1

Existence of General Political Equilibrium:The Mean Voter Theorem is an Artifact*

ByNorman Schofield

Center in Political EconomyWashington University

July 18, 2002

Contact Information:(from August 26, 2002)Fulbright Professor of American StudiesInstitut fur Sozial WissenschaftenHumboldt Universitat zu BerlinPhilosophisch Fakultat IIIUnter den Linden 6D-10099 Berlin, GermanyTel (49) 30-20-93-14-24Fax (49) 30-20-93-13-24e-mail: schofld@wueconc.wustl.edu

* This paper is based on research supported by NSF grant SBR-98-18582. Theresults presented were developed in an attempt to understand certain empiricaldata arising out of collaborative research with Gary Miller, Andrew Martin, KevinQuinn, and Itai Sened. I thank reviewers of an earlier version of the paper and anassociate editor of the Journal for helpful comments.

2

Abstract

This paper considers the first and second order conditions sufficient for

existence of a pure strategy Nash equilibrium in a vote maximizing political game, where

parties also respond to valence (or non-policy popularity). Formal stochastic voter

models of this kind usually conclude that all political agents (parties or candidates) will

converge towards the electoral mean (the origin). Here, it is shown that this result is

sensitive to two assumptions: i) that all valence coefficients are zero, and ii) that a

"domain" constraint, on the support of voter bliss points is satisfied. It is shown that

"Local Nash Equilibria" (satisfying the second order Hessian conditions) will exist for

almost all parameters, and utility functions, when the policy space is contractible (or

compact, convex), without any restriction on domain or valence. However, these Local

Nash Equilibria will almost never occur at the origin.

3

1. INTRODUCTION

Many of the decisions of a society depend on choices of electoral representatives,

over social and property rights, taxation, government regulation, etc. These

representatives, or political agents, are comparable to the entrepreneurs of an economic

system. How these political decisions are made, and the relationship between the

decisions and the nature of economic equilibrium is the subject of public economic

theory.

In parallel to the conception of an economic equilibrium as a Nash equilibrium to

a particular economic game, it would be attractive to be able to show that political

decisions are also in equilibrium. However, the most natural political game to construct

is one where representatives adopt positions to maximize votes. Since Downs (1957),

such models have led to the inference that representatives will adopt a position at the

mean of the distribution of voters' most preferred (bliss) points. This has been seen as the

only such equilibrium (Hinich, 1977). However, this conclusion seems to contradict

extensive evidence that parties do not converge in this way. In electoral systems based

on proportional representation (PR), there are typically very many parties, located at very

different positions. It is true that in the US there are two major parties, but even there,

they remain distinct (see Miller and Schofield, 2002 for a discussion of the US, and

Schofield, et al, 2002, for evidence from polities using PR).

This paper presents a general voter model involving "valence". Implicitly,

valence is attributed to political candidates because of their ability, as political

entrepreneurs, to deliver political goods to particular constituents, or activists (See

Ansolabehere and Snyder, 2000, for a one-dimensional voting model involving valence).

4

By contributing money and support to preferred candidates, these activists affect overall

voter support. The model presented is an extension of one due to Lin et al (1999).

However, the model presented here has the feature that there will generally exist multiple

equilibria. At each one of these, candidates will optimize with regard to overall non-

valence electoral support, as well as their valence function. Empirical analyses suggest

that strong valence support comes from "local maxima" of the electoral distribution.

Consequently, one would expect highly dispersed candidate positions, and many possible

local equilibria. The political equilibria so determined can be interpreted as analogues of

the general equilibria of economic theory.

It is useful to consider a very simple example to illustrate the main result of this

paper. The formal model is one involving electoral risk (or uncertainty). This is easiest

to present when the valences are stochastic. That is, the valence of party j can be

represented by an expected valence (or constant, λ j) together with a stochastic

disturbance ∈ j. In the simplest case, {∈ j}, independently, are normally distributed with

zero expectation.

Consider a situation with three voters with preferred position on the vertices of an

equilateral triangle in ℜ 2. Voter utility is defined by a negative quadratic expression.

Consequently, if each voter is located 1 unit from the origin, and each party is located at

the origin, then the latent utility for each party is -1. It is readily shown that this

situation, when each party is at the origin, satisfies the first order condition for an

equilibrium.

There are two separate cases to consider. Firstly, suppose all λ j are zero (or

equal). Then from the point of view of each voter, each party is identical. In this case,

5

the second order (Hessian) condition is satisfied, and the origin is what we shall call a

"local Nash equilibrium." A pure strategy Nash equilibrium (PSNE) can be shown to

exist at the origin if all party vote functions are concave. However, the cumulative

normal distribution is only concave if the variate has value greater than -σ/2.

Consequently, this imposes a bound on the distance of every voter position from the

origin, sufficient for concavity. Since the loss function is quadratic, the constraint on the

position, xi, of voter i has the form

|| xi ||2 < σ/√2. If there are three parties, all at the origin, then the probability that voter i

chooses party j is clearly 1/3. With the constraints satisfied, these probability functions

will be concave on arcs from the origin to each voter position, and thus on the convex

hull of the set of voter positions. Existence of a PSNE is evident. With the bound

satisfied, voter choice is essentially random.

Now suppose the valences are not zero but constants, but these valences differ

between the parties. In this case, when voter i compares parties j and k, the difference

between the valences becomes relevant.

In particular, suppose that the valences are ranked λ1 > λ2 > … > λp for example,

and suppose the bound is violated. In this case, voter 1 will discern a clear difference

between the parties even when they are all at the origin. The probability that voter i picks

party j over party k will depend on λ j - λk. In this case, the bound for concavity of ρij will

be very severe, of the form || xi ||2 < (σ/√2) - λ1 - λp. If this fails, then it immediately

implies that ρij will not satisfy the Hessian condition at the origin. If sufficient voters fail

this condition then the vote function for party j will fail the condition. Consequently, the

origin cannot be a local equilibrium for the party. Indeed, it cannot be either a local Nash

6

equilibrium or a pure strategy Nash equilibrium. Whether or not this phenomenon can

occur depends on geometry and on the variance. It is possible that, in one dimension,

sufficient voters cannot fail concavity in this fashion

(de Palma, et al, 1990). However, in the limit as σ falls to zero, the mean voter position

will not be a PSNE except for voter positions that are symmetric about the mean. A

study of this limiting condition is given in Banks and Duggan (1999). Their results

indicate that PSNE must fail for small variance.

The paper is structured as follows. Section 2 gives a brief overview of techniques

for proof of existence of equilibria, including transversality arguments for CNE. Section

3 gives the main Theorem for the vote model, showing existence of LNE subject to a

domain constraint. The relevance of varying valences is also demonstrated. Since the

theorem indicates that LNE may not exist in general, Section 3 introduces a transversality

argument for proof of existence of CNE. In the case when the space is compact and

convex, it is possible to use a gradient vector field argument to show existence of LNE.

The conclusion comments of the possibility of multiple general equilibria in the

economic realm resulting from a single political choice, and of multiple local equilibria in

the political realm for a given economic situation.

2. EQUILIBRIA IN ECONOMIC AND POLITICAL THEORY

In economic theory, there are well established procedures for proof of existence

of general competitive equilibrium. In the "topological category", these procedures use

the Brouwer fixed point theorem (Brouwer, 1910), assuming contractibility of all

preference sets. In the "vector space" category where the economic space is a compact

7

convex topological vector space and preferences convex, the Fan theorem applied to

preferred sets gives equilibrium (Fan, 1961; Bergstrom, 1992; Gale and Mas-Colell,

1975). In this context, a general equilibrium can be identified as a pure strategy Nash

equilibrium for consumers and producers. When preferences can be represented by

utility functions, then this class of theorems uses concavity and continuity of utility.

In the "differentiable" category, utilities are differentiable (or C2) and general

equilibrium depends on the solution of a system of first order equations involving the

number of agents, and commodities. When these equations were linear, then it was well

known that there would exist solutions. The more general case was dealt with by Debreu

(1972), Smale, (1974), Balasko (1975), Mas-Colell (1985), etc. These results used

transversality theory (Hirsch, 1976) to prove that the solution set generically, or almost

always, exists and consists of locally isolated points. This theory assumes only that the

economic space is a differentiable manifold without any obvious metric. In such a case,

convexity and concavity has no particular meaning. However, contractibility is well-

defined, and it is plausible to use topological notions in certain cases to construct a vector

field, or dynamic system, leading into the solution set (Dierker, 1974).

In the differentiable category, the elements of the solution set for the first order

equations can be termed "critical equilibria" (or CE). Whether or not these CE are stable

depends on the negative definiteness of the Hessian at each point (These can be defined

locally). If the Hessian is indeed negative definite at the point, then the point is termed a

local equilibrium (LE). Each of these LE can be viewed as local pure strategy Nash

equilibria. However, they cannot be interpreted as "global economic pure strategy Nash

equilibria" (or GE), in the usual sense of Nash equilibrium. This is because in the

8

differentiable category, "convexity of preference" has no meaning. (In the topological

vector space context, where convexity is defined, then existence of GE can be

demonstrated, with the appropriate assumptions). If the set of local equilibria can be

ranked under the unanimity or Pareto relation, and one of them is Pareto preferred to all

others, then the top ranked would be a candidate for a GE (under some definition of

coordinated equilibrium). However, because the LE belong to the critical Pareto set, it

would seem that the Pareto ranking cannot be used (unless there is a unique LE).

Without assuming convexity of the space and of preference, there may exist no GE (in

the general Nash equilibrium sense).

If there are multiple LE, then there is a coordination problem for the economy.

However, since the LE are by definition locally stable, each one will exhibit a basin of

attraction. If the initial position is within a particular basin, then economic exchange will

lead it to the particular LE.

In the model of political exchange presented in this paper, precisely the same

phenomenon occurs. The general result shows existence of the political equilibrium

"close to" the voter mean point, but perturbed by the valence effects. Under assumptions

on the valence function, existence is given by a first order equation, together with the

second order Hessian condition. For these calculations only differentiability is required.

If the policy space, X, is compact convex, and the utility functions of agents are concave,

then the first order conditions guarantee that this critical equilibrium (CNE) is a PSNE

(Pure strategy Nash equilibrium in the political game). This result generalizes a recent

theorem of Lin et al (1999).

9

The model as constructed involves electoral risk, characterized by a multivariate

normal distribution with some covariance structure. The second order Hessian condition

depends on a solution involving this covariance structure. Consequently, if the

covariance terms are small relative to the scale of the distribution of voter preferred

points, then concavity can fail. If concavity fails, then even though the critical point is

locally stable, there may exist different strategies for some of the agents which they

prefer to the strategy determining the critical point. Consequently, the CNE cannot be a

PSNE. Unlike the models used in economic theory (for the vector space category), there

is no intrinsic reason to suppose that political agents have convex preferences.

Indeed, there may exist no PSNE. However, in parallel with the results of

economic theory there may exist multiple local Nash equilibria (LNE) in the political

game. This gives rise to a coordination problem among the political agents. Typically,

Pareto ranking (by agents, or voters) will be incomplete, and will not give a top-ranked

PSNE. However, there will be basins of attraction for each of the political LNE. If the

agent positions commence within the basin of one of the (stable) LNE, then vote

maximization will pull them into the particular LNE.

Because the equilibria are local, perturbations of the exogenous parameters can

lead to the destruction of one or other of the LNE, resulting in rapid changes in optimal

candidate positions. Such "catastrophes" can be seen on occasion in the life of a polity.

Section 3 gives the existence Theorem 1 for the vote model, under certain

restrictions on the valence function. For general valences, the proof technique may not

be able to show existence. Consequently, Section 4 deploys genericity arguments to

argue that, generically, there will exist LNE.

10

Although the paper does not detail the nature of the policy space, X, on which

political decisions are made, it is possible to follow Konishi (1996) and view X as

incorporating economic decisions as well. The concluding remarks touch on the

interaction of political and economic theory.

3. NASH EQUILIBRIA IN THE VOTER MODEL

The situation we consider is that of a collection of political agents (whether

candidates or parties). Each chooses a policy in a space X. Let

z = (z1,…, zp) ∈ XP denote a strategy vector. The game form g : XP → W maps to a space

of w outcomes, on which each agent has a utility function Uj : W → ℜ . The game is {

Ujh(z) : XP → ℜ or Uh : XP → ℜ P, where Uj

h(z) = Uj(g(z)).

Definition 1. A pure strategy Nash equilibrium (PSNE) for the game {Ujh}P is a vector

z* ∈ XP with the property that, for each j ∈ P, there exists no zj ∈ X \ {z*} such that

Ujh(z1*,…, zj-1*, zj, zj+1*,…, zp*) > Uj

h(z1*,…, zj-1*, zj, zj+1*,…, zp*) (Eq. 1)

A more general notion, that of mixed strategy Nash equilibrium (MSNE) is

similar, but considers strategies for each agent in a space of lotteries, or mixtures, defined

over X. In the case X is a topological vector space, there are well known properties of Uh

sufficient to generate existence. Some of these properties focus on the properties of the

underlying preferences induced by Uh on X. These are based on the Fan (1961)

Theorem. For example, quasi concavity and continuity of Uih are sufficient for existence

(A function U is quasi-concave if

U(ax + (1 - a)y ≥ min[U(x), U(y)] for all x, y and α ∈ [0, 1]).

11

Concavity is a stronger property that also sufficies. (U is strictly concave if

U(ax + (1 - a)y) > aU(x) + (1 - a) U(y)).

Compactness of X is always assumed, and when convexity properties of preference are

utilized, X is assumed convex.

In the topological category where X is a topological space, a "weaker" concept is

local Nash equilibrium.

Definition 2. i.) A local pure strategy Nash equilibrium (LNE) for the game {Uhj}P is a

vector z* ∈ XP with the property that, for each j ∈ P, there exists an open neighborhood

Wj of zj* in X, such that, for no zj in Wj is it the case that

Ujh(z1*,…, zj, zj+1*,…, zp*) > Uj

h(z1*,…, zj, zj+1*,…, zp*) (Eq. 2)

ii.) An LNE, z*, is locally isolated if there exists a neighborhood W of z* in XP

such that z* is the unique LNE in W.

In the differentiable category, an even weaker concept is that of critical Nash

equilibrium (CNE). We give the definition when X is a vector space. However, the

definition is also applicable to the general case when X has a differential structure; that is,

when X is a smooth manifold (Hirsch, 1976).

Definition 3. Suppose that X is a compact topological vector space of dimension

w with smooth boundary. Let Uh : XP → ℜ P be C1-differentiable. Then z* is a

CNE iff z* ∈ XP and the first order vector equation ∂Ujh(z*) = 0 for all j ∈ P.

∂zjIf all Uj

h are C2-differentiable, then analysis of the second order Hessian

12

conditions at z* can be used to determine if the CNE z* is a LNE. Since every

PSNE is a LNE, this can be used to determine existence of PSNE.

However, there is one problem with this technique. In the case p = 1,

compactness of X is sufficient to guarantee a global optimum (the analogue of the

PSNE). In the case p ≥ 2, there may be no PSNE. To see this, consider the

situation where there is some finite set XL = {z1,…, zt} of LNE. If there is a

PSNE, then it belongs to this set. Consider the unanimity ranking >P on XL given

by za >P zb iff Ujh(za) > Uj

h(zb) for all j ∈ P. Clearly, if za is unbeaten under >P

then it will be a PSNE.

Suppose now that z* is a LNE, but that concavity of Uhj fails. Even though the

second order Hessian condition for Ujh at z* may be satisfied, it is possible that there

exists a local maximum for Uhj at some point

z′ = (z1*,…, zj, zj*+1,…,zp*) such that Uhj(z′) > Uh

j(z*). Then clearly, z* cannot be a

PSNE. This situation cannot occur if z* is Pareto unbeaten. Without this Pareto

property, the only way to determine whether z* is indeed a PSNE is by computation to

detect all LNE and then compare them directly.

The utility functions of the vote-maximizing game that we study is so complex

that this computation is generally impossible. However, the LNE that are obtained

comprise the candidate set for PSNE, so one way to seek for PSNE is the following

procedure. Select an LNE, z*, fix z*-j = (z1*,…, z*j-1, z*j+1,…) and then examine zj in the

jth strategy set. If the induced utility function for j is concave on this strategy set (with z*-

j fixed) then the Nash equilibrium property holds for j at z*. Reiteration for each j at each

LNE gives a method of determining the PSNE. Clearly, this "local" concavity property is

13

much weaker than "global" concavity. Indeed, this local concavity property will depend

on the Hessian condition, and therefore on constraints on parameters. Consequently, for

each LNE, it may prove possible to construct such a constraint. If the constraint is non-

null and is satisfied for all j, then the LNE is a PSNE. It is this technique which is

deployed here.

The theory was originally developed for two-agent competition. In this case, it is

natural to suppose that, for each agent, j, Ujh(z) = Vj(z) - Vk(z) where k is the opposing

party or agent, and Vj is j's expected vote share. Banks and Dugan (1999) give an

extensive discussion of this case. They assumed that the policy space is compact, {Uhj}

are jointly continuous in z, and concave in zj but with a further property, "aggregate strict

concavity" (a property on voter functions), and showed that then there would exist a

unique symmetric PSNE (zj*, zk*).

As mentioned in the Introduction, this paper is concerned with developing the

stochastic vote model proposed by Lin et al (1999) for the multiparty case

(p ≥ 3). In their model, utility for voter i was defined in terms of a bliss point, xi, in a

compact, convex subset X of ℜ w. Given the vector z ∈ XP of agent strategies, then uij(xi,

zj) = || xi - zj ||2 + ∈ j, where || || is the Euclidean norm, and ∈ j is a normal variate. The

number uij is interpreted as the latent utility for i from agent j. Voter i picks j with

probability

ρij(z) = Prob[uij(xi, zj) > uik(xi, zk), for all k ∈ P - {j}], and Vj(z) is the expected vote share

of agent j. Lin et al (1999) showed that there existed a CNE

(z**,…, z**) where all agents adopted the convergent strategy z** = 1/n Σ xi. Other

conditions that they determined were sufficient to prove "local" concavity on a

14

neighborhood of z**. This showed that z** was a PSNE, as long as the domain, X, of the

game was restricted to this neighborhood, for all agents, and all voters.

In empirical estimation, compatible with this model, many LNE were located

(Schofield, et al, 2002). Moreover, comparison of the estimated variance terms with the

scale factor (associated with the voter distribution) suggested that the Lin et al condition

for "local" concavity was violated. Consequently, determining whether an LNE was a

PSNE could only be done by simulation.

It was also found that valence terms {λ j} appeared relevant and differed between

the parties. There are two different equally plausible interpretations of valence. One is

that the valence of party j is a measure of the parties' attractiveness. In this case it is

appropriate to model λ j, say, as a random variable, distributed by the normal probability

density function φ, with expectation λ j, and variance σj2. This immediately gives a way

of interpreting the stochastic component of the model. With this interpretation we model

λ j by the disturbance term ∈ j with zero expectation.

The second interpretation is that λ j is a function of the party positions, and of the

ability of the party to obtain support from activists. With this assumption it is reasonable

to suppose λj is what we call "strict" (that is, not a function of zk, for k ≠ j).

For these reasons, we examine a more general form of the voter model, in order to

determine the conditions sufficient for CNE, LNE and PSNE. Since the result is in the

differentiable category, it is assumed that all functions are (C2) differentiable (second

differentials exist and are continuous).

15

Definition 4. The voting model with valence on the space X is denoted Σ and defined as

follows: X is a compact, convex topological vector space of finite dimension, w.

i.) Voter utility, for each i ∈ N, is given by

uij(xi, zj) = λ j(z) - Aij(xi)(zj) + ∈ j

where (a) Aij is a convex quadratic form, differentiable in both xi and zj, w

given by Aij(xi)(zj) = Σαt (xit - zjt)2. t=1

(b) λ j is either a differentiable valence function of z which is concave in zj, or a

constant, possibly varying between parties.

(c) {∈ j} are independent normal variates with expectation 0 and variances

{σj2}.

ii.) Given a vector z ∈ XP of agent strategies, then for each i ∈ N,

the probability function ρi = (ρi1,…, ρip) : XP → [0, 1]P is known, and each

component, ρij, is given by

ρij(z) = Prob[uij(zj) > uil(zl), for all l ∈ P - {j}]

iii.) The vote share functions are given by V : → [0, 1]P where

V(z) = (V1(z),…, Vp(z)) and Vj(z) = (1/n)Σρij(z) i

iv.) Ujh(z) = Vj(z) for each j ∈ N.

Because of the differentiability assumptions on voter utilities, and because

the normal probability density and cumulative functions are differentiable, so will be Vj,

for all j.

To prove the analogue of the Lin et al theorem, we consider additional

16

restrictions on the model. In Definition 4, the valence term λ j is allowed to be a function

of the vector z. One required restriction is that λ j is a function only of zj.

Definition 5. i.) The valence model is called strict if it is the case that each λ j is

a function only of zj. Thus λ j(z) = λ j(zj) so

λj(z1,…, zj, zj+1,…, zp) = (z1′,…, zj′, zj+1′,…, zp′)

even when zk ≠ zk′, for k ≠ j.

ii.) The model is called symmetric if it is strict, and in addition,

λj(zj) = λk(zk) wherever zj = zk.

iii.) The model is termed simple Euclidean if Aij(xi)(zj) = || xi - zj ||2 for all

xi, zj.

iv.) A LNE (or PSNE) z* is called symmetric if zj* = zk* for all j, k.

v.) A LNE (or PSNE) z* is called convergent if zj* = 1/n Σ xi, for all j. z∈ N

As noted above, Lin et al demonstrate that, if each Vj is a concave function of zj,

then there exists a convergent PSNE in the simple Euclidean model. The following

Theorem shows that for the model Σ there exists a domain X(Σ) within X, which is

defined by the vector (σ12,…, σp

2) of stochastic variances and by the valence functions

{λj}. If all agent strategies are restricted to X(Σ) then there will exist PSNE. However,

X(Σ) is determined by conditions on the Hessians of each Vj, sufficient to guarantee that

each Vj is concave. We shall say the domain constraint is satisfied for Σ w.r.t. X(Σ) if the

convex hull, Con({xi : i ∈ N}), of the voter bliss points is a subset of X(Σ). In general,

this constraint may be severe.

17

Theorem 1. For each Σ there is defined a subspace X(Σ) of X with the following

property. If the domain constraint for Σ w.r.t. X(Σ) is satisfied, then there exists a LNE

z* in the strict voting model with valence. If Σ is simple Euclidean and valence constant

for each party but differs between the parties, and the domain constraint satisfied, then

there exists a convergent LNE. However, the agent with the largest valence wins a

majority.

Without assuming the domain constraint, if Σ is symmetric, then there exists a

symmetric CNE z*, only if λ satisfies an additional first order condition at z*. If Σ is

not symmetric, then any CNE, z*, will not be symmetric. If the domain constraint is

violated, there need not exist a PSNE. In particular, if λ are constant, but vary between

the parties, and sufficient voters lie outside X(Σ) then the convergent point at the origin

cannot be a LNE. �

Theorem 1 shows that with strict valence, the first order conditions give a solution

z* which is a perturbation of the convergent point at the voter mean. For optimality, each

agent adjusts its position away from the voter mean to balance overall electoral support

(maximized at the mean) with support induced by the valence function.

It is evident that the first order condition can typically be solved. Since the {λ j}

are assumed concave, it can be inferred that the domain constraint is weaker with

valence, than without.

The strategy we deploy is to examine the Hessian of Vj with respect to zj, at a

candidate LNE. Since the Hessian is negative definite everywhere if and only if Vj is

strictly concave, this allows us to test for concavity. If the Hessian of Vj is negative

definite at zj*, then it will be so in a neighborhood of zj*, and we shall say Vj is locally

18

concave at z*. Ensuring local concavity at z* requires a domain constraint on {xi : i ∈

N}. When this test fails, local concavity at z* fails. Consequently, concavity of Vj fails,

and we cannot assert that zj* is the component of a PSNE for the polity.

To prove the Theorem we demonstrate two lemmas.

Lemma 1. Let Φ(y) be the cumulative probability function �-∞ y φ(g) dy of the

normal probability density function φ(g) =1/√(2πσ) exp -y2 with variance σ2. (2σ2)

Let g : X → ℜ be a differentiable strictly concave function defined on the vector

space X, of dimension w. Suppose that dg(0) = 0. dz

Then Φ − g : X → [0, 1] is a strictly concave function on a domain D in X, which

is defined in terms of the constraint g(z) ∈ [δ1(σ), δ2(σ)] involving σ, s.t.

g(0) ∈ [δ1(σ),δ2(σ)].

If g is symmetric, so g(z) = g(-z) for all z ∈ X, then δ1(σ) = - δ2(σ).

Proof of Lemma 1.

dΦ(g) = dΦ(g) ⋅ dg dz dg dz

= φ(g) ⋅ dg where dg ∈ ℜ w. This follows from the definition of Φ. dz dz

Thus, d2Φ(g) = d t φ(g) ⋅ dg dz2 dz dz

= φ(g) ⋅ d2g + dφ(g) dg t dg dz2 dg dz dz

But dφ(g) = - g φ(g). dg σ2

19

Thus d2Φ(g) = φ(g) - g dg t dg + d2g dz2 σ2 dz dz dz2

At z = 0, dφ(g) = 0 since dg(0) = 0 and φ(g) is finite. Now d2Φ(g) = 0 whenever dz dz dz2

d2g - 2g dg t dg = 0.dz2 σ2 dz dt

Since g is strictly concave, its Hessian is negative definite. Since Φ(g) is positive,

Φ will be strictly concave in some range of z such that the quadratic form

H(z) = d2g - g dg dg is also negative definite. dz2 σ2 ∂zj ∂zk

This is a quadratic equation in g(z) involving σ2, and will have two solutions

{δ1(σ),δ2(σ)}. Obviously, g is maximized at z = 0, (since it concave) and Φ is

strictly increasing in g. Thus, Φ is maximized at z = 0, and so δ1(σ) < 0 < δ2(σ).

This gives the constraint. The result with symmetry follows.

Q.E.D.

Example 1.

In the voting model, the function g has the general formg(z) = λ - β || x - z ||2 = λ - βy2.

Then the second order Hessian condition becomes

H(g) = φ(g) -1 λ - βy2 -2βy 2 - 2β < 0 (Eq. 1) σ2

The zeros of H(g) satisfy the quadratic expression

-4β2y2 (λ - βy2 ) = 2βσ2 or β2y4 - λβy2 - ½ σ2 = 0.

The roots are y2 = λβ ± √[λ2β2 + 2(β2σ2)] / 2β2,

= λ/2β ± (λ/2β)√[1+(2σ2/λ2)]

The positive root, y12, is of order δ1 (σ/β√2, λ/β); That is, for λ small, δ1 = σ/β√2,

20

while for σ small, δ1 = λ/β. Thus, y2 < δ1 ensures that g(z) > -σ/√2.

The second lemma studies the first order optimality conditions sufficient

to guarantee that z* is a CNE in the valence model. For the simple Euclidean

case, the second order condition is then used to show that the mean voter position

zm = 1/n Σxi, if adopted by all agents, is a LNE. If the domain constraint is

satisfied, then this LNE will be a PSNE. The analysis is then extended to the

valence case.

Lemma 2. For z* to be a CNE in the strict voting model, the sufficient first order

condition, for each j ∈ P, is

Σφ(gi(zj))dgi(zj) = 0. (Eq. 2) i dzj

Here gi(zj) = λ j(zj) - Aij(xi)(zj) - ci(z-j) where ci(z-j) is a function of

z-j = (z1,…, zj-1, zj+1,…, zp) only, and φ is the normal pdf. In the simple Euclidean

case, the mean voter position is an LNE.

Proof of Lemma 2. By definition ρij(z) = Prob[uij(zj) > uik(zk), for all k ∈ P - {j}].

For a fixed z-j, let k(i) be that index which maximizes λk(z) - Aik(xi)(zk) for k ≠ j.

Since the model is strict, λk is not a function of zj.

Let ci(z-j) = λk(i)(zk(i)) - Aik(i)(xi)(zk(i)). Clearly, uij(zj) > uik(zk) for all k ≠ j if

and only if

gi(zj) + ∈ j > ∈ k(i)

Thus, ρij(z) = Φi (gi(zj)) where Φi is the cpf of the normal variate ∈ k(i) - ∈ j, with

variance σk(i)2 + σj

2 and expectation 0. Now Vj(z) = 1/n Σ Φi(gi(zj)). i∈ N

The first order optimality condition for Vj is the vector equation

21

d (Σ Φi) = Σ d Φi dgi = Σ φ (gi) dgi = 0.dzj

i i dgi dzj i dzj

This follows because d Φi = φ(gi). Q.E.D. dgi

Proof of Theorem 1.

Consider the situation where z* is symmetric, i.e., zj* = zk*, for all j, k. Then,

Aik(i)(xi)(zk(i)*) = Aij(xi)(zj*) for all i. Thus, gi(zj*) = λk(i)(zk(i)) - λ j(zj*).

If the model is not symmetric, the functions gi differ, and so the symmetric

point z* will generally not satisfy the first order conditions. However, suppose

that the model is symmetric. Then all gi(zj*) = 0. Then the first order optimality

condition becomes Σ φ(0) dgi = 0. i dzj

For each i, this gives the equation ∂gi/∂zjt = -2(zjt - xit) + 1 ∂λ j = 0. Thus, αt ∂zjt

n(zj - 1 dλj) = Σxi or zj* = 1 dλ j* + 1/nΣ xi (Eq. 3) 2αt dzj

i 2 dzj i

where λ j*(z) = [(1/α1)(λ1(z)), (1/α2)(λ2(z)),…]. Obviously, in the case that each λ j

is a function, then this condition will not be satisfied at a symmetric point z*

unless λ also satisfies the first order condition

∂λ*jt = ∂λ*kt for all j, k, t. (Eq. 4) ∂zjt z* ∂zkt z*

(When the CNE satisfies equation 4, we can say it is "close to the convergent

point.")

If all λ j are zero or constant, then the voter mean point is a CNE.

(i) Consider this simple Euclidean model, with identical variances but varying

valences. Define a new coordinate system for X such that Σ xi = 0.

22

For each j, we now examine Φj(zj) = ΣΦ(gi(zj) to determine conditions fori

LNE.

The Hessian for g is Φj is

Hj = Σ [ φi(gi) [[ - gi [ dgi t dgi ]] + d2gi ]] (Eq. 5) i 2σ2 dzj dzj dzj

2

where the variance associated with each gi is 2σ2.

At the CNE, z* = (0,…, 0), each gi(zj*) = λ j - λk(i) = τj, a constant, while gi(zj) = τj

- || zj - xi ||2. We write τj since it depends only on λ. Since Φ is a monotonically

increasing function of gi, Φ(gi) is maximized at zj = xi. Clearly, its Hessian is therefore

negative definite at zj = xi. We can use Lemma 1 to obtain a constraint sufficient to

guarantee local concavity at zj = 0.

By Example 1, the bound is || zj - xi ||2 < τj + σ. Putting zj = 0 gives the constraint

xi ∈ Xj (0, r (τj, σ)) where Xj is a ball centered 0, of appropriate radius. Since each

constraint for i is independent of k, (k ≠ i), we obtain the domain constraint in terms of a

ball of radius r(τj, σ). Notice that τj may be positive or negative. If agent j has the largest

valence then obviously τj ≥ 0.

Clearly, if xi ∈ Xj, for all i, then all Φ(gi(zj)) at zj = 0 have negative definite

Hessian. Consequently there exists a neighborhood of zj = 0 in the jth strategy space such

that Φ(gi) is locally concave for all i in this neighborhood. Indeed, for every point, a, on

the arc [xi, zj] between xi and zj, the Hessian will be negative definite. Repeating this

process for all j gives a neighborhood of 0 in XP such that all vote share functions are

locally concave at the origin. This establishes that the origin is a LNE, subject to the

23

constraint, where the constraint is given by the intersection of Xj , over all j. If all λ j are

zero, then X(Σ) is simply an open ball about the origin of radius √σ/a, where a4 = 2.

With constant λ j's, this establishes that each Vj is concave in the ball X(Σ).

Clearly, this demonstrates that the origin is a PSNE.

The cases with constant but varying λ j's or with zero λ j's are slightly

different if the constraint fails. In the zero λ j case the origin will still be a LNE.

However, if the λ j's vary, then the origin will not be a LNE. This follows because

all agents with low valence will have positive semi definite Hessian at the origin.

ii) Consider the case with valence functions. Because of the assumption of concavity of

λj, the Hessian of λ j will also be negative definite everywhere. Suppose the first order

conditions of equation 3 are satisfied at some vector, z*, "close to the convergent point."

We can repeat the above procedure to obtain the domain constraint sufficient to guarantee

that each Vj is locally concave at z*, for each agent j.

iii) Finally, consider again the simple Euclidean case where the valences {λ j} are

constant, but differ. We have shown that the convergent point z* = (…, zj*,…)

where zj* = 1/n Σxi is an LNE. Obviously, gi(zk(i)*) = λk(i) - λ j where k(i) maximizes λ j.

These are constant but non-zero. Then gi(zk(i)*) = δ > 0. So Φi(δ) > ½ for each i. Hence

Vk(i) > ½. That is to say, k(i) wins a majority. Q.E.D.

As Lin et al (1999) observed, it is obvious from Equation 5 that if σ is

sufficiently large, then each Hessian Hj must be negative definite at the origin, no

matter what values {xi} take. Another way of expressing this, in terms of Theorem 1, is

that there exists a tighter constraintX(Σ) than X(Σ) which is sufficient to guarantee that

24

the LNE is a PSNE (when valence is constant). With concave valence, the constraint

derived from Equation 4 will be weaker. Consequently this will weaken the constraint

required to guarantee existence of PSNE. In the example that follows, it is clear that

having differing valence is sufficient to destroy the equilibrium property of the region.

Example 2. Consider a situation where the valences for 1, 2, 3 are ranked λ1, λ2, λ3 with

λ1 < λ2 < λ3, and three voters in ℜ 2, labeled {1, 2, 3} at (0, 1),

(-√3/2, -1/ 2) and (√3/2, 1/2). With all agents at the origin, g1 = λ1 - λ3. So the Hessian

for agent 1 is

H1 = Σ φ(gi) [- gi [Di ] - 2I] 2σ2

where Di is a 2×2 matrix generated by the gradients of g1 at xi. But the Hessian for voter

three with respect to agent 1 involves D3 = √3/2 -1/2

-1/2 √3/2

Hence the Hessian for voter 3 is

λ3 - λ1 [D3] - 2I 2σ2

The determinant of D3 is positive definite so, for σ sufficiently small, the determinant of

this Hessian will be positive definite at the origin. Since the situation is symmetric, z1 = 0

cannot be a LNE. Agent 3 with maximum valence does, however, have a negative

definite Hessian at the origin.

25

Note, however, that if all valences are identical, then the Hessian at the origin will

be negative definite, and so the origin will be a LNE. If the domain constraint is also

satisfied, then it will certainly be a PSNE.

One feature of the convergent equilibrium in the Euclidean model is that each

agent's optimum strategy is independent of the strategies of other agents, and is

determined only be the distribution of bliss points. This has been regarded as an

attractive feature of the spatial model. However, as Theorem 1 indicates, this convergent

equilibrium may not be a LNE if the actual domain of policy is sufficiently extensive. In

particular, the first order condition given in Lemma 2 suggests there are many CNE, and

quite possibly many LNE. For the non-convergent LNE, agent optimum strategies are

interdependent. In the case that Σ is not strict, all LNE will be interdependent. In this

case, the technique utilized in Theorem 1 cannot even be used to prove existence of LNE.

In the proof of Theorem 1, it was assumed that the disturbance terms were all

independent. Indeed, for convenience all variances were assumed identical. However,

estimation of such voter models has found it necessary to assume the more general

hypothesis that the error structure is multivariate normal (that is, allowing for non-zero

covariance terms). See Alvarez and Nagler (1998), Quinn, Martin and Whitford (1999),

and Schofield et al (1998, 2002a). Since the disturbance assumptions affect the model,

we can use Σ* to denote the covariance matrix of the model Σ. The general form of the

proof procedure of Theorem 1 then carries through. However, the first order equation 1

in Lemma 2 will then have a more complicated form. Equation 1 is a matrix equation

involving n functions {gi}N. Since there are p agents, each choosing a strategy in X (of

dimension w) we obtain wp equations. In the case of a general covariance matrix, Σ*,

26

these wp equations will involve interaction terms induced by covariance. Nonetheless,

transversality theory can be used to show that these wp equations are "generically"

independent. Since wp is the dimension of the joint strategy space, the solution is

generically of dimension 0. Consequently, the first order equations can be solved, even

for general valence terms and any differentiable multivariate covariance structure. Thus

CE will generically exist. The next section shows this, and asserts that one of these CE

will be a local equilibrium.

4. GENERIC EXISTENCE

We need to introduce the idea of a tangent bundle. At z ∈ XP, the tangent space is

Tz(XP) and the tangent bundle is defined by T(XP) = ∪ Tz(XP). The differential dUhj(z) at

z

z can be regarded as a linear map from ℜ wp to ℜ so dUhj : XP → Lin(ℜ wp, ℜ ). Since dUh

j

is C1 differentiable, this map is continuous. Moreover, there is a C1-topology on ε, under

which two profiles are close, if their components are close as linear maps at every z ∈ XP.

The differential dUhj can also be projected onto Tz(X). Then this projection

DUhj : XP → Lin(ℜ w, ℜ ) can be identified with the gradient of Uh

j in X, when zk

(for k ≠ j) are held fixed. That is to say, DUhj(z) can be regarded as an element of

a subtangent space Tz(X) ⊂ Tz(XP), where X corresponds to the jth strategy space.

This used the fact that Tz(XP) = Tz(X)×…Tz(X). We use the idea of a conic field

generated by Uh. Let D(z) = (Con Uh)(z) be the convex hull in Tz(XP) of the set of

vectors {DUhj(z): j ∈ P}. Then Con Uh : XP → TXP is a generalized vector field

over XP. Its image in TzXP is convex and the field is continuous as a correspondence.

Michael's Selection Theorem (Michael, 1956) can be deployed to construct a continuous

selection m : XP → TXP of D. That is to say, at every point z ∈ XP, m(z) ∈ D(z) ⊂ TzXP.

27

Note that by this construction each DUhj(z) lies in a different tangent space. Thus,

0 ∈ Con Uh(z) if and only if DUhj(z) = 0 for every j ∈ P. We use this construction in

Lemma 4.

We shall now use standard transversality theory to show CNE generally exist, and

are "locally isolated." We say a property of points in XP is locally isolated if it holds for

0-dimensional submanifolds of XP.

We also say a property K is C1-generic in ε if the set ε1 = {Uh ∈ ε : Uh has

property K} is open dense in the C1-topology on ε.

Definition 5. Let K1 be the property that there exists a locally isolated LNE for the

profile Uh and let K2 be the property that a CNE exists and is locally isolated.

We seek to show K1 is generic. We shall prove this by two lemmas.

Lemma 3. K2 is C1-generic.

Proof. Given Uh, let Tj(Uh) be the set in the jth strategy space such that dUhj = 0.

dzj

By the inverse function theorem, Tj(Uh) is generically a smooth submanifold of

XP of dimension (p - 1)dim(X), ie of codimension dim(X).

But then ∩ Tj(Uh) is of codimension p dim(x). Thus the property K2 is generic. j

Definition 6. A profile Uh satisfies the boundary condition if for any z in the boundary,

∂XP, the differential (DUhj(z),…, DUh

j(z)) points inward.

That is, if n(z) is the normal to the boundary at z, then DUhj(z)(n(z)) > 0, for all j.

Lemma 4. If Uh satisfies the boundary condition and if Uh exhibits a locally isolated

CNE, then there exists a LNE.

28

Proof. We have defined the conic field D = Con(Uh) XP → TXP in the preliminary to this

section. D admits a selection m: XP → TXP, that is a continuous function such that

m(z) ∈ Con(dUh(z)). Moreover, m(z*) = 0 if and only if DUhj(z*) = 0, for all j ∈ P.

Clearly, 0 ∈ Con(dUh(z)) iff DUhj(z) = 0 for all j ∈ P. The selection is a vector

field on XP. Moreover, m can be selected to be a gradient vector field on XP which

satisfies the boundary condition, i.e., m(x)(n(x)) > 0 for all x on the Boundary of XP.

Clearly, m(z*) = 0 iff z* is a CNE. Because of the Morse inequalities (Milnor, 1963), m

must exhibit a stable equilibrium, z*, say.

Let mj(z) be the projection of m at z onto the jth tangent space. Because z* is a

stable equilibrium, there exists a neighborhood Y of z* with the following property: if

z ∈ Y then for each j ∈ P mj(z) "points" towards z*. In the vector space context this

means that m(z)(z* - z) > 0. Let Yj be the projection of Y onto the jth strategy space.

Then DUj(z) points towards zj* for all zj ∈ Yj. The Cartesian product of {Yj} gives a

neighborhood of z* satisfying all second order conditions. Clearly, z* is a LNE.

(Note that this theorem is proved here for X compact and convex, but it is valid

for a gradient field on a space with non-zero Euler chraracteristic; Brown, 1970).

Theorem 2. Existence of LNE is generic in the topological subspace

ε′ = {Uh ∈ ε: Uh satisfies the boundary condition}.

Proof. By Lemma 3, there exists an open dense subset ε′′ on which K1 is satisfied. But

then ε′ ∩ ε′′ gives an open dense subspace of ε. For every profile in this set, the

procedure of lemma 4 gives a LNE.

Now let ε* = {u, λ : Xn+p → ℜ np} denote the set of all electoral systems, as discussed in

the previous section. That is to say, once bliss points in Xn and agent strategies in XP are

29

known, the n × p array of voter utilities is specified. For convenience, we shall regard the

covariance structure, Σ*, and the variances {σ12,…, σp

2} as fixed since the variances are

essentially scale factors.

Given (u, λ) ∈ ε* and the set P of political agents, then the polity Uh is well

defined. Thus there is a continuous electoral map EP : ε* → ε. By Theorem 2, existence

of LNE is generic in ε′. Since E is continuous, the inverse image of an open set in ε′ is

open in the co-image of E. Suppose EP is also proper: that is, if Y is open in ε*, then it is

open in ε′. In this case, the inverse image of a dense set is dense.

Corollary. Existence of LNE for a given set P of political agents is a generic property in

ε*, if the electoral map is proper.

The results on transversality theory used in theorem 2 can be found in Hirsch

(1976), Smale (1974), Balasko (1975), and Mas-Colell (1985). Dierker (1974) discusses

the Morse inequalities. A review of this material is in Schofield (2002).

A minor point concerns the nature of the topology. These theorems involve

topologies on utilities. Equilibrium concepts in both politics and economics should be

based on preferences. Schofield (1999) has proposed a C1-topology on preferences, when

their utility representations are smooth. This suggests that Theorem 2 can be expressed

as a genericity result for preferences.

30

5. CONCLUSION.

In this exposition, we have assumed that the policy space, X, is fixed. In fact, an

obvious extension is where there is a map ψ : X → Z, where Z is the full commodity

space. Economic agents have preferences on Z, and can, in theory, be able to induce

preferences on ψ-1(z) ⊂ X. These preferences then define bliss points in X. There are

certain subtleties of such a model which were explored by Konishi (1996). The principal

difficulty of the earlier model was in locating appropriate political equilibria. This paper

has presented one way of inferring existence of LNE in X, given electoral data about

voter bliss points.

Note, however, that there are a number of further theoretical difficulties. Firstly,

the map ψ may be multi-valued. As discussed in the Introduction, for any given

economy, there will be many possible, Pareto incomparable economic LE. Each of these

can arise from a single political decision in X. However, ψ will be locally single-valued.

That is, if the economy is initially at a particular state, e, in Z, then voters may compute

back to X to determine how changes in political decisions will affect economic outcomes

in a neighborhood of e in Z. Similarly, political decision-making will be locally single-

valued. That is, for a given economic and political situation, the local political

equilibrium will be determined by the particular basin of attraction within which the

status quo is located. As noted in the Introduction, however, the catastrophes discussed

by Balasko (1975) for the economic system can also occur in the political system.

31

References

ALVAREZ, M., and J. NAGLER (1998) When politics and models collide:Estimating models of multicandidate elections. American Journal ofPolitical Science 42, 55-96.

ANSOLABEHERE, S. and J. M. SNYDER (2000) Valence politics andequilibrium in spatial election models. Public Choice 103, 327-336.

BALASKO, Y. (1988) Foundations of the Theory of General Equilibirum. NewYork: Academic Press.

BANKS, J. and J. DUGAN (2000) A bargaining model of collective choice.American Political Science Review 94, 73-88.

BERGSTROM, T. (1992) When non-transitive relations take maxima andcompetitive equilibrium can't be beat. In Economic Theory andInternational Trade: Essays in memoriam J. Trout Rader, W. Neuefeindand R. Reizman, eds., 29-52. Berlin: Springer.

BROUWER, L. (1910) Uber Abbildung von Manningfaltigkeiten.Mathematische Annalen 1, 97-115.

BROWN, R.F. (1970) The Lefshetz Fixed Point Theorem. Glenview, Illinois:Scott and Foreman.

DEBREU, G. (1970) Economics with a finite set of equilibria. Econometrica38, 387-392.

DEBREU, G. (1976) Regular differentiable economies. American EconomicReview (Papers and Proceedings) 66, 280-287.

DE PALMA, A., HONG, G. S., and J.-F. THISSE (1990) Equilibria in multipartycompetition under uncertainty. Social Choice and Welfare 7, 247-259.

DIERKER, E. (1974) Topological Methods in Walrasian Economics. LectureNotes on Economics and Mathematical Sciences, 92. Berlin: Springer.

DOWNS, A. (1957) An Economic Theory of Democracy. New York: Harper and Row.

FAN, K. (1961) A Generalization of Tychonoff's Fixed Point Theorem.Mathematische Annalen 42, 305-310.

GALE, D. and A. MAS-COLELL (1975) An equilibrium existence theorem fora general model without ordered preference. Journal of Mathematical

32

Economics2, 9-15.

HINICH, M. (1977) Equilibrium in spatial voting: the median voter result is anartifact. Journal of Economic Theory 16, 208-219.

HIRSCH, M.W. (1976) Differential Topology. Berlin: Springer.

KONISHI, H. (1996) "Equilibrium in abstract political economies: with anapplication to a public good economy with voting," Social Choice andWelfare 13, 43-50.

LIN, TSE-MIN, J. ENELOW and H. DORUSSEN (1999) Equilibrium inmulticandidate probabilistic spatial voting. Public Choice 98, 59-82.

MAS-COLELL, A. (1985) The Theory of General Equilibrium. Cambridge:Cambridge University Press.

MICHAEL, E. (1956) Continuous Selection I. Annals of Mathematics 63, 361-382.

MILLER, G. and N. SCHOFIELD (2002) "Activists and partisan realignment inthe United States 1860-2000," Typescript, Center in Political Economy,Washington University in St. Louis.

MILNOR, J. (1963) Morse theory, Annals of Mathematics Studies 51,Princeton: Princeton University Press.

QUINN, K., A. MARTIN and A. WHITFORD (1999) Voter choice in multi-party democracies. American Journal of Political Science 43, 1231-1247.

SCHOFIELD, N. (1999) The C1-topology on the space of smooth preferenceprofiles. Social Choice and Welfare 16, 445-470.

SCHOFIELD, N., A. MARTIN, K. QUINN, and A. WHITFORD (1998)Multiparty electoral competition in the Netherlands and Germany: amodel based on multinomial probit. Public Choice 97, 257-293.

SCHOFIELD, N., A. MARTIN, K. QUINN and I. SENED (2002) "Empiricaland theoretical analyses of representative government," Typescript, Centerin Political Economy, Washington University in St. Louis.

SMALE, S. (1974) Global Analysis and Economics I: Pareto Optimum and a

Generalization of Morse Theory. In Dynamical Systems, M. Peixoto, ed.,531-544.

SMALE, S. (1960) Morse inequalities for a dynamical system. Bulletin of the

33

American Mathematical Society 66, 43-49.