Policy Divergence in Multicandidate Probabilistic Spatial Voting

Policy Divergence in Multicandidate Probabilistic Spatial VotingAuthor(s): James AdamsSource: Public Choice, Vol. 100, No. 1/2 (Jul., 1999), pp. 103-122Published by: SpringerStable URL: http://www.jstor.org/stable/30026082 .

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Public Choice 100: 103-122, 1999. 103 © 1999 Kluwer Academic Publishers. Printed in the Netherlands.

Policy divergence in multicandidate probabilistic spatial voting

JAMES ADAMS Department of Political Science, University of California at Santa Barbara, Santa Barbara, CA 93106, U.S.A.; e-mail: [email protected]

Accepted 19 November 1997

Abstract. Existing models of multicandidate spatial competition with probabilistic voting typically predict a high degree of policy convergence, yet in actual elections candidates advocate quite divergent sets of policies. What accounts for this disparity between theory and empirical observation? I introduce two variations on the basic probabilistic vote model which may account for candidate policy divergence: 1) a model which incorporates candidate-specific variables, so that candidates may enjoy nonpolicy-related electoral advantages (or disadvantages); 2) a model which allows nonzero correlations between the random terms associated with voters' candidate utilities, thereby capturing situations where voters view two or more candidates as similar on nonpolicy grounds. I report candidate equilibrium analyses for each model, which show far greater policy divergence than exists under the standard probabilistic vote model. I then analyze the strategic logic which underlies these results.

1. Introduction

Until recently, the spatial theory of multicandidate competition - i.e., competition involving three or more candidates - has been dominated by the classic Downsian model, in which each voter votes for the candidate whose platform is closest to his own preferred position (Eaton and Lipsey, 1975; Denzau, Katz and Slutsky, 1985; Hermsen and Verbeek, 1992). However, beginning with the work of de Palma, Ginsberg, Labbe, and Thisse (1989) and de Palma, Hong, and Thisse (1990), a number of recent papers model multicandidate competition in situations where voters' candidate utilities are perturbed by a random component which renders their choices probabilistic, from the candidates' perspectives (Adams, 1997a; Lin, Enelow, and Dorussen, 1997; Lom- borg, 1997; Nixon, Olomoki, Schofield, and Sened, 1995; Schofield, Martin, Quin, and Whitford, 1997).' The central quest of both deterministic and probabilistic multicandidate spatial modellers has been to determine the locations (if any) of candidate equilibria - i.e., locations in the policy space to which vote-seeking candidates will gravitate in order to win election.

Probabilistic multicandidate spatial models produce quite different equilibrium results from their deterministic counterparts. While deterministic studies conclude that multicandidate equilibria are unlikely except under con-



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trived circumstances2 (Eaton and Lipsey, 1975), equilibria appear quite likely for probabilistic vote models. The greater the variance associated with the random component in voters' candidate evaluations - i.e., the less emphasis voters attach to candidates' platforms compared to their nonpolicy motivations - the more likely probabilistic equilibria become. When the nonpolicy component is sufficiently large an equilibrium invariably exists, which locates all candidates at the "minimum-sum point", which minimizes voters' policy losses over the entire electorate (Adams, 1997a; Lin, Enelow and Dorussen, 1997). For more policy-oriented electorates equilibrium becomes more prob- lematic; however, when such equilibria exist they typically display considerable policy convergence, with the candidates coalescing into a limited number of blocs, each representing two or more candidates who present identical platforms (see de Palma, Hong, and Thisse, 1990: Figures 2-8; Nixon et al., 1995: 29). Hence policy convergence appears as a central feature of multicandidate probabilistic equilibria.

While the probabilistic voting studies cited above represent an advance over deterministic models, their conclusions are troubling on both empirical and conceptual grounds. Empirically, the prediction that candidate equilibria will involve policy convergence is incompatible with empirical evidence that in multicandidate races throughout the world, candidates do not adopt identical policies (Budge, 1994; Pierce, 1995: 70-71). This empirical finding obtains despite abundant behavioral research suggesting that voters have important nonpolicy motivations - the very condition that motivates policy convergence, according to probabilistic voting theory.3

On a conceptual level, these studies typically make two strong behavioral assumptions. The first is that all candidates are equally popular with the electorate on nonpolicy grounds, so that a voter who is indifferent between two or more candidates on policy grounds must have equal probabilities of preferring either candidate.4 This assumption simplifies the equilibrium analysis, but (as the authors of these studies recognize) it clashes with behavioral researchers' empirical findings, that voters frequently have strong, measured nonpolicy biases. These include such factors as voters' comparative assessments of candidates' personal qualities (competence, integrity, etc.) retrospective evaluations of the incumbent's performance, and so on. Such considerations render certain candidates attractive to the electorate on nonpolicy grounds (such as Eisenhower, Reagan, or de Gaulle, for instance), while other candidates suffer nonpolicy disadvantages (Converse and Dupeux, 1966; Fiorina, 1981; Page and Jones, 1979).5

The second behavioral assumption spatial modellers employ is that the error terms associated with voters' evaluations of competing candidates are uncorrelated. This specification - which amounts to the substantive assump-



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tion that voters do not perceive similarities between competing candidates - has been criticized on empirical grounds by both Alvarez, Bowler, and Nagler (1996) and Dow (1997), who find significant nonzero correlations between the error terms associated with voters' candidate and party evaluations in recent British and French elections. Given that these correlations alter voters' vote probabilities, it is plausible that they affect candidate equilibria as well.

This paper presents a model of multicandidate probabilistic voting which relaxes the strong behavioral assumptions summarized above. As in past multicandidate probabilistic models, the model incorporates a policy salience parameter, which represents the importance voters attach to policies compared to unmeasured, nonpolicy motivations. However, it also includes candidate- specific popularity terms which may bias the voters' choices towards (or against) certain candidates. In addition, the model allows correlations between voters' unmeasured candidate utility terms, thereby allowing us to model situations where voters perceive similarities between candidates.

I outline three versions of this vote model in Section 2: a "basic" model similar to those employed in previous studies; a "candidate popularity" model which incorporates candidate-specific terms; a "candidate similarity" model which relaxes the assumption of uncorrelated error terms. In Section 3 I compare candidate equilibria for each version of the model. I find that while the basic model accommodates primarily agglomerated equilibria, the popularity model and the similarity model typically accommodate dispersed equilibria. This occurs because the candidates who suffer electoral disadvantages under these more complex models - either because they are unpopular on nonpolicy grounds or because they are viewed as similar to rival candidates - are motivated to present relatively extreme policies. I discuss the strategic logic which underlies these results, and then in Section 4 I draw conclusions and suggest extensions of my approach.

2. A probabilistic vote model: Three alternative versions

In the model, a set V = {1, 2,..., m} of voters and C of candidates are located in a unidimensional policy space. For each voter i e V, i's location xi represents his position along the policy dimension, while k represents the position of candidate K e C. (Note that in what follows I refer to voters as "he" and candidates as "she".) Let voter i's utility for K's platform be given by the real-valued function p(xi, k). Let s be a nonnegative salience parameter, which varies with the importance voters attach to the candidates' platforms. Finally, let 8ik represents a random component which represents unmeasured, nonpolicy sources of the voters' candidate utility.



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2.1. The basic vote model

For the basic probabilistic voting specification voter i's utility Ui(K) for K is given as:

Ui(K) = s x p(xi, k) + Eik (1)

Theoretical results on multicandidate spatial competition for the general form (1) have been obtained by Lin, Enelow, and Dorussen (1997), while Adams (1997a, 1997b), de Palma, Hong, and Thisse (1990) and Nixon et al. (1995), obtain results for a specific functional form of (1) - the logit. For the logit, the ei are independently and identically distributed according to the double exponential, the distribution of which is F(x) = exp[-exp(-x)], with a mean equal to Euler's constant and a variance of r2/6. I adopt this specification for the basic vote model.

Because the ei are unobserved the voter's decision is probabilistic, from the candidates' perspectives. Given the double-exponential specification, the probability Pi(K) that the voter prefers K to all rival candidates in C is

Pi(K) = esxp(xi'k)/S( esxp(xi'j)) (2)

JEC

Proof: See McFadden (1978).

It follows from the functional form of (2) that if voter i is indifferent between two candidates K and L on policy grounds - i.e., p(xi, k) = p(xi, 1) - he has equal probabilities of supporting each candidate. Note further that the degree of uncertainty associated with i's decision varies with the salience parameter s. The larger the value of s, the more weight the voter places on policies, so that as s -+ oo we approach the limiting case of deterministic policy voting. By contrast, when s -+ 0 voter i attaches little or no weight to candidates' policies, and his choice appears random, from the candidates' perspectives. Finally, note that Pi(K) in Equation (2) does not depend on the behavior of other voters, which implies that voters vote sincerely for their most preferred candidate.

The central theoretical result on multicandidate equilibrium for the basic model is that when the policy salience coefficient s is sufficiently small (and positive), then an agglomerated equilibrium exists for which all candidates adopt identical policy positions (Adams, 1997a; Lin, Enelow, and Dorussen, 1997). In addition, simulation studies by Adams (1997a) and de Palma, Hong, and Thisse (1990) have found that this agglomerated equilibrium persists for quite large values of s, which exceed the parameters behavioral researchers



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have estimated for various historical elections. While dispersed equilibria have been located in elections in which voters display high degrees of policy motivations (e.g., de Palma, Hong, and Thisse, 1990: Figures 1-8; Nixon et al., 1995: 29) even these equilibria typically find the candidates clustered in a small number of blocs, with two or more candidates at each equilibrium location.

2.2. The candidate popularity model

From the perspective of behavioral researchers, Equation (1) appears under- specified, in that it ignores important and measurable nonpolicy considerations, such as voters' retrospective evaluations of incumbent performance, their evaluations of candidates' personalities, competence, integrity, and so on. For instance, empirical analyses of presidential elections in the United States and France have shown that candidates such as George Bush (versus Dukakis) and Frangois Mitterand have enjoyed nonpolicy-related advantages over their opponents (Erikson and Romero, 1990; Pierce, 1995). I incorporate the effect of these factors by adding a candidate-specific term bk to Equation (1):

Ui(K) = bk + s x p(xi, k) + 8ik, (3)

where bk represents K's nonpolicy attractiveness or popularity. Note that bk is not subscripted for voter i, indicating that I assume that nonpolicy factors have the same impact on all voters. This strong assumption is plausible for modeling the effects of such influences as national economic conditions or the effects of judgments concerning candidate competence or integrity; however, it is not appropriate for modeling the effects of variables which vary across voters, such as party identification and sociodemographic characteristics. I consider the effects of such variables elsewhere (Adams, 1998; 1997b).

For the model given in Equation (3), the probability that i votes for K is

Pi(K) = ebK+SXP(xi'k)/(S ebJ+sxp(xij)) (4) JeC

It is easily seen from Equation (4) that a voter who is indifferent between two candidates K and L on policy grounds may nonetheless have differing probabilities of supporting the candidates, due to different values of bK and bL. In what follows K is said to be more popular than L if bK > bL, while K and L are equally popular if bK = bL.



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2.3. The candidate similarity model

To my knowledge, all previous studies of equilibrium in multicandidate probabilistic spatial competition employ the assumption used in the models outlined above, that the error terms ei associated with voters' candidate evaluations are independently distributed (Adams, 1997a, 1997b; de Palma, Hong, and Thisse, 1990; Lin, Enelow, and Dorussen, 1997; Lomborg, 1997; Nixon et al., 1995).4 This amounts to the strong behavioral assumption of inde- pendence of irrelevant alternatives, i.e. that the ratio of the probabilities of voting for any two candidates not depend on the presence of other candidates in the available choice set. This assumption can lead to implausible vote probabilities in situations where voters view two or more of the candidates as similar, and therefore substitutable, on nonpolicy grounds. For instance, suppose that under the basic probabilistic vote model an election pits an incumbent candidate K against a challenger L1, and that a voter i has identical policy utilities for both candidates, i.e., p(xi,k) = p(xi,l1). Given the assumption of independent and identically distributed error terms, it is clear from Equation (2) that i has equal probabilities of voting for K and L. Now suppose the field of candidates is expanded to include a second challenger L2 and that the voter is indifferent between the policies proposed by L1 and L2 - i.e., p(xi, 11) = p(xi, 12). It seems plausible that L2 might split Ll's vote in the three-way race, and in particular, that the probability of voter i's choosing K should be less affected than the probability of choosing L1. However, under the assumption of independent error terms, Equation (2) shows that the model would predict:

p(xi, k) = p(xi, 11) = p(xi, 12) = Pi(K) = Pi(L1) = Pi(L2) = 1/3.

Dow (1997) has shown that the IIA assumption is violated with respect to voters' utilities for the candidates competing in the 1995 French presidential election, while Alvarez, Bowler, and Nagler (1996), report the same finding with respect to voters' preferences for the competing parties in Britain in 1987.

To model situations in which voters perceive similarities between candidates, I employ the generalized extreme value distribution (GEV), of which the double exponential described above is a special case. Using the GEV specification (the details of which are presented in the Appendix), it is pos- sible to model situations in which the error terms associated with voters' candidate evaluations are correlated with each other. Specifically, let the set C of candidates be partitioned into n subsets labeled S1i,...,Sn where 9f = {Si, ..., Sn). Within each subset SF, let rF represent the correlation between the error terms Sip and eiQ associated with voters' unmeasured utilities for any



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two candidates P e SF and Q e SF. Furthermore, let the correlation between all eip and 8iR for which candidates P and R are in different subsets equal zero. Under these conditions, the probability that voter i prefers candidate K e SF

is:

Pi(K) =

e[s xp(xi,k)l/(1-rF) (CJESF e[SXp(xi,j)]/(1 -rF))-rf

LsGEf(-JESG esxp(xi,j)]/(1-rG))l1-rG (5)

Proof: See McFadden (1978).

Note that when rF equals zero for all SF E 9ti, indicating no correlation between the unmeasured components of candidate utilities, Equation (5) reduces to the basic logit model of Equation (2). To illustrate the effect of in- troducing correlated error terms, suppose that in the election described above candidates L1 and L2 comprise one subset S1 and K comprises the second subset S2, and that the correlation between the error terms associated with voters' evaluations of L1 and L2 is rl = 0.5. In this case Equation (5) gives voter i's vote probabilities as Pi(K)= .42, and Pi(L1) = Pi(L2) = .29. By contrast, for the case of uncorrelated error terms Pi(K) = Pi(L1) = Pi(L2) = 1/3.

3. Candidate equilibria for alternative vote models

To compare candidate equilibria under the three vote models outlined above, I consider a simple scenario in which three vote-maximizing candidates W, Y, and Z compete for votes from an electorate uniformly distributed along the policy continuum [0, 10]. I assume that each voter i e V evaluates the policy positions of the candidates according to quadratic losses, with the error terms associated with voters' candidate evaluations distributed according to the double exponential.

Because deriving multicandidate equilibria analytically is a problem of great complexity (see Lin, Enelow, and Dorussen, 1997), I instead report the results of computerized searches designed to locate equilibria. I report these results for the three models of voter motivations described above: the basic vote model given by Equation (2); the candidate popularity model given by (4); the candidate similarity model given by (5). For each model I ask whether candidate equilibria exist (for varying values of the policy salience coefficient s), and if so, where they are located. I then analyze the strategic logic which underlies these results.



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3.1. Candidate equilibria for the basic vote model

Using (2), the expected vote E(W) of candidate W for an electorate uniformly distributed on [0, 10], which evaluates candidates' policies according to quadratic losses, is

E(W) = JPi(W)

=

S010e(sx[-(xi-w)2]/(SjeC esx[-(xi-j)2]), (6)

with comparable expressions for E(Y) and E(Z). Using (6), the simulation proceeded as follows. First, a value of s was

specified, and all candidates were located at the voter mean 5.0. Next, with the positions of Y and Z fixed, W's expected vote share E(W) was calculated at each location from the set of 101 positions {0, 0.1, ..., 10}, and W was then located at her expected vote-maximizing position. Next, with the positions of W and Z fixed, candidate Y's vote-maximizing policy position was similarly computed.7 This process continued until either: 1) the candidates reached a Nash equilibrium - defined as a configuration such that no candidate could increase her expected vote share by relocating, or, 2) each candidate moved 200 times without the candidates reaching an equilibrium configuration. In the latter case the scenario was defined as being in disequilibrium.

Figure 1, which shows the results of simulations for varying values of s, indicates that equilibria exists for s < .16. An agglomerated equilibrium exists over this entire range of values, in which all candidates locate at the voter mean 5.0. Furthermore, over the range .10 < s < .13 a second, dispersed equilibrium exists, in which two candidates locate symmetrically on either side of the voter mean, while the third locates at the mean; note that these dispersed equilibria involve rather small degrees of policy differentiation, with all candidates locating in the policy interval [4,6].8 Finally, no equilibrium exists for s > .16.

The finding of agglomerated equilibria for small values of s is consis- tent with earlier theoretical results (Adams, 1997a; de Palma et al., 1989, 1990; Lin, Enelow, and Dorussen, 1997), that for low values of s a unique agglomerated equilibrium must exist. What is more significant is that agglomerated equilibria persist even for significantly nonzero values of s, which exceed the parameter estimates behavioral researchers have reported for various historical elections. For instance, Alvarez and Nagler (1995: Table 3) estimate a voter salience parameter of approximately s = .06, with respect to the three-candidate 1992 U.S. presidential election, while Rivers (1988: Table 2) estimates s = .05 and s = .10 for alternative models of voters' utility functions.9 Each of these parameters estimates supports an agglomerated three-candidate equilibrium, for the stylized situation analyzed here. Finally,



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locationss

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5

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1

0 s=1 s=3 s=5 s=7 s=9 s=11 s=13 s=15 s=17 s=19

Degree of policy voting

Figure 1. Candidate equilibria for the basic vote model

note that equilibria break down for s > .17 because the shape of candidates' expected vote functions begins to resemble the function for deterministic policy voting, for which we have seen that candidate equilibria rarely exist.

The simulation results for the basic model thereby confirm previous theoretical work, that probabilistic voting leads to a high degree of candidate agglomeration, which holds for realistic values of s. Completely convergent equilibria are the norm, while even those rare dispersed equilibria display only modest degrees of policy divergence.

3.2. Equilibria for the candidate popularity model

Using (4) the expected vote E(W) for candidate W under the candidate popularity model is

E(W) = JPi(W) = 0

10

J ebw+s x

[-(xi-w)]2/(Z ebj+sx[-(xi-i)])2), (7)

JeC

with equivalent expressions for E(Y) and E(Z). In simulating candidate competition for this model while varying s, we

must also specify values for the candidate-specific parameters bw, by, and bz. There are three mutually exclusive cases to examine: 1) the case where all candidates have different popularity parameters; 2) the case where one candidate is more popular than her two rivals, who are equally popular; 3) the



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case where one candidate is less popular than her two equally popular rivals. (The fourth case, in which all candidates have equal popularity parameters, reduces to the basic vote model examined above.)

3.2.1. Case 1: All candidates have different popularity parameters Here I examine the case where bw = 1, by = 0, and bz = -1 - i.e., W is more popular than Y, who is more popular than Z. For these parameters, the vote probabilities for a voter indifferent between the candidates on policy grounds are given by (4) as Pi(W) = .64, Pi(Y) = .28, and Pi(Z) = .08. Using (7), the simulations followed the same procedure used for the basic vote model, with the candidates initially locating at the voter mean, and then moving sequentially in search of their vote-maximizing positions.

Figure 2A, which shows the results of simulations for varying values of s, indicates that candidate equilibria are dramatically different from the results for the basic model. For s < .06 agglomerated equilibria exist, as was the case for the basic model. However, for .07 < s < .18 only dispersed equilibria exist. These equilibria locate W, the most popular candidate, near the center of the policy space, while the less popular candidates Y and Z locate at progressively greater distances from the center as s increases. (Note that a second, equivalent set of dispersed equilibria exist for this model, which are symmetric with the first set with respect to the voter mean.) Furthermore, these dispersed equilibria involve far greater degrees of policy differentiation than was the case for the unbiased vote model, with the two extreme candidates Y and Z separated by up to 5 units on the 10-point policy scale.

The strategic logic that motivates Y and Z to differentiate their policies from W revolves around W's greater popularity. To grasp the underlying intuition, note that from Z's perspective, if Z locates near W in the policy space, then all voters will have higher measured utilities for W than for Z. While voters' unmeasured motivations will prompt some support for Z, her expected vote will be quite low. By adopting an extreme position that differentiates her policy from W, Z becomes significantly more attractive to extreme voters (on her side of the policy continuum), whose vote probabilities for Z increase sharply. While this policy shift depresses Z's appeal to centrist voters (as well as voters on the opposite end of the policy continuum), these voters were unlikely to support Z in any case, which limits the electoral cost of shifting to an extreme position. A similar logic explains Y's motivation to differentiate herself from W; however, since Y is more popular than Z, her need to differentiate herself from W is less acute, and consequently she moves a shorter distance away from the mean.

Note that the above argument is similar to the argument advanced by Feld and Grofman (1991) with respect to two-candidate elections in which



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10 -

9

V 7 E

0 0 o 6

E 5 m

2

C


Figure 2A. Equilibria for the candidate popularity model: bw = by = 1, bz = 0

voters give the incumbent the "benefit of the doubt" - i.e., give the incumbent candidate a nonpolicy-related advantage in comparison with her rival.1l The authors argue that a "benefit of the doubt can force challengers to locate relatively far away from the incumbent in order to beat him ...because a challenger with nearly identical issue positions to the incumbent cannot win (1991: 28)."

3.2.2. Case 2: Two candidates are equally popular; the third is more popular

Here I examine the case where bw = 1 and by = bz = 0. For these parameters, the vote probabilities for a voter indifferent between the candidates on policy grounds are Pi1(W) = .58, and Pi(Y) = Pi(Z) = .21.

Figure 2B, which shows candidate equilibria for varying values of s, shows a pattern quite similar to the results for case 1. For s < .06, only agglomerated equilibria exist, while for .07 < s < .08, both dispersed and agglomerated equilibria exist. For .09 < s < .16 only dispersed equilibria exist. As in the previous case, it is the less popular candidates Y and Z who adopt extreme policy positions, for the reason outlined above: candidate W's popularity motivates her less popular rivals to differentiate themselves from W on policy grounds.



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0

w

Y

Z

s=1 s=3 s=5 s=7 s=9 s=11 s=13 s=15 s=17 s=19


Figure 2B. Equilibria for the candidate popularity model: bw = 1, by = bz = 0

3.2.3. Case 3: Two candidates are equally popular; the third is less popular Here I examine the case where bw = by = 1, and bz = 0. For these parameters, the vote probabilities for a voter indifferent between the candidates on policy grounds are Pi(W) = .Pi(Y) = .42, and Pi(Z) = .17.

The equilibrium results for these candidate-specific parameters are reported in Figure 2C. As was the case for the earlier simulations, at low levels of policy salience (s < .07) a unique agglomerated equilibrium exists. How- ever, in the interval .08 < s < .16 dispersed equilibria exist in which the least popular candidate Z takes relatively extreme positions, while W and Y are paired near the center. The strategic logic which motivates Z's noncentrist positioning is - as in cases 1 and 2- the desire to differentiate her policy from those of her more popular rivals.

In summary, the introduction of candidate-specific parameters motivates considerable policy dispersion, compared with the results for the basic model. This occurs because less popular candidates shift towards policy extremes in order to differentiate themselves from their more popular rivals. However, both the basic model and the candidate popularity model support equilibria over similar ranges of values of the policy salience coefficient s. Hence the candidate popularity model alters the nature of multicandidate equilibria, but does not fundamentally alter the conditions for the existence of equilibria, compared with the basic vote model.



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W and Y

Z

s=1 s=3 s=5 s=7 s=9 s=11 s=13 s=15 s=17 s=19


Figure 2C. Equilibria for the candidate popularity model: bw = by = 1, bz = 0

3.3. Equilibrium for the candidate similarity model

Using (5), the expected vote E(W) for candidate W under the candidate similarity model is

E(W) = 1

eSXp(xi'w)]/(1-rF)( (exp(xij)]/(1-rF)-rF) / 1100 JeS-F

(e[S p(xij)]/(1-r))

1

)

(8) JeSG

with equivalent expressions for E(Y) and E(Z). To explore the effect of candidate similarity, it is necessary to specify the

correlation structure of voters' nonpolicy utilities for the candidates. Here I consider two cases: 1) a scenario in which the error terms associated with voters' utilities for two of the candidates are perfectly correlated with each other, which amounts to the substantive assumption that these candidates are viewed as perfectly substitutable on nonpolicy grounds;1" 2) a scenario in which the correlation between voters' nonpolicy utilities for two of the candidates is .69, and the correlation with the third candidates is zero. I examine the latter specification because it reflects empirical estimates reported by Dow (1997), with respect to the correlation structure for voter utilities in the 1995 French presidential election.



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W

Y

Z

s=1 s=3 s=5 s=7 s=9 s=11 s=13 s=15 s=17 s=19


Figure 3A. Equilibria for the candidate similarity model, for ryz = 1.0

3.3.1. Case 1: Two candidates are viewed as completely similar on nonpolicy grounds

Here I assume that candidate W constitutes subset S1 while Y and Z constitute S2, and that ryz =1.0 - i.e., Y and Z are viewed as perfectly similar on nonpolicy grounds. Under this scenario Equation (5) implies that a voter who is indifferent between the candidates on policy grounds has vote probabilities Pi(W)= .50, and Pi(Y)= .25. Thus Y and Z suffer an electoral disadvantage by virtue of the correlation between voters' nonpolicy utilities. This occurs because a voter who has high nonpolicy utility for Y must have the same high nonpolicy utility for Z, so that Y and Z split the votes of potential supporters.

Figure 3A, which shows simulation results for this scenario, reveals that candidate equilibria are agglomerated only for s = .01, while dispersed equilibria exist over the interval .02 < s < .12. For the dispersed equilibria candidates Y and Z move towards opposite extremes, while Z remains at the voter mean. The amount of policy divergence at equilibrium is considerable, while Y and Z separated by at least three policy units over the entire interval .06 < s < .12. For s > .13 equilibrium breaks down.

Note that these results share a common feature with the simulations for the candidate popularity model reported above: in both cases, the candidates who suffer an electoral disadvantage take relatively extreme policy positions while the advantaged candidate remains at the center. However, the strategic logic which underlies these two sets of results is quite different. In the simulations on candidate popularity, the less popular candidates moved to the extremes in



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order to differentiate themselves from the more popular centrist candidate. In this case, Y and Z move to opposite sides of the policy spectrum in order to differentiate themselves from each other.

This motivation for candidates Y and Z - who are viewed as completely similar on nonpolicy grounds - to advocate dissimilar policies can be under- stood by considering the alternative strategy of policy convergence. If Y and Z were to locate with W at the voter mean, then Y and Z would split the votes of exactly the same group of supporters, in that every voter who prefers Y to W also prefers Z to W. The result is that W wins 50% of the vote, and Y and Z receive only 25% each. However, by moving to different regions of the policy space Y and Z no longer split the votes of the same group of supporters, since voters located near Y will strongly prefer Y to Z on policy grounds, and those near Z will strongly prefer Z to Y. The fact that Y and Z now appeal to different constituencies more than balances the fact that their policies are more extreme than would be desirable under the basic vote model.

3.3.2. Case 2: Two candidates are viewed as substantially similar on nonpolicy grounds

Here I again assume that candidate W constitutes subset S1 while Y and Z constitute S2, but that ryz = 0.69. The correlation 0.69 reflects Dow's (1997) empirical estimate of the correlation between the error terms associated with voters' utilities for Jacques Chirac and Edouard Balladur, two of the four ma- jor candidates for the French presidency in first-ballot voting in 1995. Under this scenario, Equation (5) implies that a voter who is indifferent between the candidates on policy grounds has vote probabilities Pi(W) = .44, and Pi(Y) = PiZ = .28.

Figure 3B, which shows simulation results for this scenario, reveals a pattern which is similar to the results for ryz = 1, though less extreme. For .01 < s < .09 an agglomerated equilibrium exists at the voter mean. Fur- thermore, over the range .07 < s < .14 dispersed equilibria exist, for which candidates Y and Z again move towards opposite extremes. The amount of policy divergence for these dispersed equilibria is considerably less than was the case for ryz = 1, with Y and Z locating at about 3.8 and 6.2. However, the degree of divergence still exceeds the policy divergence obtained for the basic vote model.

In summary, when voters perceive nonpolicy-related similarities between candidates, this motivates the "similar" candidates to differentiate their policies, thereby leading to policy divergence at equilibrium. Furthermore, the candidate similarity model supports equilibrium over similar ranges of the policy salience coefficient s, compared with the results for the basic model and the candidate popularity model.



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8

7

6

5

4

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1

0

W

Y

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s=1 s=3 s=5 s=7 s=9 s=11 s=13 s=15 s=17 s=19


Figure 3B. Equilibria for the candidate similarity model: ryz = .69

4. Conclusion

Three general principles concerning multicandidate spatial competition emerge from this analysis: 1) probabilistic vote models which incorporate candidate popularity and similarity provide increased incentives for policy dispersion, compared with the basic model; 2) each probabilistic model accommodates policy equilibria over similar ranges of policy salience coefficients, so that the likelihood that policy equilibria exist is not greatly affected by the choice of models; 3) in more complex probabilistic models, it is the candidates who suffer electoral disadvantages (either due to their lack of popularity or because of their similarities to rival candidates) who typically take extreme policy positions at equilibrium.

Most important, the introduction of more complex models of voter motivations changes markedly the results established for basic probabilistic vote models, thus confirming a similar observation made by Erikson and Romero (1990) in the two-candidate context. What we have seen here is how these results are affected.

Against my approach, one could argue that my results rest upon a particular functional form - the logit. However, the logit has been extensively employed for earlier work on multicandidate spatial competition under the basic version of the probabilistic vote model (Adams, 1997a, 1997b; de Palma, Hong, and Thisse, 1990; Nixon et al., 1995), so it is interesting to derive its implications for more complex models. In addition, the analytical diffi-



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culty of these equilibrium analyses have compelled me to resort to computer simulations, rather than taking the more satisfactory analytical route.

The approach developed here can be extended in several directions in order to study more realistic situations. These include: alternative voter dis- tributions; elections involving more than three candidates (de Palma et al., 1989); alternative candidate objectives such as rank-maximization (Denzau, Katz, and Slutsky, 1985); multidimensional policy spaces (Lomborg, 1997; Schofield et al., 1997); still more complex models of voter motivations, which incorporate such factors as political partisanship and socioeconomic characteristics (Adams, 1997b, 1998). This final suggestion seems especially promis- ing in light of the central point developed here: that the application of a simple behavioral vote model to multicandidate competition changes the results of the basic probabilistic model. It therefore appears plausible that a more com- plete behavioral specification will produce different policy equilibria - and further insights into the nature of multicandidate spatial competition.

Notes

1. Although there are sometimes important conceptual differences between spatial models involving political parties and those involving candidates - see especially note three - in discussing prior research I group both types of papers together.

2. This summary applies to models involving vote- or rank-maximizing candidates and sin- cere voters. Several studies obtain candidate equilibria by relaxing these assumptions, including studies by Austen-Smith and Banks (1988), Palfrey (1984), and Feddersen, Sened, and Wright (1990).

3. Nixon et al. (1995) and Schofield et al. (1997) explain the divergence between theoretically- derived convergent equilibria and parties' policy divergence in historical elections by arguing that in parliamentary democracies parties do not necessarily maximize votes, since they must also consider postelection bargaining over the governing coalition. This strategic consideration differentiates models of party competition from candidate-centered models, and for this reason my approach may not be appropriate for analyzing party competition in parliamentary democracies. It is for this reason that I focus on candidate competition in this paper.

4. Schofield et al. (1997) relax this assumption - as well as the assumptions of uncorrelated error terms described below - in their analysis of party spatial competition in elections in Germany and the Netherlands. I discuss their approach below (see note 6).

5. Note that in addition to the influences considered here, behavioral researchers have identi- fied voter-specific variables, including party identification and sociodemographic characteristics, as influences on the vote (Campbell, Converse, Miller, and Stokes, 1960; Alvarez and Nagler, 1995). These variables will not be considered here, for reasons outlined in Section 2.

6. Three partial exceptions to this generalization are papers by Alvarez and Nagler (1995), Alvarez, Bowler, and Nagler (1996), and Schofield et al. (1997), which examine the effect on expected vote of varying the policy locations of candidates/parties competing in recent American, British, German and Dutch elections. Each of these papers uses vote



120

probability functions estimated via multinomial probit analysis (of national election study data), which incorporates correlated error terms. However, these papers do not examine the possibility of candidate/party equilibria, but instead calculate each competitor's vote-maximizing position with the positions of their rivals fixed.

7. I performed alternative sets of simulations in which the candidates were initially located at the extremes of the policy continuum, and which varied the order of candidate policy movement. These simulations produced identical results to those reported here.

8. Note that these three-candidate equilibria are considerably less dispersed than those reported in the simulation study conducted by de Palma, Hong, and Thisse (1990: Figure 1), which also examined a uniform voter distribution. The difference between their results and

my own are explained by the fact that de Palma et al. employ a linear loss function, while I employ quadratic losses. Apparently the quadratic formulation creates greater incentives for candidate agglomeration.

9. I have converted the parameter estimate reported by Alvarez and Nagler to the value s would assume if the 6-point policy scale employed in their study were converted to the 10- point scale employed here. Furthermore, Alvarez and Nagler's estimate is based upon the probit model rather than logit; because logit coefficients are generally about 1.5 times as large as probit coefficients (due to different variances for the double exponential compared with the normal distribution used in probit) I have multiplied this rescaled parameter by 1.5. A number of additional studies (notably Dow, 1997; Nixon et al., 1995; Schofield et al., 1997) present parameter estimates of voters' policy salience coefficients for historical multicandidate elections. However, these studies employ a Euclidean distance metric rather than quadratic losses, so that their estimates are not directly comparable.

10. Note that Feld and Grofman's analysis also encompasses advantages which may be related to policies; for instance, voters may possess greater certainty concerning the incumbent's policies than those of her rivals, which causes risk-averse voters to evaluate the incumbent's policies more positively. The central point for Feld and Grofman is that this benefit of the doubt - whatever its source - causes voters to treat the policy distance between themselves and the incumbent as less than it actually is.

11. The stipulation that the candidates are viewed as similar on nonpolicy grounds is important. If the correlation between the error terms associated with voters' candidate utilities were due to policy similarities between the candidates (see for instance Alvarez and Na- gler, 1996: Figure 1), then this correlation would change as the candidates move about in the policy space. However, for nonpolicy similarities it seems reasonable to assume that the correlation structure remains constant for changes in candidates' proposed policies.

References

Adams, J. (1997a) Multicandidate spatial competition with probabilistic voting. Forthcoming in Public Choice.

Adams, J. (1997b). Spatial competition for multicandidate probabilistic voting with biased voters. Presented at the Annual Meeting of the Midwest Political Science Association, Chicago, April 1997.

Adams, J. (1998). Partisan voting and multiparty spatial competition. Journal of Theoretical Politics 10: 5-31.

Alvarez, M., Bowler, S. and Nagler, J. (1996). Issues, economics, and the dynamics of multiparty elections: The British 1987 general election. Unpublished manuscript.



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Alvarez, M. and Nagler, J. (1995). Economics, issues, and the Perot candidacy: Voter choice in the 1992 presidential election. American Journal of Political Science 39: 744-000.

Alvarez, M. and Nagler, J. (1998). When politics and models collide: Estimating models of multiparty elections. American Journal of Political Science 42: 55-96

Austen-Smith, D. and Banks, J. (1988). Elections, coalitions, and legislative outcomes. American Political Science Review 82: 405-422.

Budge, I. (1994). A new theory of party competition: Uncertainty, ideology, and policy equilibria viewed comparatively and temporally. British Journal of Political Science 24: 443-467.

Campbell, A., Converse, P., Miller, W. and Stokes, D. The American voter. New York: Wiley. Converse, P. and Dupeux, G. (1996). De Gaulle and Eisenhower: The public image of the

victorious general. In A. Campbell et al., Elections and the political order. New York: Wiley.

de Palma, A., Ginsberg, V., Labbe, M. and Thisse, J. (1989). Competitive location with random utilities. Transportation Science 23: 244-252.

de Palma, A., Hong, G. and Thisse, J. (1990). Equilibria in multiparty competition under uncertainty. Social Choice and Welfare 7: 247-259.

Denzau, A., Katz, R. and Slutsky, S. (1985). Multi-agent equilibria with market share and ranking objectives. Social Choice and Welfare 2: 95-117.

Dow, J. (1997). Voter choice and strategies in French presidential elections: The 1995 first ballot election. Presented at the 1997 Annual Meeting of the Midwest Political Science Association.

Eaton, B. and Lipsey, C. (1975). The principle of minimum differentiation reconsidered: New developments in the theory of spatial competition. Review of Economic Studies 42: 27-49.

Erikson, R. and Romero, D. (1990). Candidate equilibrium and the behavioral model of the vote. American Political Science Review 84: 1103-1125.

Feddersen, T., Sened, I. and Wright, S. (1990). Rational voting and candidate entry under plurality rule. American Journal of Political Science 34: 1005-1016.

Feld, S. and Grofman, B. (1991). Incumbancy advantage, voter loyalty, and the benifit of the doubt. Journal of Theoretical Politics 3:115-137.

Fiorina, M. (1981). Retrospective voting in American national elections. New Haven: Yale University Press.

Hermsen, H. and Verbeek, A. (1992). Equilibria in multiparty systems. Public Choice 73: 147-166.

Iversen, T. (1994). Political leadership and representation in Western European democracies: A test of three models of voting. American Journal of Political Science 38: 45-74.

Lin, T., Enelow, J. and Dorussen, H. (1997). Equilibrium in multicandidate probabilistic spatial voting. Public Choice, forthcoming.

Lomborg, B. (1997). Adaptive parties in a multiparty, multidimensional system with imperfect information. Unpublished manuscript.

Luce, D. (1959). Individual choice behavior. New York: Wiley. Markus, G. and Converse, P. (1979). A dynamic simultaneous equation model of electoral

choice. American Political Science Review 73: 1055-1070. McFadden, D. (1978). Modelling the choice of residential location. In: A. Karquist et

al. (Eds.), Spatial interaction theory and planning models. Amsterdam: North-Holland Publishing Company.

Nixon, D., Olomoki, D., Schofield, N. and Sened, I. (1995). Multiparty probabilistic voting: An application to the Knesset. Political Economy working paper. Washington: University in St. Louis.



122

Page, B. and Jones, C. (1979). Reciprocal influences of policy preferences, party loyalties, and the vote. American Political Science Review 73: 1071-1089.

Palfrey, T. (1984). Spatial equilibrium with entry. Review of Economic Studies 51: 139-156. Pierce, R. (1995). Choosing the chief Presidential elections in France and the United States.

Ann Arbor: University of Michigan Press. Rivers, D. (1988). Heterogeneity in models of electoral choice. American Journal of Political

Science 32: 737-760. Schofield, N., Martin, A., Quin, K. and Whitford, A. (1997). Multiparty competition in the

Netherlands and Germany: A model based on multinomial probit. Presented at the Annual Meeting of the Midwest Political Science Association, Chicago, April 1997.

Appendix

The generalized extreme value model is specified as follows. Let the set C of candidates be partitioned into n subsets labeled S1,..., Sn, where 91 = {S1i,..., Sn}). The utility that voter i obtains from candidate P in subset SF is denoted Ui(P) = s x p(xi, p) + 8iP, as in the basic vote model. The GEV model is obtained by assuming that 8iP for all P e SF, are distributed according to the GEV distribution. For the GEV, the joint cumulative distribution of the random variables 8ip, for all P e SF, is assumed to be

exp(-SSceBaS(SIeSF e[-eip]/(1-rF))1-Ff.

This distribution is a generalization of the distribution that gives rise to the logit model. For logit, each EiP is independent with a univariate extreme value distribution. For GEV, the marginal distribution of each EiP within each subset are correlated with each other. Specifically, rF is the correlation of the error terms with each subset SF. For any two candidates P and Q in different subsets, there is no correlation between

8iP and EiQ.



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Policy Divergence in Multicandidate Probabilistic Spatial Voting

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