A (Non)-Downsian Model of Political Competition
Simon Loertscher∗
April 29, 2015
Abstract
This paper provides a spatial model of multi-party competition under proportional representation. Each party’s equilibrium position balances incentives to steal votes and the
need to deter additional entry. The model simultaneously determines the number of active
parties and their platforms and permits comparative statics with respect to the distributions of voter preferences. Central predictions of the theory are that modal positions induce
fiercer competition but are typically not occupied in equilibrium and that multi-modal distributions give rise to waves of entry or exit.
Keywords: Multi-party competition, political economics, entry deterrence, party formation, waves of entry.
JEL-Classification: C72, D72.
∗
Department of Economics, FBE Building, Level 4, University of Melbourne, Victoria 3010, Australia. Email:
simonl@unimelb.edu.au.
1
2
1 INTRODUCTION
1
Introduction
In a democracy, the number of active parties and their political platforms are an endogenous
outcome that depend on its political institutions and the preferences of its voters. The emergence and success of the Tea Party shows that party formation is an important issue even for
countries like the United States that may historically been considered two-party systems. Understanding the process that determines how many parties are active and what platforms they
choose thus requires a model in which these variables are determined endogenously. However,
developing such a model has proved challenging.1
To illustrate the difficulties, consider the standard Hotelling-Downs model and assume that
the political system has proportional representation, so that parties maximize votes. The
following figure, which reproduces a figure in Downs (1957), does not describe an equilibrium
outcome in a simultaneous moves game, notwithstanding assurances to the contrary. The figure
contains both a uniform distribution and a symmetric multi-modal density with four modes.
The parties’ positions as depicted are {1/8, 3/8, 5/8, 7/8}. Although the locations 3/8 and 5/8
1/8
3/8
5/8
7/8
are mutually best responses and best responses to the extremists’ locations, the parties at the
extremes are not best responding to their neighbors’ locations because each of them would be
better off moving toward the center.2
1
The main obstacle is that, with a fixed number of agents, a pure strategy equilibrium need not exist, and
that the existing mixed strategy equilibrium is not tractable even for the seemingly simple case of uniform
distributions (Osborne and Pitchik, 1986). Given that the equilibrium payoffs are hard to pin down for a
given number of active agents, determining the number of active agents endogenously has proved even more
challenging.
2
As observed by Downs, for the uniform distribution any player with an interior location is completely
1 INTRODUCTION
3
Interestingly, for the uniform distribution the configuration {1/8, 3/8, 5/8, 7/8} is a subgame
perfect equilibrium outcome in the sequential location game proposed by Prescott and Visscher
(1977, PV hereafter) if and only if the cost of entry is 1/8. In a sequential location game, a
large number of players decide in a predetermined sequence whether to enter, and if so, which
location to occupy. Once chosen, locations cannot be changed subsequently, and entry involves
the same positive cost for all players. The problem with sequential location games, however,
is that their analysis becomes intractable for all intents and purposes for large families of
distributions, including all those with interior modes, such as “normal”-looking densities, which
unfortunately seem to be empirically the most relevant ones. In a nutshell, as first observed by
Loertscher and Muehlheusser (2011), the difficulties stem from the fact that optimal locations
close to a mode may be given by a first-order conditions. With the sequential nature of moves,
this gives almost inevitably rise to the possibility that, on or off the equilibrium path, some
player’s optimal location may be determined by a first-order condition, which creates a tedious
and intractable Stackelberg-style problem.3
The present paper provides a model of multi-party competition under proportional representation, which subject to a mild regularity condition is tractable for arbitrary distributions.
The model permits comparative statics with respect to distributions on parties’ equilibrium
locations and the number of parties who are active in equilibrium. It builds on the ideas and
insights of sequential locations games but circumvents the Stackelberg problem by making all
location choices on the equilibrium path simultaneous rather than sequential. At the heart
of this is a fundamental tension between parties’ incentives to steal votes from competitors
and the need to deter detrimental entry by new parties. This tension is central in Downs’s
book, which refers to the vote-stealing incentive as a centripetal pull, which may be strongly
counteracted by the need to prevent additional entry. For the examples underlying the figure
above, the present model predicts that in equilibrium parties will positions themselves exactly
as depicted by Downs when the fixed cost of entry is 1/8 regardless of whether the density is
the uniform or the symmetric multi-modal density. As the cost of entry decreases marginally,
indifferent across all the locations in between its closest competitors: by moving it would win just as many votes
on one end as it would lose on the other. For locations of the closest neighbors that are symmetric relative to
the minima, the same is true for the symmetric multi-modal density.
3
Absent interior modes, equilibrium outcomes for sequential location games can be determined with a simple
algorithm, as shown by Loertscher and Muehlheusser (2011), with the added qualification that densities be
symmetric around interior minima, if such minima exist. Our analysis makes use of this algorithm without
imposing any such symmetry.
4
1 INTRODUCTION
the number of entrants in the equilibrium PV focus on is five under the uniform, which is also
an equilibrium outcome in the present paper. Interestingly, this contrasts with the symmetric
multi-modal density, for which the number of equilibrium entrants doubles from four to eight
as the cost of entry becomes slightly less than 1/8, which is illustrated in the following figure.
As we will show, such waves of entry are a generic feature of symmetric multi-modal densities.
Uniform
Symmetric multi-modal
Having said what our model does, let us now say what it is. It has two stages. In stage
one, a large number of parties decide simultaneously whether they want to participate in the
upcoming election. If a party decides to enter, it also determines the political platform with
which it runs. Once chosen, the platform cannot be altered. In stage two, after observing all
the entry and location choices of the parties in stage one, another party decides whether to
enter, and if so, which location to occupy.4 For all parties, running is associated with the same
fixed entry cost, which gives the minimum vote share a party has to receive in order to break
even. The election takes place after stage two with a continuum of voters whose preferences
are distributed according to a commonly known distribution. Each voter casts a vote for the
4
Alternative ways of modeling the second stage include infinitely many sub-periods, with a large number of
potential entrants in each sub-period.
1 INTRODUCTION
5
nearest party. Representation is proportional.
Not only can this game generate the configuration envisioned by Downs for the uniform and
the symmetric multi-modal density as an equilibrium outcome, by assuming that locations are
fixed it also provides a game-theoretic formalization of Downs’s idea that parties cannot leapfrog
each other (which is hard to incorporate into a simultaneous moves game), and it resonates
well with his notion of ideological inertia. The two-stage structure is a way of capturing some
of the dynamics inherent to political competition. From a methodological point, its merits is
that it makes sure that in equilibrium all entry occurs in stage one and simultaneously. Stage
two makes sure that all equilibrium locations are characterized by balancing the incentives to
steal votes and the desire or need to deter subsequent entry.
While the model shares many prominently Downsian features, it is distinctly non-Downsian
in other important dimensions. Apart from the obvious difference in setups, perhaps the most
substantive difference is that the model is tractable and predictive under general conditions.
Central predictions of the theory are that waves of entry and exit – large changes in the number
of active parties after small parameter changes – are part of the equilibrium comparative statics.
A second key insight from our model is that modal locations are typically not occupied in
equilibrium. Modal locations may appear intuitively attractive, and certainly fare prominently
in Downs’ book. However, the equilibrium analysis here reveals that modal locations are
unlikely to be occupied because, if occupied, they are vulnerable to entry insofar as an entrant
locating adjacently to its left or right will get at least half of the vote share of the party at the
mode. Consequently, a modal location can only be defended against additional entry with the
help of the locations of the lefthand and righthand neighbors. Of course, modal locations are
occupied in equilibrium in the introductory example when then the cost of entry is 1/8, but
this only re-emphasizes the point: They are occupied because coincidentally the mass between
any two modes is twice the cost of entry for this parameterization. As soon as the cost of entry
is less, modal locations are no longer occupied in equilibrium.
Methodologically, the paper shows that under a regularity condition that will be explained
next any game with multiple modes can be dissected into disjoint unimodal part-games, with
the number of part-games being given by the number of modes of the entire game. As an
illustration, re-consider the introductory example, When the fixed cost is 1/8, the mass between
any two minima is just equal to two times the cost of entry. As the cost of entry decreases, the
game can be dissected at the minima into four identical part-games, each of which can be solved
2 MODEL
6
as a unimodal game, with the equilibrium locations being such that they leave each a mass equal
to the entry cost between themselves and the nearest minimum. The afore mentioned regularity
condition ensures that even non-symmetric multi-modal densities can be partitioned into partgames. For the symmetric multi-modal density in the example, the regularity condition is met
when the entry cost 1/8 or smaller, but not otherwise.
The framework for analyzing spatial competition assuming a uniform distribution was initiated by the seminal work of Hotelling (1929), with subsequent contributions by Smithies
(1941), Prescott and Visscher (1977), d’Aspremont, Gabszewicz, and Thisse (1979), Osborne
and Pitchik (1986), and Vogel (2008). The application of the spatial framework to political
economics, suggested by Hotelling, was explored by Downs (1957); see Enelow and Hinich
(1984) and Osborne (1995) for surveys of this literature. Notable exceptions with non-uniform
distribution include Palfrey (1984) and Weber (1992) in political economics and Anderson,
Goeree, and Ramer (1997) with two given firms who first choose locations simultaneously and
then prices. More recently, Callander (2005) and Loertscher and Muehlheusser (2008, 2011)
have studied models of political competition with non-uniform distributions. Seminal empirical applications include those of Poole and Rosenthal (1991, 1997) and Gentzkow and Shapiro
(2010).
The remainder of this paper is organized as follows. Section 2 provides the model, defines
various operators, and provides the regularity condition. The main result is stated and proved
in Section 3. Section 4 performs comparative statics. Section 5 concludes.
2
Model
2.1
Setup
Voters’ preferences are distributed along the [0, 1]-interval according to the density f (y) with
f (y) > 0 for all y ∈ [0, 1] with the cumulative distribution being denoted F (y). Each voter casts
a vote for the party that is located closest to her bliss point location. The payoff generated
by winning all votes is normalized to 1, and the payoff of receiving a vote share s ∈ [0, 1] is
simply s. Additional and fairly weak regularity restrictions on f will be imposed at the end of
the following section.
There are two stages before the election takes place. In stage one, there are N regular
parties. All i = 1, .., N simultaneously decide whether they want to enter the electoral contest
7
2 MODEL
and, if they do so, on which platform xi ∈ [0, 1] party i will run its campaign. Once chosen,
a platform xi cannot subsequently be altered. The cost of participating in the race is K > 0
for every party. The payoff of not running is normalized to zero. In stage two, after observing
the entry and location choices in stage one, one other party decides whether it wants to enter
and, if it does, which location to occupy. Like “normal” parties, the stage two party bears a
campaign cost of K and has a payoff of zero if it does not run. Assume that N is greater than
the largest integer not bigger than 1/K.
2.2
Operators
For any x ∈ [0, 1], define L(x) and R(x) as follows
L(x) := F −1 (K + F (x))
and
R(x) := F −1 (F (x) − K).
These are well-defined if we adhere to the convention that F −1 (y) = 1 for any y ≥ 1 and
F −1 (y) = 0 for any y < 0. Observe that L(x) is such that a party located at L(x) reaps a
payoff of K from the voters to its left, provided that its marginal voter to the left is at x < L(x).
Analogously, a party located at R(x) receives K votes from voters to its right if its marginal
voter on the right is at x > R(x).
Next, for any x ∈ [L(0), 1] define λ(x) as the largest location to the right of x such that
if both x and λ(x) are occupied a single additional entrant who chooses the optimal location
inside (x, λ(x)) would get a vote share of K. Analogously, for any x ∈ [0, R(1)], let ρ(x) be the
location to its left such that if ρ(x) and x with ρ(x) < x are occupied, an entrant who locates
optimally within (ρ(x), x) nets a vote share of exactly K. Therefore, if x and λ(x) respectively
ρ(y) and y are occupied, then in equilibrium no additional entry will occur between x and
λ(x) and ρ(y) and y, respectively.5 Observe that both λ(x) and ρ(x) and L(x) and R(x)
are increasing in x and that ρ(λ(x)) = x; that is, λ(x) = ρ−1 (x). Moreover, L(x) and λ(x)
are increasing in K while R(x) and ρ(x) are decreasing in K. For example, for the uniform
distribution we have L(x) = x + K and λ(x) = x + 2K and R(x) = x − K and ρ(x) = x − 2K.
5
An optimal location inside an interval (x, y) with x and y occupied need not be unique. This is, for example,
the case for the uniform distribution for which the whole interval (x, y) are optimal locations. Of course, strictly
speaking, an optimal location need not exist, which is for example the case for monotonically increasing or
decreasing densities. For monotonically increasing densities, the optimal location inside (x, y) will be the largest
number less than y while for decreasing densities it would be the smallest number greater than x, which in real
numbers does not exist. But interpreting the model as the limit of one in which locations occur on a finite grid
as the grid size goes to zero fixes the problem. Occasionally, we will make use of the notation x+ to denote the
smallest number to the right of x and y − to denote the largest number to the left of y. The operators λ(x) and
ρ(x) were first used by Loertscher and Muehlheusser (2011) while L(x) and R(x) are generalizations of concepts
introduced there.
8
2 MODEL
Of course, because λ(x) and L(x) are to the right of x while ρ(x) and R(x) are to the left
of x, one might as well have chosen exactly opposite labels. However, the argument cuts both
ways: Because λ(x) and L(x) (and ρ(x) and R(x)) both refer to something to their left (right),
and so the present labeling seems as good as its mnemonic alternative.
It is also useful to have some concise notation for the j-th iterated application of the
operators λ(x) and ρ(x) on themselves. To that end, let λ1 (x) := λ(x), and for j ≥ 2, set
λj (x) := λ(λj−1 (x)). Similarly, define ρ1 (x) := ρ(x) and ρj (x) := ρ(ρj−1 (x)) for j ≥ 2.
2.3
Regularity
A point M ∈ (0, 1) is said to be a minimum of the density f if it is such that f (x) is decreasing
for all x smaller than M in some neighborhood of M and increasing in x for all x greater than
M in some neighborhood of M . If the density exhibits multiple minima, these are denoted Mi
and indexed in increasing order with i = 1, ..., m.
The following lemma is the key for rendering the characterization of equilibrium locations
tractable by allowing us to define a suitable regularity condition. Lemma 1 establishes the
existence of a unique pair of locations around any minimum point that satisfy the following
generalized symmetry notion: The mass between the two locations is 2K and the mass between
each location and the midpoint between them is K. The lemma assumes that f (y) is decreasing
for all y ∈ [ρ(M ), M ) and increasing for all y ∈ (M, λ(M )]. We will comment on this assumption
and how it can be relaxed below.
Lemma 1 For every minimum point M ∈ (0, 1), there exists a unique pair of locations x∗0 (M )
and x∗1 (M ) with x∗0 (M ) < M < x∗1 (M ) such that
F ((x∗0 (M ) + x∗1 (M ))/2) − F (x∗0 (M )) = K
(1)
F (x∗1 (M )) − F ((x∗0 (M ) + x∗1 (M ))/2) = K.
(2)
Moreover, f (x∗i (M )) > f ((x∗i (M ) + x∗j (M ))/2) for i, j = 0, 1 and j 6= i.
Notice that these two locations are such that if each is occupied and no player locates in
between, the player at x∗0 (M ) will earn K from its right and the player at x∗1 (M ) earns K
from its left. Observe also that if f (x) is locally symmetric around M in the sense that
f (M + x) = f (M − x) for all x ∈ [0, L(M ) − M ], then x∗1 (M ) = L(M ) and x∗0 (M ) = R(M )
and (x∗0 (M ) + x∗1 (M ))/2 = M . That x∗1 (M ) = L(M ) and x∗0 (M ) = R(M ) satisfy (1) and
9
2 MODEL
(2) is immediate if f (x) is locally symmetric. The observation then follows from the fact that
x∗0 (M ) and x∗1 (M ) is the unique pair of locations that satisfy (1) and (2). Notice further
that x∗1 (M ) = λ(x∗0 (M )) and, equivalently, x∗0 (M ) = ρ(x∗1 (M )). Because x∗0 (M ) < M and
x∗1 (M ) > M , this implies
λ(M ) > x∗1 (M )
and
ρ(M ) < x∗0 (M ).
(3)
Moreover, denoting by x∗ (i) := (x∗0 (Mi ) + x∗1 (Mi ))/2 the midpoint around the ith minimum
point Mi , we have
x∗0 (Mi ) = R(x∗ (i))
and x∗1 (Mi ) = L(x∗ (i)).
(4)
Figure 1 illustrates Lemma 1. The functions x1 (x0 ) and x0 (x1 ) are the functions that are
implicitly defined by (1) and (2), respectively, and are defined more formally in the proof.
x0
M
x*1
L(M)
ʄ(M)
x1
x1(x0)
x*0
R(M)
ʌ(M)
x0(x1)
Figure 1: Illustration of Lemma 1
Proof of Lemma 1: Existence is proved first, and throughout this proof dependence of x∗i (M )
on M is suppressed. Rearrange terms and invert (1) to get the function x1 : [ρ(M ), λ(M )] →
[M, ∞)
x1 (x0 ) := 2F −1 (F (x0 ) + K) − x0 .
(5)
Analogously, the function x0 : [ρ(M ), λ(M )] → (−∞, M ]
x0 (x1 ) := 2F −1 (F (x1 ) − K) − x1
(6)
10
2 MODEL
can be derived manipulating (2) in a similar vein. Both functions are continuous and increasing
on their domains. Moreover, they satisfy
x0 (λ(M )) = M
x1 (ρ(M )) = M
and
x0 (M ) < ρ(M )
and x1 (M ) > λ(M ).
The key insights leading to these observations are that for an increasing density λ(M ) is by
construction the optimal location to the right of M that generates a vote share of K from the
left, given that the closest lefthand competitor is at x0 = M . Consequently, x0 (λ(M )) = M
for the player located at λ(M ) to receive a vote share of K. In contrast, because the density
over [ρ(M ), M ] is decreasing, the optimal location inside (ρ(M ), M ) is ρ(M )+ , a player located
at M will net strictly less than K to its left when its closest lefthand neighbor is at ρ(M ).
Consequently, x0 (M ) must be less than ρ(M ) for the party at M to receive of vote share of K
from its lefthand side. Symmetric arguments apply for x1 (x0 ).
Because of continuity, there will be a point of intersection, that is, a point (x∗0 , x∗1 ) such
that x0 (x∗1 ) = x∗0 and x1 (x∗0 ) = x∗1 . Moreover, any such fixed point satisfies x∗0 ∈ [ρ(M ), M ] and
x∗1 ∈ [M, λ(M )] despite the fact that the range of x0 (x1 ) and x1 (x0 ) may partly lie outside the
interval [0, 1]. To see this, notice that the domain of x0 is [ρ(M ), M ] while the range of x0 (x1 )
is (∞, M ]. Therefore, the intersection of range and domain, in which any fixed point x∗0 must
lie, is [ρ(M ), M ]. Analogously, the intersection of range and domain for x1 will be [M, λ(M )],
within which any fixed point x∗1 will lie.
For a given fixed point, denote by x∗ := (x∗0 + x∗1 )/2 the midpoint associated with it. It
will next be shown that (x∗0 , x∗1 ) is the unique fixed point. We first show that f (x∗i ) > f (x∗ )
for i = 0, 1 holds for the following reasons. Rather obviously, f (x∗i ) ≤ f (x∗ ) for both i = 0, 1
cannot hold given that x∗0 < M < x∗1 . So, suppose for example that f (x∗0 ) ≤ f (x∗ ). But this
implies that over the interval [x∗0 , x∗ ] the density is smaller everywhere than over the interval
[x∗ , x∗1 ], which is a contradiction to (1) and (2).
To prove uniqueness, it will now be shown that in x0 -x1 -space with x0 on the horizontal axis
and x1 on the vertical axis, the function x1 (x0 ) is strictly steeper at any point of intersection
than (the inverse of) x0 (x1 ) at this point. The derivatives are
dx1 (x0 )
2f (x0 ) − f (F −1 (F (x0 ) + K))
=
dx0
f (F −1 (F (x0 ) + K))
(7)
11
2 MODEL
and
dx0 (x1 )
2f (x1 ) − f (F −1 (F (x1 ) − K))
=
.
dx1
f (F −1 (F (x1 ) − K))
(8)
At a point of intersection (x∗o , x∗1 ), F −1 (F (x∗1 ) − K) = F −1 (F (x∗0 ) + K) = x∗ , so that
2f (x∗0 ) − f (x∗ )
dx1 (x0 )
|x0 =x∗0 =
dx0
f (x∗ )
(9)
dx0 (x1 )
2f (x∗1 ) − f (x∗ )
.
|x1 =x∗1 =
dx1
f (x∗ )
(10)
and
So x1 (x0 ) being steeper than the inverse of x0 (x1 ) at every point of intersection is equivalent
to
f (x∗ )
2f (x∗0 ) − f (x∗ )
>
,
f (x∗ )
2f (x∗1 ) − f (x∗ )
(11)
where the denominator on the right hand side (and the numerator on the left hand side) is
positive because f (x∗i ) > f (x∗ ) for i = 0, 1. Therefore, (11) is equivalent to 2f (x∗0 )[2f (x∗1 ) −
f (x∗ )] + 2f (x∗1 )[2f (x∗0 ) − f (x∗ )] > 0, which is true because f (x∗i ) > f (x∗ ) for i = 0, 1. For every Mi with i = 1, .., m, let
x∗ (i) :=
x∗0 (Mi ) + x∗1 (Mi )
2
be the midpoint. Let x∗ (0) = 0 and x∗ (m + 1) = 1. Before imposing the regularity condition,
the following observation is useful. If the two densities, call them f and g, satisfy f (y) = g(y) for
y ∈ [x∗0 (Mi ), x∗1 (Mi )], where (x∗0 (Mi ), x∗1 (Mi )) is the fixed point (or the generalized symmetry
point) around Mi under f , then it is also a fixed point around Mi under g. The reason is
simply that x∗0 (Mi ) and x∗1 (Mi ) are defined exclusively with respect to the part of the density on
[x∗0 (Mi ), x∗1 (Mi )]. By hypothesis, the two densities are identical there. Although the assumption
that f be monotone on [ρ(Mi ), Mi ) and (M, λ(Mi )] was used to prove that such a fixed point
exists, the preceding argument together with (3) and (4) shows that the same fixed point
continues to exist when f is only monotone over [R(x∗ (i)), Mi ) and (Mi , L(x∗ (i))].
This now permits us to state our regularity condition:
Assumption 1 [Regularity] Every minimum point Mi is such that
R(Mi ) ≥ 0
and
L(Mi ) ≤ 1.
(12)
Moreover, f (y) is decreasing for all y ∈ [R(x∗ (i)), Mi ) and increasing for all y ∈ (Mi , L(x∗ (i))]
for all i = 1, .., m.
12
2 MODEL
Unless stated otherwise, Assumption 1 will hold. In words, it requires K to be small in
relation to f in the sense that troughs and humps of f do not follow each other at too high a
frequency. It also precludes interior minima from being too close to the lower and upper bound
of the unit interval. If f is symmetric around Mi , then x∗ (i) = Mi and the monotonicity
condition becomes that f (y) be decreasing for all y ∈ [R(Mi ), Mi ) and increasing for all y ∈
(Mi , L(Mi )]. The regularity permits a surprisingly simple characterization and determination
of equilibrium locations without being essential for deriving equilibrium locations. Section 3.3
provides an example of how equilibrium can be derived absent regularity. Observe also that
the assumption is only non-vacuous for multi-modal distributions. If a density has no interior
minimum, as if for example the case with unimodal distributions, then the regularity condition
is always satisfied.
The operators L(x) and R(x) are most naturally applied to minimum points Mi if densities
are locally symmetric around minimum (and to x∗ (i) otherwise), and to the boundary points
0 and 1. Under the regularity condition and local symmetry, we have for every minimum point
Mi
ρ(Mi ) < R(Mi ) < Mi < L(Mi ) < λ(Mi ).
Moreover, defining xρ := (ρ(Mi ) + Mi )/2 and xλ := (λ(Mi ) + Mi )/2 for a minimum point Mi ,
we have
R(xρ ) = ρ(Mi )
and L(xλ ) = λ(Mi ).
These properties do, however, not generalize. For example, if f (x) is decreasing over the
interval [y, λ(y)], we have L(y) = (y + λ(y))/2. For monotone densities, they follow because
when f (x) is either decreasing or increasing over the interval [y, λ(y)], the optimal location
inside the interval (y, λ(y)) is y or λ(y), respectively. In contrast, if f (x) is quasiconcave over
the interval [y, λ(y)] with f ′ (y0 ) for some y0 ∈ (y, λ(y)), L(y) can be larger or smaller than
(y + λ(y))/2 while, if L(y) < (y + λ(y))/2 holds, we have L((y + λ(y))/2) > λ(y). The latter
follows because the mass in between y and λ(y) is never more than 2K and strictly less than
2K when f has a unique maximum on [y, λ(y)]. Because by construction, F (L(y)) − F (y) = K,
F (λ(y)) − F (L(y)) < K follows. Symmetric properties hold for ρ and R and the relationship
between them.
3 EQUILIBRIUM
3
13
Equilibrium
3.1
Preliminaries
A property that simplifies the analysis is the following:
Lemma 2 On the equilibrium path all entry occurs in stage one.
Proof of Lemma 2: Suppose not. Then it must be profitable for some player i in stage two
to enter at some location xi . But if this pays off for i in stage two, then it would have paid off
to enter for any of the players who were present in stage one but who did not enter, which is
a contradiction. Lemma 1 has established the existence and uniqueness of a fixed point (x∗0 (Mi ), x∗1 (Mi ))
for every minimum point Mi under the regularity condition. Notice that x∗1 = λ(x∗0 ), which is
equivalent to saying that ρ(x∗1 ) = x∗0 . Consequently, if both x∗0 (Mi ) and x∗1 (Mi ) are occupied in
some equilibrium, then no location inside (x∗0 (Mi ), x∗1 (Mi )) can be occupied in this equilibrium
as the party (or parties) locating in between would not break even. Moreover, because x∗0
and x∗1 satisfy the generalized symmetry conditions (1) and (2), if x∗0 and x∗1 are occupied in
equilibrium, then the party at each of these two locations receives exactly K votes from the
segment in between x∗0 and x∗1 . This strongly suggests that any such two locations will be
occupied in equilibrium. The following proposition formalizes this idea by showing that under
the regularity conditions for all Mi , x∗0 (Mi ) and x∗1 (Mi ) will be occupied in equilibrium.
Proposition 1 Assume the regularity condition is satisfied. Then, in any equilibrium and for
any minimum point Mi the locations x∗0 (Mi ) and x∗1 (Mi ) will be occupied and no location inside
(x∗0 (Mi ), x∗1 (Mi )) will be occupied.
Proof of Proposition 1: It has already been established in the preceding text that if x∗0 (Mi )
and x∗1 (Mi ) are occupied, then no party can profitably enter inside (x∗0 (Mi ), x∗1 (Mi )), which
proves the second part of the statement. Moreover, by construction x∗0 (Mi ) and x∗1 (Mi ) are
such that if both are occupied (and no one is located in between), each party receives a vote
share of K from in between. Consequently, the parties occupying these locations break even,
which is a necessary condition for equilibrium. We are thus left to show that in equilibrium
no locations can be occupied closest to Mi from the left and the right other than x∗0 (Mi ) and
x∗1 (Mi ).
3 EQUILIBRIUM
14
The following cases need to be considered:
Case 1 : The locations yi and yi+1 closest to Mi – that is, satisfying yi ≤ Mi ≤ yi+1 with no
location occupied in between – satisfy x∗0 (Mi ) < yi and yi+1 < x∗1 (Mi ). Notice first that for
this to occur, it has to be the case that the next smallest and next largest locations occupied in
equilibrium, denoted yi−1 and yi+2 , satisfy yi−1 < x∗0 (Mi ) for otherwise the party at yi would
not break even and yi+2 > x∗1 (Mi ) because otherwise the party at yi+1 would not attract enough
votes (recall that x∗0 (Mi ) = ρ(x∗1 (Mi ))). Without inducing addition entry, either the party at
yi could change its location to x∗0 (Mi ) or the party at yi+1 could increase its location to x∗1 (Mi )
and thereby increase its vote share because max{f (x∗0 (Mi )), f (x∗1 (Mi )} > f ((yi + yi+1 )/2).
Case 2 : Assume then that in equilibrium exactly one party locates at y ∈ (x∗0 (Mi ), x∗1 (Mi ))
with x∗0 (Mi ) or x∗1 (Mi ) or both not occupied in equilibrium. If neither is occupied, the party at
y would be have been better off by choosing the location x ∈ {x∗0 (Mi ), x∗1 (Mi )} that generates
a larger vote share from the left of x∗0 (Mi ) or the right of x∗1 (Mi ), respectively. (The party’s
optimization problem is convex over this interval.) If an additional entrant locates next to it
in stage 2, its unique best response will be to take the other location in {x∗0 (Mi ), x∗1 (Mi )}. If
only one of the two locations x∗0 (Mi ) or x∗1 (Mi ) is not occupied, the firm at y would be strictly
better off moving to this location, which would not induce any addition entry.
Case 3 : No location inside [x∗0 (Mi ), x∗1 (Mi )] is occupied in equilibrium. But then one of the
parties that did not enter in stage 1 could have entered and chosen either one of them, thereby
generating a net profit because even it it induces additional entry in stage 2, this entry will
occur at the location x∗1 (Mi ) if it chooses x∗0 (Mi ) and at the location x∗0 (Mi ) if it chooses x∗1 (Mi ).
If such additional entry occurs, the party who enters in stage 1 will receive K votes from the
voters in between plus additional votes from the other side. If no stage 2 entry occurs, then it
will receive even more votes. In either case, it will pay off to enter. .
A variety of auxiliary results that carry over from sequential locations games are summarized
in the following lemma:
Lemma 3 Let xi and xi+1 be two locations that are occupied in some equilibrium satisfying
xi < xi+1 .
(i) ρ(xi+1 ) ≤ xi , which is equivalent to λ(xi ) ≥ xi+1 , implies that no player locates between xi
and xi+1 in equilibrium.
(ii) ρ(xi+1 ) > xi , which is equivalent to λ(xi ) < xi+1 , implies that at least one more player
3 EQUILIBRIUM
15
locates between xi and xi+1 in equilibrium.
(iii) (a) Assume that f is increasing over [xi , xi+1 ] and that xi < ρ(xi+1 ) < λ(xi ) < xi+1 .
Then the optimal entry deterring location between xi and xi+1 is λ(xi ). (b) Assume that f
is decreasing over [xi , xi+1 ] and that xi < ρ(xi+1 ) < λ(xi ) < xi+1 . Then the optimal entry
deterring location between xi and xi+1 is ρ(xi ). Moreover, in either case deterring entry is
optimal for the player locating between xi and xi+1 .
(iv) Assume xi and xi+1 are such that ρ(xi+1 ) < M < λ(xi ), which is equivalent to λ(xi ) > M ,
for some minimum point M . Then the optimal entry deterring location between xi and xi+1 is
either ρ(xi+1 ) or λ(xi ).
Sketch of Proof of Lemma 3: (i) is true by the definition of λ and ρ. (ii) is proved in
Lemma 4(ii)) in Loertscher and Muehlheusser (2011) while (iii) is an implication of Lemma
3(i) there. (iv) is true because the (unconstrained) optimal location between xi and xi+1
satisfying xi < M < xi+1 is either adjacently to the right of xi or adjacently to the left of
xi+1 . This property carries over to the case when the locations are restricted to lie within
[ρ(xi+1 ), λ(xi )]. Importantly, whether or not entry deterrence is optimal in this case cannot be
said without imposing further restrictions. 3.2
Equilibrium Configuration
It is instructive to first determine the equilibrium configuration under the assumption of a
unimodal distribution whose density is maximized on the interior of [0, 1]. Let λi := λi (L(0))
and ρi := ρi (R(1)), and let l be the (generically unique) integer such that f (x) is increasing at
λl and decreasing at λl+1 . Analogously, define r as the (generically unique) integer such that
f (x) is increasing at ρri +1 and decreasing at ρri .6 Depending on parameter values, either (i)
λl+1 < ρr or (ii) λl+1 ≥ ρr can be the case. In case (i), denote by
l
λ +x
x + ρr
−F
µ := arg
max
F
2
2
x∈[ρr+1 ,λl+1 ]
(13)
the optimal location within the (in this case non-empty) interval [ρr+1 , λl+1 ], under the condition that the lefthand and righthand neighbor of the player locating at x is at λl and ρr+1 ,
respectively. For case (ii), define µ := ∅, indicating that the location µ does not exist in this
case.
The non-generic cases in which l and r are not unique only occur if λl (L(0)) or ρr (R(1)) coincides with a
local maximum.
6
16
3 EQUILIBRIUM
Proposition 2 Assume the density is unimodal with an interior maximum on [0, 1]. Then,
under the regularity condition the set of equilibrium locations is
X ∗ := {L(0), λ1 , .., λl , µ, ρr , ., , .ρ1 , R(1)}.
Proof of Proposition 2: The leftmost and rightmost equilibrium locations are L(0) and
R(1) for any K < 1/2. To see this, consider the leftmost equilibrium location and suppose
it were smaller than L(0). The closest neighbor to the right of the leftmost party must be
at some x > L(0) for otherwise the leftmost party would not break even. Consequently, the
leftmost party could increases its vote share without attracting any entry to its left my moving
to the L(0). This proves that the leftmost location cannot be smaller than L(0). To see that
it cannot be larger, it suffices to observe that if it were larger, another party could profitably
enter. Symmetric arguments apply to R(1) and the rightmost location.
With the only possible exception of µ, the interior equilibrium locations on the increasing
part of the density are given by {λ1 , , .., λl }. To see this, consider two locations xi and xi+1
on the increasing part of the density with xi+1 ≤ λl . Assume that both locations are occupied
in equilibrium and that no location inside (xi , xi+1 ) is occupied in this equilibrium, but that
xi+1 6= λ(xi ). If xi+1 > λ(xi ), an additional party could profitably enter at λ(xi ), contradicting
that no other location inside (xi , xi+1 ) is occupied in this equilibrium. Assume then that
xi+1 < λ(xi ). For the party at xi+1 to break even, it has to be the case that its righthand
neighbor is at a location xi+2 > λ(xi ). But because the density is increasing, the party at xi+1
would increase its vote share by choosing λ(xi ) without attracting additional entry (additional
entry is forestalled to its left by the definition of λ(xi ); entry to its right cannot occur either
because xi+2 was presumed to be the equilibrium location closest to xi+1 when xi+1 < λ(xi ); as
xi+1 increases, the incentives to enter the interval (xi+1 , xi+2 ) decrease). Labeling equilibrium
locations in increasing order, on the increasing part of the density equilibrium locations must
thus satisfy xi+1 = λ(xi ). Together with x1 = L(0) this proves the claim. Analogous arguments
apply to the decreasing part of the density and {ρr , , .., ρ1 }.
The only remaining question is whether all equilibrium locations are given by L(0), λ1 ,
.., λl , ρr , , .., ρ1 , R(1), which boils down to the question whether, if occupied, λl and ρr deter
entry in between. If so, the answer is yes. Otherwise, an additional party and locates optimally
at µ. 17
3 EQUILIBRIUM
Figures 2 illustrate Proposition 2 for a symmetric hum-shaped Beta-distribution. In both
panels, l = 2 = r. As the cost decreases from K = 0.09, which is the case in the lefthand panel,
to K = 0.8, an additional entrant comes in and occupies the location µ, which is 1/2 because
of symmetry.
L(0)
ʄ1
ʄ2
ʌ2
ʌ1
(a) K = 0.09
R(1)
L(0)
ʄ1
ʄ2
ʅ
ʌ2
ʌ1
R(1)
(b) K = 0.08
Figure 2: Illustration of Proposition 2
Part-Games There are m interior points x∗ (i) with i = 1, .., m. These points allow us to
partition the unit interval into m + 1 non-overlapping subintervals Pi := [x∗ (i), x∗ (i + 1)])
satisfying ∪m
i=0 Pi = [0, 1]. The subinterval Pi is also referred to as the i-th part-game. Notice
that under the regularity in any such part-game Pi with i = 1, .., m, the density f (y) is first
increasing for y close to but greater than max{Mi , x∗ (i)} and then decreasing for y close to but
smaller than min{Mi+1 , x∗ (i + 1)}.
Compared to sequential location games, the simultaneous moves nature in the present setup
does not only simplify the analysis considerably when it comes to determining equilibrium
locations with hump-shaped densities. The simultaneous moves also allow us to determine the
fixed points x∗0 (Mi ) and x∗1 (Mi ) around any minimum point Mi in a direct and simple way. If all
locations were chosen sequentially, one would have to determine similar fixed points accounting
for the order in which players enter, with limited hope that such fixed points would then permit
a similarly simple dissection of the full game into essentially independent part-games as the
present setup does.
For any i = 1, ..m and j = 1, .., define λji := λj (x∗1 (Mi )). Similarly, for any i = 0, .., m − 1
18
3 EQUILIBRIUM
and any j = 1, .., let ρji := ρj (x∗0 (Mi+1 )) and j = 1, ... Let li be the (generically unique)
integer such that f (x) is increasing at λlii and decreasing at λili +1 . Analogously, define ri as
the (generically unique) integer such that f (x) is increasing at ρiri +1 and f (x) is decreasing at
ρri i .7
l
Depending on parameter values, either (i) λii+1 < ρri i or (ii) λili +1 ≥ ρri i can be the case. In
case (i), denote by
µi := arg
max
r +1
x∈[ρi i
l +1
,λii
F
]
x + ρiri
2
−F
λlii + x
2
!
(14)
the optimal location within the (in this case non-empty) interval [ρiri +1 , λlii +1 ], under the condition that the lefthand and righthand neighbor of the player locating at x is at λlii and ρiri +1 ,
respectively. For case (ii), define µi := ∅, which is meant to indicate that the location µi does
not exist in this case. It will be useful to be able to say, for example, that in equilibrium
µi will be occupied without explicitly distinguishing between case (i) and case (ii), with the
r
l
understanding that if (ii) is the case, no player locates inside the interval [ρi i+1 , λii+1 ]. Lastly,
let r0 be the (generically) unique integer such that ρr00 ≤ L(0) < ρr00 −1 . Analogously, let lm be
the generically unique integer such that λlmm < R(1) < λlmm +1 .
For i = 1, .., m − 1, let
Xi := {x∗1 (Mi ), λ1i , .., λlii , µi , ρri i , .., ρ1i , x∗0 (Mi+1 )}.
(15)
Also, set X0 := ∅ if f (x) is increasing over [0,1] and Xm := ∅ if f (x) is decreasing over [0, 1].
(Notice that in either case, no interior minimum point exists.) If f (x) is increasing for x close
to 1, let Xm := {x∗1 (Mm ), λ1m , .., λlmm , R(1)}, and similarly, for f (x) decreasing for x close to 0,
let X0 := {L(0), ρr00 , .., ρ10 , x∗0 (M1 )}. Lastly, let
X := ∪m
i=0 Xi .
A main result of this paper can now be stated.
Theorem 1 The set of equilibrium locations is X if the regularity condition holds.
Proof of Theorem 1: From Proposition 1, we know that x∗0 (Mi ) and x∗1 (Mi ) will be occupied
in any equilibrium. This implies that the i-th part-game is isomorphic to the game with a
7
The non-generic cases in which li and ri are not unique only occur if λj (x∗1 (Mi )) or ρj (x∗0 (Mi+1 )) coincides
with a local maximum.
19
3 EQUILIBRIUM
hump-shaped density that was analyzed in Proposition 2 with L(0) being replaced by x∗1 (Mi )
and R(1) being replaced by x∗0 (Mi+1 ) and the cost of entry being adjusted to
Ki :=
K
F (x∗ (i + 1)) − F (x∗ (i))
and the density being given by
fi (y) :=
F (x∗ (i +
f (x)
1)) − F (x∗ (i))
for x ∈ [x∗ (i), x∗ (i + 1)] and y := (x − x∗ (i))/(x∗ (i + 1) − x∗ (i)). This means that in part-game
i the set of equilibrium locations is Xi . Aggregating over all part-games, we get X. 3.3
Equilibrium without the Regularity Condition
Absent the regularity condition, determining equilibrium outcomes is still possible, of course,
but at this stage proves cumbersome and can only be done on a case-by-case basis. As an
illustration and to complete the treatment of the example in the introduction, let us re-examine
the model with a multi-modal symmetric density. For the sake of concreteness, consider the
“sine-density” fS (x) = 1 + sin(8πx − π/2)/5, which is illustrated in Figure 3. For K = 1/8, the
set of equilibrium locations is {1/8, 3/8, 5/8, 7/8} and each player who is active in equilibrium
receives a vote share of 1/4 in equilibrium, so that, numbering active players in the order of
equilibrium locations, their payoffs are Π∗(i) = 1/4−K for i = 1, .., 4. As K increases marginally,
four players will still enter in equilibrium, but the equilibrium locations will change basically
because the extremist parties are driven towards more moderate platforms because entry in
their hinterlands has become less attractive. The left- and rightmost locations xL and xR will,
as always, be xL = F −1 (K) and xR = F −1 (1 − K). For the sine density, F −1 (K) ≈ K and
F −1 (1 − K) ≈ 1 − K. As long as K < 1/4, which implies xL < 1/4 and xR > 3/4, there
is always an equilibrium in which two additional players enter and locate at x2 = 1/2 − xL
and x3 = 3/2 − xR . The reason is simple: Equilibrium vote shares will be 1/4 for each party.
Hence, they will all break even. Moreover, x2 and x3 are best responses to each other and
to the location of the neighboring extremist party. (In fact, any x2 ∈ (xL , x3 ) will be a best
response absent additional entry given the perfect symmetry of the density and symmetry of
the locations xL , x3 and xR ; however the consistency requirement of mutual best responses
absent additional entry is only satisfied at x2 = 1/2 − xL and x3 = 3/2 − xR ; because a party
−
who optimally locates in the the interval (x2 , x3 ) will choose a location x∗ ∈ {x+
2 , x3 }, such an
20
3 EQUILIBRIUM
entrant will receive the same vote share as an entrant who optimally locates to, say, the left
of xL , which is less than K - consequently, the locations x2 and x3 are not only mutually best
responses absent additional entry given the extremist neighbors’ locations, but they also deter
additional entry.)
Figure 3: Sine-Density
Observe that as K increases on (1/8, 1/4), xL and x3 continuously increase and x2 and xR
continuously decrease. As K approaches 1/4, xL and x2 approach 1/4 from below and from
above, respectively while x3 and xR approach 3/4 in the same manner. When K = 1/4, two
players will exit and the remaining equilibrium locations will be {xL , xR } for any K ∈ [1/4, 1/2).
Thus, this sequence of equilibria is characterized by a wave of exit – of two players becoming
inactive at once – when K converges to 1/4.
However, for K ∈ (0.178, 1/4) there is another sequence of equilibria, in which only three
players enter. The equilibrium locations in such a three-party equilibrium are {F −1 (K), 1/2, F −1 (1−
K)}. The reason for this is that absent further entry, 1/2 is the uniquely optimal location in between xL and xR satisfying xL = 1−xR and xL < 1/4. Moreover, this location is entry-deterring
(on either side, by symmetry) if and only if K > 0.178. It is maximally entry-deterring in the
sense that no other single location inside (xL , xR ) can deter additional entry if the location 1/2
cannot.
21
4 COMPARATIVE STATICS
4
Comparative Statics
We now analyze how the equilibrium outcomes vary with distributions. We first provide a result
on the number additional entrants that can occur when the entry cost decreases marginally.
Then we explain in what sense modal locations are vulnerable and therefore “unlikely” to be
occupied in equilibrium.Lastly, we show that the equilibrium outcome PV focus on for the
uniform is identical to the one obtained in the limit as a symmetric unimodal density converges
to the uniform.
4.1
Waves of Entry and Exit
As noted in the introductory example, the model with multi-modal densities can give rise to
waves of entry insofar as a small parameter change can induce a large number of parties to
enter in equilibrium. To investigate this phenomenon more systematically, we now consider a
symmetric multi-modal density with m + 1 modes (and m minima), where symmetry means
1
1
1
1
, f (x) = f x + m+1
, f (x) = f x − m+1
and for any x > m+1
.
that for any x < 1 − m+1
Assume that K is such that exactly m + 1 parties enter in equilibrium, which because of
symmetry implies K =
1
2(m+1) ,
and then consider a small decrease of the entry cost. The
following corollary to Theorem 1 says that the number of parties who enter in equilibrium
doubles.
Corollary 1 With a symmetric multi-modal density with m + 1 modes and an entry cost of
K =
1
2(m+1) ,
1
2(m+1)
the number of entrants in equilibrium is m + 1. As the entry cost decreases to
− ε with ε > 0 small, the number of entrants in equilibrium doubles to 2(m + 1).
A natural conjecture that remains to be validated (or rebutted) is that doubling the number
of parties that enter in equilibrium is an upper bound on waves of entry when the entry cost
decreases marginally, provided K < 1, so that at least one party enter before the cost decrease.
Proof of Corollary 1: At K =
1
(2(m+1) ,
the regularity condition condition is just satisfied.
Consequently, it will be satisfied as K decreases, so that Theorem 1 applies and the game
can be dissected into part-game. Each part-game will be symmetric and have two entrants in
equilibrium (who will locate at x∗1 (Mi ) and x∗0 (Mi+1 ) in part-game i). 22
4 COMPARATIVE STATICS
4.2
Modal Locations and Entry Deterrence
Although there may be something intuitively appealing about modal locations, the equilibrium
analysis of the present model reveals that with the need to deter entry, this is not necessarily
so. In fact, modal locations are particularly vulnerable and difficult to defend against entrants,
which makes it, loosely speaking, unlikely that they will be occupied in equilibrium as explained
next. When modal locations are occupied in equilibrium, as is for example the case for the
symmetric multi-modal densities used in Corollary 1 when the entry cost
1
2(m+1) ,
this is an
artefact of the specific parameterization. For K = 1/8, L(0), x∗0 (Mi ) and x∗1 (Mi ) and R(1)
coincide with the modes. For any slightly smaller cost, this coincidence disappears, and the
equilibrium locations are different from the modal locations.
To understand the sense in which modal locations, or locations close to the modes, are
vulnerable and difficult to defend, consider first the locations L(0) and R(1) that are occupied
in equilibrium for any K, provided only K < 1/2 of course. These locations are maximally
entry-deterring insofar as, if occupied, they deter any additional entry to the left and right,
respectively, independently of the location and entry choices of any other player. At the same
time, they allow a party who occupies one of these two locations to break even regardless of
the (rational) choices of any other player. (Of course, this is the very reason they are occupied
in any equilibrium.) Next consider equilibrium locations x∗0 (Mi ), x∗1 (Mi ), λhi and ρji . For a
player at one of these locations to deter entry and to break even, a little more is required than
for the locations L(0) and R(1): It requires that exactly one neighbor plays as prescribed by
equilibrium: For x∗1 (Mi ) and λhi , this is the lefthand neighbor while for x∗0 (Mi ) and ρji it is
the righthand neighbor’s behavior that is critical. Where exactly the player on the other side
locates is then immaterial for the question of whether the party at any such location breaks
even. No similar thing is true for modal locations or the equilibrium locations µi that are
closest to a mode because these can only deter entry if both the lefthand and the righthand
neighbor choose locations that are neither too close nor too far away.
4.3
Uniform as the Limit Case
The equilibrium locations obtained as the outcome of the subgame perfect equilibrium PV
focus on in their sequential location game with a uniform distribution coincide with the limit
of the unique equilibrium locations of the simultaneous location game with any symmetric
5 CONCLUSIONS
23
unimodal density as this density converges to a uniform density. Formally, denote by fSk (x)
the k-th element in a sequence of symmetric unimodal density {fSk (x}∞
k=0 , where symmetric
means that for any k = 0, .. and for any x ∈ [0, 1], fSk (x) = fSk (1−x) and unimodal means that
fSk (x) is increasing for x < 1/2 (and decreasing for x > 1/2, which of course is an implication
of symmetry and the preceding assumption). The sequence is such that fSk +1 (x) > fSk (x) if
fSk (x) < 1 and fSk +1 (x) < fSk (x) if fSk (x) > 1 and limk→∞ fSk (x) = 1 for all x.
Because the regularity condition is vacuously satisfied for this family of densities for any k
and K, the set of equilibrium locations is unique and given by those described in Proposition
2. Because µ = 1/2 by symmetry, one gets indeed the equilibrium PV selected whenever the
number of entrants in equilibrium is odd.
5
Conclusions
This paper develops a model that permits party formation decisions and their political platforms
to be determined endogenously. Formally, the model is a game in which location and entry
decisions are made simultaneously, with the threat of entry being maintained by a player who
could possibly enter at a later stage. Equilibrium locations balance the incentives to steal votes
against the need to deter entry. Core predictions of the model are that modal locations are
vulnerable to entry and thus unlikely to be occupied in equilibrium, with unlikely meaning
that they can only be occupied for knife-edge parameterizations. At the same time, it is of
course true that areas around a mode will attract more parties and induce them to position
closer to each other than areas around local minima. An additional feature uncovered by our
equilibrium analysis is that small changes in parameter values can induce large changes in the
number of active parties. For example, for symmetric multi-modal densities, the number of
active parties can double as the entry cost decreases marginally. Various extensions of the
model seem valuable, including allowing for multiple, possibly ideologically constrained players
to enter in stage two who may also differ with respect to their entry costs.
24
A ALTERNATIVE SETUP WITH AN INFINITE TIME HORIZON
Appendix
A
Alternative Setup with an Infinite Time Horizon
There is an alternative, largely equivalent setup with an infinite horizon in which time is
discrete. In every period t, t = 1, .., n players or parties simultaneously decide whether or not
to enter, where n is large in a sense that will be made precise shortly. All players are risk
neutral and maximize expected discounted payoffs, the common discount factor being δ. Of
particular interest is the case when δ is close to 1.
Entry involves a fixed cost K > 0 while the value of staying out is 0 for all players. If
a player who is given the opportunity to enter in period t decides to enter, it also chooses a
location x ∈ [0, 1], which remains fixed for all periods t, t + 1, .. .8 It is assumed that that
K<
1
.
2(1 − δ)
(16)
It can be shown that condition (16) ensures that at least two players enter in equilibrium.
Voters’ preferences are distributed along the [0, 1]-interval according to the time-invariant
density f (y) with f (y) > 0 for all y ∈ [0, 1] and with cdf F (y). In every period, each voter
casts a vote for the party that is located closest to her bliss point location. The per period
payoff generated by winning all votes is normalized to 1, and the per period payoff of receiving
a vote share s ∈ [0, 1] is simply s.9
8
In other words, players who have entered cannot exit. But given that there are no costs of operation, this
is without loss of generality because any player that has entered and paid the entry cost is weakly better off
staying in the game than leaving it.
9
Notice that the discounted total value is 1/(1 − δ). Therefore, the political economy cannot support more
active parties than the largest integer smaller than 1/((1 − δ)K). The number of possible entrants per period,
n, is sufficiently large if it is no less than this integer.
25
B ILLUSTRATIONS FOR THE UNIFORM
B
Illustrations for the Uniform
3/9
1/9
6/9
1/2
8/9
Figure 4: Uniform, entry cost of 1/9.
1/9
1/9
1/9
1/12
1/12
1/9
1/9
1/9
1/12
1/12
1/9
3/9
1/2
6/9
8/9
Figure 5: Uniform, entry cost of 1/9 and vote shares
REFERENCES
26
References
Anderson, S., J. Goeree, and R. Ramer (1997): “Location, Location, Location,” Journal
of Economic Theory, 77, 102–127.
Callander, S. (2005): “Electoral Competition in Heterogenous Districts,” Journal of Political
Economy, 113(5), 1116–1145.
d’Aspremont, C., J. J. Gabszewicz, and J.-F. Thisse (1979): “On Hotelling’s ’Stability
in Competition’,” Econometrica, 47(5), 1145–50.
Downs, A. (1957): An Economic Theory of Democracy. Harper Collins.
Enelow, J., and M. J. Hinich (1984): The Spatial Theory of Voting: An Introduction. New
York: Cambridge University Press.
Gentzkow, M., and J. Shapiro (2010): “What Drives Media Slant? Evidence from U.S.
Daily Newspapers,” Econometrica, 78(1), 35–71.
Hotelling, H. (1929): “Stability in Competition,” Economic Journal, 39, 41–57.
Loertscher, S., and G. Muehlheusser (2008): “Global and local players in a model of
spatial competition,” Economics Letters, 98(1), 100–106.
(2011): “Sequential Location Games,” RAND Journal of Economics, 42(3), 639–663.
Osborne, M. J. (1995): “Spatial Models of Political Competition under Plurality Rule,”
Canadian Journal of Economics, 28(2), 261–301.
Osborne, M. J., and C. Pitchik (1986): “Price Competition in Duopoly,” Journal of Economic Theory, 38(2), 238–61.
Palfrey, T. (1984): “Spatial Equilibrium with Entry,” Review of Economic Studies, 51(1),
139–156.
Poole, K., and H. Rosenthal (1997): Congress: a political-economic history of roll call
voting. Oxford University Press.
Poole, K. T., and H. Rosenthal (1991): “Patterns of Congressional Voting,” American
Journal of Political Science, 35(1), 228 – 278.
REFERENCES
27
Prescott, E. C., and M. Visscher (1977): “Sequential location among firms with foresight,”
Bell Journal of Economics, 8(2), 378–393.
Smithies, A. (1941): “Optimal Location in Spatial Competition,” Journal of Political Economy, 49(3), 423–439.
Vogel, J. (2008): “Spatial Competition with Heterogenous Firms,” Journal of Political Economy, 116(3), 423–466.
Weber, S. (1992): “On Hierarchical Spatial Competition,” Review of Economic Studies, 59,
407–425.