Public Choice (2011) 146: 469–499 DOI 10.1007/s11127-010-9599-4

Campaign allocations under probabilistic voting Deborah Fletcher · Steven Slutsky

Received: 5 September 2008 / Accepted: 12 January 2010 / Published online: 30 January 2010 © Springer Science+Business Media, LLC 2010

Abstract We develop a probabilistic voting model where candidates compete by advertising in different media markets. Ads are viewed by everyone within a market and cannot be targeted to subgroups such as one candidate’s partisans. Candidates estimate the distribution of voter preference intensities in a market, and campaign ads then shift this distribution. Individuals with any intensity vote with some probability for each candidate. We derive comparative static implications of changes in a variety of factors on the advertising decisions of each candidate. Using campaign advertising data from 2002, we find these results to be consistent with actual campaign allocation behavior. Keywords Campaign allocations · Probabilistic voting JEL Classification D72

1 Introduction Political advertising expenditures are considerable and continue to grow. For example, $600 million were spent on broadcast television and radio advertising in the 2004 race for president, nearly triple the spending in 2000 (Associated Press 2004). Similarly, it costs at least $1 million to win a House seat and several million dollars to win a Senate seat (Christian Science Monitor 2006). Despite this trend, factors affecting advertising allocations across media markets have not been widely studied. Snyder (1989) and Strömberg (2008) consider allocation decisions in an Electoral College setting, where parties or candidates decide how much to allocate to the separate contests in different states. Allocations in an Electoral College (presidential) race differ significantly D. Fletcher () Department of Economics, Miami University, Oxford, OH 45056, USA e-mail: [email protected] S. Slutsky Department of Economics, University of Florida, Gainesville, FL 32611, USA

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from races in a single state or district containing multiple media markets, such as races for governor or Congress. In the former, the magnitude of victory in any state is irrelevant, while in the latter, candidates care about the vote margin in each media market, as only the vote totals in the entire contest are important. In this paper, we develop an empirically testable model of campaign allocations for single-state or single-district contests when there are multiple media markets in the state or district. To do this, we build on, but must modify, the existing analyses of two-party competition. Two-party competition has typically been modeled in a Downsian context, where the candidates compete by choosing platform positions in some issue space. If the space is multidimensional and individuals vote deterministically for the candidate whose platform they prefer, then pure strategy Nash equilibria rarely exist. An alternative is to assume probabilistic voting, where candidates are uncertain about how an individual will vote.1 Even in multidimensional contexts this can lead to pure strategy equilibria if the probability that an individual votes for a candidate is a concave function of that candidate’s strategy and a convex function of the opponent’s strategy.2 For many political campaigns, the standard Downsian approach does not seem appropriate for analyzing advertising allocation decisions. Candidates rarely announce platform changes in ads for fear of being damaged by charges of flip-flopping. Instead, they compete by trying to change voter attitudes about the salience of issues or about valence attributes such as character and competence.3 We develop a non-Downsian probabilistic voting model in which individuals have preference intensities for one candidate over the other, based on a variety of factors including the candidates’ issue positions. Candidates estimate these intensities for voters of particular types based on observable factors. These estimates differ from individuals’ true intensities due to unobservable traits characterizing each individual. Individuals with any estimated intensity will vote for each candidate with some probability. The candidates engage in campaign activities to seek to shift the intensities and hence the expected vote they receive.4 Although candidates attempt to do so, advertising cannot be perfectly targeted to specific, narrow groups of viewers. Instead, an ad in any media market may be seen by individuals with different political preferences. Models should include this kind of imperfect targeting when testing their empirical relevance. Our specific formulation of probabilistic voting is most closely related to that of Lindbeck and Weibull (1987), in which candidates compete by proposing alternative redistributions of income.5 The basic structure of probabilistic voting due to candidates’ incomplete 1 See Mueller (2003) for a survey of the literature on probabilistic voting. 2 Coughlin (1992) proves that if candidate uncertainty about voters is based on the presence of unobserv-

able traits which follow a binomial logit distribution, then the concave-convex assumption will be satisfied. Kirchgassner (2000) argues that such concavity-convexity cannot hold if the strategy spaces are unbounded. 3 See Enelow and Hinich (1984) for one approach to introducing such factors in a Downsian framework. 4 Other non-Downsian models similar to the one here are Skaperdas and Grofman (1995) and Strömberg

(2008). The relations between their approaches and the model here are discussed below. 5 Coughlin (1986, 1991) considers candidates competing by proposing redistribution policies in a probabilis-

tic voting framework based on a logit distribution. The analyses of Snyder (1989) and Strömberg (2008) also utilize probabilistic voting. Snyder (1989) assumes a particular form of the probabilistic voting function, where a candidate who spends nothing in a district has a zero probability of winning even if the opponent spends only one dollar in that district. Strömberg (2008) includes significant complicating factors, such as having a type of uncertainty about voter preferences that leads to random outcomes. Candidates then maximize the probability of winning instead of expected vote. For tractability, he then must make some strong structural assumptions which are not needed in our framework.

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information on voter preferences is similar, as are the sufficient conditions we develop for the existence of pure strategy Nash equilibria. However, while the analyses of redistributive policies and advertising allocations overlap, there are some significant differences. One is that in the redistribution contest, candidates have the same budget constraints, whereas candidates often have very different advertising budgets. Another difference is that individual voters react identically on the margin to promised redistributions by the different candidates, whereas a marginal advertising dollar from each candidate can have a very different effect on voters.6 These two differences have important consequences. For redistribution, Lindbeck and Weibull (1987) show that candidates choose identical policies, while for advertising, the equilibrium strategies can differ. Thus, interesting comparative static questions arise in the advertising setting that do not arise for redistribution. Although Lindbeck and Weibull (1987) assume a form of imperfect targeting, in which individuals are combined into groups with members of a group receiving the same redistribution, they do not focus on how changing the composition of the groups would change the candidates’ equilibrium policies, except in some polar cases. Our main focus relates to the effects of changes in market demographics on equilibrium advertising allocations. Political competition based on advertising allocations instead of policy proposals is not only realistic, but also helps to empirically test the probabilistic voting model. Direct tests of this model in the Downsian context have been relatively rare. In part, this is due to the fact that if changes in candidate platforms during the course of the campaign occur at all, they are difficult to observe or quantify except in very broad measures. The substantial empirical literature addressing campaign spending differs fundamentally from our approach. It looks at how total campaign expenditures affect election outcomes, and does not consider how these expenditures are allocated across media markets.7 The main issue that has arisen is the potential simultaneity of campaign contributions, and thus expenditures, with respect to vote share.8 Because we explore campaign allocations across media markets, we are able to escape this problem. First, how candidates divide advertising across media markets is likely to have only second-order effects on the candidates’ budgets. Second, electoral outcomes are not included in our reduced-form equations, so we can look at the candidates’ activities directly without having to estimate how these activities affect votes in the different markets. We derive empirically testable comparative static implications of the model and then use campaign advertising data from U.S. senate and gubernatorial races in 2002 to test the model’s predictions. We also shed light on some debates about voter behavior. In particular, we consider whether union members have become swing voters or remain, at least in the candidates’ views, intensely partisan Democrats, and whether voting behavior differs significantly in the South from the rest of the country. The evidence seems consistent with the 6 Another difference is in the underlying structure of the contest: advertising is analogous to an all-pay auction,

whereas in redistributive contests, only the winner pays. This might affect welfare analysis but does not seem significant for a positive analysis of allocation decisions. 7 This literature spans more than 35 years, going at least as far back as Dawson and Zinser (1971), who

estimate the winner’s vote share in House and Senate elections as a function of expenditures, incumbency, and party affiliation of the candidates. 8 This problem was first recognized by Giertz and Sullivan (1977), and a variety of authors including Jacobson

(1985), Grier (1989), Levitt (1994), Gerber (1998), and Erikson and Palfrey (1998, 2000) have attempted to address the issue in a number of different ways. These include making assumptions about the relation between the expected and actual vote, using simultaneous equation systems or instruments for campaign contributions and expected vote, and only considering repeated contests between the same pairs of candidates.

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probabilistic voting model, especially in the non-Southern states. We find that candidates seem to treat union members as partisan Democrats, and that in some crucial respects, the South continues to differ from the rest of the country. Section 2 presents the model, including discussions of the error distributions, the campaign effectiveness function, and our assumption that candidates can only imperfectly target voters with their advertisements. Section 3 looks in more depth at the role of imperfect voter characterization and considers when the equilibrium exists and is unique. The comparative static implications are derived in Sect. 4, with some specific testable implications given in Sect. 5. The empirical tests are presented in Sect. 6, and conclusions are given in Sect. 7. All proofs are contained in an Appendix.

2 The model Consider an election campaign between two candidates L and R, whose policy positions have been set prior to the start of the campaign. The candidates engage in campaign activities to alter voter preferences in their favor. Since no voter is likely to prefer one candidate’s position on every issue, candidates may affect voter preferences by trying to change the saliency of different issues in the voter’s mind. Independent of policies, candidates may also focus on valence factors such as character and competence. Each voter is characterized by a preference intensity whose post-campaign value is denoted as I . The initial pre-campaign value is I 0 . Positive values of I indicate a preference toward R, negative values a preference toward L, and 0 indifference. To further specify the model, three questions must be considered: First, what information do candidates have about voter preferences, and what do they then believe about voter behavior? Second, how do campaign activities affect voter preferences? Finally, what restrictions exist on candidate efforts to target voter types? 2.1 Candidate information and beliefs We assume that some voter traits are only imperfectly observed by candidates. Such errors make voters appear to have a random element in their behavior, even though individuals vote with certainty for their preferred candidate.9 We call this approach one of imperfect voter characterization.10 In this model, candidates will typically partition individuals into finite sets of types based on observable traits, with each type having a representative value of I . A typical voter of type I will differ from the representative value. Let I a be the actual intensity, with I a = I + ε, for ε, an unobserved random variable whose probability distribution function (pdf) is ϕ(ε). We make the following general assumptions about ϕ(ε), which are comparable to those of Lindbeck and Weibull (1987): ϕ(ε) = ϕ(−ε) 

ϕ (ε) ≥ 0

(1a)

if ε < 0

∃c, ϕ(ε) > 0

and



ϕ (ε) ≤ 0 if ε > 0

if − c < ε < c

and

ϕ(ε) = 0

(1b) if ε < −c or ε > c

(1c)

9 Among the spatial models of probabilistic voting, our approach is most similar to the additive bias model in

Banks and Duggan (2004). 10 An alternative justification for probabilistic voting would be that individuals make mistakes in voting given

their true preferences. The possibility of an error might decline with intensity since more intense voters might take more care when casting their ballots.

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Fig. 1 The probability of voting for candidate R

That is, ϕ(ε) is symmetric, weakly single-peaked, continuously differentiable and strictly positive on its support (−c, c). For any observed value of I , the candidates believe that the probability of voting for candidate R is given by a function V (I ), defined by:  ∞ V (I ) ≡ Pr{ε > −I } = ϕ(ε)d ε. (2) −I

Then V is differentiable with ∂V /∂I = ϕ(−I ) so that V is nondecreasing in I and strictly increasing if −c < −I < c. The second derivative of V is ∂ 2 V /∂I 2 = −ϕ  (−I ). Given assumption (1b) that ϕ is weakly single-peaked, V is locally concave in I when I is positive and is locally convex when I is negative. That is, the vote function for R is locally concave in the neighborhood of the partisans for R and locally convex in the neighborhood of the partisans for L.11 Then V (I ) will have a shape like that in Fig. 1. This model of voter behavior is the same as that underlying Strömberg (2008): voters have preference intensities toward candidates. Those intensities are swayed by campaign activities and are affected by random factors. Stromberg uses a reduced-form approach to describing the distribution of voter intensities. He makes the strong simplifying assumption that the overall distribution of voter intensities is normal with a constant variance and a mean that may shift over time. Here, instead of directly specifying the reduced form, we make assumptions on the distribution of voter types and on the errors with which a type is observed. A reduced-form distribution can be calculated by combining the distribution of types with ϕ(ε). However, this distribution will generally not be normal or unimodal even when nice properties are assumed about ϕ(ε). The two approaches complement each other. Stromberg’s approach makes analyzing the complications of the Electoral College and the additional random elements in his model more tractable. Our approach allows us to consider important questions such as how greater polarization of the electorate in a market affects resource allocation. This in effect adds weight in the tails of the reduced-form distribution and takes it from the middle, creating a new distribution that may not be unimodal. This type of change cannot be considered if the reduced-form distribution is restricted to being normal. 2.2 Effectiveness of candidate activities The candidates engage in activities designed to alter the sign or magnitude of voters’ preference intensities. Candidate R seeks to increase intensity for an individual of precampaign intensity I 0 with expenditures x while L seeks to lower I 0 with expenditures y. 11 If ϕ is strictly single-peaked, then V exhibits strict concavity and strict convexity.

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Let h(I 0 , x, y) denote the effectiveness of spending by the candidates in changing the preference intensity of an individual with pre-campaign intensity I 0 , where h may be positive or negative depending on the magnitudes of x and y. Hence, I = I 0 + h(I 0 , x, y). Certain properties of h are significant for the results. First, what types of individuals can most be swayed by campaign advertising: those with low intensities (with I 0 near 0), those with strong intensities favorable to the candidate, or those with strong opposing intensities?12 Second, what is the shape of the advertising response function: do ads have diminishing effectiveness, or are there at least some initial increasing returns at low levels of advertising?13 The literature on the effectiveness of advertising in general and of political advertising in particular is contradictory or silent about these properties. Finally, there are also questions about clutter: does ad effectiveness depend only on the level of the candidate’s own ads, or also on the level of the opponent’s efforts, or on the level of advertising in other races as well? For empirical analyses, clutter across races may be significant. For example, clutter may create differences between presidential and nonpresidential election years in the United States. For the theoretical model here with a single race, the crucial issue is whether there is cross-candidate clutter. In general, we assume diminishing effectiveness, which is independent of voter intensity, but initially make no assumption about clutter.14 Assuming h is twice continuously differentiable, these imply: hx > 0,

hxx < 0,

hy < 0,

hyy > 0,

and

hI ≡ 0

(3)

In addition, a useful benchmark case adds no cross-candidate clutter15 and has the same effectiveness function for both candidates: I = I 0 + g(x) − g(y),

with g  > 0, g  < 0

(4)

2.3 Targeting In an ideal world for candidates, they would be able to target their expenditures precisely, with different messages and expenditure amounts for each voter type. Candidates certainly attempt to accomplish this by running different ads on television programs whose viewers 12 Most of the campaign advertising literature has concluded that ads reinforce partisanship (see Iyengar and

Simon 2000 for a discussion of this literature). However, Freedman et al. (2004) present evidence that political ads have the strongest effect on those with the lowest levels of political information, who would presumably be those voters with weak intensities. 13 This question has been a source of debate in the advertising literature for some time. In an extensive

survey article, Simon and Arndt (1980) conclude that the advertising response function is concave rather than S-shaped, but this question has not been settled in the literature. Bronnenberg (1998) suggests that the empirical evidence supports a concave function because it is not optimal for a firm to advertise in the rising part of the response function. He also finds evidence that long-term and short-term response functions may have different shapes, and that the functions may even differ over different consumer types. In the campaign advertising literature, Mueller (2003) discusses the S-shaped curve, but most of the literature incorporates either linear or concave response functions; for examples, see Grier (1989) and Levitt (1994). See Morton (2006) for a general discussion of how political advertising affects voters. 14 The assumption that h is independent of voter intensity is not used in the existence or uniqueness results of Theorem 1 but helps simplify the comparative statics results. 15 One specification that incorporates clutter is I = I 0 + h[(x − y)/(x + y)]. If h has significant curvature, then strict concavity in x may fail if h > 0 and strict convexity in y may fail if h < 0. If −1 − y/x < h / h < 1 + x/y holds, then h is concave in x and convex in y.

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Table 1 Voter intensities before and after the campaign Intensities and number Partisans for L

Neutrals

Partisans for R

γn

Market n pre-campaign

−θn

0

post-campaign

−θn + h(xn , yn )

h(xn , yn )

γn + h(xn , yn )

number of each type

n1

n2

n3

pre-campaign

−θm

0

γm

post-campaign

−θm + h(ER − xn , EL − yn ) h(ER − xn , EL − yn ) γm + h(ER − xn , EL − yn )

number of each type

m1

Market m

m2

m3

have different demographics, or by giving different speeches depending on the nature of the audience. In reality, perfect targeting is not possible, and spillovers occur where a message intended for one type is received by another. To model this formally, we assume that candidates face two media markets, denoted n and m. Each market contains three types of voters: partisans for each candidate and neutrals. The pre-campaign intensities of preference for voters preferring candidate L are −θn and −θm , while those for R are γn and γm . The neutral types have intensities of 0 in both markets. Let N and M be the total number of voters in communities n and m, respectively. In n, there are n1 partisans of L, n2 neutrals, and n3 partisans of R. In m, these numbers are m1 , m2 , and m3 . We will call those individuals initially preferring L submarkets 1, those initially neutral submarkets 2, and those initially preferring R submarkets 3. Candidates can target the markets differently but cannot separately target “niches,” or submarkets, within a market. Thus, R can allocate resources with xn going to market n and xm to market m, while L can set yn and ym for the two communities, respectively. The campaign expenditures by the two candidates lead to the same shift in the intensities of all voter types within a market. The initial and post-campaign intensities in the two communities are shown in Table 1.16 The two candidates face resource constraints x n + xm = E R

and

yn + ym = E L

(5)

where ER and EL are their available resources, which we assume are fixed.17 Without loss of generality, we assume that R has at least as many resources as L, so that EL ≤ ER . We also assume that the resource difference is not too great, relative to voter intensities:  h

 1 1 ER , EL < min[θn , θm ] 2 2

(6)

16 Skaperdas and Grofman (1995) similarly assume three types of voters, partisans of each candidate and

undecideds. They have one market with the candidates deciding the mix of positive and negative campaign activity. Intensities of partisanship and the effectiveness of campaign activities are combined in reduced form functions specifying the extent to which voters move from being partisans to undecided or the reverse. 17 Initially we assume that advertising prices are the same in both markets but later we allow for different

prices related to differences in market size.

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This means that if both candidates spend half their resources in each market, the resource advantage to R is not enough to turn L’s partisans into partisans for R. Thus, V remains locally convex around the post-campaign intensities of the voters most favorable to L. The candidates simultaneously make their allocation decisions with the goal of maximizing their expected net votes.18 In reality, campaigns have a dynamic element, where over the course of the campaign, candidates may respond to their opponents’ allocations. In fact, the Campaign Media Analysis Group, the source for the data used in our empirical analysis, makes its profits by selling real-time information about campaign advertising allocations to political candidates as well as the media.19 In our simultaneous model, then, candidate R chooses xn and xm to maximize n1 V [−θn + h(xn , yn )] + n2 V [h(xn , yn )] + n3 V [γn + h(xn , yn )] + m1 V [−θm + h(xm , ym )] + m2 V [h(xm , ym )] + m3 V [γm + h(xm , ym ))]

(7)

while L chooses yn and ym to minimize this. Denote the function in (7) as Vˆ (xn , xm , yn , ym ). 3 Existence and uniqueness of equilibria A standard sufficient condition for existence of a pure strategy equilibrium under probabilistic voting is that the probability that an individual votes for a candidate must be a concave function of that candidate’s strategy and a convex function of the opponent’s strategy (see Coughlin 1992 for a discussion of this result). Although this was initially derived for candidates choosing platforms in an issue space, it applies for any strategy space with similar topological properties.20 The specific assumptions that ensure the concavity and convexity depend on the context. Here, Vˆ must be concave in xn and xm and convex in yn and ym . As given in (7), Vˆ is essentially the composite function V (h). Recall that h is concave in x and convex in y. Thus, Vˆ would immediately satisfy the appropriate properties if V did, but as argued after (2), V (I ) is not globally concave. Thus, further restrictions are required to ensure that Vˆ satisfies the appropriate properties. Two different problems arise given the shape of V . One is that for I < 0, V (I ) is locally convex. Vˆ is locally concave in x, however, if h is sufficiently concave to outweigh the convexity of V . The following condition is sufficient for this:       ϕ (ε)  −hxx hyy < max  , (8) min x∈[0,ER ],y∈[0,EL ] (hx )2 (hy )2 ε∈[−λ,λ] ϕ(ε)  18 Maximizing expected plurality is commonly assumed as the objectives of the candidates in probabilistic

voting models in the spatial context. This objective is easier to analyze than the standard alternative of maximizing the probability of winning, as done in Strömberg (2008). A number of papers, including Aranson et al. (1974), Ledyard (1984), Snyder (1989), Duggan (2000) and Patty (2007), have considered the relation between outcomes under the two objectives. In some, perhaps special, circumstances, they are equivalent. In the redistribution context, Lindbeck and Weibull (1987) develop some differences in outcomes when candidates maximize their probability of winning the contest rather than expected plurality. It should be noted that expected plurality maximization is not only more convenient but in some circumstances can be justified as more realistic. Candidates with a low probability of winning may desire to lose by as small a margin as possible. 19 Developing a dynamic analysis would be valuable but extremely complex. Examination of the data suggests

that the pattern of allocations across media markets is relatively stable over time. Thus, modeling simultaneous allocation decisions is, at the very least, a good starting point. 20 In their spatial voting model, Banks and Duggan (2004) give other sufficient conditions on the expected

vote share functions for existence of an equilibrium.

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Fig. 2 Global nonconcavity of V (x, y): Assume I < −c and let x  be such that I + h(x  , y) = −c. Even though V is locally concave on the support of the error distribution ϕ, it is not globally concave

Note that ϕ  (ε)/ϕ(ε) = (∂ 2 V /∂I 2 )/(∂V /∂I ) is a curvature measure of V (I ) and −(hxx )/(hx )2 and (hyy )/(hy )2 are partial curvature measures of h. These are mathematically analogous to measures of absolute risk aversion, in that both measure the degree of concavity.21 Thus (8) in a sense asserts that the curvature of h exceeds the variation in ϕ. Models of probabilistic voting over an issue space utilize conditions analogous to (8). In those models, restrictions on utility functions analogous to the concavity of h here are needed to overcome the local convexity of V . Condition C1 in Lindbeck and Weibull (1987) is directly analogous to our (8). The right-hand side of their C1 depends on curvature conditions of the individual voter’s utility function instead of the advertising effectiveness function. Note that their weaker condition C2 does not directly apply in this context, since C2 is based on the candidates having identical equilibrium strategies, which is not true here. Lin et al. (1999) construct a multicandidate model in which preferences are based on distances between a candidate’s position and the voter’s ideal point, and the errors are normally distributed. Their condition (4), relating distance to variance, guarantees local concavity and is analogous to (8) here. Even when (8) ensures local concavity, a second problem arises. When c is finite, since Vˆ has a lower bound of 0, it cannot be globally concave in x for values of I 0 below −c. This problem is illustrated in Fig. 2. Similarly, for values of I 0 above c, Vˆ cannot be globally convex in y because of its upper bound of 1. This can lead to nonexistence of pure strategy equilibria. For example, consider the limiting case as c goes to 0. V (I ) converges to a correspondence that looks like a step function, except that at I = 0, V can take any value. This is essentially the case of deterministic voting, where candidates believe that individuals vote with certainty for their preferred candidate.22 This leads to a Colonel Blotto game modified by imperfect targeting. Fletcher and Slutsky (2009) show that pure strategy equilibria generally do not exist in this case. Similar results will hold for cases near deterministic voting when c is positive but sufficiently small. For c large relative to the possible values of I , the upper and lower bounds of one and zero for V do not come into play, and the function Vˆ satisfies the appropriate convexity 21 The terms for h here as compared to the coefficient of absolute risk aversion have an additional first partial

derivative in the denominator, and thus are not invariant to linear transformations. Since h is being compared to another function ϕ, such transformations are not admissible. Condition (8) must be evaluated for each specific transformation. Most directly, the right-hand side of (8) relates to the concavity index of Debreu and Koopmans (1982), which for a twice continuously differentiable function f (x) equals infx [f  (x)/(f  (x))2 ]. They show that the sum of two functions is quasiconvex if the sum of their concavity indices is nonnegative. Here, condition (8) relates to a composite of two functions instead of their sum. 22 Deterministic voting is usually specified with V (0) = 1 so that an indifferent voter has equal probability 2

of supporting either candidate, whereas in the limiting case any probability of voting for either candidate is allowed when I = 0. This distinction would make little difference in the analysis of deterministic voting.

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or concavity property over the entire range of x and y. To specify this bound formally, let λ ≡ max[θn − h(0, EL ), θm − h(0, EL ), γn + h(ER , 0), γm + h(ER , 0)]. The following condition is a sufficient bound on c: (9)

λ
In effect, (9) asserts that no candidate has enough resources to win a submarket with certainty even if the other candidate does not compete in that market. Under (9), for all xi and yi , −θi + h(xi , yi ) + ε > 0 and γi + h(xi , yi ) + εˆ < 0 both hold for some ε or εˆ arising with positive probability. Clearly (9) holds if ϕ has unbounded support with c = ∞. A similar problem arises in probabilistic voting over issue spaces, but it has only been addressed implicitly in that literature. For example, both Lindbeck and Weibull (1987) and Lin et al. (1999) assume positive density everywhere so that c = ∞, guaranteeing that a condition analogous to (9) holds in both of their frameworks. In McKelvey and Patty’s (2006) model, voters consider which candidate to vote for and whether to abstain, and act strategically with respect to other voters’ actions. Voters have a random payoff disturbance for each action they take, independent of who wins the election. Although this model differs significantly from ours, their assumption that these random disturbances have full support seems analogous to our (9). Special cases of h and ϕ demonstrate that (8) and (9) can both be satisfied. For (8) to hold, the right-hand side expression related to h must be strictly positive. For the benchmark case in (4), consider the following three examples of the function g, where k is a positive constant. For g(z) = k ln z, −(hxx )/(hx )2 = (hyy )/(hy )2 = −g  /(g  )2 = 1/k for all x or y. 1 Clearly, then, the right-hand side of (8) is strictly positive. For g(z) = kz 2 , −(hxx )/(hx )2 = 1

1

−1

(1/k)x − 2 and (hyy )/(hy )2 = (1/k)y − 2 . The right-hand side of (8) is then (1/k)ER 2 > 0. On the other hand, the constant marginal effectiveness case of g(z) = kz is ruled out since the right-hand side of (8) would be 0 and the condition could not be satisfied. Next consider three particular distributions for ϕ(ε): uniform, normal, and quadratic. For the uniform distribution, ϕ(ε) = 1/(2c), and (8) always holds since ϕ  = 0. Condition (9) then imposes a sufficiently large value of c given the values of the other parameters. For the 2 2 normal distribution, ϕ(ε) = (1/((2π)1/2 σ ))e−(ε /(2σ )) , where σ is the standard deviation of ε. Since this has unbounded support, (9) holds automatically. For this distribution, ϕ  /ϕ = −ε/σ 2 . On the range −λ ≤ ε ≤ λ, the maximum of its absolute value is λ/σ 2 . Given any allowable function h, (8) will hold for sufficiently large σ . For the quadratic distribution, ϕ(ε) = 3(1 − (ε/c)2 )/4c.23 Then ϕ  /ϕ = −2ε/(c2 − ε 2 ). If (9) holds, then the left hand side of (8) equals 2λ/(c2 − λ2 ). For a given λ, both (8) and (9) hold for sufficiently large c. In all three cases, the requirements for these conditions to hold are that the error distribution be sufficiently diffuse either by having a large support in the uniform and quadratic cases or a large variance for the normal distribution. Given sufficiently diffuse observation errors, a unique equilibrium exists in this model. Theorem 1 Under conditions (8) and (9), the objective function Vˆ (xn , xm , yn , ym ) in (7) is strictly concave in xn and xm and strictly convex in yn and ym , and therefore a unique Nash equilibrium exists, which is in pure strategies. Conditions (8) and (9) impose that the error term is sufficiently diffuse to guarantee existence of an equilibrium. Probabilistic and deterministic voting, then, are both subcases  23 For this function, ϕ(c) = ϕ(−c) = 0 and c ϕ(ε) dε = 1. −c

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of imperfect voter characterization, the former occurring when the error term is sufficiently diffuse, and the latter when it is sufficiently concentrated.24 The conditions given above for each case are sufficient but not necessary. Deterministic-like voting behavior may arise even when c is not near zero, and probabilistic voting behavior may arise even with some nonconcavity in the objective function, if the nonconcavity is for strategies not near the equilibrium ones. Determining better boundaries for each case can be difficult. When neither sufficient condition holds, the equilibrium may be similar to one of the cases or may differ from both, having a mix of equilibrium types.25 Remark 1 The formal model has only two media markets and three submarkets within each market, but the existence result is not limited to that special case. Theorem 1 extends straightforwardly to situations with any number of media markets or submarkets. Condition (7) would appear to become more complex, but would simply include more terms, each of which would satisfy the appropriate properties. We consider the special case to facilitate the comparative statics below. Remark 2 In particular cases, Kuhn-Tucker corner conditions may be used to specify conditions under which the equilibrium will be at an interior and when it will be at a boundary, with at least one candidate spending nothing in a market. Assume h satisfies (4). Having g  (0) = ∞ is sufficient for the equilibrium to be at an interior.26 If that does not hold, then the only possible corner solutions have yi = 0 and 0 ≤ xi < ER , where yi is the expenditure in market i of the resource-disadvantaged candidate. If xi > 0 so that only the resourcedisadvantaged candidate is at a corner, then a necessary condition for such a corner is that xi is less than the difference between the candidates’ resources.

4 Comparative statics To gain insight into the nature of probabilistic voting in this context and to develop some testable implications of the model, we derive some comparative statics results. Consider an interior equilibrium starting from a situation where the two markets are identical: ni = mi , i = 1, 2, 3, θn = θm , and γn = γm . Then the equilibrium is xn = xm = 12 ER and yn = ym = 12 EL .27 Let μ be any parameter entering only market n whose change, therefore, makes the markets different from each other. The comparative statics with respect to μ give insights into how candidates would tend to allocate resources when facing asymmetric markets.28 Therefore, in what follows, derivatives are generally taken in market n so x and y are presumed to be xn and yn . For simplicity, we will define the values of ϕ in the 24 In the spatial context, Banks and Duggan (2004) analyze convergence of the probabilistic voting model to

the deterministic by having the variance of the bias term in the additive bias model go to zero. This relates to violation of our condition (8) rather than (9). 25 In the spatial context, under relatively general conditions, Banks and Duggan (2004) show existence of an

equilibrium in mixed strategies in the candidates’ policy choices. 26 Lindbeck and Weibull (1987) make an analogous assumption on marginal utilities of income to rule out

corner solutions. 27 If x = x or y = y multiple equilibria would exist, contradicting Theorem 1. n m n m 28 These comparative statics give the effects of small changes starting from a situation of complete symmetry

between the markets. The results will hold for a neighborhood around symmetry, but may not be valid if the markets are very different. Deriving the effects for other situations is difficult, involving ambiguous factors.

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three submarkets of market n by ϕ1 = ϕ[θn − h( 12 ER , 12 EL )], ϕ2 = ϕ[−h( 12 ER , 12 EL )], and ϕ3 = ϕ[−γn − h( 12 ER , 12 EL )]. Using (5) to eliminate xm and ym from (7), the first-order conditions of the resulting expression with respect to xn and yn are Vˆ1 − Vˆ2 = 0 and −Vˆ3 + Vˆ4 = 0. Totally differentiating with respect to μ and solving yields the comparative statics terms ∂xn /∂μ and ∂yn /∂μ. From the concavity and convexity of Vˆ in xn and yn , respectively, the denominators of these expressions are positive. It then follows that: sign(∂xn /∂μ) = sign[Vˆ33 (∂ Vˆ1 /∂μ) − Vˆ13 (∂ Vˆ3 /∂μ)]

(10)

sign(∂yn /∂μ) = sign[Vˆ11 (∂ Vˆ3 /∂μ) − Vˆ13 (∂ Vˆ1 /∂μ)]

(11)

In order to sign some of the comparative statics, it is helpful to place bounds on the magnitude of ∂ 2 h( 12 ER , 12 EL )/∂xn ∂yn . These bounds essentially specify the magnitude of clutter effects when candidates divide their resources equally. Consider two possible conditions bounding hxy : (hx / hy )hyy < hxy < (hy / hx )hxx hx hy ϕ3 /ϕ3

< hxy <

hx hy ϕ1 /ϕ1

(12) (13)

Given (8), the bounds in (13) imply those in (12). The condition in (13) is valid only if ϕ is strictly single-peaked, strengthening the weak single-peakedness specified in (1b). Given strict single-peakedness, ϕ3 > 0 since it is evaluated at an ε < 0 while given (6), ϕ1 < 0 since it is evaluated at an ε > 0. Condition (12) is satisfied for a wide variety of special cases. The case specified in (4) with hxy = 0 satisfies the condition since the lower bound is negative and the upper bound positive. It is straightforward to check that the example with clutter given in footnote 15 also satisfies (12). 4.1 Parameters not entering through the advertising effectiveness function An important set of parameters to consider are those that do not directly influence the effectiveness of advertising. Theorem 2 Assume that μ does not enter market n through h(xn , yn ). If (12) holds, then sign(∂xn /∂μ) = sign(∂yn /∂μ) = sign(∂ Vˆ1 /∂μ) There are two crucial results contained in Theorem 2. First, for parameters which do not enter the function h, the two candidates respond to differences between markets in the same way. If one candidate spends more on ads in market n than in market m, then so will the other candidate. Second, determining the market in which both candidates spend more reduces to a relatively simple condition. If ∂ Vˆ1 /∂μ is positive then both spend more in market n while if it is negative, they spend more in m. Based on this result, the effects on allocation of a variety of parameters not directly entering h are presented in Table 2. In the following subsections, we discuss the comparative statics for polarization, partisan leanings, nonmonetary resources, and market size.

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Table 2 Comparative statics predictions for parameters μ not entering h μ

Interpretation

∂ Vˆ1 /∂μ

Sign of ∂xn /∂μ and ∂yn /∂μ

I

μ = θn or μ = γn

partisans become more intense

II n1 = n01 + μ/2, n2 = n02 − μ,

n3 = n03 + μ/2

∂ Vˆ1 /∂θn = n1 hx ϕ1 0 if ϕ is uniform ∂ Vˆ1 /∂γn = −n3 hx ϕ3 – if ϕ is strictly single-peaked

symmetric increase in the [ 12 (ϕ1 + ϕ3 ) − ϕ2 ]hx 0 if ϕ is uniform – if ϕ is strictly concavea

number of partisans

– if − 12 θn + h < 0 and ϕ

relative to neutrals,

is strictly single-peakedb

keeping market size constant III n1 = n01 + μ n3 = n03 − μ

market demographics

[ϕ1 − ϕ3 ]hx

0 if ϕ is uniform

shift in favor of the

Sign[2h + γn − θn ] if ϕ is

resource-disadvantaged

strictly single-peakedc

candidate IV n1 V (−θn + h + k1 μ) either differential shifts in −[k1 n1 ϕ1 + k2 n2 ϕ2 +n2 V (h + k2 μ)

the intensities of all

+n3 V (γn + h

submarkets, or

+k3 μ)

differential non-

+k3 n3 ϕ3 ]hx

0 if ϕ is uniform – if k1 = k2 = 0 and ϕ is strictly single-peaked

advertising resources benefiting one candidate V ni = n0i + μ, some i

increasing population in a ϕi hx

+

market by increasing the size of one submarket

VI ni = μn0i , all i

proportional increase in

Vˆ1

+

the size of all submarkets μ represents parameters entering market n. Each row considers different possibilities for the parameter represented. From Theorem 2, sign(∂ Vˆ1 /∂μ) is the same as the signs of ∂xn /∂μ and ∂yn /∂μ. A positive sign means that both candidates shift advertising from market m to market n in response to the parameter change. Note that h and hx are evaluated at ( 12 ER , 12 EL ) and ϕ1 ≡ ϕ(θn − h), ϕ2 ≡ ϕ(−h), and ϕ3 ≡ ϕ(−γn − h) a Let α ≡ min[θ , γ ]. Strict concavity and single-peakedness imply strict single-peakedness. Then ϕ + ϕ = n n 1 3 ϕ(−θn + h) + ϕ(γn + h) < ϕ(−α + h) + ϕ(α + h) < 2ϕ(h) where the last inequality follows from strict

concavity. ∂ Vˆ1 /∂μ < 0 then follows b If − 1 θ + h < 0 then h < θ − h which implies that ϕ(h) > ϕ(θ − h). From symmetry, ϕ ≡ ϕ(−h) = n n 2 2 n ϕ(h) > ϕ(θn − h) ≡ ϕ1 . Since h < γn + h, ϕ2 > ϕ3 . These imply ϕ1 + ϕ3 < 2ϕ2 with ∂ Vˆ1 /∂μ < 0 c From symmetry and strict single-peakedness, ϕ > ϕ iff θ − h is closer to 0 than γ + h, which in turn is n n 1 3 equivalent to 2h + γn − θn > 0

4.1.1 Increased polarization One market can be more polarized than the other in two ways. In one, the corresponding submarkets are the same size in both markets, but partisan intensities are greater in the more polarized market. Result (I) in Table 2 considers this. If the error distribution is not uniform, then both candidates spend less in the market with more intense partisans.

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Fig. 3 An example where the condidates spend more in the more polarized market: h = h( 12 ER , 12 EL ) and θn = γn . h is sufficiently large to make θn − h close to 0, and thus larger than ϕ2 and ϕ3 . If polarization is greater in community n because n1 and n2 are bigger than m1 and m3 while n2 is smaller than m3 , then the candidates spend more in market n

In the other, voters in the corresponding submarkets have the same intensities, but the more polarized market has fewer swing voters and more partisans. This is considered in Result (II). Increasing polarization in this way has ambiguous effects. For a uniform error distribution, both candidates allocate resources equally to the two markets. For a strictly concave error distribution or for a strictly single-peaked error distribution when the difference in the resources of the candidates is not too large, both spend less in the more polarized market.29 If the resource difference between the candidates is sufficiently large and ϕ has regions of nonconcavity, then the market with more partisans can be a better target for candidate activity. See Fig. 3 for an example of this. 4.1.2 Differential partisanship One way for one market to lean more toward a candidate than does the other is if it has relatively more partisans of that candidate. Result (III) considers this by having more partisans of L and fewer partisans of R in market n than in m, keeping the total populations in the two markets the same. The effect depends on whether ϕ1 is larger or smaller than ϕ3 . For uniformly distributed errors, these are equal so despite the different population mixes, candidates spend the same in both markets. When the error distribution is strictly single-peaked, 2h + γn − θn is positive unless the intensity of individuals favoring L sufficiently exceeds that of those favoring R. The difference in these intensities increases in the resource difference between the candidates. Both candidates tend to spend more in the market that leans relatively toward the poorer candidate unless the poorer candidate’s partisans are sufficiently more intense. 4.1.3 Nonmonetary resources In addition to their paid advertising, candidates often have a variety of types of nonmonetary resources that sway voters, such as unpaid volunteers and media endorsements. Result (IV), through different values of the parameters ki , considers some ways in which these nonmonetary resources interact with the paid ads. First, consider a candidate receiving a newspaper endorsement read by the entire market, where all submarkets are affected equally. This can be modeled by assuming that k1 = k2 = k3 . When ϕ is strictly single-peaked, a positive μ shifts all submarkets in market n in favor 29 Note that the sufficient condition bounding the resource difference, h( 1 E , 1 E ) < 1 θ is stronger than 2 R 2 L 2 n

(6) but is clearly not necessary.

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of R. The effect on advertising allocations is ambiguous. If ER = EL , θn = γn , and n1 = n3 , then an initial increase in μ from 0 would not change either candidate’s allocation. Large changes in μ would reduce xn and yn if ϕ were strictly concave. If ϕ is bell-shaped, the effect depends on the values of θn and γn relative to the concave and convex regions of ϕ. Deviations from market symmetry could cause the initial changes in xn and yn to be of either sign. Second, consider one candidate having an additional resource that is not a substitute for advertising and that particularly affects that candidate’s partisans. Union membership for a Democratic candidate and evangelical Christian membership for a Republican candidate could be examples of this. Such members could be directly contacted by mail, phone calls or personal visits without spillovers to others. These contacts might have different effects than that of an additional ad. This can be modeled by setting k1 = k2 = 0, with only submarket 3 affected. In this case, both candidates spend less in the market in which one of them has this advantage.30 As discussed in Sect. 4.2.1 below, an alternative way to specify these additional resources would be that they enter the function h. 4.1.4 Market size Consider the effect of market size on candidate allocations when everything else, including advertising price, is held fixed. Results (V) and (VI) show that both candidates spend more in the larger market in two cases: if only some of the submarkets are larger, or if all submarkets are proportionally larger when ad prices are independent of community size. In fact, ad prices tend to increase with market size, which means that changes in market size will indirectly enter into the function h. The results in that specification are discussed in Sect. 4.2.2 below. 4.2 Parameters entering through the advertising effectiveness function Other parameters, include some types of nonmonetary resources and market size when it affects ad prices, enter the function h and directly interact with advertising levels. 4.2.1 Nonmonetary resources Assume that in market n, unions or churches allow candidates to approach their partisans directly at no cost. If these approaches are perfect substitutes for ordinary advertising, then the expected vote for R in market n is n1 V (−θn + h(xn , yn + μL )) + n2 V (h(xn , yn )) + n3 V (γn + h(xn + μR , yn )). Theorem 3 Starting from initially identical markets, the following results hold: (A) ∂xn /∂μR < 0 and ∂yn /∂μL < 0 either if (13) holds or if ϕ is uniform. (B) ∂xn /∂μL = 0 if ϕ is uniform and ∂xn /∂μL < 0 if (12) holds and ϕ is strictly singlepeaked. 30 This analysis was based upon the candidates having additional resources in only one market. However, it

could be extended to their having additional resources in both, with μ indicating the difference in resources between the two markets. This would require a slight reinterpretation of the bound in (6) since the additional resources are formally equivalent to a change in the intensities of support.

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(C) ∂yn /∂μR can have any sign. (i) ∂yn /∂μR = 0 if ϕ is uniform. (ii) ∂yn /∂μR < 0 if (12) holds and either if EL = ER and ϕ is strictly single-peaked or if ER > EL and ϕ is quadratic or normal, or if n2 is sufficiently small relative to n1 . (iii) ∂yn /∂μR > 0 if (12) holds, ϕ3 /ϕ3 < ϕ2 /ϕ2 , and n1 is small relative to n2 . Overall, in a wide range of circumstances, as given in condition (A), if a candidate has nonmonetary resources in one market that can be targeted toward that candidate’s base, then that candidate will spend less in that market and more in the other.31 If the poorer candidate has extra nonmonetary resources, then the richer candidate also spends less in that market relative to the other.32 For the most part, these results are similar to those in result (IV) of Table 2, when the nonmonetary resources do not enter the function h, since typically the candidates spend less in the market with the nonmonetary resources. There are some subtle differences, however. If the error distribution is uniform, nonmonetary resources not entering h do not induce unequal expenditures across markets by either candidate. When the nonmonetary resources enter h, the candidate with nonmonetary resources does deviate from equal expenditures across markets but the other candidate does not. 4.2.2 Market size and ad prices Advertising prices may be proportional to market size so that the cost per person is constant, or they might rise less than proportionally so that cost per person declines. First consider the direct effect of differential ad prices in communities that are identical except that their ad prices are not the same. Since xi and yi are the dollar expenditures on ads, xi /μi and yi /μi would be the number of ads in market i. The function h should depend upon the number of ads and not the dollars spent with h(xi /μi , yi /μi ). Assume that μm = 1 as a normalization and consider increases in μn above this level. Theorem 4 Starting from identical markets and normalizing the price of an ad to 1 in market m, if (12) is satisfied then: 1 1 ∂xn /∂μn − xn = ∂(xn /μn )/∂μn + xn < 0 2 2 1 1 (ii) ∂yn /∂μn − yn = ∂(yn /μn )/∂μn + yn < 0 2 2 (i)

31 The two sufficient conditions in (A) seem to go in different directions. One (ϕ uniform) involves small values of ϕ  with no restrictions on hxy while the other through condition (13) involves limits on the magnitude of the cross effect hxy which are weaker when ϕ  is larger. They are in fact consistent, since the expressions signing these derivatives are quadratic in hxy . For the expressions to be positive, reversing the comparative statics requires hxy to violate (13) and for the peak of the quadratic to be above 0. When ϕ  is small the peak is reduced, while when ϕ  is big the range of hxy for which the expression might be positive is small. Hence, it will be only for a limited set of values of ϕ  and hxy , if any, for which the signs of the comparative statics

will be reversed. 32 It is possible that the poorer candidate may actually spend more in a market where the richer candidate also

has nonmonetary resources. The conditions for this seem unlikely to hold, however. From Theorem 3C (iii), for candidate L to spend more in the market where R has extra nonmonetary resources, R must also have strictly more monetary resources, the error distribution must be relatively flat in the tail, and n1 cannot be large relative to n2 .

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In the community with the higher ad price, the candidates will run fewer ads but may spend more or less in total.33 Second, assume that ad prices are proportional to market size. Again starting from otherwise identical markets, ni = μmi and, in market n, the ad effectiveness function is h(xn /μ, yn /μ). Changes in μ now incorporate ad price changes as well as the direct effect of population size. Theorem 5 Starting with identical markets, let μ denote a proportional increase in the populations of all the submarkets of n, causing a proportional ad price increase. Then 1 ∂xn /∂μ = −∂(xn /μ)/∂μ = xn 2 1 ∂yn /∂μ = −∂(yn /μ)/∂μ = yn 2 In contrast to Table 2’s result when ad prices are independent of market size, candidates run fewer, not more, ads in larger markets when ad prices are proportional to market size. They spend more in the larger market, but spending rises less than population and hence ad price. If ad prices vary with population but not proportionally, then candidates could run more, fewer, or about the same number of ads in larger markets. Even though the effect of market size on the number of ads is ambiguous in these cases, dollar spending on ads will always be higher in larger markets.

5 Empirical approach 5.1 Testable implications In the formal analysis above, candidates only allocated resources between two markets. If this assumption were crucial, our ability to test the model empirically would be limited since few campaigns are over exactly two media markets. However, because the equilibrium is in pure strategies, similar results hold for any pair of markets in multi-market situations. Fix expenditures in all but two of the multiple markets. Spending in one market does not spill over to others, and the objective functions of maximizing expected vote are additive across markets. Decisions over any pair of markets are then independent of resource allocation in other markets, given the total spending allocated to the pair of markets. Thus we can test the results in contests with more than one market. Significant additional difficulties exist for empirical tests of the comparative statics results. Not all the parameters in the theoretical model are easily observable and, because of the complexity of the model, some easily observable factors do not relate to unambiguous comparative static predictions. The only directly observable parameters are the total sizes of the different media markets N and M. Parameters such as voter intensities θi and γi and the numbers of the different voter types ni and mi are not directly observable. One approach is L = ER or if ϕ is uniform, then −Vˆ11 Vˆ33 + (Vˆ13 )2 = A2 g  (xn )g  (yn ). Substituting yields ∂xn /∂μn = (g  (xn )/(2g  (xn )))[1 + xn g  (xn )/g  (xn )] and ∂yn /∂μn = (g  (yn )/(2g  (yn )))[1 + yn g  (yn )/g  (yn )]. The terms zg  (z)/g  (z) are curvature measures of g analogous to the coefficient of relative risk aversion and could be greater or less than −1, showing that in this case the ambiguity in the signs of ∂xn /∂μn and ∂yn /∂μn depend in part on the degree of curvature of g. 33 To explore this further, assume the special case of h in (4). If E

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to use observable proxies for the unobservable parameters. For example, the fractions of a market’s population that are Black or evangelical Christian are significantly correlated with voter intensities and submarket sizes. Observables for which the model does not make unambiguous predictions, such as partisan leaning, cannot be used to test the model. However, determining which of the counteracting tendencies is dominant is of interest. One clear prediction is that a change in any observable should affect both candidates in the same way, even if the direction of the effect is ambiguous. From Theorem 2, this is always true for factors that do not enter the effectiveness function h. Even when a factor enters h, such as the type of nonmonetary resources considered in Theorem 3, the candidates differ in how their allocations compare across markets only in exceptional cases. That is, if one market differs from another in some dimension, both parties will tend to spend more in the same market relative to the other market. This is a nonobvious prediction of the model since it might be thought that a Democrat would spend more, and the Republican less, in the market more favorable to Democrats. Two additional hypotheses follow from the proxies fraction Black and fraction evangelical Christian. African-Americans vote overwhelmingly for Democrats, with little variability across elections. Controlling for the overall fraction of the market’s population supporting the Democratic candidate, a higher fraction Black would indicate that the media market has a relatively large number of voters with a high intensity of support for Democrats. Given a reasonably shaped error distribution, the theory suggests that both candidates should spend less in such markets as there is a smaller chance of swaying the voters there. Thus, a second testable implication is that both candidates should devote fewer resources to the market with a higher fraction of African-Americans. In recent years, evangelical Christians have played a role for Republicans similar to that of African-Americans and unions for Democrats. They are intense supporters and a source of resources. Either of these leads to the third testable implication that both candidates should spend less in a market with a higher fraction of evangelical voters.34 To summarize, our empirical model will be used to test the following hypotheses: 1. Any difference in a market characteristic will affect both the Republican and Democratic candidate in the same direction. That is, the effect of any market characteristic should have the same sign for both candidates. 2. Both candidates should advertise less in markets with a larger fraction of AfricanAmericans, ceteris paribus. 3. Both candidates should advertise less in markets with a larger fraction of evangelical Christians, ceteris paribus. 5.2 Empirical insights from the model The above three implications allow us to test the validity of the model. Some of the model’s predictions are ambiguous because of countervailing factors. The empirical results give insight as to which of these factors dominate. As noted above, the effect of market size on advertising strategies depends upon how ad prices vary with population. The dataset we use to measure candidate advertising contains an ad price variable, but it is not considered very reliable (see Goldstein and Freedman 2002). 34 See Page (2004) in USA Today for a discussion of the connection between church membership and church

attendance on voting in the 2000 presidential election and on the organizational use the 2004 Bush campaign made of “friendly congregations” in the 2004 campaign.

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We consider it here to get some idea, even if a crude one, of the relation between advertising price and population across markets. We calculate the average price per ad second for each candidate in each market, and regress the natural log of this price on the log of the market population. State dummies are included to control for pricing differences across states. For the entire sample, the elasticity of market price with respect to population is 0.63 (with a robust t -statistic of 24), suggesting that advertising price is not independent of population size, but varies less than proportionally with population.35 This result is consistent with Stratmann (2009). Using price data that is superior to ours, Stratmann finds that, while price differences cannot be explained entirely by differences in population, the simple correlation between these variables is 0.97. This variation in ad price is between the case in Table 2 result (VI) where candidates advertise more in the larger market, and that in Theorem 5 where they advertise less in the larger market. However, the theory does not determine at what elasticity the two effects would cancel each other out. Thus, it does not provide an unambiguous prediction for how market size affects candidates’ advertising decisions. Some other observable variables, including the partisan leaning of a market or the closeness of the contest, have been used in empirical analyses.36 These variables clearly relate to the model here but not in any simple way, and their effects are more ambiguous than might appear on the surface. Consider closeness. It would seem straightforward that the candidates would spend more where the contest is closer. Under probabilistic voting, however, this is not necessarily true since votes can be gained from individuals who vote for the opponent with probability greater than one half.37 There is no consistent prediction for the effect of closeness on the spending patterns of the candidates. If closeness is correlated with spending decisions, it is because of correlations between closeness and unobserved intensities or subgroup populations. For example, if the number of initially neutral individuals happens to be larger in more equally divided markets, the candidates would spend more in markets where the contest is closer. Although we have no predictions for its effect, we include closeness in some of the empirical specifications below as it has been an important component of other 35 The adjusted R-squared statistic for this regression is 0.82. The result is robust to several changes in spec-

ification. In addition to this regression, we use the average over candidates and races so the sample size is smaller, and the average price only for prime time ads to reduce simultaneity by candidates choosing more ads in cheaper time slots. In all cases the estimated elasticities fall between 0.61 and 0.64, with t -statistics of at least 13. In addition, because we will consider the South and the rest of the country separately below, we also estimate this relation separately for each region. There does not appear to be a large difference between the Southern and non-Southern elasticities of ad price with respect to population. The estimated elasticities range from 0.55 in the non-South to 0.65 in the South, and are both statistically different from one at the 99.9% confidence level. Thus, in both cases, ad price appears to vary less than proportionally with population. 36 To avoid simultaneity problems when estimating the effectiveness of campaign expenditures on vote shares,

these variables are often measured by proxies such as the vote outcome in the presidential election closest in time to the election being analyzed, as in Erikson and Palfrey (1998, 2000). Other authors (for example, Glantz et al. 1976 and Gerber 1998) have used proxies such as party registration in the district or state. 37 To see this, consider two identical markets in which R gets a higher expected vote. This could be because

initially γi > θi ; that is, because partisans for R are more intense than those for L. It could also be because n3 = m3 > m1 = n1 , so that there are more partisans for R than for L in both markets. In the first case, the contest in market n could be made closer by raising θn or by lowering γn relative to the values in market n. From result (I) of Table 2, both candidates will spend more in market n if γn is decreased but less if θn is increased. Similarly, they will spend more if the contest is made closer through an increase in n1 but less if it is through a decrease in n3 . In all cases, the result is not due to an effect through changing the closeness of the election, but to the direct effect of changing the parameters. An increase in n1 makes N bigger than M and thus market n gets more advertising by the candidates and the reverse for a decrease in n3 . Similarly, increasing θn reduces spending because market n is more polarized while lowering γn reduces polarization.

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analyses of campaign spending. If candidates spend more in more evenly divided markets, it would indicate that closer markets also happen to be less partisan. A final relevant demographic is union membership. Two potential effects exist, which may work in opposite directions. One is that union resources tend to go to Democratic candidates. This would be like having extra resources available to Democratic candidates, such as those considered in result (IV) of Table 2 or in Theorem 3. This would tend to lead both candidates to spend less in markets with larger union memberships. A second effect relates to the nature of union voters. Historically, they were viewed as strong Democratic voters, but more recently, some view them as swing voters whose views on various issues align with different parties. They might favor Democrats on economic issues and Republicans on social and security issues, and thus might be persuaded to support either party depending on which issues they consider more salient. If they actually are swing voters, both candidates would tend to spend more in markets with higher union membership.38 Because of these potentially countervailing effects, the model does not have an unambiguous prediction about the effect of union membership. The sign of this effect sheds light on the candidates’ beliefs about the current behavior of union members. A negative sign would indicate that candidates do not view union members as swing voters or, if they do, that the resource effect dominates. As discussed below, we will also use the model to see if the South differs significantly from the rest of the country in its political behavior. 6 Empirical evidence 6.1 Data We will test these hypotheses by examining candidate advertising behavior in the 2002 races for governor and U.S. Senate. Our advertising data come from the Campaign Media Analysis Group (CMAG), which collects individual ad-level data for political advertisements on broadcast and cable television in the 100 largest media markets. These data were made available to us by the Wisconsin Advertising Project, and include information on the content and length of each ad, as well as the station on which it was aired and at which time.39 We use ads only from the 2002 elections, because this is the only non-presidential election year for which the data are currently available.40 In years where there are presidential elections, coattail effects may mean that the extent of clutter exceeds the bounds in conditions 12 and 13 that affect many of our comparative statics results. Examination of candidate allocations suggests that candidates are almost always at an interior solution.41 To further test our model of probabilistic voting, we consider whether 38 There is not a consensus among politicians and commentators as to the nature of union voters. In post

2002 election commentary, Dunham (2002) in Business Week argues that Karl Rove had tried with some success to make them swing voters. On the other hand, the Pew Research Center (2005) analyzes polling data and concludes that while Republicans have made gains generally among middle income voters, “the labor movement has done a very good job of getting those members it has to the polls, and keeping them loyal to the Democratic Party.” 39 See Goldstein and Rivlin (2005) for the CMAG data and Goldstein and Freedman (2002) for a detailed

description of it. 40 CMAG data are not available from the Wisconsin Advertising Project for 1998 or before 1996, and the

most recent year for which data are available is 2002. 41 In our sample of two-party races in states with at least two markets, there is only one case in which a

candidate allocated no advertisements to a market. In the Alabama race for U.S. Senate, Parker—who ulti-

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various market characteristics affect resource allocation in ways that are consistent with our comparative statics predictions. Our general empirical strategy is to examine the relation between the quantity of political advertising for a given candidate in the markets he faces and the relevant characteristics of those markets. Our model was developed only for races with two candidates. For this reason, we consider only advertisements that ran after the primary. Elections in Louisiana in 2002 were held by “nonpartisan blanket primary,” where at the first stage, all candidates of all parties ran in the same contest. Then, if no candidate received more than 50% of the vote, the top two vote-getters ran against each other in the general election, even if they were in the same party. Because this means there are many candidates in each race, we drop Louisiana races from our analysis. We also drop the governor’s races in Arizona, New York, and Oklahoma, as there were three serious candidates in each of these races. Several other races had a third candidate who ran only a few ads or received 1% or less of the vote; these candidates are dropped but those races are otherwise included. We drop the U.S. Senate races in Michigan and Illinois because CMAG lists ads by only one candidate. We also drop several races because the state has only one media market. Finally, many media markets cross state lines, so that states may contain some markets in their entirety and varying fractions of other markets. Suppose that the price of advertising varies proportionally with population, and a state contains all of one market and ten percent of another market. The effective price of an ad will be ten times higher in the second market than in the first. Given that candidates would be likely to make different advertising decisions in these kinds of markets, we drop a market from a state if less than one third of the market lies in that state.42 This leaves us with 34 markets in 11 states for Senate candidates, and 54 markets in 14 states for gubernatorial candidates. As there are two candidates in each race and we consider the allocations of each candidate, this gives us 176 observations. Based upon the testable implications in Sect. 5, we will consider how the number of advertisements for a candidate in the markets he faces are affected by the markets’ fractions Black, fractions evangelical Christian, voting age population, fractions belonging to a union, fractions that voted for Gore in 2000 and, in some specifications, the “competitive distances” of the races. Voting age population is the population aged 18 and older in the market. Data for this variable as well as fraction Black are from the 2000 Census of Population and Housing (U.S. Census Bureau 2000) at the MSA level. Union membership data at the MSA level are from Hirsch and Macpherson (2003). We divide the number of union members by the population aged 18 and older for the fraction union.43

mately lost the race—played 302 ads in Birmingham, 201 in Huntsville, and none in Mobile. His opponent Sessions, on the other hand, played 1809 ads in Birmingham, 1705 in Huntsville, and 1314 in Mobile. In fact, the allocations are surprisingly symmetric across markets. If we consider the pairs of markets faced by a candidate, in fewer than 8% of the cases do candidates play twice as many ads in one market in the pair as in the other. This relative symmetry is inconsistent with a model of deterministic voting but is consistent with the probabilistic voting model considered here (see Fletcher and Slutsky 2009). 42 Chattanooga (GA), St. Louis (IL), Charleston (KY), Cincinnati (KY), Greenville (NC), Norfolk (NC),

Charleston (OH), Charlotte (SC), Savannah (SC), and Shreveport (TX) are dropped. The portion of the Shreveport market in Arkansas is dropped for the same reason, leaving that state with only one market, Little Rock. Because we consider only states with two or more media markets, Little Rock is dropped from Arkansas as well. Finally, Davenport is dropped from Illinois because only four percent of the population of Illinois lies in the Davenport market. 43 Another measure of union presence would be to divide union members by the employed population. Our

results are robust to this alternative measure. All signs in Table 4 are unchanged and significance levels change only slightly.

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Religious affiliation data at the county level are available from Jones et al. (2002), who survey churches in each county for the number of their adherents. As the data are surveybased and the definition of membership and adherence varies across denominations and congregations,44 substantial noise in this variable is likely. The Association of Religion Archives (TheARDA) (2006) categorizes denominations by type including Protestant evangelical and presents the Jones et al. (2002) data at the MSA level. We derive fraction evangelical by dividing the number of Protestant evangelicals by the total population in the MSA as given by TheARDA.45 The fraction that voted for Gore in 2000 controls for the level of Democratic Party support. With this control, fraction Black and fraction evangelical indicate the intensity of support (analogous to θ and γ , respectively, in the theory), holding the number of partisans of each candidate (the submarket sizes in the theory) constant. Finally, in some specifications we include a variable for the competitive distance in the 2000 presidential race, which is equal to the absolute value of 0.5 minus the fraction of the market’s vote for Gore.46 Data for these two variables are from Leip (2006) at the county level, and we aggregate from the county to the MSA.47,48 Historically, Southern voting patterns have differed from those in the rest of the country and as a consequence of this, many studies in the voting literature include a Southern dummy.49 The main explanation for the special nature of Southern voting is race, which seems to have trumped economic and other factors. Recently, however, there has been debate as to whether the South continues to differ or whether voting patterns there can now be explained by the same factors as elsewhere, with race no longer playing the same defining role.50 By including a Southern indicator variable interacted with the variables for market characteristics, we may be able to gain some insight into this question.51 For example, the fraction that voted for Gore can be interpreted as a measure of voter preferences that should 44 In fact, membership and adherence of traditionally Black churches were available for 1990 but not for

2000. The authors state that they stopped reporting Black church membership because those figures are only rough estimates and not very reliable. 45 We also construct fraction evangelical by aggregating the Jones et al. (2002) county data to the MSA level

and dividing by the total population in the MSA as provided by the U.S. Census Bureau from the 2002 Census of Population and Housing. With the exception of the single market of Greenville, SC, whose fraction evangelical we calculated to be greater than one, the difference between the two measures is at most three decimal places out, and the correlation between them is 1.000. As a result, the signs and significance of our results are the same with either measure if Greenville is excluded as an outlier. 46 Note that the value of this variable will be small in close elections and will become larger as the race

becomes less competitive. 47 We use the U.S. Census Bureau’s county and MSA codes, which can be found at http://www.census.gov/

population/estimates/metro-city/99mfips.txt. 48 Because these two variables could exhibit significant multicollinearity, we also have also estimated regres-

sions with a dummy variable equaling 1 if Gore received more than 50% of the vote in the market, and 0 otherwise. Very little is different from the results presented here: no coefficient changes sign, and there are only minor differences in statistical significance. 49 See, for example, Erikson and Palfrey (1998, 2000). 50 The traditional view of the predominance of race in Southern voting was initiated by the classic book by

Key (1949). Knuckey (2005) argues that racial factors reasserted themselves in southern politics in the 1990’s after becoming not a significant predictor of voting in the period prior to that. The main recent challenge to this traditional view is Johnston and Shafer (2006), who argue that class and economic factors other than race explain southern voting patterns after the civil rights era of the 1960s. 51 Recall that there is only one year (2002) of data available from a non-presidential year. Due to the resulting

small number of observations, we have not included other demographic variables such as income and educa-

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potentially be interpreted differently by candidates for senate and governor in the South than in the rest of the country. As such, we would expect the fraction Gore to have a different effect on resource allocation in the South than in the rest of the country, which would be shown by a statistically significant coefficient on the interaction between South and fraction Gore. Table 3 shows summary statistics for each of the variables used in our analysis for each market.52 We present summary statistics for the full sample, as well as separating the sample into Southern and non-Southern markets. Some differences are immediately apparent. The mean size of the voting age population in the non-Southern states is 1.5 times that in the Southern states, but much of this is due to the large population of Los Angeles. The median values are 672,427 and 689,881, respectively. Both the mean and median fraction Black as well as the mean and median fraction evangelical Christian in the South are more than twice that in the non-South. The mean and median fraction union in the non-South, however, are about 2.6 times that in the South. 6.2 Estimation The theoretical model predicts how a given candidate will allocate resources across markets, given the different characteristics of those markets. We estimate the following ordinary least squares equation: (ADS)i = β(X)i + α(SENATE)i + δ(SOUTH)i + ν(SOUTH∗ X)i + η(STATE)i + ui

(14)

where (ADS)i is the number of advertising occurrences shown in market i, and Xi is a vector of market characteristics, as discussed above and shown in Table 3. SENATE is a dummy variable equal to one if the race is for the U.S. Senate to account for possible spending differences between Senate and gubernatorial races. In order to test whether the Southern states are significantly different from the rest of the nation, we include a dummy variable SOUTH, as well as a vector of interactions between the market characteristics and the Southern dummy, SOUTH*X. Finally, STATE is a full set of state dummies, which is intended to capture spending differences across states. Recall that the advertising variable is for the television broadcast market and the market characteristics are based on the MSA; these two areas do not perfectly overlap. As a result, there will be some measurement error for each market, which will persist for all candidates whose race includes that market. Standard errors are clustered by market to capture these correlations in the error term. The results of our ordinary least squares regressions are shown in Table 4. The first column shows results for all candidates; the second is for Democrats only and the third is for Republicans only. Columns 4–6 are the same as 1–3 except that the variable for competitive distance is added. The three hypotheses given in Sect. 5 generally are supported by our empirical results, although some of the estimates are not statistically significant. Given our small sample size, the results conform closely to the predictions of the model, yielding strong support for the probabilistic voting model. tional attainment that might serve as controls. In any case, we would have no priors about the direction of the effect of such variables. 52 We assume that ads placed on behalf of a candidate by another group are perfect substitutes for advertising

by the candidate, and thus include all advertising for a candidate, regardless of its source. However, our results are unchanged if only ads sponsored by the candidate’s campaign are included.

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Table 3 Summary statistics: means and standard deviations for advertising and market characteristics Variable

All states

Non-Southern

Southern

states only

states only 2406

Total number of ads

2318

2222

shown in the market

(1511)

(1575)

(1453)

Fraction Black

0.143

0.090

0.191

(0.102)

(0.056)

(0.111)

Fraction evangelical

0.199

0.127

0.265

Christians

(0.118)

(0.081)

(0.108)

Population aged 18

1273

1544

1025

and older (thousands)

(1712)

(2202)

(1040)

Fraction belonging to

0.064

0.094

0.036

a union

(0.041)

(0.039)

(0.015)

Absolute value of 0.5–

0.094

0.068

0.116 (0.052)

fraction Gore in 2000

(0.057)

(0.052)

Fraction that voted for

0.426

0.456

0.399

Gore in 2000

(0.081)

(0.074)

(0.078)

Senate dummy

0.386

0.310

0.457

(0.488)

(0.465)

(0.501)

176

84

92

Observations

Our first prediction is that any difference in a market characteristic will cause both candidates to devote more resources to the same market, so that the sign of each coefficient should be the same in the Democrat-only specifications (columns 2 and 5) as in the Republican-only specifications (columns 3 and 6). This is true for all coefficients except the fraction that voted for Gore in 2000, for which the model does not provide a sign prediction. This coefficient has a mixed sign across specifications and is never statistically significant. The coefficient for voting age population and the Southern dummy have the same sign across specifications, but are statistically significant in only a few of them. Thus, the results for these variables offer weak support for hypothesis 1. The coefficients for the fractions Black and evangelical are negative and statistically significant across all specifications, offering stronger support for hypothesis 1, and also for hypotheses 2 and 3, that both candidates should devote fewer resources to markets with larger fractions of Black voters and evangelicals. The coefficients without the interactions reflect allocations in the non-Southern states. When the interaction with the southern dummy is included, the net effect of fraction Black in the South remains negative, but the overall effect of fraction evangelical switches sign, becoming positive. The results shown in the Table do not indicate whether the overall effects in the South are statistically different from zero. We also perform regressions where the same equations are estimated separately for the South and non-South.53 While the effect of the fraction Black remains negative in all specifications for the South, it is statistically significant in only two 53 These results are available on request. Note that the estimated effects from the equation with the Southern

dummy interacted with all variables, as shown in the Table, will be the same as those from estimating the regressions separately for the South and non-South; the only difference is in what specific hypotheses can be tested with the standard errors given by the estimation.

(1,625)

a union

(2,507)

evangelical

(3,625)

8,895∗∗

(2,402)

5,650∗∗

(1,696)

(2,069)

(1,588)

1,830

(1,829)

1,057

(872)

1,664∗

(931)

–788

(1,456)

–1,744

(0.02)

0.022

(2,052)

–5,479∗∗∗

2,421

7,273∗∗∗

2,126

South*fraction

South*fraction Black

(1,169)

(1,706)

(1,169) 2,452

(910)

Senate

2,461

2,542∗∗

(1,001)

2,103∗∗

Gore in 2000

South

(1,474)

670

Fraction that voted for

2,128

(2,702)

–4,746∗

Competitive distance

(0.03)

(0.02)

–3,245∗

0.070∗∗

0.046∗∗

Voting age population

Fraction belonging to

(3,341)

(2,267)

Christian

(in thousands)

–7,128∗∗

(1,439)

(1,843)

(1,397)

–6,303∗∗∗

–2,822∗

–3,364∗

–3,093∗∗

Fraction evangelical

Fraction Black

(3)

All candidates

Republicans

(2) Democrats

(1)

Table 4 Determinants of campaign advertising levels (4)

(5)

(2,456)

(1,575)

(2,547)

6,225∗∗

(1,464)

1,192

(1,748)

(3,654)

8,100∗∗

(2,007)

1,767

(2,372)

(1,190) 4,664∗

(916)

2,542∗∗∗

(1,587)

3,973∗∗

2,103∗∗

(1,044)

1,995

–1,960

491

(3,250)

–2,643∗

–5,792∗

(0.04)

0.059

(3,407)

–6,172∗

(1,773)

–3,046∗

Democrats

(1,860)

–4,656∗∗

(0.02)

0.031

(2,368)

–5,015∗∗

(1,290)

–2,664∗∗

All candidates

(6)

(2,447)

4,350∗

(1,530)

616

(1,914)

2,940

(888)

1,664∗

(873)

–1,014

(1,406)

–3,325∗∗

(1,581)

–3,520∗∗

(0.02)

0.003

(2,173)

–3,857∗

(1,319)

–2,282∗

Republicans

Public Choice (2011) 146: 469–499 493

(7,951)

(0.06)

8,503

(6,929)

pop (thousands)

South*fraction union

(3)

Observations

0.8823 88

88

0.6467

(1,447)

1,907

176

0.5669

(1,486)

88

0.8843

(2,071)

352

(2,958)

–1,895

(1,261)

–5,182∗∗∗

(2,098)

–930

(6)

88

0.6748

(1,409)

1,902

(2,830)

–4,402

(967)

–2,571∗∗∗

(1,906)

–1,556

(7,483)

12,393

(0.07)

0.317∗∗∗

Republicans

∗ Statistically significant at the 90% level

∗∗∗ Statistically significant at the 99% level ∗∗ Statistically significant at the 95% level

Voting age population and fraction Black are from the 2000 Census of Population and Housing. Union membership data are from Hirsch and Macpherson (2003). Religious affiliation at the county level is from Jones et al. (2002). Competitive distance is equal to the absolute value of (0.5—the fraction of the market’s vote for Gore). Data for this variable and the fraction that voted for Gore in 2000 were provided by David Leip at the county level. We aggregate county-level variables to the MSA level

Robust standard errors are in parentheses, and are clustered by market. A full set of state dummies is also included. The unit of observation is the market. Columns 1 and 4 include observations for both Democratic and Republican candidates, while columns 2 and 5 include only Democrats and columns 3 and 6 include only Republicans. The Southern states are defined as those states that were in the Confederacy

0.5601

176

R squared

(2,003)

1,127

355

(1,472)

Intercept

1,131

(2,271)

(970)

–3,877∗∗∗

(1,527)

–1,243

(7,996)

15,338∗

(6,256)

13,865∗∗

0.244∗∗∗ (0.09)

0.281∗∗∗

(5) Democrats

(0.06)

distance

(950)

(4) All candidates

–3,149

(1,239)

(964)

–2,571∗∗∗

(1,774)

(2,028) –5,182∗∗∗

(1,525)

–3,877∗∗∗

1,418

533

(8,831)

5,345

(0.08)

0.299∗∗∗

Republicans

South*competitive

South*Senate

975

11,661

0.266∗∗∗

South*voting age

South*fraction Gore

0.233∗∗∗

All candidates

(0.08)

(2) Democrats

(1)

Table 4 (Continued)

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of them. The effect of evangelicals on spending in the South is always positive, but it is statistically significant even at the 80% confidence level in only half of the specifications. Thus, hypothesis 2 is supported in the non-South and weakly supported in the South, but hypothesis 3 is supported only in the non-South. In addition to testing the model, our empirical analysis offers some insight to the effect of market size, given its possible relation to the price of advertising, to the way candidates view union members, and to differences between the Southern states and the rest of the country. Market size has a statistically insignificant effect in four of the six specifications without the interaction, and the effect in the non-South is small. A one standard deviation increase in voting age population (which more than doubles the population of the average-sized market) is associated with an increase of at most 154 advertising occurrences, a 7% increase from the mean number of ads. The small magnitude of this effect is consistent with the evidence that advertising prices do vary with population but at a significantly less than proportional rate. In the South, the effect is larger: a one standard deviation increase in voting age population is associated with an increase of about 334 ads, or about 14% of the mean number of ads. Recall that the theory suggests that dollar spending on advertising should always be higher in larger markets, even though the number of ads shown in the larger market may be higher, lower or the same as in the smaller market. Our empirical results show that candidates do run slightly more ads in larger markets, especially in the South. Candidates spend more in larger markets, both because they show more ads and because each ad is more expensive. This reinforces Stratmann’s (2009) argument that campaign spending is the wrong variable to use to estimate the returns to campaign advertising, and that the number of ads is a better measure. Stratmann suggests that this explains why previous literature has found additional incumbent spending either to have no effect, or to hurt the incumbent, while he finds that, when the price of advertising is allowed to vary across markets, spending is productive for both incumbents and challengers. In the non-South, union membership has a negative effect in all specifications. The effect is statistically significant at the 90% confidence level in four of the six specifications, and at 89% in one more. In the South, the effect of union membership becomes positive, but the difference is significant at the 90% confidence level only in two specifications, and at 80% confidence in two more. This may suggest that in the non-South, candidates do not view union members as swing voters, or the resource effect dominates. In the South, however, union members appear to be viewed more as swing voters than as a resource to Democrats. The specification shown in Table 4 allows us to test directly whether differences between the South and non-South are statistically significant; this is simply a t -test of the interaction coefficient. The most striking difference is in the effect of fraction evangelical. The difference is statistically significant and, as discussed above, so large that the overall effect switches sign. Another large difference can be seen in the Senate variable. While Senate candidates allocate significantly more resources than gubernatorial candidates in the nonSouth, this is reversed in the South. In fact, when the regressions are estimated separately for the two regions, the negative effect in the Southern states is statistically significant at the 95% level in all six specifications. There is also a statistically significant difference in the effect of voting age population, strengthening the positive effect of market size on the number of ads. There are weakly significant differences in some specifications in the effects of fraction Black, fraction union and fraction that voted for Gore. These factors clearly indicate the existence of fundamental differences between the South and the rest of the country. There is some evidence that the treatment of race continues to be an important difference in how

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campaign resources are allocated. The most striking difference, though, is in how evangelical Christians increase candidates’ spending in the South. How to explain this in the context of the probabilistic voting model is an open question.

7 Conclusions We have modified the probabilistic voting model usually specified in a spatial context to apply in a non-Downsian context where the candidates’ strategies are their campaign activities instead of their issue positions. As candidates rarely switch platform positions during a campaign, this modification is more realistic than the standard model, and has important empirical implications. Empirically, focusing on observable campaign activities such as the level of advertising in different media markets instead of on issue positions that are harder to observe or quantify opens up the model to tests of its validity. Looking at statewide races in the United States during the 2002 election cycle, we find that candidate behavior, at least in the non-South, seems to correspond in large part to the comparative statics predictions of the model. Probabilistic voting is not only an intuitively plausible approach; it is also consistent with campaign decisions. One crucial assumption of our model is that all individuals vote. An important and realistic consideration left out is that sometimes individuals abstain, a factor underlying many probabilistic voting models in the spatial context. Having advertising influence voter turnout, especially when candidates are uncertain as to who the voters will be, would be an important extension although one which would considerably complicate the analysis. Turnout considerations might help explain the significant differences we find between candidate behavior in the South and in the rest of the country. Historically, the South has been the region with the lowest turnout of voters; while this is still true, the differential is decreasing with time (McDonald and Popkin 2001). Whether the reason candidates spend more, instead of less, in media markets with intense partisans is to affect turnout is a question worth further exploration. Acknowledgements We are grateful for the comments of Damon Clark, Sarah Hamersma, Jonathan Hamilton, and seminar participants at New York University and the University of Florida, and for the research assistance of Michelle Marcus. Earlier versions of this paper were presented at the 2006 meetings of the Society for Social Choice and Welfare and at the 2007 World Public Choice Society Meetings. The data use agreement from the Wisconsin Advertising Project requires the following acknowledgement: “The data were obtained from a project of the University of Wisconsin Advertising Project includes media tracking data from TNSMI/Campaign Media Analysis Group in Washington, D.C. The University of Wisconsin Advertising Project was sponsored by a grant from The Pew Charitable Trusts. The opinions expressed in this article are those of the author(s) and do not necessarily reflect the views of the University of Wisconsin Advertising Project or The Pew Charitable Trusts.”

Appendix: Proofs Proof of Theorem 1 Vˆ is the sum of six terms. From (9), each term lies strictly between 0 and 1 for all values of the strategies for the two players. Given condition (8), the second partial derivatives of V are negative with respect to xn and xm , and positive with respect to yn and ym . Vˆ is then strictly concave in xn and xm and strictly convex in yn and ym . From this and the assumptions that the strategy spaces are compact and convex, existence of a pure strategy Nash equilibrium follows.

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To show uniqueness, use (5) to eliminate xm and ym from (7). The Jacobian of the system of first-order conditions of the resulting Vˆ with respect to xn and yn can be shown to be negative quasidefinite. All the conditions of the Rosen uniqueness theorem as given in Friedman (1990, page 86) are satisfied and the equilibrium is unique.  Proof of Theorem 2 Substituting the appropriate derivatives into (10) and (11) yields: sign(∂xn /∂μ) = sign[A(hyy − (hy / hx )hxy )∂ Vˆ1 /∂μ] sign(∂yn /∂μ) = sign[A((hy / hx )hxx − hxy )∂ Vˆ1 /∂μ] Since A > 0, condition (12) ensures that the expressions multiplying ∂ Vˆ1 /∂μ in both con ditions are positive. Hence, the signs equal the sign of ∂ Vˆ1 /∂μ, as asserted. Proof of Theorem 3 (A) Substituting expressions for the derivatives in (10) and manipulating yields sign(∂xn /∂μR ) = sign[A(hxy − (hx / hy )hyy )(hxy /(hx hy ) − ϕ3 /ϕ3 ) + ((hxy − (hy / hx )hxx )(A + hyy /(hy )2 )A)]. If (13) is assumed, then hxy /(hx hy ) − ϕ3 /ϕ3 < 0 and since (12) is then implied, hxy − (hx / hy )hyy > 0 and hxy − (hy / hx )hxx < 0. From (8), A + (hyy /(hy )2 )A > 0. Hence, sign(∂xn /∂μR ) < 0 in this case, as asserted. If ϕ is uniform, then ϕ3 = A = 0 and sign(∂xn /∂μR ) = sign[A((hxy )2 − hxx hyy )/(hx hy )]. Since hxx < 0, hyy > 0, hx > 0 and hy < 0, then sign(∂xn /∂μR ) < 0 also holds in this case. Substituting into (11) yields sign(∂yn /∂μL ) = sign[A(hxy − hy / hx )hxx )(hxy / hx hy ) − ϕ1 /ϕ1 ) + (hxy − (hx / hy )hyy )(A + (hxx /(hx )2 )A)]. From (13), (hxy / hx hy ) − (ϕ1 /ϕ1 ) > 0, from (12), hxy − (hy / hx )hxx < 0 and hxy − (hx / hy )hyy > 0, and from (8), A + (hxx /(hx )2 )A < 0 making sign (∂yn /∂μL ) < 0. If instead, ϕ is uniformly distributed so that ϕ1 = A = 0, then again sign(∂yn /∂μL ) < 0. (B) Substituting into (10) yields sign(∂xn /∂μL ) = sign[(hxy − (hx / hy )hyy )(A ϕ1 + Aϕ1 )]. From (12), hxy − (hx / hy )hyy > 0 so sign(∂xn /∂μL ) = sign(A ϕ1 + A ϕ1 ). Sign (∂xn /∂μL ) = sign[(n2 ϕ2 )((ϕ1 /ϕ1 ) − (ϕ2 /ϕ2 )) + (n3 ϕ3 )((ϕ1 /ϕ1 ) − (ϕ3 /ϕ3 ))] after substituting for A and A. For a uniform distribution, ϕi = 0, all i, so ∂xn /∂μL = 0. Given (6), strict single-peakedness implies that ϕ1 ≡ ϕ  (θn − h( 12 ER , 12 EL )) < 0, that ϕ2 ≥ 0 with strict inequality if ER > EL , and that ϕ3 ≡ ϕ  (−γn − h( 12 ER , 12 EL )) > 0. ∂xn /∂μL < 0 follows. (C) Sign (∂yn /∂μR ) = sign[n1 ϕ1 ((ϕ1 /ϕ1 ) − (ϕ3 /ϕ3 )) + n2 ϕ2 ((ϕ2 /ϕ2 ) − (ϕ3 /ϕ3 ))] from analogous substitutions into (11). Again, for ϕ uniform, the right-hand side is 0. If ϕ is strictly single-peaked, then ϕ1 < 0 and ϕ3 > 0. Hence, all terms are negative except for ϕ2 /ϕ2 . If EL = ER , ϕ2 is taken at the peak of the error distribution and must equal 0, making the right-hand side negative. If EL < ER , then ϕ2 > 0. The right-hand side is still negative if ϕ3 /ϕ3 > ϕ2 /ϕ2 . For the quadratic distribution, ϕ  (ε)/ϕ(ε) = −2ε/(c2 − ε 2 ) which for −c < ε < 0 is decreasing in ε so this holds. For the normal distribution ϕ  (ε)/ϕ(ε) = −ε/σ 2 which again decreases in ε as required. However, there can exist single-peaked distributions for which ϕ2 /ϕ2 > ϕ3 /ϕ3 provided γn and ER −EL are sufficiently large and that ϕ  becomes small for ε far from 0. The two terms on the right-hand side are then of opposite sign with the positive term dominating for n2 sufficiently bigger than n1 and being dominated for n2 sufficiently smaller than n1 .  Proof of Theorem 4 Totally differentiating the first-order conditions Vˆ1 − Vˆ2 = 0 and −Vˆ3 + Vˆ4 = 0, substituting the expressions for the various derivatives and simplifying yields ∂xn /∂μn = 12 xn + 12 (A)2 (hy hxy − hx hyy )/(−Vˆ11 Vˆ33 + (Vˆ13 )2 ). Since Vˆ11 < 0 and Vˆ33 > 0, the denominator is positive. Since (12) implies that hy hxy − hx hyy < 0, then ∂xn /∂μn −

498

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< 0. Since ∂(xn /μn )/∂μn = ∂xn /∂μn − xn at μn = 1, then ∂(xn /μn )/∂μn + 12 xn < 0 as required for (i). Similarly, ∂yn /∂μn = 12 yn + 12 (A)2 (hx hxy − hy hxx )/(−Vˆ11 Vˆ33 + (Vˆ13 )2 ) =  ∂(yn /μn )/∂μn + y n . From (12), hx hxy − hy hxx < 0 yielding (ii). 1 x 2 n

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