Campaign Contributions as Valence Tim Lambie-Hanson November 2011

Abstract In this paper, I study how the possibility of voters contributing to candidates in response to the candidates’ policy proposals affects the Downsian model. By introducing campaign contributions as the production mechanism of valence, I am able to allow contributions to intuitively influence the outcome of the election. Under a large class of voter and wealth densities, as long as voter utility due to policy is continuous and single-peaked, I find that if an equilibrium exists, both candidates propose the ideal policy of the median voter. This result is robust to a two-period election model, where voters donate in the first period and vote in the second.

1

Introduction

No matter how similar two candidates standing for election may appear, their policies are not identical.1 In the standard HotellingDowns model of political competition2 however, both candidates propose the ideal policy of the median voter in equilibrium. Campaign contributions are one factor that could contribute to diver1 For

further anecdotal and empirical evidence, see Grofman (2004). (1957)

2 Downs

1

gence in policy platforms as candidates in search of contributions may seek to distinguish themselves from rivals. Recent empirical work suggests both that extreme voters are more likely to be large donors (Berinsky and Lewis, 2007), and that more extreme candidates garner more campaign contributions (Ensley 2009). By differentiating oneself from her opponent, might a candidate be able to secure enough in campaign contributions to overcome the policy disadvantage that comes from moving away from the ideal policy position of the median voter? I examine whether introducing campaign contributions from individuals in a two-candidate Downsian model results in equilibria where candidates’ policy platforms diverge (referred to below as divergent equilibria). Despite introducing donations in such a way that allows for the most extreme voters being the most eager to donate, I find that a divergent equilibrium never exists. When an equilibrium exists, both candidates propose the ideal policy of the median voter. Voters in my model are either informed or uninformed. An informed voter cares only about the policy proposals of the candidates and votes accordingly. An uninformed voter cares about the policy proposals of the candidates, but can also be influenced by campaigning.3 In this model, only informed voters contribute. Candidates 3I

characterize uninformed voters differently from previous literature by allowing them

to have some policy preference, which I feel is more realistic. See Baron (1994) for a more traditional discussion of informed and uninformed voters.

2

use the contributions to create valence in order to lobby uninformed voters and increase their winning chances.4 A voter may contribute to a candidate because she really likes the candidate’s policies or she really dislikes the opponent’s policies. Additionally, some voters are inherently more likely to donate than others. I attempt to capture this intuition by modeling the informed voters’ decision to contribute as a function of the difference in utility generated by each candidate’s policy proposal and a personal threshold that attempts to capture the voter’s inherent willingness to contribute. Intuitively, the main result relies on the idea that by proposing a policy very similar to her opponent’s policy, a candidate can ensure that her opponent receives no campaign contributions. Thus, as long as she also proposes a policy that is preferred by the median voter, the candidate will win the election. This result is robust to any continuous voter and wealth densities, regardless of the proportion of informed and uninformed voters. I also consider a two-stage election model and show that the main result of no divergent equilibrium is robust. Theoretical literature on campaign contributions from individuals is far less extensive than literature investigating the influence of interest groups in politics. Glazer and Gradstein (2005) model Downsian competition with donation-maximizing candidates and show that policy platforms diverge. They assume that voters’ utility 4 See

Andina-Diaz (2006) for a model with endogenously created valence.

3

depends in part on the policy enacted. Thus, under certain conditions, voters may donate to increase the vote share of her preferred candidate. I also find policy platform divergence when candidates are contribution-maximizing.5 Shieh and Pan (2009) show that if voters are motivated by expressive considerations; that is, if voters maximize utility derived only from the position of the preferred candidate (and not the expected utility derived from the outcomes of the election), then introducing voter donations will lead to policy divergence only if candidates care, at least in part, about the total donations each garners.6 Additionally they assume both voters and wealth are uniformly distributed. I assume more realistic voter behavior and only restrict the voter and wealth distributions to be continuous in deriving a similar result. This paper also fits into endogenous valence literature. Recently, several papers have proposed multiple stage games, where first candidates choose their policy position and then, after observing all candidates’ policy proposals, each candidate chooses how much costly valence to buy.7 The candidates’ objective is to maximize their expected utility (the difference between their expected rents and the cost of the valence). Platform divergence is a characteristic of the resulting equilibria. This paper also views valence as costly, but differs in that it restricts the amount a candidate can spend to the 5 Appendix

B

6 Specifically,

they find that if candidates can steal at least

1 N

in donations where N is the

mass of voters (and donors) then a divergent equilibrium will exist. 7 Zakharov (2009), Ashworth and de Mesquita (2009)

4

amount she raises through donations (which is determined by the proposed policies of both candidates).

2

The Model

2.1

Essentials

Two office-motivated candidates, L and R, engage in Downsian competition by proposing policies l and r on [0,1]. There are two groups of voters: informed and uninformed. Informed voters care only about the policy proposals of the candidates and have single-peaked, symmetric preferences. I assume that these preferences can be represented by some continuous function W (|i − x|) where i is the voter i’s ideal policy and x candidate X’s policy proposal. An informed voters may also decide to contribute to a candidate in order to increase the probability that her preferred candidate wins. I assume that an informed voter’s decision to contribute depends only on the difference in utility the voter derives from the two candidates’ policy proposals and the voter’s inherent willingness to donate, measured by an individual threshold, t(i). The size of potential contributions is distributed across the voters’ ideal policy points by some differentiable cumulative distribution function G(x) with corresponding density g(x).8 I assume that voter i’s probability of contributing to candidate L 8 One

could think of g() as being the density of voter wealth or at least highly correlated

with voter wealth.

5

can be described by a contribution function πL (W (|i − l|) − W (|i − r|), ti ), where πL is non-decreasing in utility difference, and ti is a voter’s personal threshold independently drawn from a probability distribution with positive support. If the difference in utility that voter i gets from the candidates’ policy proposals is below ti , the voter will not contribute. Thus, if W (|i − l) − W (|i − r|) < ti , then πL (W (|i − l|) − W (|i − r|), ti ) = 0. Voter i’s decision to contribute to candidate R is determined through a symmetric process. I assume that the candidates use the contributions to generate valence through advertising or other campaign-related activities such that each candidate’s valence is equal to the amount of contributions each candidate garners. The total value of donations to candidate J, and thus candidate J’s valence, is: Z 1 πiJ g(i)di vJ =

(1)

0

Uninformed voters’ preferences over policy can also be described by W (|i − x|). However, uninformed voters are susceptible to campaigning, and thus also care about the candidates’ valence. The degree to which each uninformed voter cares about valence varies, and is described by a salience term, si ∈ R+ . For informed voters, since they care only about policy, si = 0. Thus, voter i (either informed or uninformed) votes for candidate L (R) if: W (|i − l|) + si vL > (<)W (|i − r|) + si vR

(2)

where vX is candidate X’s valence and si ∈ R+ ∪ {0} is voter i’s 6

salience to the candidates’ valence. If W (|i − l|) + si vL = W (|i − r|) + si vR , voter i randomizes between the two candidates. The distribution of all voters over their ideal policies is described by some differentiable cumulative distribution function F (x) with corresponding density f (x). Some of the voters are informed and some are uninformed, but I allow the fraction and distribution of each group to vary without restriction. 2.2

Which voters donate?

An informed voter’s decision to donate, given the candidates’ policy proposals, depends on only the difference in utility derived from the two candidates’ policies and her personal threshold ti . The concavity of the specific utility function, W (|i−x|), determines, in part, which voters are more likely to donate. Holding all else constant, if W (|i − x|) is strictly concave, more extreme voters (those with ideal policy points further from the candidates’ mean policy position) will be more likely to donate, as the difference in utility generated by policy positions l and r, W (|i−l|)−W (|i−r|), is largest for extreme voters.9 Alternatively, if W (|i−x|) is strictly convex, the most extreme voters would be the last to donate. Concave voter utility from policy (also referred to as a convex 9 Additionaly,

concave W (|i − x|) gives rise to some possibly counterfactual predictions.

For example, an extreme right wing voter would be more likely to donate to a moderate leftist candidate facing an extreme leftist candidate than to an extreme rightest candidate who proposes the voter’s ideal policy facing a moderate rightest candidate.

7

loss function) is more commonly used in the literature,10 but there is some debate as to which class of utility gives more realistic predictions.11 Empirical results on donations suggest that more extreme voters are more likely to donate,12 implying that a concave W (|i−x|) may be more appropriate when considering a voter’s decision to donate. However other empirical studies find that voters tend to be risk neutral in voting when uncertain about candidates’ policies, suggesting a utility such as W (|i − x|) = −|i − x| is more reasonable for modeling voting decisions.13 Aside from Proposition 3, I do not restrict W (|i − x|) other than to require continuity. A simple example demonstrating which voters donate can be easily constructed. Assume all voter thresholds are identical, ti = T , define voter i’s utility from policy as W (|i − x|) = −(i − x)2 , and specify πJ (the probability of donating to candidate J) in the following way:   1 if W (|i − j|) − W (|i − j − |) > T, πJ =  0 otherwise.

(3)

In such a parameterization voter i donates to candidate L if: −(i − l)2 − (−(i − r)2 ) > T l+r −b i < 2 where b =

T 2(r−l)

(4) (5)

denotes a buffer region. Symmetrically, it can be

shown that voter i donates to candidate R if i < 10 See,

for example, Groseclose (2001). Osborne (1995) p. 22 for further discussion. 12 Francia et al. (2005) 13 Berinsky and Lewis (2007) 11 See

8

l+r 2

+ b.

Thus all voters to the left of the mean position of the candidates, l+r , 2

minus a buffer region, b, will donate to candidate L, and all

voters to the right of the mean position of the candidates plus an identical buffer region will donate to candidate R (see Figure 1). Density 2.0

xl + xr 2

1.5

1.0 b

b

0.5 xl

xr

L's donations 0.2

R's donations 0.4

0.6

0.8

Policy Position 1.0

Figure 1: In this example W (|i − x|) = −(i − x)2 , ti = T = 0.05, and π = 1 if utility difference is below threshold.

Due to the concavity of Wi (x), voters with ideal policies at the edges of the policy spectrum will be the first to donate and a voter with an ideal policy position at the mean of the candidates’ policies will never donate. Donations to each candidate will come only from the voters outside of the buffer region,b = policy. 9

T , 2(r−l)

from the mean

3

Equilibrium

Either no equilibrium will exist when the candidates’ objective is to win the election or in the unique equilibrium both candidates propose the median policy position, depending on the particular functional forms, densities, and parameterization of the model.14 Proposition 1. If the political competition game has an equilibrium, both candidates propose the median ideal policy of f (x). The intuition for this result relies on the fact that since π = 0 if W (|i − l|) − W (|i − r|) < t(i), by proposing a policy very similar to her opponent’s policy, a candidate can ensure that no donations are garnered by her opponent. By playing a policy that is both close to her opponent and preferred by the median voter, the candidate can ensure that she will win the election. Thus, regardless of the voter and donor densities, if an equilibrium exists, both candidates propose the ideal policy of the median voter in equilibrium. Under special circumstances (some such circumstances involving the symmetry of the voter and donation densities are discussed in the subsequent section), however, can the existence of an equilibrium be guaranteed. For an example when no equilibrium exists, see Appendix A.15 14 The

proof of this result requires that the candidates have perfect information about the

lower bound of the voters’ thresholds, ti . 15 Note that in equilibrium no contributions are garnered from voters. Thus, voters vote according to policy preferences only. As a result, the analysis is insensitive to si , the salience of voter i to advertising, as long as, as has been assumed, si is finite and all voters care

10

3.1

Implications of Symmetric Voter and Donor Densities

The robustness of Proposition 1 to any continuous voter density, f (x), and continuous donation density, g(x), is a desirable property; but, so is the existence of an equilibrium. By further restricting those densities, an equilibrium can be guaranteed. Proposition 2. If the ideal policy of the median voter is at the midpoint of the support of the donation density, g(x), and g(x) is symmetric, continuous, and single-peaked, then both candidates propose the ideal policy of the median voter. To understand the intuition of the result, first assume that L proposes a policy to the left of the median voter and R proposes the median policy. Consider a voter with an ideal policy a distance z to the left of the candidates’ mean policy and a voter with an ideal policy a distance z to the right of the mean policy. Both, a priori the realization of ti for each voter, have the same expected probability of donating. But the voter to the right of the mean policy will have a higher donation density (she will donate more). So for every donation that L expects to receive, R expects to receive a donation as well, and the donation to R will be larger (see Figure 2). Thus, no incentive to deviate from the ideal policy of the median voter in part about the policy positions of the candidates. However, if si converges to ∞ for a measurable set of voters (meaning this voters will vote for the candidate with higher valence with certainty), the analysis changes slightly. In such a case, the only equilibrium is now both candidates proposing the median ideal policy of only the voters with finite si . The intuition is identical to the main case, but as some voters do not care about policy, their mass is not considered when determining equilibrium.

11

exists. Density 2.0 l+r 2 1.5

1.0 z

z

0.5 l

g

r

Policy 0.2

0.4

0.6

0.8

1.0

Figure 2: If L deviates, she will receive fewer contributions on average. For every voter who contributes to L with probability p, there is a voter who contributes more to R with the same probability.

Bearing in mind that empirical studies have found that more extreme voters are more likely to donate large amounts, perhaps it is unrealistic to assume that g(x) is single-peaked. But if, instead, the voter’s utility due to policy, W (|i − x|), is strictly concave, then the assumption that g(x) be single-peaked can be relaxed and the result still holds. Proposition 3. If the median voter determined by f (x) is located 12

at the midpoint of the voter continuum, the donation density g(x) is symmetric and continuous, and the voter utility, W (|i − x|), is strictly concave, both candidates propose the ideal policy of the median voter. To understand the intuition of Proposition 3, suppose that candidate L proposes a policy to the left of the median voter while candidate R proposes the ideal policy of the median voter. Recall that for concave W (|i − x|), since π L (·) is non-decreasing in W (i − l) − W (i − r), the most extreme voters will have the highest probability of donating. Due to the symmetry of g, g(x) = g(1−x).16 Therefore, for any voter who is expected to contribute to candidate L with probability pL , there is a corresponding voter who contributes the same amount to candidate R with probability pR , such that pR ≥ pL (see Figure 3). Thus, the candidates do not have incentive to propose a policy other than the median policy. 16 Assumes

voter ideal points are distributed on [0, 1].

13

Density 2.0

1.5

mean policy

1.0

g 0.5 x

x i

l

j

r

Policy 0.2

0.4

0.6

0.8

1.0

Figure 3: For concave (convex) W (·), voter i’s expected probability of donating to candidate L will be less than (greater than) or equal to voter j’s expected probability of donating to candidate R.

By allowing W (|i−x|) to be convex, it is possible that a candidate could garner enough donations to overcome her policy disadvantage by proposing a policy different from the ideal policy of the median voter, since in such a case extreme voters have the lowest probability of donating.

14

4

Discussion

In this section, the implications of two tweaks to the model are considered and discussed. First, I discuss the implications of holding the election in two stages. In this case, the median voter theory holds. Second, I find that donation-maximizing candidates will, as also shown in existing literature, diverge in policy position. 4.1

A Two-Stage Election Model

A plausible alternative to the one-stage model proposed in this paper is to consider the election occurring in two periods. In the first period, candidates vie for donations and in the second period stand for election. Again it seems logical that the centrifugal force introduced by donations in the first period may lead to policy platform divergence; however, it turns out that the median voter theorem holds for a two-stage election with the same donation rule. Consider a two-stage election where during the first stage the candidates propose policies and voters donate according to the rule described by Equation 2. Then, in the second stage, candidates propose a second policy and voters vote for their preferred candidate. Assume that if a candidate proposes a different policy in the second period than in the first, voters are uncertain about which policy will be enacted. As a result voters expect the first period policy enacted with probability p and the second policy enacted with probability 1 − p. Thus in the second period Voter i votes for Candidate L (R)

15

if: pW (|i−l1 |)+(1−p)W (|i−l2 |)+si vL > (<)pW (|i−r1 |)+(1−p)W (|i−r2 |)+si vR (6) If pW (|i−l1 |)+(1−p)W (|i−l2 |)+vL = pW (|i−r1 |)+(1−p)W (|i− r2 |) + vR , Voter i randomizes between the two candidates. Proposition 4. In the two stage game of political competition, both candidates propose the ideal policy of the median voter in both periods. The intuition of the proof is similar to that of the one-stage election. In the second stage of the election, the candidate has no incentive to propose a policy different from the ideal policy of the median voter, since all contributions were made in the first period. Suppose in the first stage one candidate decides to propose a policy to the different from the ideal policy of the median voter to try to garner contributions. Then the candidate’s opponent can assure herself victory by proposing a policy slightly closer than the candidate to the ideal policy of the median voter. By doing so, as shown in the one-stage case, she can prevent the candidate from garnering any contributions, and as she proposes the preferred policy in the first period, the median voter will have higher expected utility by voting for her. Therefore, no candidate will deviate from the median policy in the first period.17 17 Another

possible modeling technique for this two stage game is to assume that candidates

face a penalty for dishonesty if the policy they propose in the second period differs from the

16

4.2

Candidates maximize donations

Assume the electorate has uniform donation density over [0,1], and consider the parameterization shown in the example given in section 2.2. Candidate X receives donations equal to the measure of voters who donate to X. Thus, candidate L receives donations equal to l+r 2

−

T 2(r−l)

and candidate R receives donations equal to 1 −

l+r 2

−

T .18 2(r−l)

Thus each candidate’s problem is clear. L chooses l to maximize l+r 2

−

T 2(r−l)

and R chooses r to maximize 1 −

l+r 2

−

T . 2(r−l)

The

first order conditions give rise to the condition that in equilibrium √ L chooses l and R chooses r such that l = r − T ,19 , and the size of donations that L(R) receives is at least as large as L(R) would receive by locating on the opposite side of R(L).20 This equilibrium results in all voters with ideal policy points to the left of l donating to L and all voters with ideal policy points to the right of r donating √ to R. Total donations are of value 1 − T , with the amount each policy they propose in the first period, C(|x1 − x2 |). Assume also that the more the second period policy proposal deviates from the first period policy proposal, the larger the penalty, thus

dC d|x1 −x2 |

> 0. In an earlier version of this paper, I show that both candidates still

propose the ideal policy of the median voter in both periods. 18 Recall

right of

R’s

l+r 2

+

T 2(r−l)

l+r 2

−

T 2(r−l)

donate to candidate L and voters to the

donate to candidate R.

T = 0. 2(r−l)2 T second order condition holds: - (r−l)3 < 0. T first order condition: − 21 + 2(r−l) 2 = 0.

19 L’s

L’s

that voters to the left of

first order conditon:

1 2

−

R’s second order condition is the same as L’s. 20 Formally

this condition is

l+r 2

−

T 2(r−l)

≥ 1−

r 0 +r 2

−

T 2(r 0 −r)

∀r0 ∈ (r, 1], but this reduces

to limiting l to the left of the median, and r to the right of the median.

17

candidate receives dependent on their relative positions.

5

21

Conclusion

The results from this paper add to the evidence that Downsian models with opportunistic candidates fail to produce equilibria with divergent policy platforms. Under a large class of voter and donation densities, for any continuous, single-peaked voter utility derived from policy, no equilibria with divergent policy platforms exist. When the election is extended to two periods, the implications are even stronger and the unique equilibrium where both candidates propose the ideal policy of the median voter will always exist under well-behaved donation densities. These results imply that the search for explanations of divergent policy platforms must look elsewhere. One such direction would be to investigate the motivations of voters. Many of the results in this paper are based on the assumption that voters donate only based on the relative policy positions of the candidates. Other factors may contribute to voters’ decisions to donate while at the same time driving more realistic results in electoral models. 21 Similar

to the results from firms in duopoly, candidates do not maximize overall donations

in equilibrium. To maximize total donations, candidates would position themselves at extreme opposite ends of the policy spectrum. This minimizes measure of voters who are donors. are of value 1 −

T .22

T , 2(r−l)

which in turn maximizes the

Total donations from candidates adopting this strategy

This is not an equilibrium, because by unilaterally deviating towards

the center, a candidate can increase the donations she receives.

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Appendix A: A simple example when no equilibrium exists Assume the donation density, g(x), and the voter density, f (x), both are the triangular density with support [0,1] and mode at 3/4. The median voter is located at the policy position that corresponds to q 3 ≈ .612. Further, assume W (|i − x|) = −(i − x)2 , all voters have 8 the same donation threshold ti = T , and the parameterization of π() given in section 2.2. As indicated in Equation 5, in such a case all voters with ideal policy points less than

l+r 2

−

T 2(r−l)

will donate to L and all voters

with ideal policy points greater than

l+r 2

+

T 2(r−l)

will donate to R.

Suppose we set the threshold level T = 0.05. Then if R proposes the median policy platform, L can win the election by proposing, for example, l = 0.55. By doing so, she will gain a valence advantage of 0.04 which will overcome the policy disadvantage of 0.004 in the median voter’s calculation (see Figure 5). Thus, both candidates proposing the ideal policy of the median voter is not an equilibrium.

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Figure 4: By deviating L garners more donations than R, giving L a valence advantage large enough to overcome her policy disadvantage when Wi (x) = −(i − x)2 .

Not all asymmetry in g(x) and f (x) leads to a case where no equilibrium exists. For example, if T > 0.111 in the previous example, it can be shown (through slightly arduous algebra), that in equilibrium both candidates will locate at the ideal policy of the median voter.

Appendix B: Proofs Proof of Proposition 1. This proof is by contradiction. Assume that one candidate will propose a policy. That candidate will either pro20

pose the ideal policy of the median voter or she will not.

Case 1: Suppose one candidate proposes a policy different from the ideal policy of the median voter.

Then the candidate’s opponent wins by proposing a policy slightly closer to the ideal policy of the median voter than the candidate. By doing so, the opponent ensures that the opponent receives no contributions. Since ti is drawn from a distribution with strictly positive support, min ti , is some positive real number. Thus by proposing a policy such that the difference in utility is less than min ti for all informed voters, neither candidate will garner contributions. Such a policy is guaranteed to exist by the continuity of voter utility, W (|i − x|). Since the candidate’s policy position is viewed as inferior by the median voter and no donations, thus no valence, are garnered, the candidate will lose the election. Thus she will not propose a policy different from the ideal policy of median voter.

Case 2: Suppose the opponent will propose the median policy position of f(x).

The best response in this case depends specification of the model. In some cases, the candidate can win the election by deviating from the median and garnering enough donations, hence valence, to overcome her policy disadvantage. In such cases, no equilibrium exists. 21

In other cases, no profitable deviation exists and the candidate’s best response is to play the median policy position and tie, as predicted by the median voter theory. Proof of Proposition 2. This proof is by contradiction. Assume that L proposes a policy to the left of the median voter and R proposes the median policy. A voter i with ideal policy a distance z to the l+r , 2

left of the mean policy proposal,

donates with probability pLi =

π(W (| l+r − z − l|) − W (| l+r − z − r|), ti ) to candidate L. The voter 2 2 j with ideal policy a distance z to the right of the mean policy l+r proposal will donate with probability pR j = π(W (| 2 + z − r|) −

+ z − l|), tj ). W (| l+r 2 Since both voters are the same distance from the mean proposed policy, both voters have the same difference in utility due to the candidates policies. Further, since ex ante ti = tj , the donor located at

l+r 2

− z donates, on average, to candidate L with the same

probability as the donor located at

l+r 2

+ z donates to candidate R

(E|pLi | = E|pR j |). Due to the symmetry and single-peakedness of g, g( l+r − z) < g( l+r + z), since 2 2

l+r 2

< 0.5.

Thus for each informed voter who is expected to donate with probability p to candidate L, there is a corresponding informed voter who is expected to donate t with probability p to candidate R and who will donate more. Thus, the expected donations that candidate R garners, as well as the corresponding expected valence generated, will be at least large as those of candidate L.

22

Since candidate R will have larger expected valence and the policy preferred by the median voter (guaranteed by the symmetry of f (x)), R will have a larger probability of winning. Since candidate L can win with probability

1 2

by choosing the ideal policy of

the median voter, she will not deviate. Thus, in equilibrium both candidates propose the ideal policy of the median voter. Proof of Proposition 3. This proof is by contradiction. Suppose that candidate L proposes a policy to the left of the median voter while candidate R proposes the ideal policy of the median voter. For concave W (|i − x|), since π L (·) is non-decreasing in W (i − l) − W (i − r), the most extreme voters will have the highest probability of donating. Further, since L proposes a policy to the left of the median, the most extreme leftist voters will be less extreme (their ideal policies will be closer to the mean of the policies proposed by the candidates), than the most extreme rightest voters. Therefore, assume the leftist voter i with an ideal policy at x donates to L with probability pLi and the rightest voter j with ideal policy 1 − x will donate with probability pR j . Since the voter with ideal policy x is closer to the mean policy proposal of the candidates, a priori the realization of ti and tj , E|pLi | < E|pR j |. Due to the symmetry of g, g(x) = g(1 − x). Therefore, for any voter who is expected to contribute to candidate L with probability pL , there is a corresponding voter who contributes the same amount to candidate R with probability pR , such that pR ≥ pL . Thus, the candidates

23

do not have incentive to propose a policy other than the median policy.

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Grofman, B. (2004). Downs and two-party convergence. Annual Review of Political Science, 7:25–46. Groseclose, T. (2001). A model of candidate location when one candidate has a valence advantage. American Journal of Political Science, 45:862–886. Osborne, M. J. (1995). Spatial models of political competition under plurality rule: A survey of some explanations of the number of candidates and the positions they take. Canadian Journal of Economics, 27:261–301. Shieh, S. and Pan, W.-H. (2009). Individual campaign contributions in a downsian model: expressive and instrumental motives. Public Choice, forthcoming. Zakharov, A. (2009). A model of candidate location with endogenous valence. Public Choice, 138:347–366.

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