
An implication of (a) is that the condition for opposition with another
agent is relatively weak, all that is required is that there is an agent in
the opposite orthant. An implication of (b) is that a small change in
the location of an agent’s ideal point–consider for example voter C in
the figure–does not induce a change in shape of the voter’s indifference
curve at the equilibrium point. This local invariance of the shape of
indifference curves to the location of ideal points explains why “radial
17
symmetry” exceptions to non-existence results (identified for example by
Plott) are more stable whenever they obtain. The classic radial symmetry counterexample requires a very strong notion of opposition between
pairs of voters whose ideals do not lie at the origin. Under Euclidean
preferences, opposition requires that voters have ideal points arranged
precisely such that each voter i whose ideal is not at the origin can
be matched with some other voter with an ideal point lying on a line
that passes through the origin and through i’s own ideal point. Such
precisely balanced arrangements of ideal points are upset by the most
infinitessimal perturbation. Under uniform city block preference profiles,
in contrast, small shifts in the location of voter ideal points do not upset the much more tolerant relations of opposition between ideal points
that underlie the majority-rule equilibrium. In this important sense, the
class of preference profiles (within the set of uniform city block preference
profiles) for which the DDM has a non-empty winset is not generic.
For an illustration of this stability effect consider the locations of
points A, B, C, D and E in Figure 3. Under Euclidean preferences,
voters D and B are opposed in this example. But from this radially
symmetric distribution, any tiny perturbation in one agent’s ideal (for
example a shift in agent D’s ideal to D0 ), is sufficient to remove the
opposition between agents D and B, and displace the equilibrium. Under
city block preferences however such a shift in the location of ideal points
does not remove the opposition between D and B. To illustrate this,
in Figure 3 we mark the indifference curves of agents A - E under the
assumption of a uniform city block preference profile. A perturbation
from D to D0 does not in this case produce a rival point to the DDM
at E. The reason is that B and D remain opposed to each other: even
following the perturbation from D to D0 , E remains “between” B and D0 .
This is because d(B, E)+d(E, D) = d(B, D) under the city block metric,
although not under the Euclidean metric. Indeed all of the shaded area
between B and D0 is, under the city-block metric, between B and D0 .
We thus get a clear sense of the perturbations of agent D’s ideal point
that are needed to destroy the opposition between B and D and in this
way upset the equilibrium. The perturbation must put D’s ideal point
north and/or east of E, in other words into another orthant, for this to
happen. Clearly, this a very significant perturbation.
3.4
Probability of Equilibrium
Having established the possibility of majority rule voting equilibria under
city block preferences in spaces of any dimension, we now move on to
consider the probability of these. To generate a baseline estimate of
18
Figure 3: Robustness of Radial Symmetry Under a Unifrom City Block
Preference Profile
the frequency of MRE existence we employ simulation techniques to
estimate the likelihood of MRE existence given a random distribution
of ideal points, as the number of voters and dimensions increases, where
randomness means that any voter is equally likely to be median on any
dimension, and this likelihood is independent of any voter’s position on
any other dimension. The results of the simulations are reported in
Figure 4.
Dimensions
Players
1
3
5
7
9
11
13
15
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
3
1
1
0.66
0.47
0.37
0.33
0.30
0.25
4
1
1
0.33
0.14
0.06
0.03
0.02
0.02
5
1
1
0.15
0.02
0.00
0.00
0.00
0.00
6
1
1
0.05
0.00
0.00
0.00
0.00
0.00
Figure 4: The table reports the estimated probability of observing a
core point at the DDM in m dimensions when n players have ideals
drawn from a multivariate normal distribution with a diagonal variance
co-variance matrix. Estimates are based on 5000 independent draws of
n person polities; 95% confidence intervals are in all cases within ±1%
of our estimates.
The results in Figure 4 confirm that an MRE exists with probability
1 for dimensions 1 and 2 and for committees with 1 and 3 voters, but the
19
probability of MRE existence declines rapidly thereafter, reaching close
to 0 with nine voters in five dimensions. Hence although even in high
dimensional space there is no generic absence of an MRE, the existence
of an equilibrium point may be very unlikely, at least if agent ideal points
are distributed randomly.
We close this section by considering two important ways in which
distributions of the ideal points of human agents are likely to be nonrandom in ways that increase the likelihood of equilibrium existence in
higher dimensional settings.
The first has to do with correlations of agents’ positions on different
dimensions — so that an agent’s position on stem cell research, for example, can be predicted from her position on gay marriage. If we think
of the empirical cigar shaped scatterplots of ideal points produced when
agent preferences on different dimensions are correlated, we can see that
these are much more likely than random distributions of ideals to locate
pairs of voters in opposing orthants of the space. For this reason, configurations of ideal points yielding majority voting equilibria under city
block preferences are more likely with correlated preferences than they
are with random preferences. To assess the importance of this effect we
report in Figure 5 the likelihood of MRE existence when agents have
ideal points drawn from a multivariate normal distribution with m × m
variance covariance matrix {σ i,j } where σ i,j = 1 if i = j and σ i,j = .9.
otherwise. The results indicate that the probability that an MRE exists
is considerably higher in this example of correlated ideal points than in
the baseline case of random ideal points. We also note, however, that
the probability of a majority voting equilibrium continues to fall sharply
as the number of agents and dimensions increases.
Players
1
3
5
7
9
11
13
15
Dimensions
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
1
1
1
3
1
1
0.93
0.83
0.75
0.69
0.60
0.55
4
1
1
0.86
0.70
0.55
0.42
0.31
0.25
5
1
1
0.79
0.58
0.40
0.27
0.17
0.11
6
1
1
0.75
0.50
0.32
0.18
0.12
0.06
Figure 5: The table reports the estimated probability of observing a core
point at the DDM based on 5000 independent draws of n ideal points
from a m-dimensional multivariate normal distribution with a variance
covariance matrix given by {σ i,j } where σ i,j = 1 if i = j and σ i,j = .9.
otherwise. 95% confidence intervals are in all cases within ±1% of our
estimates.
20
A second source of structure in the set of ideal points that can increase the probability of equilibrium under city-block preferences arises
in weighted voting games, such as those played in a legislature by welldisciplined political parties. In such games, each party has weight that
reflects a situation in which a subset of voters belong to the same party
and behave in a disciplined manner, “as if” they share the same ideal
point. Two things happen in this setting. First, a large number of
agents coalesce into a small number of disciplined parties with different
voting weights; the number of effective agents is dramatically reduced
and the above arguments apply. Second, weighted voting dramatically
changes the probability that a large voting weight is located at the DDM
and hence is contained in every rugged halfspace. This is because the
probability that a party will be median on some dimension is strongly
conditioned by that party’s weight, as well as by its position in the ordering of parties on the dimension in question.
Trivially, a party with a majority weight will be median on every
dimension and that party’s ideal point will be a majority voting equilibrium in a space of any dimension under any conceivable assumption
about preferences. Extending work on “dominant players” in majority voting games with non-uniform weights (Peleg 1981; Einy 1985; van
Deemen 1989), define a “system-dominant” party in a weighted voting
game with four or more parties as a party which does not itself control
a majority, but which can form a majority with any other single party,
even the smallest. This implies no other two-party coalition can form
a majority.11 Given Proposition 1 this further implies that, a sufficient
condition for an equilibrium under a uniform city block preference profile is that a system dominant party is at the dimension-by-dimension
median (which, we argue below, is “likely”) and at least one party other
than the system dominant party is in each rugged halfspace–a distribution that could obtain for example if any two non dominant parties
had ideals in opposite orthants.
Since it can form a winning coalition with any other party, it is easy
to see that a system dominant party occupies the median position on
every dimension for which it is not at one of the two extreme positions in
the ordering. Thus, for example, in an eight-party system with random
party positions, the probability that a system dominant party is median
on an arbitrary dimension is 0.75. More generally the probability that
a system-dominant party is at the DDM of an n-party m-dimensional
11
The largest two-party rival to a winning coalition between the largest and smallest parties is a coalition between the second and third largest parties, which must
be losing since it is in the complement of the winning coaltion between largest and
smallest parties.
21
£
¤m
space is n−2
. If we combine this with the Downsian intuition in ren
lation to majority voting in legislatures that the largest party is less
likely than others to be at an extreme position on salient policy dimensions, then this probability will increase dramatically in real-world
policy spaces. More generally in weighted voting games, the probability
a largest party with a random ideal point is at the median on an arbitrary dimension, and hence at the DDM, is greater than the probability
an arbitrary voter is at the DDM in an unweighted voting game with
random ideal points. In this sense the baseline probability of a majority voting equilibrium under city block preferences increases in weighted
voting games.
Note that none of the above enhancements in the probability of majority voting equilibria applies under the classical non-existence results
with smooth preferences, since such equilibria generically do not obtain.
4
Non uniform city block preference profiles
Our proposition depends heavily not simply on the fact that agents have
city block preferences but also that there exists a uniform city block
preference profile. When agents attach heterogenous weights to each
dimension, the conditions for equilibrium are considerably more severe.
In this case we have that a non-median voter i’s gradient vector,
evaluated at yj = 0 is proportional to:
[αi,1 sgn(xi,1 ), αi,2 sgn(xi,2 ), ..., αi,3 sgn(xi,m )]
In the case where there is a single satiated voter at the DDM, we
can apply Plott’s theorem for necessary conditions for MRE existence.
In this instance since each voter’s gradient vector depends on the m
scalars, (αi,j )j=1,2,...,m , the space of all agents’ gradients is given by Rm .
However, the set of profiles of α vectors in Rm that do not satisfy Plott’s
conditions is open and dense and hence we return to a problem of generic
non-existence of equilibria (with Plott points). The problem here is that
agents with ideals in opposite orthants are not necessarily opposed in
the sense of having opposing gradients.
Recall we noted that equilibrium is more likely under city block preferences for two reasons. First, agents that are median on some dimension have kinked indifference curves and accept, as a result, changes in
the status quo in fewer directions than agents with smooth indifference
curves (at the equilibrium point). Second, parallel indifference curves
render relations of opposition much more common.
Relaxing the uniformity constraint on agent weights removes the second but not the first of these effects. The result is that while equilibrium
22
is much less likely with non-uniform than with uniform city block preferences, it is not generically absent in dimensions greater than 1 and
can obtain in situations where multiple agents are median on different
dimensions.
Given a random distribution of ideals, the chances of an MRE obtaining in a 3 person game in R2 is 19 .12 For an example of such a situation
in R3 , consider a case of three agents with ideal points that have a
sign structure given by: s1 = (0, 1, −1), s2 = (−1, 0, 1), s3 = (1, −1, 0).
Assume further that each agent i places weights on loss in these dimensions given by (αi,j )j=1,2,3 with αα1,1
> αα3,1
> αα2,1
, αα2,2
> αα1,2
> αα3,2
,
1,2
3,2
2,2
2,3
1,3
3,3
α3,3
α2,3
α1,3 13
and α3,1 > α2,1 > α1,1 .
In this case the origin is an equilibrium point.
To see that the origin is an equilibrium, consider a trade-off between
agents 1 and 2 on dimensions 1 and 3. With αα2,3
> αα1,3
any negative
2,1
1,1
movement on dimension 3 that is sufficiently large to compensate agent
1 for a negative movement on dimension 1 is so large a movement from
agent 2’s perspective that it outweighs any benefits from the negative
movement on dimension 1. In a similar way all deals are unacceptable
for any two agents and so the origin is an equilibrium (we show this for
all cases in the appendix).
Such possibilities exist then in two and three dimensions. They are
however rare. And the problem of MRE existence becomes considerably
more severe with either more agents or more dimensions. We can identify
a stable MRE with 5 agents in 3 dimensions, but the conditions for this
are very restrictive and our simulation work suggests that they obtain
for a random polity with probability close to 0%. Finally, it is easy to see
analytically that there is generically no MRE in four dimensions even
with only three agents. To see this note that in such cases there will
always be some pair of agents for whom there are two dimensions on
which neither of the two are median. (In particular, one of three agents
is necessarily median on two dimensions, hence with ideals in general
position, neither of the remaining two agents is median on those two
dimensions). With random weights this pair (a majority) will find some
trade-off mutually beneficial on these two dimensions.
The importance of these results lies in the fact that, if non-uniform
weights are empirically appropriate, then we may discount our results
based on the assumption of uniform weights outright; in the combined set
12
For 1/3 of the cases one player is at the DDM and no core exists; for the remaining
2/3 only 1/6 of have weights that guarantee a core.
13
These inequalities ensure that each player cares (relatively) more about the dimension that she is median on than do the other players. This makes it difficult for
two players to strike a deal involving changes in either of the (two) dimensions on
which they are median.
23
of uniform and non-uniform preference profiles, the set of non-uniform
profiles is generic.14
5
Conclusion
For a very general class of smooth preference profiles, long-standing results show that a majority rule equilibrium fails to exist whenever policy
choices are defined in a space of dimension greater than one. However,
we have shown here that if agents have a uniform city-block preference
profile the non-existence of equilibrium is not generic: an equilibrium is
possible in spaces with an arbitrarily large number of dimensions and is
“likely” in situations with relatively few dimensions and agents. These
results extend in part to non-uniform city-block preference profiles, but
only for spaces of dimensionality no greater than three and only under
more restrictive conditions. Hence, city-block preference profiles, though
“close” to a smooth preference profile for which generic non-existence of
equilibrium obtains, are different in crucial ways.
These differences matter for the study of policy stability: our results
provide new conditions for when we should expect complex policies to
be more or less likely to be stable. But they also matter for students
employing non-cooperative approaches to study more highly specified
political processes; for such models, equilibria of voting games are often
required as inputs to more complex processes and for these our results
provide a rationale–and conditions–for using the median voter result
even in higher dimensional settings.
The significance of our results depends ultimately on the extent to
which humans do in fact act as if they employ a city block metric in political settings, uniformly. This is an open empirical question. We have
seen that city block preferences accord closely with extensive empirical
research on human perception in settings where analytical dimensions
of perceived similarity and difference are separable. But while experimental research in cognitive science shows that real humans use city
block metrics when analyzing similarities and differences between nonpolitical objects, and while experimental research in political science
(such as Halfpenny and Taylor 1973) shows that, once endowed with a
city-block metric, voters choose the DDM, we know of no research to date
that establishes whether or not people act as if they employ city block
metrics when evaluating similarities and differences between potential
14
Note however many strongly spatial models and many empirical models do in
practice assume that agents have a common perception of the space. Thus Likert
scaling, factor analysis and multidimensional scaling typically all assume a common
perception of the policy space.
24
political outcomes. We have shown however that the implications of
such research are important. If results from the cognitive psychology of
human perceptions of similarity and difference do extend to perceptions
of political stimuli, then the possibility of equilibrium under majority
rule re-emerges without the need to confine the domain of choice to a
single dimension or to impose restrictive institutional assumptions.
25
6
Appendix: Proof of Propositions 1
Restatement of Proposition 1: In a majority rule game with a uniform city block preference profile and an odd number of agents, an MRE
exists (and corresponds to the dimension by dimension median) if and
only if a majority of ideal points lies in every rugged halfspace.
Proof. (i) Necessity: To see that an MRE exists only if a majority of ideal points lies in every rugged halfspace assume, contrary to
the claim, that 0 is an MRE but there exists some rugged half space
HC –corresponding to a collection C of orthants–that does not contain
a majority of agent ideal points. Let XC denote the elements of X corresponding to the orthants in C and let X 0 = X \XC denote the elements
of X not in XC .
Note that since HC is a rugged halfspace, there exists an (arbitrarily
short) vector v with x.v > 0 for each x in XC . Note furthermore, that
for all x0 ∈ X 0 , x0 .v < 0 and hence x0 .(−v) > 0.
We aim to show that all agents with ideal points in Rm \HC prefer
−v to 0; since a majority of agents have ideals in Rm \HC , we have then
that 0 is not an MRE.
Consider an arbitrary agent i with ideal point y in Rm \HC and let
0
y denote the element of X given by:
⎧
sgn(yj ) if yj 6= 0
⎨
¡ 0¢
0
0
y = yj j=1,2,...,m where yj = sgn(vj ) if yj = 0 and vj 6= 0
⎩
1 if yj = 0 and vj = 0
Since y is an element of the closure of f (y 0 ) but lies outside of HC , we
have that y 0 is in X 0 and in particular that y 0 .v < 0. For sufficiently
short v (such that for yj =
6 0, sgn(yj ± vj )=sgn(yj )), we have:
−v - i 0
Xm
Xm
|yj + vj | ≥
|yj |
⇔
j=1
j=1
X
X
X
sgn(yj )(yj + vj ) +
|vj | ≥
sgn(yj )yj
⇔
{j:yj 6=0}
{j:yj =0}
{j:yj 6=0}
X
X
⇔
sgn(yj )vj +
|vj | ≥ 0
{j:yj 6=0}
{j:yj =0}
But since, by construction, sgn(yj ) = yj0 whenever yj 6= 0, and |vj | =
sgn(vj )vj = yj0 vj whenever yj = 0, we have −v -i 0 ⇔ y 0 .v ≥ 0, and
conversely: −v Âi 0 ⇔ y 0 .v < 0. Since by construction, y 0 .v < 0 we then
have that −v is preferred to 0 for all i with ideals in Rm \HC .
(ii) Sufficiency: To see that an MRE exists if a majority of ideal
points lies in every rugged halfspace assume, contrary to the claim, that
a majority lies in every rugged halfspace but that a movement v 6= 0 is
26
preferred by some majority. Assume without loss of generality that for
no x ∈ X, x.v = 0 (to see that this is without loss of generality note
that since each agent’s preferred-to set is open, the intersection of these
preferred to sets is also open and hence the set of points that beat 0 is
open; hence if x.v = 0 we can freely select a v 0 arbitrarily close to v that
also beats 0 but for which x.v 6= 0). Since for no x ∈ X, x.v = 0, there
exists a collection of 2m−1 orthants, C, and a corresponding subset of X,
XC for which x.v < 0 for all x ∈ XC . Since for this collection x.(−v) > 0,
the closure of the union of the elements of C (or, equivalently, the union
of the closure of these orthants), HC , constitutes a rugged half space and
hence, by assumption contains a majority of elements.
We now show that since x.v < 0, v ≺i 0 for all i in the closure of
each orthant in C. From a similar argument to that given in part (i),
for sufficiently small v we have:
v % i0
Xm
Xm
|yj − vj | ≤
|yj |
⇔
j=1
j=1
X
X
X
sgn(yj )(yj − vj ) +
|vj | ≤
sgn(yj )yj
⇔
{j:yj 6=0}
{j:yj =0}
{j:yj 6=0}
X
X
⇔
sgn(yj )vj −
|vj | ≥ 0
{j:yj 6=0}
{j:yj =0}
If y ∈ clos(f (x)), then we have sgn(yj ) = xj whenever sgn(yj ) 6= 0;
in cases where sgn(yj ) = 0, we have |vj | = xj vj if sgn(xj ) =sgn(vj ) and
|vj | = −xj vj otherwise. Hence
X
X
X
sgn(yj )vj −
|vj | = x.v−2
xj vj
{j:yj 6=0}
{j:yj =0}
{j:yj =0,sgn(xj )=sgn(vj )}
and so we have:
v % i0
⇔ x.v − 2
⇔ x.v ≥ 2
and hence, the converse:
X
{j:yj =0,sgn(xj )=sgn(vj )}
X
{j:yj =0,sgn(xj )=sgn(vj )}
v ≺i 0 ⇔ x.v < 2
X
xj vj ≥ 0
xj vj
{j:yj =0,sgn(xj )=sgn(vj )}
xj vj
P
Since the term 2 {j:yj =0,sgn(xj )=sgn(vj )} xj vj is positive we have x.v <
P
0 ⇒ x.v < 2 {j:yj =0,sgn(xj )=sgn(vj )} xj vj and hence x.v < 0 ⇒ v ≺i 0.
But this implies that all agents i with ideal in the rugged half space
HC , oppose the movement in direction v. Since this group constitutes a
majority, 0 cannot be beaten by v. This establishes sufficiency.
27
Claim 4 Consider a situation with three agents with ideal points in R3
that have sign structure s given by: s1 = (0, 1, −1), s2 = (−1, 0, 1), s3 =
(1, −1, 0). Assume further that each agent i places weights (αi,j )j=1,2,3
such that αα1,1
> αα3,1
> αα2,1
, αα2,2
> αα1,2
> αα3,2
, and αα3,3
> αα2,3
> αα1,3
. In
1,2
3,2
2,2
2,3
1,3
3,3
3,1
2,1
1,1
this case the origin is an MRE.
Proof. A movement v is acceptable to agents 1 and 2 only if the
following two inequalities hold:
−α1,1 |v1 | + α1,2 v2 − α1,3 v3 > 0
((1))
−α2,1 v1 − α2,2 |v2 | + α2,3 v3 > 0
((2))
If conditions (1) and (2) hold they must also hold for some v1 ≤ 0 and
v2 ≥ 0. It is sufficient then to show that these can not hold for any v1 ≤ 0
1,2 v2
and v2 ≥ 0. To satisfy condition (1) we require that: v3 < α1,1 vα1 +α
;
1,3
2,2 v2
2,2 v2
1,2 v2
. Together: α2,1 vα1 +α
< α1,1 vα1 +α
,
for (2) we need v3 > α2,1 vα1 +α
2,3
1,3
h
i
h 2,3
i
or: v1 αα2,1
< v2 αα1,2
; but since αα1,1
− αα1,1
− αα2,2
> αα2,1
and αα2,2
> αα1,2
2,3
1,3
1,3
2,3
1,3
2,3
2,3
1,3
the LHS is positive and the RHS is negative. From the symmetry of
the conditions we observe that identical reasoning establishes that no
coalition between 2 and 3 or between 1 and 3 can form.
28
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