Local Political Equilibria

Local Political Equilibria Norman Schofield

∗

December 8, 2003

Abstract This article uses the notion of a “Local Nash Equilibrium” (LNE) to model a vote maximizing political game that incorporates valence (the electorally perceived quality of the political leaders.) Formal stochastic voting models without valence typically conclude that all political agents (parties or candidates) will converge towards the electoral mean (the origin of the policy space.) The main theorem presented here obtains the necessary and sufficient conditions for the validity of the “mean voter theorem”. The conditions involve the party valences, and the electoral and stochastic variances. Since a pure strategy Nash Equilibrium (PNE), if it exists, must be a LNE the failure of the necessary condition for an LNE at the origin also implies that PNE cannot be at the origin. To account fot the non-convergent location of parties, the model is extended to include activist valence (the effect on party popularity due to the efforts of activist groups. It is shown that optimal party locations depend on balancing the “marginal electoral pull” or gradient against the gradient characterizing activist influence. The condition for optimality indicates that as the “exogenous” valence of a party leader falls, then activist influence becomes more significant. It is shown that LNE (satisfying the second order Hessian conditions) will exist for almost all parameters, and utility functions, when the policy space is compact, convex, without any restriction on the variance of the voter ideal points or on the party valence functions. However, these Local Nash Equilibria will almost never occur at the origin. These theoretical conclusions appear to be borne out by empirical evidence from a number of countries.

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 political economy. This article attempts to model the choices of the leaders of political parties as equilibria to a vote maximizing electoral game (Austen-Smith & Banks 1998, Austen-Smith & Banks 1999). Since Downs (1957), such models have tended to lead to the inference that representatives will adopt a position at the electoral mean (Hinich 1977, Banks & Duggan 2004) or near the electoral median (McKelvey 1986, Banks, Duggan & Lebreton 2002, Banks, Duggan & Lebreton 2003). 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 may be a large number of parties, located at very different non-centrist positions (Schofield & Sened 2002, Schofield & Sened 2003). In electoral systems based on plurality (or first past the post), either there is no party near the center, as in the US (Poole & Rosenthal 1984, Miller & Schofield 2003, Schofield, Miller & Martin 2003), or, as in Britain, the centrist party is not a likely candidate for government (Schofield 2003c). ∗ This article is based on research supported by NSF grant SES0241732. I would also like to acknowledge the debt that I owe to Jeff Banks, for inspiration and motivation in my modest attempt to outline an equilibrium theory of politics. I thank Ben Klemens and Lexi Shankster for preparing the manuscript and figures. I am very grateful to John Duggan who kept asking difficult questions about the proof of the theorem as given in the earlier version of this paper.

1

The model presented here attempts to account for non-centrist choices of parties by extending the stochastic model of voting (Lin, Enelow & Dorussen 1999) to include “valence” (Stokes 1992). Valence here is assumed to have two different sources. It may either be an exogeneous effect due to the differences in the average perceived quality of the party leaders (Ansolabehere & Snyder 2000, Groseclose 2001, Aragones & Palfrey 2003, Schofield 2003c), or more generally, valence can be endogeneous, due to the indirect effect of party leader choices on the willingness of activists to support the party by fund raising, et cetera. This effect allows party leaders to increase the positive eletoral response by using the media (Aldrich 1983b, Aldrich 1983a, Aldrich & McGinnis 1989). To be more explicit, in the first model each party (or candidate, or agent), j, is described by an average level of (exogenous) perceived competence (or quality), λj . That is to say, a typical voter, i, when making comparisons between agents j and k will compare their policies, zj and zk , and also their valences λj and λk . Each agent will adjust his or her policy zj so as to maximize the vote share. Because the agents do not know precisely what weight each voter gives to the perceived quality of the agent, the agent uses the expected vote share as the measure of political response. This version is an extension of the standard “probabilistic” or stochastic vote model (Coughlin 1992, Banks & Duggan 2004). The addition of exogenous valence changes the result known as the “mean voter theorem.” Although the first-order condition is satisfied when all agents adopt the mean voter position, it may be the case that one of the agents with a low valence will, in fact, minimize vote share at the mean. Consequently, the second-order (or Hessian) condition has to be considered. A standard condition, or assumption, that is usually made for proof of existence of pure strategy Nash Equilibrium (PNE) is “concavity” of the payoff function (in this case the vote share). Concavity is equivalent to the requirement that the Hessian, for the vote share function of each agent, be negative semi-definite on the entire strategy domain. Because this condition is unlikely to be met, I consider a weaker equilibrium concept, that of local pure strategy Nash Equilibrium(LNE). A vector is a LNE whenever the appropriate Hessian for each agent is negative semi-definite at that vector. I shall refer to this condition as local concavity. By definition a PNE, if it exists, is a LNE but not conversely. Because the probabilistic vote model intrinsically involves the variance, σ 2 , of the stochastic component, it turns out that necessary and sufficient conditions for local concavity at the joint electoral mean position can be expressed in terms of constraints involving the parameters of the model, together with σ, and the variance of the set of voter ideal points. Analyses have shown that this necessary condition fails, in empirical studies of Netherlands (Schofield, Martin, Quinn & Whitford 1998a) and Israel (Schofield & Sened 2003). Consequently, there is no reason to expect convergence by parties to the electoral mean in these countries. In the case when the policy space is unidimensional, then it is shown here that there is a single necessary and sufficient condition on the Hessian which can be used to verify whether a LNE exists at the voter mean. This condition can be shown to be satisfied in elections in Britain in 1992 and 1997 (Schofield 2003c). It may not prove surprising that the mean voter theorem is invalid for multiparty competition. In the U.S. for example, candidates must win primaries in order to compete in presidential elections, and it may be the case that a policy position that wins a primary is much more radical than a position that wins the presidential election. However, political parties in Britain do not have to face primaries. Moreover, the fact that there are two principal parties in Britain, competing under an electoral system based on plurality rule, implies that maximizing vote share is almost identical to maximizing the probability of winning the election (Duggan 2000). According to the empirical analysis, under the assumption of exogenous valence, all parties would have maximized their vote share by adopting an identical position at the voter mean. The evidence offered in Schofield (2003c) and supported by Alvarez, Nagler & Bowler (2000) is that the British parties were perceived by the electorate to have very different positions on the single economic policy axis. To account for this divergence of party position, the model of exogenous valence is extended to include activist support (Aldrich & McGinnis 1989). By contributing time and support to a preferred candidate, activists enhance overall voter support. Such activist support for a candidate is a function of the party or candidate position. Moreover, activist contributions to a party can be expected to exhibit decreasing marginal returns. It is plausible therefore that each party’s activist valence function will be concave in the party strategy. 2

The most general model that I consider is one where both exogenous valence and activist valence are included. It is shown that the first-order condition for the optimal party position involves a trade-off or balance between what I shall call the marginal “electoral pull” and the marginal “activist pull.” In other words, the electoral gradient and the activist gradient vectors must be opposite but equal in magnitude at an equilibrium position for the party. Another way of interpreting this result is that when exogenous valence falls, for whatever reason, then activist valence becomes more significant in determining the optimal position of the party. Moreover, if we assume that activist valence is indeed a concave function of strategy, then it is possible to determine conditions on the eigenvalues of each of the parties’ activist valence functions, sufficient to guarantee that PNE do indeed exist. The purpose of this article is to present the argument that, in complex political environments, existence of PNE need not be guaranteed. However, genericity arguments can be used to show that LNE will typically exist. Since the set of LNE will include the set of PNE, it is possible to use simulation techniques in actual empirical situations to determine the set of LNE, and by further analysis determine whether any of these LNE are indeed PNE. The notion of LNE has not generally been used in modeling political phenomena. It is much more common when existence of PNE cannot be guaranteed, to use the notion of mixed strategy Nash equilibrium (MNE). There seem to be two reasons to reject the applicability of the notion of MNE. First of all, as Banks et al. (2002) demonstrate, in two-party competition, the support of MNE will belong to a subset of the uncovered set (McKelvey 1986). It is conjectured that the uncovered set will tend to be small and centrally located with regard to the electoral distribution (see also Banks et al. (2003); Schofield (1999).) Consequently, results using the MNE concept suggest that two party competition will lead to convergent party locations. There is no empirical evidence for such a conclusion. Secondly, party leaders are obliged to make policy pronouncements in the form of manifestos, etc. Empirical analyses with current techniques require that parties be precisely located. Empirical models based on precise estimates of party position typically give statistically significant empirical models of voter behavior in the US and Britain (Poole & Rosenthal 1984, Schofield et al. 2003, Alvarez et al. 2000, Schofield 2003c). This suggests that the MNE concept, while theoretically appealing, is of little relevance for the understanding of political bahavior. The next section of the article introduces the stochastic model, and states the main theorem on existence of electoral equilibrium under the various assumptions on valence. Section 3 offers some examples based on Theorem 1. Section 4 extends the model to include activist valence and presents Theorem 4. An empirical example from Britain is given to illustrate this Theorem. A short Section 5 attempts to address possible shortcomings of the model, and in particular, argues for the usefulness of the notion of local Nash Equilibrium. Since there are many possible alternatives to the assumption of vote share maximization, section 6 considers the more general question of generic existence of local Nash equilibria when agent payoffs are differentiable. Section 7 concludes.

2

Nash equilibria in the voter model

The situation we consider is that of a collection P = {1, . . . , j, . . . , p}, of political agents (whether candidates or parties). Each agent, j, chooses a policy zj in a set X. As usual in such models we assume X is a compact, convex subset of Euclidean space, of finite dimension w. Let z = (z1 , . . . , zp ) ∈ X P denote a strategy vector for the set of agents. The game form h : X P → W maps from the set of strategy vectors to a space, W , of outcomes, on which the jth agent has a utility function Uj : W → R. The game is {Ujh (z) : X P → R} or U h : X P → RP , where Ujh (z) = Uj (h(z)). Definition 1 A pure strategy Nash equilibrium (PNE) for the game {Ujh }P is a vector z∗ ∈ X P with the property that, for each j ∈ P , there exists no zj ∈ X such that ∗ ∗ ∗ ∗ Ujh (z1∗ , . . . , zj−1 , zj , zj+1 , . . . , zp∗ ) > Ujh (z1∗ , . . . , zj−1 , zj∗ , zj+1 , . . . , zp∗ )

A more general notion, that of mixed strategy Nash equilibrium (MNE) is similar, but considers strategies for each agent in a space of lotteries, or mixtures, defined over X. In the case where X is a compact, convex 3

subset of a topological vector space, there are well known properties of U h sufficient to generate existence (Austen-Smith & Banks 1998). Some of these focus on the properties of the underlying preferences induced by U h on X. These are based on the Fan (1961) Theorem. For example, quasi concavity and continuity of Ujh are sufficient for existence of PNE. (A function U is quasi-concave if U (ax + (1 − a)y) ≥ min[U (x), U (y)] for all x, y ∈ W and a ∈ [0, 1].) Concavity is a stronger property that also suffices. U is strictly concave if U (ax + (1 − a)y) > aU (x) + (1 − a)U (y)), for any real a. 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 {U hj }P is a vector z∗ ∈ X P with the property that, for each j ∈ P , there exists an open neighborhood W j of zj∗ in X, such that, for no zj in Wj is it the case that ∗ ∗ Ujh (z1∗ , . . . , zj , zj+1 , . . . , zp∗ ) > Ujh (z1∗ , . . . , zj∗ , zj+1 , . . . , zp∗ ).

ii.) An LNE, z∗ , is locally isolated if there exists a neighborhood W of z ∗ in X P 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 U h : X P → RP be C 1 -differentiable. Then z∗ is a CNE iff z∗ ∈ X P and the first order vector equation ∂Ujh ∗ ∂zj (z )

= 0 is satisfied for all j ∈ P .

If all Ujh are C 2 -differentiable, then analysis of the second order Hessian conditions at z∗ can be used to determine if the CNE z∗ is a LNE. (Clearly every LNE must be a CNE). Since every PNE must be a LNE, this can be used to determine existence of PNE. This is the technique used in this article. We determine the condition for existence of a CNE then select ∗ ∗ one CNE z∗ , and fix z∗−j = (z1∗ , . . . , zj−1 , zj+1 , . . . ). We then determine the conditions under which the Hessian of the party j’s vote share function is negative semi-definite at zj∗ 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 gives a method of determining the PNE. I shall call the Hessian condition for z∗ to be a LNE “local concavity.” Clearly, this condition is much weaker than “global” concavity, since global concavity is equivalent to the requirement that the Hessian be everywhere negative semi-definite. The exercise is to determine those constraints on the parameters of the model which are necessary and sufficient for local concavity to hold when all parties adopt the mean electoral position. As observed, the Hessian condition is that all eigenvalues be non-positive. Formally, this includes the situation where one or more of the eigenvalues of the Hessian are zero, and this equivalent to the property that the determinant of the Hessian is zero. This implies that the critical point of the vote share function is degenerate. However it is well known that the property of having a degenerate critical point is a non-generic property (Hirsch 1976). Indeed as we shall see, the degeneracy of the Hessian is associated with a knife edge property of the parameters of the model. It therefore is convenient to focus the analysis on an examination of the conditions under which the Hessian is negative definite, with all eigenvalues negative. If this property is satisfied, for all j, then the resulting LNE we shall call strict. Equivalently, we may say that strict local concavity is satisfied. The stochastic voter model 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 4

is j’s “expected vote share.” Banks & Duggan (2004) give an extensive discussion of this case. In addition to compactness and convexity of X, and joint continuity of Vj , they assumed a further property, “aggregate strict concavity” (a property on voter utility functions), and showed that then there would exist a unique symmetric PNE (zj∗ , zk∗ ) with zj∗ =zk∗ equal to the mean of the voter ideal points. The purpose of this article is to attempt an extension of the stochastic model of Lin et al. (1999) and Banks & Duggan (2004) so as to bring its conclusions more into line with the empirical evidence as regards non-convergence. I shall do this by exploring the conditions on the parameters of the model which are necessary and sufficient for strict local concavity and thus for existence of strict LNE. These conditions will then be necessary for existence of strict PNE. The model is appropriate for multiparty situations, with p ≥ 3, so by definition the political game is not zero sum. The key idea underlying the formal model is that party leaders attempt to estimate the electoral effects of party declarations, or manifestos, and choose their own positions as best responses to other party declarations, in order to maximize their own vote share. There may, of course, be different motivations for party leaders. Nonetheless, the formal model based on vote maximization has the clearest theoretical structure. Because party leaders cannot predict vote response precisely, the voter model has an inherent stochastic element. In the model with “exogenous” valence, the stochastic element is associated with the weight given by each voter, i, to the average perceived quality or valence of the party (leader). The data of the model is a set of points, {xi }i∈N (where N is of size n) of voter ideal points for the electorate, N . Each xi lies in a compact convex subset, X, of Euclidean space, Rw . Each of the parties P = {1, . . . , j, . . . , p} chooses a policy, zj ∈ X, to declare. Let z = (z1 , . . . , zp ) ∈ X P be a typical vector of party policy positions. Given z, each voter, i, is described by a vector ui (xi , z) = (ui1 (xi , z1 ), . . . , uip (xi , zp )), where uij (xi , zj ) = λj + µj (zj ) − β||xi − zj ||2 + j . Here, λj is the “exogenous” valence of party j, µj (zj ) is the “activist” valence, β is a positive constant and || · || is the usual Euclidean norm on X. The terms {j } are the stochastic components, assumed to be normally distributed with zero expectation and identical standard deviation σ. In logit estimations, these error terms are also assumed independent. . This independence assumption is used in the proof of Theorem 1, although it is obvious from the theorem proof that analogous results hold if the more general assumption is made that the stochastic errors are multivariate normal with general variance / covariance matrix. I shall also assume that the exogeneous valences are ranked λp ≥ λp−1 ≥ · · · ≥ λ2 ≥ λ1 . Because of the stochastic assumption, voter behavior is modeled by a probability vector. The probability that a voter i chooses party j is ρij (z) = P ROB[[uij (xi , zj ) > uil (xi , zl )], for all l 6= j]. Here P P ROB stands for the probability operator. The expected vote share of party j is V j (z) = (1/n) i∈N ρij (z) where N is the voter population of size n. I shall assume that each agent j chooses zj to maximize Vj . The first result will focus on “exogenous” valence and assume µj ≡ 0. I denote this model by Σ. In this model it is natural to regard λj as the “average” weight given by a member of the electorate to the perceived competence or quality of candidate j. The “weight” will in fact vary throughout the electorate, in a way which is described by the normal distribution. Because of the differentiability of the cumulative normal distribution, the individual probability functions {ρij } are C 2 -differentiable in the strategies {zj }. Thus, the vote share functions will also be C 2 -differentiable. The technique of Lin et al. (1999) in asserting the mean voter theorem makes use of C 2 -differentiability. Let x∗ = (1/n)Σi xi . Then the mean voter theorem for the formal stochastic model asserts that the “joint mean vector” z∗0 = (x∗ , . . . , x∗ ) is a PNE. Lin et al. (1999) argued that the restriction sufficient for the validity of the theorem was relatively weak, and simply required that σ 2 was “sufficiently large.” However a precise condition was not given,

5

To determine whether this result can be applied to empirical cases, I shall obtain the necessary and sufficient conditions for existence of LNE of the form (x∗ , . . . , x∗ ). First, we may transform coordinates so that in the new coordinates, x∗ = 0. To characterize the LNE, we must represent in a simple form the variance covariance matrix (or data matrix, ∇) of the voter ideal points. Let X be endowed with a system of coordinate axes (1, .t, ..s, ..w). For each coordinate axis let ξ t = (x1t , x2t , . . . , xnt ) be the vector of the tth coordinates of the set of n voter ideal points. The symmetric w × w voter variance/covariance data matrix ∇ is then defined to be   (ξ 1 , ξ 1 ) · · · (ξ t , ξ 1 ) (ξ s , ξ 1 ) · · ·   (ξ 2 , ξ 2 ) (ξ 3 , ξ 2 )     (ξ , ξ ) (ξ , ξ ) 2 3 3 3     .. . .   . . . ∇=  (ξ 1 , ξ t )  (ξ t , ξ t ) (ξ s , ξ t )    (ξ 1 , ξ s )  (ξ t , ξ s ) (ξ s , ξ s )     .. ..   . . (ξ w , ξ w )

Here we use (ξ s , ξ t ) to denote scalar product. Now define the “convergence coefficients”{ Aj } for the contest of party j against party p by Aj = β σ 2 (λp − λj ). Then the Hessian matrix C is defined to be C = (A1 /n)∇ − I, where I is the w × w identity matrix. Theorem 1 The necessary and sufficient condition that z∗0 be a LNE is that the eigenvalues of the Hessian matrix C be all non-positive. It is obvious that an analogous results holds for a strict LNE when all the eigenvalues are required to be negative. As mentioned above, the situation where one or more of the eigenvalues are zero is non-generic. The proof of Theorem 1 depends on considering the first and second order conditions at z ∗ for each vote share function. The first order condition is obtained by setting dVj /dzj = 0 (where we use this notation for full differentiation, keeping z1 , . . . , zj−1 , zj+1 , . . . , zp constant). The second order condition is that the Hessian d2 Vj /dzj2 is negative semi-definite at the origin. If this holds for all j at z∗0 , then z∗0 is a LNE. However, we need only examine this condition for the vote function V1 for the lowest valence party. As we shall show, if the condition holds for V1 , then it holds for every Vj . As usual conditions on C for the eigenvalues to be non positive depend on the trace, trace(C), and determinant, det(C), of P C. These turn on the value of A1 and on the electoral variance/covariance matrix,∇ . Let 2 ) vt2 = (1/n) i∈N xit 2 = (1/n)(ξ t , ξ t ) be the variance on the tth coordinate axis, and let v 2 = (v12 + · · · + vw be the total electoral variance. Then we can show that a sufficient condition for the eigenvalues of the Hessian for V1 to be non-positive is that A1 v 2 ≤ 1. (This condition is obtained from considering the determinant of C.) The necessary condition is that A1 v 2 ≤ w, where this condition is obtained from examining the trace of C. Note that the case λp = λ1 was studied by Lin et al. (1999). In that case, the convergence coefficient A1 is zero so the joint origin, z0∗ , is a LNE. However the examples presented below to illustrate the theorem suggest that even when the joint origin is a LNE, it will not be a PNE. This indicates that PNE are unlikely to exist at the origin. Outline of Proof. Consider the lowest valence party, 1, and any voter i. Now, the probability that i picks party 1 is a multivariate integral involving all other parties. However, in the case p ≥ 3, we can simplify the problem by considering an auxiliary game where we take the limiting case of λk equal to minus infinity, for all k = 2, . . . , p − 1. In this game the probability that i picks 1 over p is   ρi1p (z) = P ROB λ1 − β||xi − z1 ||2 + 1 > λp − β||xi − zp ||2 + p . 6

Thus, ρi1p (z) = Φ(gi1p (z)), where the “comparison function” for voter i between party 1 and party p is given by gi1p (z) = λ1 − β||xi − z1 ||2 − λp + β||xi − zp ||2 . Here Φ is the cumulative probability function (CPF) of the variate p − 1 , with variance 2σ 2 , and I use the notation Φ(d) = P ROB[p − 1 < d]). Now let V1p (z) = n1 Σi Φ(gi1p (z)). It obviously follows that V1 (z) ≤ V1p (z), so the first and second order conditions on V1p (z) can be used to infer the same conditions on V1 (z). Taking the first differential gives the first order condition on V1p (z) as Σi φ(gi1p (z))dgi1p /dz1 = 0, where φ(gi1p ) is the normal probability density function for the variate p − 1 at the value gi1p . Now, dgi1p /dz1 = −2β(z1 − xi ). Consider the vector z∗0 = (0, . . . , 0) ,(where Σxi = 0). Then gi1p (z∗0 ) = g = λ1 − λp , for all i. Thus the first order condition for a CNE is Σi φ(g)(z1 − xi ) = 0. Since the same equation must hold for {z2 , . . . , zp } we see that z∗0 = (0, . . . , 0) solves the first order condition. The second order condition on the Hessian H1 = d2 V1p /dz12 is that this be negative semi-definite. Computation of H1 shows that it is given by    2   1 d gi1p −gi1p (z)[∇i1 ] 1 . (1) + H1 (z) = Σi Hi1 (z) = Σi φ(gi1p (z)) n n 2σ2 dz12 The term [∇i1 ] is a w × w symmetric matrix involving the differentials dgi1p /dz1 , while [d2 gi1p /dz12 ] is the negative definite Hessian of gi1p P (namely, −2βI, where I is the identity matrix.) Substituting z1 = zp = 0, and noting that the summation i∈N ∇i1 = 4β 2 ∇, where ∇ is the voter variance/covariance data matrix, as above, we obtain H1 (z)

   2β β φ(λ1 − λp ) [λ − λ ] ∇ − nI p 1 n σ2   A1 = 2βφ(λ1 − λp ) ∇−I n

=

Obviously, the first term involving φ(λ1 − λp ) in this expression is positive. So the condition that z1 = 0 constitutes a best response to zp = 0 depends on the eigenvalues of C = [ An1 ∇ − I]. If C has non positive eigenvalues, then the same argument can be repeated to show that the matrix [ An2 ∇−I] also has non positive eigenvalues, where A2 = σβ2 (λp − λ2 ), as above. The same argument can be repeated for j = 2, . . . , p − 1. Finally, for the case j = p, note that the coefficient Ap = 0. Thus the relevant Hessian for party p must be negative definite. Consequently z1 = z2 = · · · = zp = 0 will all be mutual best responses. This shows the condition is sufficient. To prove necessity, suppose that C has a positive eigenvalue. Then there exists a number λp+1 > λp with the following properties: First, V1 (z) ≥ V1p+1 (z), in a neighborhood of the origin. Here we define 1X V1p+1 (z) = Φ(gi1p+1 (z)), and gi1p+1 (z) = λ1 − β||xi − z1 ||2 − λp+1 + β||xi − zp ||2 . n i Secondly, V1p+1 (z) also has a Hessian with a positive eigenvalue. It obviously follows that V 1 (z) cannot have a local maximum at z1 = 0.  To illustrate this theorem, we can give corollaries for the one and two dimensional cases. Corollary 2 If X is one dimensional then the necessary and sufficient condition for z ∗0 to be a LNE is that β 2 2 σ 2 (λp − λ1 )v ≤ 1. (Following the definition above, v is the electoral variance on the single axis.) Proof. Just as in the above, gi1 (0) = λ1 − λp is the same, for all i. Moreover, ∇i1 = (dgi1 /dz1 )2 = 4x2i β 2 . Thus, X  ((λp − λ1 )/σ 2 )(x2i β 2 ) − β . (2) H1 (0) = φ(λ1 − λp ) i

7

Clearly, H1 (0) is non-positive if and only if β(λp − λ1 )v 2 ≤ σ 2 . If this condition fails, then z1 = 0 is a minimum of the vote share function, V1 , given (z2 , . . . , zp ) = (0, . . . , 0). Consequently, z∗0 cannot be an LNE.  Notice that as σ 2 → 0, then it becomes impossible for z∗0 to be an LNE if the valences differ. See also Banks & Duggan (2004). Corollary 3 If X is two dimensional then z∗0 is a LNE if β(λp − λ1 )v 2 ≤ σ 2 and only if β(λp − λ1 )v 2 ≤ 2σ 2 . Proof. The condition that both eigenvectors be negative is equivalent to the condition that det(C) is positive and trace(C) is negative. Now   det(C) = (A1 /n)2 (ξ 1 , ξ 1 ) · (ξ 2 , ξ 2 ) − (ξ 1 , ξ 2 )2 + 1 − (A1 /n) [(ξ 1 , ξ 1 ) + (ξ 2 , ξ 2 )] .

The first bracket is non-negative, by the triangle inequality, so det(C) is non-negative if (A1 /n)[(ξ 1 , ξ 1 ) + (ξ 2 , ξ 2 )] ≤ 1, or β(λp − λ1 )v 2 ≤ σ 2 .

(3)

Now trace(C) = (A1 /n)[(ξ 1 , ξ 1 ) + (ξ 2 , ξ 2 )] − 2, and this is non-positive if β(λp − λ1 )v 2 ≤ 2σ 2 .

(4)

If the first inequality (given in Eq. 3) fails, but the second (given in Eq. 4) is satisfied, then the eigenvalues may still be non-positive, and can be explicitly computed in terms of the model parameters and data. If the second condition fails then obviously at least one of the eigenvalues must be strictly positive, and so z ∗0 cannot be a LNE.  In the two dimensional case there is one situation where computation of eigenvalues is particularly easy. If the covariance (ξ 1 , ξ 2 ) of the electoral data on the two axes is (close to) zero, then the data matrix is (approximately) diagonal and the two axes can be treated separately. In this case we obtain two separate necessary and sufficient conditions: β(λp − λ1 )vt2 ≤ σ 2 for t = 1, 2. In the examples below we deal with the one dimensional case first, and then give a two dimensional illustration of the result.

3

Examples

Example 1. Consider the model Σ but with zero valence, in one dimension, with two voters {1, 2} at x1 = −1 and x2 = +1, and two parties {1, 2} at z1 , z2 . As above, the probability voter i picks party 1 over 2 is   P ROB −β(xi − z1 )2 + 1 + β(xi − z2 )2 − 2 > 0 . As before, gi = −β(xi − z1 )2 + β(xi − z2 )2 is the “comparison function” for voter i in comparing party 1 with party 2 when the parties are at (z1 , z2 ). This probability is Φ(gi ), where Φ is the cumulative normal distribution for the variate 2 − 1 , with variance 2σ 2 , and expectation 0. Suppose now that z2∗ = 0 (the mean of the voter ideal points). We seek to determine whether z1 = 0 comprises a best response to z2∗ = 0, thus making up a component of a Nash equilibrium. To do so we seek to determine the first and second order conditions on V1 . Now V1 = 12 [Φ(g1 ) + Φ(g2 )], so dV1 /dz1 = φ(g1 )dg1 /dz1 + φ(g2 )dg2 /dz1 , where φ(gi ) is the value of the normal probability density function, φ, at gi . Since z1 = z2 , we see g1 = g2 and the first order condition is dg1 /dz1 +dg2 /dz1 = 0 or z1∗ = (1/2)(x1 +x2 ) = 0. As before, z1∗ = x∗ . From Equation 1, "  2 # d2 Φ(gi ) dg gi d2 gi . = φ(gi ) − 2 dz12 dz12 2σ dz1 Clearly, if z1 = 0, then gi = 0 for i = 1, 2. Moreover, d2 gi /dz12 = −2β is negative definite. Consequently, V1 has a negative definite Hessian at z1 = 0. Although V1 has a local maximum at z1 = 0, this does not 8

imply that (0, 0) is a Nash equilibrium. (Obviously, the vector (0, 0) is what we have termed a LNE.) The sufficient condition for z1 = 0 to be a best response to z2 = 0, when z1 can lie in the domain [−1, +1], is that V1 is a concave function in z1 (for z2 fixed at 0), in the domain z1 ∈ [−1, +1]. As before, we use the standard result in convex analysis that a necessary and sufficient condition for concavity of a differentiable function on some domain is that its Hessian be negative semi-definite on the domain. To see whether this is so, we evaluate the Hessian of Φ(g2 ) at z1 = −1.     We obtain g2 = −β(2)2 + β(1)2 = −3β. So, d2 Φ(g2 )/dz12 = φ(−3β) −2β + 3β/2σ 2 (4β)2 . Similarly 2 2 d2 Φ(g1 )/dz √ 1 =φ(β)[−2β], which is clearly negative. The first expression is negative only if σ > 12β . Clearly, if σ < β 12, then the Hessian of Φ(g2 ) at z1 = −1 will be positive. Thus, for σ “sufficiently” small, the Hessian of V1 at z1 = −1 will also be positive. In this example, therefore, for some value of σ, with σ ≤ 12β 2 , the expected vote function, V1 will not be concave on [−1, +1]. Consequently, the vector (0, 0) need not be a PNE. It is of course possible for the vote function to be concave on a smaller domain about the origin, but this will not guarantee that 0 is a best response by party 1. As the example should make clear, the requirement of concavity imposes a relationship between the variance of the voter ideal points and the stochastic variance. As expressed in Theorem 1, if the stochastic variance is given, then the constraint is imposed on the variance of the set of bliss points. Alternatively, as in the example, if the bliss points are given then the constraint is imposed on the stochastic variance.  Example 2. We now modify Example 1, by introducing valence terms λ 2 , λ1 with λ2 > λ1 . We first use Corollary 2 to determine the conditions for (0, 0) to be P a LNE. Now dgi /dz1 = 2βxi = ±2β depending on whether i=1 or 2. By Equation 2, H1 (0) = 21 φ(λ1 − λp ) i [((λp − λ1 )/2σ 2 )(2βxi )2 − 2β], so the necessary and sufficient condition is (λp − λ1 )β ≤ σ 2 , as stated in Corollary 2. (In this case the electoral variance is 1.) Suppose this condition is satisfied, so that (0, 0) is an LNE. Then we can obtain a condition for concavity so as to ensure that (0, 0) is a PNE. Let z2 = 0. A sufficient condition for z1 = 0 to be a best response in the domain [−1, +1], is the that the Hessian of V1 be non-positive on this domain. Consider z1 = −1. The comparison functions g2 and g1 are now g2

= λ1 − β(x2 − z1 )2 − λ2 + β(x2 − z2 )2 = λ1 − λ2 − 3β

g1

= λ1 − β(x1 − z1 )2 − λ2 + β(x1 − z2 )2

and

= λ1 − λ2 + β.

The Hessian at z1 = −1 is then o n   φ(λ1 − λ2 − 3β) −2β + (3β + λ2 − λ1 )/2σ 2 [4β]2 + φ(λ1 − λ2 + β){−2β}.

While the second term is clearly negative, the first term may be positive. A sufficient condition for the first term to be negative at z1 = −1 is then σ 2 > 4(3β + λ2 − λ1 )β. Since λ2 > λ1 this condition is more severe than the one obtained in Example 1. Obviously a sufficient condition for the Hessian of V 1 to be negative at z1 = −1 is much more restrictive than the one obtained in Example 1. Moreover, even when the condition for (0, 0) to be a LNE is satisfied, there is no formal reason to expect (0, 0) to be a PNE.  Example 3. To illustrate the computation of eigenvalues in higher dimension, consider a situation where the valences for 1, 2, 3 are ranked 2 < λ3 , and three voters, labeled {1, 2, 3}, have the ideal p with λ1 < λp points in R2 given by (0, 1), (− 3/4, − 21 ) and ( 3/4, − 21 ). With all agents at the origin, the comparison function gi for each voter is given by λ1 − λ3 (in comparing agent 1 against agent 3). So the Hessian for agent 1 involves the matrix 9



−gi [∇i ]β − I H1 = Σi φ(gi ) σ2



Here ∇i is the 2 × 2 matrix generated by the gradients of gi at xi . Summing the terms [∇i ] gives the voter variance/covariance data matrix  3  0 2 ∇= . 0 32 q q As before we let ξ 1 = (− 34 , 0, 34 ) and ξ 2 = (− 21 , 1, − 21 ) be the vectors of voter ideal points on the first

and second coordinates. The the diagonal terms in ∇ are given by the quadratic forms (ξ 1 , ξ 1 ) = (ξ 2 , ξ 2 ) = 32 and the off diagonal terms are given by the scalar product (ξ 1 , ξ 2 ) = 0.Thus the product moment correlation coefficient or covariance between ξ 1 and ξ 2 is zero. The Hessian for agent 1 is given by λ3 − λ 1 [∇]β − 3I. σ2 A necessary condition for this matrix to have negative eigenvalues is that its trace be negative. If the trace is zero or positive, then one of the eigenvalues must be positive (in the degenerate case, both eigenvalues are zero). Now let v 2 = 31 [(ξ 1 , ξ 1 ) + (ξ 2 , ξ 2 )] = 13 [v12 + v22 ] = 1 be the empirical variance of the voter ideal points on the two axes. Thus the necessary condition that the two eigenvalues be negative is that the trace λ3 −λ1 2 σ 2 [v ]β − 2 be negative. Because ∇ is diagonal this condition reduces to the necessary and sufficient condition that 12 (λ3 − λ1 )β is bounded above by σ 2 .If this fails then both of the eigenvalues of the Hessian must be positive at the origin. That is to say, if σ 2 is sufficiently small then z1 = 0 cannot be a best response and so the origin cannot be a LNE. Agent 3 with maximum valence does, however, have a negative definite Hessian at the origin.Note that due to the symmetry of the voter ideal points,if the above condition fails, then agent 1 can increase vote share by moving in any direction away from the origin. In the non symmetric case,with non-zero covariance,when the trace condition fails then one eigenvalue will exceed the other,and agent 1 should move away from the origin in the eigenspace associated with this eigenvalue. 

4

Activist Valence

We now consider a more general model, Σ1 , involving both exogenous valence {λj } and activist valence {µj }. To keep the analysis relatively simple we shall focus below on the competition between two parties, called j and p. We can then apply the results to the case of Britain, where there are two principal parties. As in the proof of Theorem 1, the first order condition for agent j is X

φ(gij (z))

i

dgij = 0. dzj

However, now gij (z) = λj + µj (zj ) − β||xi − zj ||2 − λp − µp (zp ) + β||xi − zp ||2 . Hence the first order solution for agent j is zj∗ =

1 dµi + Σi αij x. 2β dzj

In this equation, the coefficients {αij } involve z and the exogenous valence terms {λj }. Moreover, each αij is strictly increasing in λj and decreasing in λp . Let us denote the vector Σi αij xi by dVj∗ /dzj and call it the “(marginal) electoral pull” due to exogenous valence. 10

Then the first order condition can be written dVj∗ 1 dµj + − zj∗ = 0. dzj 2β dzj

(5)

Say the electoral pull and activist pull are “balanced” if this equation is satisfied. The first term in this expression (the “marginal or gradient electoral pull”) is a gradient vector pointing towards the “weighted electoral mean.” (This weighted electoral mean is simply that point where the electoral pull is zero.) As λj is exogenously increased, this vector increases in magnitude. The vector dµ j /dzj “points towards” the position at which the total of activist “contributions” is maximized. We may term this vector the “(marginal or gradient) activist pull.” Moreover, if the activist function is “sufficiently concave” (with negative eigenvalues of large modulus), then the second order condition (the negative semi-definiteness of the Hessian of the “activist pull”) will guarantee that the vector z∗ given by the solution of the system of equations of the form of (Eq. 5), for all j, will be a LNE. This can be seen by examining the Hessian as in (Eq. 1). The following Theorem states these conclusions (see Schofield (2003b), for a proof). Theorem 4 Consider a formal or empirical vote maximization game, with both exogenous popularity valences {λj } and activist valences {µj }. The first order condition for z∗ to be an equilibrium is that, for each j, the electoral and activist pulls must be balanced. Other things being equal, the position z j∗ will be closer to a weighted electoral mean the greater is the exogenous valence, λ j . Conversely, if the activist valence function µj is increased (due to the greater willingness of activists to contribute to the party) then the nearer will z j∗ be to the activist preferred position. If all activist valence functions are sufficiently concave (in the sense of having negative eigenvalues of sufficiently great magnitude) then the solution given by (Eq. 5) will be a PNE.

Example 4. Figure 1 illustrates this result, in a two-dimensional policy space derived from an empirical model for Britain (Schofield 2003c). In this figure, the positions of the Conservative Party (CONS), Labour (LAB), Liberal Democrats (LIB), Plaid Cymru (PC), Ulster Unionists (UU) and Scottish Nationalists (SNP) were estimated from MP surveys. The two dimensional policy space was estimated from factor analysis of a sample survey. The left right axis is the usual economic axis, while the north south axis represents attitudes to the European Union (with south being pro-Europe, and north pro-Britain.) Figure 1 shows the estimated density function of the distribution of voter ideal points. It is obvious from this distribution that the covariance (ξ 1 , ξ 2 ) between the two axes is close to zero. We can therefore estimate the eigenvalues of the Hessians on these two axes separately. On the single economic axis, the empirical estimate of the exogenous valence of the Conservative Party dropped from 1.58 at the election of 1992 to 1.24 at the 1997 election, while the Labour valence increased from 0.58 to 0.96 over the same period. The Liberal Party valence was set at zero for both elections. Because the scale of the figure is indeterminate, we are at liberty to use the standard deviation as the unit of measurement. For 1997, the spatial coefficient, β, was estimated to be 0.50, while the difference 2 between the Conservative and Liberal Democratic valences was 1.24. The ratio σv 2 of the electoral variance (on the economic axis) to the stochastic variance was one. Therefore the necessary and sufficient condition for the mean voter theorem was satisfied (because (0.5) × (1.24) ≤ 1.) We can infer that when only one axis of policy is relevant, then all parties should converge to the electoral mean. However this conclusion changes when the second policy axis, involving attitudes to Europe, is included. When the eigenvalues for the Liberal Democratic Party are computed, this party’s eigenvalue at the origin on the European axis is positive. As Figure 1 indicates, the Liberal Democratic party vacated the electoral origin, and adopted a pro-European position that was more acceptable to its party activists. While this accounts for the Lib Dem position, it still does not explain the positions taken up by the Conservative Party. According to Theorem 1, this party’s high valence should have given it a vote maximizing position at the electoral origin. However, we can provide an explanation for the locations of the Labour and Conservative Parties by developing the idea that the high valences for these parties were due both to exogenous valence terms for the parties as 11

Figure 1: Estimated party positions in the British Parliament in 1997 (based on MP survey data and a National Election Survey), and the highest density plots of the voter sample distribution at 95%, 75%, 50% and 10% levels.

12

Pro-Britain

B

Activist Pull Economic leftist indifference curve

Optimal Conservative Position Economic conservative indifference curve

Electoral Pull

ECONOMIC DIMENSION

L

C

Labour

Capital

Political cleavage line

Contract curve between economic leftists and proEurope activist

Optimal Labour Position E Pro-Europe Indifference curve

Pro-Europe

Figure 2: Illustration of voter maximizing positions of Conservative and Labor party leaders in a twodimensional party space.

13

well as valence due to activist support. The data presented in Clarke, Stewart & Whiteley (1998) suggest that Labour exogenous valence (λLAB due to Blair) rose in this period. Conversely, the relative exogenous term, λCON , for the Conservatives fell. Since the coefficients (in the equation for the electoral pull) for the Conservative party depend on (λCON − λLAB ), these must all fall in this period. This has the effect of increasing the marginal effect of activism. Indeed, it is possible to include the effect of two potential activist groups for the Conservative Party—one “pro-British” and one “pro-Capital.” The optimal Conservative position will be determined by a version of (Eq. 5) which equates the “electoral pull” against the two “activist pulls.” Since the electoral pull fell ∗ ∗ between the elections, the optimal position zCON , will be one where zCON is “closer” to the locus of points where the marginal activist pull is zero (i.e., where dµCON /dzCON = 0). This locus of points I shall refer to as the “activist contract curve” for the Conservative party. Note that in Figure 2, the indifference curves of representative activists for the parties are described by ellipses. This is meant to indicate that preferences of different activists on the two dimensions may accord different saliences to the policy axes. The “activist contract curve” given in the figure, for Labour say, is the locus of points satisfying the equation dµLAB /dzLAB = 0. This curve represents the balance of power between Labour supporters most interested in economic issues and those more interested in Europe. The optimal positions for the two parties will be at appropriate points on the locus between the respective “activist contract curves” and a point “near” the origin (where the electoral pull is zero). As relative exogenous valence for a party falls, then the optimal party position will approach the activist contract curve. Moreover, the optimal position on this contract curve will depend on the relative intensity of political preferences of the activists of each party. For example, if grass roots “pro-British” Conservative party activists have intense preferences on this dimension, then this feature will be reflected in the activist contract curve and thus in the optimal Conservative position. For the Labour party, it seems clear that two effects are apparent. Blair’s high exogenous popularity gave an optimal position closer to the electoral center than the optimal position of the Conservative party. Moreover, this affected the balance between pro-Labour or “old left” activists in the party, and “new Labour” activists, concerned to modernize the party through a European style “social democratic” perspective. This conclusion is compatible with Blair’s successful attempts to bring “New Labour” members into the party (Seyd & Whitely 2002).

5

Comments on the Strategic Voter Model

A number of objections can be raised against the model presented here. I believe it worthwhile addressing these. a. Voter rationality Why is it appropriate to build a stochastic element into the model? Is it the case that voters choose probabilistically? The purpose of the model is to determine how political agents choose. Opinion poll data will give approximate vote shares for each agent. Each agent can build a rough model of electorate response to declarations. This is proxied by the econometric models of electorate response. These have built in to them a stochastic element, and have proved quite successful at post- or pre-dicting voter response (Poole & Rosenthal 1984, Alvarez & Nagler 1998, Alvarez et al. 2000, Quinn, Martin & Whitford 1999, Schofield, Sened & Nixon 1998b, Schofield et al. 1998a). b. Agent rationality Why should political agents (party leaders or candidates) attempt to maximize expected vote share? In two party competitions maximizing expected plurality or probability of winning is an obvious alternative (Duggan 2000). In two-party competition, empirical estimates of the variance of the vote share functions are generally low, so probability of winning is a close proxy to expected vote share.

14

In multiparty competition, the objections are more serious. A party on the electoral periphery with a large vote share is unlikely to be asked to join a government coalition. Consequently, the correlation between policy control and vote share is weak. Attempts to model strategic attempts to gain votes with control of policy have proved very difficult. It is necessary to combine a model of post-election bargaining (Banks & Duggan 2000) with pre-election maneuvering (Schofield & Sened 2002). The contention of this article is that a simple vote maximization model can lead to the very different political configurations that are found empirically in different countries. c. Activists Early models (Aldrich 1983b, Aldrich 1983a, Aldrich & McGinnis 1989) tended to assume that activists controlled candidate policies. The activist model presented here is based on the assumption that different potential activist groups compete with one another to influence candidates or parties. Moreover, activists are likely to hold policy positions far from the electoral origin. Theorem 4 asserts that this poses a complex optimization problem for agents. If a party leader’s exogenous valence falls, then activists become more important for the party. Thus a party led by a low valence politician may be forced away from the electoral origin. d. The effect of activists on valence It seems most natural in modeling plurality electoral systems (such as Britain or the US) to assume activists affect valence. In such systems, a small advantage in vote shares results in a large seat advantage. Money and activist effort can therefore pay off handsomely. I have not attempted to model the activist calculus, since there are many possible ways to do this. It is plausible that activist calculations have a large role to play in the predominantly two-party system of the US (Miller & Schofield 2003, Schofield et al. 2003). e. Local Nash equilibria In empirical studies these can be determined by simulation. They accord quite well with actual policy positions of parties (Schofield & Sened 2003). It is clear from the simulation that exogenous valence terms have considerable impact on equilibrium positions in multiparty systems based on proportional representation. Since exogenous valence terms fluctuate, these changes induce small changes in local equilibria. Indeed, the local equilibrium notion is compatible with small policy changes by parties as they attempt to adapt to the changing electoral environment.

6

Extensions of the model and generic existence of LNE

One feature of the convergent equilibrium of the “mean voter theorem” is that each agent’s optimum strategy is independent of the strategies of other agents, and is determined only by the distribution of voter ideal points. This has been regarded as an attractive feature of the spatial model. However, as Theorem 1 indicates, this convergent equilibrium will not be a LNE, and therefore cannot be a PNE if the necessary condition is violated. As Example 4 indicates, a key element of the condition is the ratio of the variance of the voter ideal points to the stochastic variance. When there are many parties with differing valences, as in Israel, then simulation of the vote maximzing model has found multiple non convergent LNE (Schofield & Sened 2003). For these non-convergent LNE, agent equilibrium strategies are interdependent. In the proof of Theorem 1, it was assumed that the errors were independent and identically normally distributed (iind). However, estimation of such voter models has found it necessary to adopt the more general hypothesis that the error structure is multivariate normal (that is, allowing for non-zero covariance terms). See Alvarez & Nagler (1998); Quinn et al. (1999); and Schofield et al. (1998a). The proof procedure of Theorem 1 does carry through in this case. However, the first order equation for CNE and the determination of the Hessians at these CNE will then have a more complicated form. The first order equations are matrix equations involving n smooth 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, these wp equations will involve interaction terms induced by covariance. Nonetheless, transversality theory (Banks 1995, Saari 1997, Austen-Smith & Banks 1999) can be used to show that these wp equations are “generically” 15

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 CNE will generically exist. Indeed, it can be shown that one of these CNE will be a LNE. To outline a proof of this assertion, we need to introduce the idea of a tangent bundle (Hirsch 1976). At z ∈ X P , the tangent space is Tz (X P ) and the tangent bundle is defined by T (X P ) = ∪z T z(X P ). The differential dUjh (z) at z can be regarded as a linear map from Rwp to R so dUjh : X P → Lin(Rwp , R). Since dUjh is C 1 differentiable, this map is continuous. Moreover, there is a C 1 -topology on the set of such maps, , under which two profiles are close, if their components are close as linear maps at every z ∈ X P . The differential dUjh can also be projected onto Tz (X). Then this projection DUjh : X P → Lin(Rw , R) can be identified with the gradient of Ujh in X, when zk (for k 6= j) are held fixed. That is to say, DUjh (z) can be regarded as an element of a subtangent space Tz (X) ⊂ Tz (X P ), where X corresponds to the jth strategy space. This uses the fact that Tz (X P ) = Tz (X) × · · · × Tz (X). We use the idea of a conic field generated by U h . Let D(z) = (ConU h )(z) be the convex hull in Tz (X P ) of the set of vectors {DUjh (z) : j ∈ P }. Then ConU h : X P → T X P is a generalized vector field over X P . Its image in Tz X P 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 : X P → T X P of D. That is to say, at every point z ∈ X P , m(z) ∈ D(z) ⊂ Tz X P . Note that by this construction each DUjh (z) lies in a different tangent space. Thus, 0 ∈ ConU h (z) if and only if DUjh (z) = 0 for every j ∈ P . We shall now use standard transversality theory to show CNE generally exist, and are “locally isolated.” We say a property of points in X P is locally isolated if it holds for 0-dimensional submanifolds of X P . Say a property K is C 1 -generic in  if the set 1 = {U h ∈  : U h has property K} is open dense in the 1 C -topology on . Definition 4 Let K1 be the property that there exists a locally isolated LNE for the profile U h 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 5 K2 is C 1 -generic. dU j

Proof. Given U h , let Tj (U h ) be the set in the jth strategy space such that dzjh = 0. By the inverse function theorem, Tj (U h ) is generically a smooth submanifold of X P of dimension (p − 1)dim(X), i.e., of codimension dim(X). But then ∩j Tj (U h ) is of codimension pdim(X). Thus the property K2 is generic.  Definition 5 A profile U h satisfies the boundary condition if for any z in the boundary, ∂X P , the differential (DUjh (z), . . . , DUjh (z)) points inward. That is, if n(z) is the normal to the boundary at z, then DUjh (z)(n(z)) > 0, for all j. Lemma 6 If U h satisfies the boundary condition and if U h exhibits a locally isolated CNE, then there exists a LNE. Proof. We have defined the conic field D = Con(U h )X P → T X P in the preliminary to this section. The conic field D admits a selection m : X P → T X P , that is a continuous function such that m(z) ∈ Con(dU h (z)). Moreover, m(z ∗ ) = 0 if and only if DUjh (z ∗ ) = 0, for all j ∈ P . Clearly, 0 ∈ Con(dU h (z)) iff DUjh (z) = 0 for all j ∈ P . The selection is a vector field on X P . Moreover, m can be selected to be a gradient vector field on X P which satisfies the boundary condition, i.e., m(x)(n(x)) > 0 for all x on the boundary of X P . Clearly, m(z ∗ ) = 0 iff z ∗ is a CNE. Because of the Morse inequalities (Milnor 1963), m must exhibit a stable equilibrium, z ∗ , say. 16

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 characteristic; see Brown (1970).) Theorem 7 Existence of LNE is generic in the topological subspace 0 = {U h ∈  : U h satisfies the boundary condition}. Proof. By Lemma 5, there exists an open dense subset 00 on which K2 is satisfied. But then 0 ∩ 00 gives an open dense subspace of . For every profile in this set, the procedure of Lemma 6 gives a LNE.  Now let ∗ = {(u, λ) : X n+p → Rnp } denote the set of all electoral systems. That is to say, once bliss points in X n and agent strategies in X P are known, the n × p array of voter utilities is specified. For convenience, we shall regard the covariance structure, and the variances {σ 21 , . . . , σ 2p } ,as fixed since the variances are essentially scale factors. Given (u, λ) ∈ ∗ and the set P of political agents, then the political game U h is well defined. Thus there is a continuous electoral map EP : ∗ → . By Theorem 7, existence of LNE is generic in 0 . Since EP is continuous, the inverse image of an open set in 0 is open in the co-image of EP . Suppose EP is also proper: that is, if Y is open in ∗ , then its image is open in 0 . In this case, the inverse image of a dense set is dense. Theorem 8 Existence of LNE for a given set P of political agents is a generic property in  ∗ , if the electoral map is proper. The results on transversality theory used in Theorem 7 can be found in Hirsch (1976), Smale (1960), and Smale (1974). Dierker (1974) discusses the Morse inequalities. A review of this material can be found in Schofield (2003a). A minor point concerns the nature of the topology. Theorems 7 and 8 involve topologies on utilities. Equilibrium concepts in both politics and economics should be based on preferences. Schofield (1999) has proposed a C 1 -topology on preferences, when their utility representations are smooth. This suggests that Theorem 8 can be expressed as a genericity result for preferences.

7

Conclusion

In this exposition, I 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 economic commodity space. Economic agents have preferences on Z, and can, in theory, 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 with Konishi’s 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 ideal points. Note, however, that there are a number of further theoretical difficulties. Firstly, the map ψ may be multivalued. For any given economy, there will be many possible Pareto incomparable economic local equilibria. 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.

17

Although the article has focused on party leaders who attempt to maximize vote share, the general model can, in principle, be extended to include more general candidate or leader motivations, as long as these can be assumed to be C 2 -differentiable functions.

References Aldrich, J. (1983a), ‘A Downsian spatial model with party activists’, Amer Pol Sci Rev 77, 974–990. Aldrich, J. (1983b), ‘A spatial model with party activists: Implications for electoral dynamics’, Public Choice 41, 63–100. Aldrich, J. & McGinnis, M. (1989), ‘A model of party constraints on optimal candidate positions’, Math and Com Mod 42, 437–450. Alvarez, M. R. & Nagler, J. (1998), ‘When politics and models collide: estimating models of multicandidate elections’, American Journal of Political Science 42, 55–96. Alvarez, M. R., Nagler, J. & Bowler, S. (2000), ‘Issues, economics, and the dynamics of multiparty elections: The British 1987 general election’, Amer Pol Sci Rev 94, 131–150. Ansolabehere, S. & Snyder, J. M. (2000), ‘Valence politics and equilibrium in spatial election models’, Public Choice 103, 327–336. Aragones, E. & Palfrey, T. (2003), Spatial competition beteen two candidates of different quality:the effects of candidate ideology and private information, Technical report, California Institute of Technology. Austen-Smith, D. & Banks, J. (1998), ‘Social choice theory, game theory, and positive political theory’, Ann Rev Pol Sci 1, 259–287. Austen-Smith, D. & Banks, J. (1999), Positive Political Theory I: Collective Preference, Michigan University Press. Austen-Smith, D. & Duggan, J., eds (2004), Social Choice and Strategic Behavior, Springer. Banks, J. (1995), ‘Singularity theory and core existence in the spatial model’, J Math Econ 24, 523–536. Banks, J. & Duggan, J. (2000), ‘A bargaining model of collective choice’, Amer Pol Sci Rev 103, 73–88. Banks, J. & Duggan, J. (2004), ‘The theory of probabilistic voting in the spatial model of elections: The theory of office-motivated candidates’, in Austen-Smith & Duggan (2004). Banks, J., Duggan, J. & Lebreton, M. (2002), ‘Bounds for mixed strategy equilibria and the spatial model of elections’, Jour of Econ Theory 103(1), 88–105. Banks, J., Duggan, J. & Lebreton, M. (2003), Social choice in the general model of politics, Technical report, University of Rochester. Brown, R. F. (1970), The Lefshetz Fixed Point Theorem, Scott and Foreman. Clarke, H., Stewart, M. & Whiteley, P. (1998), ‘New models for new labour: the political economy of labour support: January 1992 – April 1997’, Amer Pol Sci Rev 92, 559–575. Coughlin, P. (1992), Probabilistic Voting Theory, Cambridge University Press. Dierker, E. (1974), Topological Methods in Walrasian Economics, number 92 in ‘Lecture Notes on Economics and Mathematical Sciences’, Springer. 18

Downs, A. (1957), An Economic Theory of Democracy, Harper and Row. Duggan, J. (2000), Equilibrium equivalence under expected plurality and probability of winning maximization. Fan, K. (1961), ‘A generalization of Tychonoff’s fixed point theorem’, Mathematische Annalen 42, 305–310. Groseclose, T. (2001), ‘A model of candidate location when one candidate has valence advantage’, American Journal of Political Science 45, 862–886. Hinich, M. (1977), ‘Equilibrium in spatial voting: The median voter result is an artifact’, Jour of Econ Theory 16, 208–219. Hirsch, M. W. (1976), Differential Topology, Springer. Kavanagh, D., ed. (1992), Electoral Politics, Clarendon Press. Konishi, H. (1996), ‘Equilibrium in abstract political economies: With an application to a public good economy with voting’, Social Choice and Welfare 13, 43–50. Lin, T.-M., Enelow, J. & Dorussen, H. (1999), ‘Equilibrium in multicandidate probabilistic spatial voting’, Public Choice 98, 59–82. McKelvey, R. D. (1986), ‘Covering, dominance, and institution-free properties of social choice’, American Journal of Political Science 30, 283–314. Michael, E. (1956), ‘Continuous selection I’, Annals of Mathematics 63, 361–382. Miller, G. & Schofield, N. (2003), ‘Activists and partisan realignment in the United States’, Amer Pol Sci Rev 97, 245–260. Milnor, J. (1963), Morse Theory, number 51 in ‘Annals of Mathematics Studies’, Princeton University Press. Peixoto, M., ed. (1974), Dynamical Systems. Poole, K. & Rosenthal, H. (1984), ‘Us presidential elections 1968-1980: A spatial analysis’, American Journal of Political Science 28, 283–312. Quinn, K., Martin, A. & Whitford, A. (1999), ‘Voter choice in multiparty democracies’, American Journal of Political Science 43, 1231–1247. Saari, D. G. (1997), ‘Generic existence of a core for q-rules’, Economic Theory 9, 219–260. Schofield, N. (1999), ‘The C1 topology on the space of smooth preference profiles’, Social Choice and Welfare 16, 445–470. Schofield, N. (2003a), Mathematical Methods in Economics and Social Choice, Springer. Schofield, N. (2003b), ‘Valence competition in the spatial stochastic model’, Journal of Theoretical Politics 15, 371–383. Schofield, N. (2003c), A valence model of political competition in Britain, 1992–1997, Technical report, Washington University at Saint Louis. Schofield, N. (2004), ‘Equilibrium in the spatial valence model of politics’, Journal of Theoretical Politics. Schofield, N. & Sened, I. (2002), ‘Local Nash equilibrium in multiparty politics’, Annals of Operations Research 109, 193–210. 19

Schofield, N. & Sened, I. (2003), Multiparty competition in Israel, Technical report, Washington University at Saint Louis. Schofield, N., Martin, A., Quinn, K. & Whitford, A. (1998a), ‘Multiparty electoral competition in the Netherlands and Germany: A model based on multinomial probit’, Public Choice 97, 257–293. Schofield, N., Miller, G. & Martin, A. (2003), ‘Critical elections and political realignment in the us:1860–2000’, Political Studies 51, 217–240. Schofield, N., Sened, I. & Nixon, D. (1998b), ‘Nash equilibrium in multiparty competition with ‘stochastic’ voters’, Annals of Operations Research 84, 3–27. Seyd, P. & Whitely, P. (2002), New Labour’s Grassroots, Macmillan. Smale, S. (1960), ‘Morse inequalities for a dynamical system’, Bull Am Math Soc 66, 43–49. Smale, S. (1974), ‘Global analysis and economics I: Pareto optimum and a generalization of Morse theory’, in Peixoto (1974), pp. 531–544. Stokes, D. (1992), ‘Valence politics’, in Kavanagh (1992), pp. 141–162.

20