Public Choice98: 269–286, 1999. 269c 1999Kluwer Academic Publishers. Printed in the Netherlands.

Opposition backlash and platform convergence in a spatial votingmodel with campaign contributions

RICHARD BALL �

Department of Economics, Haverford College, Haverford, PA 19041, U.S.A.

Accepted 8 January 1997

Abstract. This paper investigates the effects of campaign contributions on candidate behaviorin elections. The particular focus is on how candidates choose their platforms when they knowthat the positions they take will influence the level of campaign contributions that they (andtheir opponents) receive from concerned interest groups. The analysis is carried out in thecontext of a simple one-dimensional spatial voting model with two candidates and two interestgroups. Since the earliest Hotelling-Downs formulations, a central issue in the literature onspatial voting has been the degree to which, under various sets of assumptions, the candidates’platforms converge in equilibrium. This paper extends that literature by examining how theintroduction of interest groups making campaign contributions affects the degree of platformconvergence. The paper shows that when choosing their platforms, candidates face a trade-offbetween generating increased support from opponents and provoking a backlash from theopposition. An example is developed to illustrate a surprising result that can occur because ofthe backlash effect: the introduction of twoextremistinterest groups may lead the candidatesto moderatetheir platforms, resulting in a greater degree of platform convergence than wouldbe observed in the absence of any campaign contributions.

1. Introduction

This paper investigates the effects of campaign contributions on candidatebehavior in elections. The particular focus is on how candidates choose theirplatforms when they know that the positions they take will influence thelevel of campaign contributions that they (and their opponents) receive fromconcerned interest groups. The analysis is carried out in the context of a simpleone-dimensional spatial voting model with two candidates and two interestgroups. Since the earliest Hotelling-Downs formulations, a central issue in theliterature on spatial voting has been the degree to which, under various sets ofassumptions, the candidates’ platforms converge in equilibrium. This paper

� This paper was presented at the 1996 annual meeting of the Public Choice Society.The comments received there, particularly from the discussants John Matsusaka and MichaelMunger, are gratefully acknowledged. Helpful discussion and assistance were also provided byLynne Butler, Ellsworth Dagg, Marissa Golden, Picard Janne, Miller Maley and Sid Waldman.Any remaining errors are of course the responsibility of the author.

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extends that literature by examining how the introduction of interest groupsmaking campaign contributions affects the degree of platform convergence.

The general qualitative result of the paper is that when choosing their plat-forms candidates must anticipate not only the response of the allied interestgroups from which they receive contributions, but also the response of theopposition interest groups that contribute to their opponents. The incentivesto please allies on the one hand, but not to alienate opponents on the other,will in general pull the candidates in opposite directions. Which effect dom-inates – and so whether the introduction of campaign contributions leads togreater or less platform convergence – depends on the particular specificationof the model. An example developed in this paper illustrates one surprisingpossibility: the introduction of twoextremistinterest groups may lead thecandidates tomoderatetheir platforms, resulting in a greater degree of plat-form convergence than would be observed in the absence of any campaigncontributions.

The trade-off candidates face between choosing platforms that will currythe favor of like-minded interest groups versus moderating their positions toreduce the backlash from opposition interest groups has not been explicitlyanalyzed in previous spatial models of electoral competition with campaigncontributions. This paper demonstrates, in fact, that in one well-known study(Magee, Brock and Young, 1989) the failure to incorporate the oppositionbacklash effect represents a serious omission in the analysis.

Section 2 of this paper presents a simple version of the probabilistic spatialvoting model (PSVM)without the influence of campaign contributions byinterest groups. This preliminary model serves as a benchmark for comparisonwith the results obtained when campaign contributions by interest groupsare introduced. The version of the PSVM presented in this section closelyfollows the structure developed by Wittman (1983, 1990), Hansson and Stuart(1984), Calvert (1985) and Mitchell (1987).1 This model is based on theclassic Hotelling-Downs formulation, with two modifications: candidates areallowed to care not only about winning the election, but also about whatpolicy is implemented after the election; and (from the point of view of thecandidates) the outcome of the election depends stochastically rather thandeterministically on the platforms chosen by the candidates. A general resultin the PSVM literature is that, with these modifications of the Hotelling-Downs model, the candidates’ platforms need not completely converge inequilibrium, and a simple example illustrating such non-convergence is alsopresented in Section 2.

Section 3 develops a two-stage version of the model in which the candidates’simultaneous choices of platforms are followed by simultaneous choices ofcampaign contribution levels by two interest groups. An example developed in

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Section 3 illustrates the possibility of increased platform convergence arisingas a result of the influence of extremist interest groups, and Section 4 exploresin general the trade-off candidates face between seeking the support of alliedinterest groups and alienating opposition interest groups. Section 5 reviewsseveral previous works that have studied two-stage spatial voting models withinterest group campaign contributions, and Section 6 concludes.

2. A probabilistic spatial voting model without interest group influence

Consider an election between two candidates, A and B. During the campaignCandidate A announces a platform a and Candidate B announces a platformb. These platforms are simply the policies that the candidates commit toimplement if elected.2 The unidimensional policy space from which a and bare chosen is normalized to the interval [0,1]. The outcome of the electiondepends stochastically on the candidates’ platforms, with the probability thatCandidate A wins the election given by a function P(a,b). (Candidate B’sprobability of winning is of course 1–P(a,b).) Two assumptions are madeabout the probability of winning function:

(A1) 0� P(a;b) � 1

(A2) For a< b; P(a;b) is non-decreasing in a and non-decreasing in b, andfor a> b; P(a;b) is non-increasing in a and non-increasing in b

(A1) simply states that the probability represented by P(a,b) must always bebetween 0 and 1. (A2) reflects the spatial nature of the competition betweenthe candidates. It says that if one candidate moves his platform toward thatof his opponent, then he does not decrease (and may increase) his probabilityof winning the election; if he moves his platform away from his opponent’s,then he does not increase (and may decrease) his probability of winning.A particular form of P(a,b), derived from explicit micro-foundations andsatisfying both of these assumptions, is introduced below in example 1.

The candidates’ objective functions depend on two components. First, eachplaces an intrinsic value on holding office; these office-motivation parametersare denoted kA and kB. In addition, each cares about the policy implementedafter the election (regardless of who is in office); the policy preferencesof Candidates A and B are denoted respectively as u(�) and v(�). The overallobjective functions are the sum of the expected policy payoff and the expectedoffice payoff. For Candidates A and B respectively these are

U(a;b; kA) = P(a;b)u(a) + [1� P(a;b)]u(b) + P(a;b)kA

V(a;b; kB) = P(a;b)v(a) + [1� P(a;b)]v(b) + [1� P(a;b)]kB (1)

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In the one-period game with no campaign contributions, Candidates A andB simultaneously choose their platforms a and b from the interval [0,1], andtheir payoffs are given by U(a,b,kA) and V(a,b,kB).

A central result of the previous literature is that in this probabilistic ver-sion of the spatial voting model in which the candidates are at least partlypolicy-motivated, the candidates’ platforms need not completely converge inequilibrium.3 This paper investigates how the degree of platform convergenceobserved in equilibrium is affected by the introduction of special interestgroups that influence the outcome of the election by making campaign contri-butions. As a benchmark for comparison with the model involving campaigncontributions, the following example illustrates an equilibrium with less thancomplete platform convergence that arises in the basic PSVM without interestgroup influence.

Example 1. Consider the normal form of the PSVM developed above.For simplicity of exposition, assume that the candidates are purely policy-motivated (kA = kB = 0).4 Assume that the candidates’ preferences overpolicies z in the interval [0,1] are given by u(z) = �1

2(z� 1)2 and v(z) =

�12z2, so that the ideal points of Candidates A and B are 1 and 0, respectively.To derive the probability of winning function used in this example,5 sup-

pose that the voters’ ideal points are distributed along the policy space [0,1].Assuming that every voter’s policy preferences are single-peaked (and sym-metric), then each individual will vote for the candidate whose platform isclosest to his ideal point.6 In this case, the winner of the election will be thecandidate whose platform is closest to the ideal point of the median voter.7

The source of randomness in the model is that candidates do not know withcertainty what the median voter’s most preferred policy is.8 Their (common)beliefs about the location of median voter’s ideal point, m, are represented bya density f(m) over the policy space [0,1]. Given the announced platforms aand b, each candidate perceives his probability of winning simply to be theprobability that m is closer to his platform than to his opponent’s platform(and each candidate’s probability of winning is one half if the two platformsare equidistant from m). The probability of winning function will then be ofthe general form

P(a;b) =

8>>>>>>>><>>>>>>>>:

a+b2R

0f(m)dm if 0� a< b� 1

12 if 0 � a= b� 1

1Ra+b

2

f(m)dm if 0� b< a� 1

(2)

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Suppose also that the candidates’ beliefs are that the median voter’s idealpoint is distributed uniformly on the policy space [0,1], so that f(m) =�

1 for 0� m� 10 otherwise

. In this case, the probability of winning function (2) will

have the particular form

P(a;b) =

8>><>>:

a+b2 for 0� a< b� 1

12 for 0� a= b� 1

1� a+b2 for 0� b< a� 1

(3)

Both the general form of the probability of winning function given in (2), aswell as the special case given in (3), satisfy assumptions (A1) and (A2).

As discussed in footnote (3), we know that each candidate will choose aplatform that is closer to his ideal point than is his opponent’s platform, sothat we must have 0� b < a � 1. Given this condition, the probability ofwinning function will be of the form P(a;b) = 1� a+b

2 , so the candidates’objective functions can be written as

U(a;b;0) = (1� a+b2 )[�1

2(a� 1)2] + (a+b2 )[�1

2(b� 1)2]

V(a;b;0) = (1� a+b2 )(�1

2a2) + (a+b2 )(�1

2b2)(4)

The players’ reaction functions are then implicitly defined by the first-orderconditions9

@U(a;b;0)@a = (�1

2)[�12(a� 1)2]� (1� a+b

2 )(a� 1) + (12)[�

12(b� 1)2] = 0

@V(a;b;0)@b = (�1

2)(�12a2) + (1

2)(�12b2)� (a+b

2 )(b) = 0 (5)

These equations are solved by the pair(a� = 34;b

� = 14), which constitutes

a Nash equilibrium for this game. This equilibrium illustrates the result ofless than complete platform convergence in this version of the model withoutcampaign contributions. �

Some intuition into the candidates’ choices of platforms in this example canbe gained by examining the general form of the first-order condition definingCandidate A’s equilibrium platform. (A symmetric analysis of Candidate B’sequilibrium choice of platform would yield the same intuition.) For an interiorsolution with 0< b� < a� < 1, this first-order condition will be of the form

@U(a�;b�; kA)

@a=@P(a�;b�)

@a[u(a�)� u(b�) + kA ] +

@u(a�)@a

P(a�;b�) = 0

(6)

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The first term on the right-hand-side of (6) is the marginal cost to Candidate Aof an increase in his platform. It consists of the decrease10 in his probabilityof winning resulting from diverging from Candidate B’s platform, timeshis net benefit of winning the election (the difference between the policypayoff he receives if he wins and the policy payoff he receives if CandidateB wins, plus the intrinsic value he places on holding office). The secondterm shows the marginal benefit of an increase in a. This term consists ofthe amount by which the increase in a increases Candidate A’s preferencefor his own platform, times the probability that Candidate A wins and thatplatform is actually implemented. Condition (6) states that, given CandidateB’s equilibrium platform b�, Candidate A’s optimal platform a� equates thesemarginal costs and benefits.

3. The PSVM with campaign contributions

This section extends the one-stage version of the PSVM developed in Section2 to include a second stage in which special interest groups make campaigncontributions. In the first stage, candidates A and B simultaneously choosetheir platforms a and b from [0,1]. In the second stage, after observing theplatforms selected by the candidates, two interest groups, labeled IG A and IGB, simultaneously decide how much money to contribute to each candidate.The amount of money that IG A contributes to candidate A will be denoted�, and the amount of money that IG B contributes to candidate B will bedenoted�.11

Each candidate’s probability of winning now depends not only on thelocations of the platforms, but also on the amounts of money contributedto each candidate. The function~P(a;b; �; �) will represent candidate A’sprobability of winning (which again is equal to one minus Candidate B’sprobability of winning). Three assumptions are made about this generalizedprobability of winning function:

(~A1) 0� ~P(a;b; �; �) � 1

(~A2) For a<b, ~P(a,b,�,�) is non-decreasing in a and non-decreasing in b,and for a>b, ~P(a,b,�,�) is non-increasing in a and non-increasing in b

(~A3) ~P(a,b,�,�) is non-decreasing in� and non-increasing in�

(~A1) ensures that~P(a;b; �; �) always gives a probability between zero andone. (~A2) is a direct extension of (A2), and again reflects the spatial natureof the electoral competition. (~A3) simply states that campaign contributionsto a candidate increase (or at least do not decrease) his chance of winning

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the election. The simplest way to think of this is that the candidates usethe contributions they receive to buy publicity such as television advertising,and that by advertising a candidate increases his probability of winning theelection. This assumption is consistent with the empirical work of Jacobson,who argues that, at least for non-incumbents in congressional elections, “theamount of money [they] spend has a large effect on the proportion of votesthey receive and on their probability of winning the election” (Jacobson,1992:50).12 Similarly, one could think of the expenditure made by the interest groupnot as a contribution to the candidate’s campaign, but as an “independentexpenditure” made on behalf of the candidate to help him get elected.13

The major alternative explanation for why interest groups make campaigncontributions is that they use them to “‘buy’ access to the successful candi-date” (Austen-Smith, 1987: 123).14 This is the approach taken by Aransonand Hinich (1979) in their study of the effects of legislation limiting or requir-ing disclosure of campaign contributions. They make the polar assumptionthat contributions haveno effect on the outcome of the election, and that“the contributor’s sole purpose in giving is to invest in a kind of insurancepolicy: : : [T]hat insurance either provides a benefit if the recipient candidatewins, or forestalls punishment if that candidate is less than magnanimous invictory” (p. 440). Similar assumptions are maintained by Hinich and Munger(1994), who, following Welch (1974), focus on the trade-off a contributorfaces between “quid pro quo” and “ideological” motivations.

The assumption maintained in the present paper, that contributions aremade to affect the outcome of the election, was also adopted by Austen-Smith(1987) and Magee, Brock and Young (1989). It is appropriate here because theissue being studied is how candidates modify their positions in anticipation ofcontributions by interest groups; it is the fact that these contributions influencewho is most likely to win the election that makes the candidates care aboutthem and anticipate them when choosing their platforms.15

As before, the policy preferences of Candidates A and B are representedby functions u(�) and v(�), and their office-motivation parameters are kA andkB, so their payoff functions for the two-stage version of the PSVM can bewritten as

~U(a;b; �; �; kA) =

~P(a;b; �; �)u(a) + [1� ~P(a;b; �; �)]u(b) + ~P(a;b; �; �)kA

~V(a;b; �; �; kB) =

~P(a;b; �; �)v(a) + [1� ~P(a;b; �; �)]v(b) + [1� ~P(a;b; �; �)]kB

(7)

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IG A has policy preferences represented by a function!(�), and IG B’spolicy preferences are represented by a function (�). As in Austen-Smith(1987) and Magee, Brock and Young (1989), the interest groups’ net payoffsare given by the expectation of their policy preference functions, less theircampaign expenditures. For interest groups A and B, respectively, these are

(a;b; �; �) = ~P(a;b; �; �)!(a) + [1� ~P(a;b; �; �)]!(b) � �

(a;b; �; �) = ~P(a;b; �; �) (a) + [1� ~P(a;b; �; �)] (b) � �(8)

The timing of the game is as follows. Candidates A and B simultaneouslychoose their platforms a and b from the interval; after observing the platformschosen by the candidates, IGs A and B simultaneously choose non-negativecontribution levels� and�; and the payoffs are as given in (7) and (8).

The following example extends the one-shot game solved in example 1to the two-stage version with campaign contributions outlined above. Allparameter values and functional forms are consistent with those used inexample 1, so that the results from the two examples are comparable. Thissecond example illustrates the possibility of increased platform convergenceresulting from the introduction of extremist interest groups.

Example 2. Assume as in example 1 that the candidates are purely policy-motivated (kA = kB = 0) and that their policy preferences are given byu(z) = �1

2(2 � 1)2 and v(z) = �1222. In the absence of any campaign

contributions(� = � = 0), the probability of winning function will beidentical to that used in example 1:~P(a;b;0;0) = P(a;b), where P(a;b) is ofthe form given in (3). When at least one of the contribution levels is positive,assume that the probability of winning function is of the form

~P(a;b; �; �) = max�

0; min�

P(a;b) +

��� �

�+ �

�;1��

(9)

where P(a;b) is again of the form given in (3). It is easy to check that thisfunctional form satisfies (~A1), (~A2) and (~A3). It implies that if the interestgroups make equal (positive) contributions, they have no influence on theoutcome of the election, and the probability of winning reduces to P(a;b). Ifthe contribution to Candidate A is greater than the contribution to CandidateB (� > �), then Candidate A’s probability of winning is increased. Theamount by which this probability increases depends not only on the absolutedifference in the two contribution levels, but on this difference relative tothe total amount of money contributed to both candidates. When� > �,Candidate A’s probability of winning is decreased in a similar manner. Theparameter is assumed to be positive; the larger is , the greater is the impact

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of the campaign contributions. Themaxandminnotation simply ensures that~P(a;b; �; �) always takes on a value between 0 and 1.

A polar assumption is made about the policy preferences of the interestgroups. IG A is assumed to be “allied” with Candidate A in the sense that thepolicy preferences of IG A are identical to those of Candidate A.16 IG B issimilarly “allied” with Candidate B, so we have!(z) = u(z) = �1

2(z� 1)2

and (z) = v(z) = �12z2. The interest groups (like the candidates) are thus

extremists, in the sense that their ideal points are at the end-points of thepolicy space.

A sub-game perfect Nash equilibrium for this example can be found withthe standard backward induction procedure for two-stage games.17 Givenannounced platforms a and b from the first stage, the objective functions ofthe interest groups when choosing their campaign contributions in the secondperiod will be18

(a;b; �; �) =h1� a+b

2 + �����+�

�ih�1

2(a� 1)2i+h

a+b2 �

�����+�

�i h�

12(b� 1)2

i� �

(a;b; �; �) =h1� a+b

2 + �����+�

�i��1

2a2�+h�+�

2 � �����+�

�i ��1

2b2�� �

(10)

Taking the corresponding first-order conditions and solving them simulta-neously19 yields the equilibrium campaign contributions for the second stageas functions of the platforms chosen in the first stage

��(a;b) = 2 (a� b)�1� a+b

2

�2

��(a;b) = 2 (a� b)�1� a+b

2

��a+b

2

� (11)

In the first stage, anticipating this second-stage response from the interestgroups, the candidates play an induced game with payoffs20

~U(a;b; ��(a;b); ��(a;b);0) = [1� a+b2 + (1� a� b)]

[�12(a� 1)2] + [a+b

2 � (1� a� b)][�12(b� 1)2]

~V(a;b; ��(a;b); ��(a;b);0) = [1� a+b2 + (1� a� b)]

(�12a2) + [a+b

2 � (1� a� b)](�12(b

2)

(12)

The candidates’ first-order conditions for the induced game are

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@~U(a;b;��(a;b);��(a;b);0)@a = (�1

2)(12 + )

[(b� 1)2 � (a� 1)2]� (a� 1)[1� a+b2 + (1� a� b)] = 0

@~V(a;b;��(a;b);��(a;b);0)@b = (1

2)(12 + )

[a2 � b2]� b[a+b2 � (1� a� b)] = 0

(13)

These equations are solved by the pair

a�� =3+ 2 4+ 4

b�� =1+ 2 4+ 4

(14)

These expressions show the platforms chosen by the candidates in a sub-gameperfect Nash equilibrium of the game with campaign contributions. �

This example illustrates the central result of this paper: the equilibrium plat-forms found in the game with campaign contributions (example 2) exhibit agreater degree of platform convergence than those found in the game with-out campaign contributions (example 1). Indeed, the equilibrium platformsfound in example 2 have the interesting property that as approaches infinity,the candidates’ platforms converge toward1

2 (with Candidate A’s platformconverging from above and Candidate B’s platform converging from below).That is, as the probability of winning becomes very sensitive to the influenceof campaign contributions, the equilibrium tends toward (but never reaches)complete platform convergence. On the other hand, as the sensitivity of theprobability of winning function vanishes (i.e., as goes to zero), the candi-dates’ platforms diverge from each other toward(a� = 3

4; b� = 14), which are

the equilibrium platforms of the game with no campaign contributions foundin example 1.

4. The heart of the matter

The increased platform convergence observed when interest group influenceis added to the model is somewhat counterintuitive. The interest groups intro-duced in example 2 are extremists, in the sense that their ideal policies of 0and 1 are at the end-points of the policy space. One might conjecture thatCandidate A would move his platform closer to 1 in anticipation of a greatercampaign contribution from his allied IG A, and that Candidate B wouldmove his platform closer to 0 in anticipation of a greater campaign contribu-tion from his allied IG B. If this intuition were correct, then the introductionof interest group influence would result in increased platform divergence.

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Example 2, however, shows that the introduction of interest group influencecan have just the opposite effect.

The intuition for this result can be gained by examining the trade-offsdefining Candidate A’s equilibrium choice of platform. (A symmetric analy-sis of Candidate B’s equilibrium choice of platform would yield the sameintuition.) In the induced game between Candidates A and B (where the can-didates anticipate the equilibrium outcome of stage 2 that will follow theirplatform choices), Candidate A’s objective function has the general form

~U(a;b; ��(a;b); ��(a;b); kA) =

~P(a;b; ��(a;b); ��(a;b))u(a) + [1� ~P(a;b; ��(a;b); ��(a;b))]u(b)(15)

+~P(a;b; ��(a;b); ��(a;b))kA

With Candidate B’s platform at its equilibrium value b��, Candidate A’sequilibrium choice of platform a�� is defined by the first-order condition

@~U(a��;b��;��(a��;b��);��(a��;b��);kA)@a =h

@~P�@a + @~P�

@�

@��(a��;b��)@a + @~P�

@�

@��(a��;b��)@a

i[u(a��)� u(b��) + kA ]

+@u(a��)@a

~P� = 0

(16)

where the notation~P� � ~P(a��;b��; ��(a��;b��); ��(a��;b��)) is introducedfor brevity. This first-order condition is a direct analog of condition (6) dis-cussed following example 1. Just as in (6), the marginal benefit to CandidateA of an increase in his platform,@u

@a~P�, is equal to the amount by which the

increase in a increases Candidate A’s preference for his own platform, timesthe probability that Candidate A wins and that platform is actually imple-mented. The marginal cost of an increase in a is again equal to the change inthe probability of winning induced by this increase in platform times the netbenefit of winning the election. In the case of (16), however, the change in theprobability of winning depends not only on the direct effect of the change inthe location of Candidate A’s platform,@

~P�@a , but also on two indirect effects.

The terms@~P�

@�@��

@a and @~P@�

@��

@a show how an increase in a changes the contri-bution of each interest group to its allied candidate, and how these changes inthe contributions in turn affect Candidate A’s probability of winning. Thesetwo effects work in opposite directions. Since IG A will want to contributemore to Candidate A as his platform gets closer to IG A’s ideal point of 1,@��

@a

will be positive; since this increase in�will increase~P, the entire term@~P�@�

@��

@awill also be positive. Simultaneously, however, as Candidate A increases his

280

platform, he becomes more unattractive to IG B, whose ideal point is 0. It willtherefore become more important to IG B that Candidate B win the election.IG B will consequently increase its contribution to Candidate B, so@��

@a willbe positive. Since this increase in� will decrease Candidate A’s probabilityof winning, the entire term@

~P@�

@��

@a will be negative.The introduction of interest group influence thus creates both an incentive

and a disincentive for a candidate to choose a more extreme platform. Bymaking himself more attractive to his allied interest group, he would increasethe campaign contribution he receives and thereby increase his probability ofwinning. Simultaneously, however, he would make his position more objec-tionable to the opposition interest group, increasing the contribution made tohis opponent, and decreasing his probability of winning. In general, eitherof these effects could dominate: the candidates could feel greater pressureto choose more extreme platforms to curry the favor of their allied interestgroups, resulting in greater platform divergence; or they could feel greaterpressure to choose more moderate platforms to avoid alienating the oppositioninterest group, resulting in increased platform convergence .

The fact that the latter effect dominates in example 2 can be traced to thefact that the policy preference functions specified for the interest groups arestrictly concave. Since each candidate’s platform is relatively close to theideal point of his allied interest group, the concavity of the interest group’spreferences mean that a movement of the platform in the direction of theinterest group’s ideal point will make the candidate only a little more attrac-tive to the interest group, inducing only a small increase in the campaigncontribution. On the other hand, since each candidate’s platform is relativelyfar from the ideal point of the opposition interest group, concavity implies thata small movement of the platform away from that interest group’s ideal pointwill make the candidate much more objectionable to the interest group, andinduce a large increase in the contribution to the opponent. In equilibrium,therefore, each candidate chooses a more moderate platform, losing a bit ofsupport from his allied interest group, but decreasing more substantially thecontribution made to his opponent.

The result of increased convergence found in example 2 is a consequenceof the particular functional forms introduced there. The general point, how-ever, is simply that in choosing their platforms, candidates face a trade-offbetween rallying support from their allies and fanning the flames of oppo-sition backlash. Which of these effects will dominate will depend on theparticular specification of the model, but both effects need to be accountedfor. The following section shows that in at least one well-known study ofinterest group influence on elections (Magee, Brock and Young, 1989), theeffect of opposition backlash was entirely overlooked.

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5. Related literature

Several previous works have studied models similar in structure to the gamedeveloped in this paper. Austen-Smith (1987), Hinich and Munger (1994),and Magee, Brock and Young (1989) all analyze games in which candidates’simultaneous choices of platforms are followed by interest groups’ simulta-neous choices of contribution levels. None of these works, however, explicitlyanalyze the issue that is central to this paper, the trade-off candidates facebetween pleasing their allies and alienating their opponents. In the cases ofAusten-Smith and Hinich and Munger, this is a consequence not of any erroror omission, but simply reflects the fact that those studies focus on differentsubstantive issues. As shown below, however, the fact Magee, Brock andYoung did not account for this trade-off reflects a critical omission in theiranalysis.

Austen-Smith (1987) assumes that voters have noisy perceptions about thecandidates’ true positions, and money received from contributors is used toclarify those positions to the public. Although that paper does consider howthe candidates would adjust their positions in anticipation of the responsesof the contributors, there is no explicit analysis of the trade-off betweencurrying the favor of allies versus increasing the alienation of the opposition.In an important recent work that recasts the traditional spatial voting modelin terms of ideology (as opposed to simple policy positions), Hinich andMunger (1994) extend the analysis of Austen-Smith to allow candidates to usecampaign contributions not only to clarify their own positions to the public,but also to add noise to the public’s perception of the opponent’s platform.21

Their focus is on the negative consequences of such “mud-slinging”, ratherthan on the question of how campaign contributions affect the candidates’choices of platforms. Indeed (for reasons appropriate in the context of theissues they study), Hinich and Munger make the polar assumptions thatcandidates’ positions are fixed (all they can do in the campaign is clarifythem to the public), and that contributors believe that their donations have noimpact on the outcome of the election.

The probabilistic voting model of Magee, Brock and Young (1989; hereafterMBY) is couched in terms of trade policy, but is formally very similar to themodel developed in this paper. The policy space from which the candidateschoose platforms is the level of a tariff or subsidy. One of the candidates (MBYcall them parties) and one of the interest groups (MBY call them lobbies)are inherently protectionist (favoring a tariff), and the other candidate andinterest group are inherently proexport (favoring a subsidy). The candidatessimultaneously choose platforms; the interest groups simultaneously choosecontributions levels; and the candidates’ probabilities of winning depend bothon their platforms and on the contributions of the interest groups.

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The analysis of MBY, however, contains a critical omission: there is animplicit (and unjustifiable) assumption that changes in a candidate’s platformchange the contribution that he receives from his allied IG, but do not changethe contribution made to his opponent by the opposition IG. This omission canbe seen most clearly in the formal developmentof their model in the Appendixto Chapter 3 (pp. 267–270). Consider in particular their equations (3.11) and(3.12), which show the objective functions22 of the candidates in the firstperiod induced game, incorporating the equilibrium second stage responsesof the IGs. In each candidate’s objective function, the contribution of the alliedIG is shown to depend on the candidates’ platforms, but the contribution of theopposition IG is just given as a constant. The first-order conditions (3.15) and(3.16) also reflect this omission: although these conditions take into accounthow the allied IG’s contribution changes as a candidate changes his platform,the change in the opposition IG’s contribution is implicitly assumed to bezero.

MBY thus implicitly assume away the effect of opposition backlash that iscentral to this paper. What they fail to consider is that although changes ina candidate’s platform do not change the contribution he receives from theopposition interest group (since that will be zero in any case), changes inhis platformwill change the contribution that the opposition interest groupmakes to the opposition candidate, and that in turn will change the candidate’sprobability of winning the election.

6. Conclusion

In a study of PAC influence on legislative behavior, Sabato (1984: 130) arguesthat “the perception on Capitol Hill is that one wrong move on a vote of majorimportance to a large PAC could trigger a ‘max-out’ to one’s rival”. He goeson to quote a House of Representatives member on this issue: “We’re forcedalways to look over our shoulders and figure out whether this vote will costus $10,000 from a PAC, or worse yet, whether it will provoke a PAC intogiving $10,000 to an opponent”. In an analysis of the first Clinton presidency,Ferguson (1995) describes a similar phenomenon when he states that Clintongot into “trouble by straying too far to theleft of center” (p. 6). He was savedby “senior ‘centrist’ advisers [like David Gergen] who could help him getback on track in the middle of the road” (p. 6). Ferguson argues that thismove back to the center was a result of the pressures of campaign financing:“That Clinton strongly resembles a registered republican and might well godown in history as the most conservative Democratic president since GroverCleveland was entirely predictable. Anyone should have seen it who hadfollowed what might be termed the ‘Golden Rule’ of political analysis – to

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discover who rules, follow the gold (i.e., trace the origins and financing ofthe campaign: : : )” (p. 8).

These statements capture precisely the phenomena analyzed in this paper.When choosing how to present themselves to the public, candidates must con-sider not only how their own contributors will react, but how their opponents’contributors will react as well. This trade-off has not been explicitly analyzedin the previous literature, and the importance of the opposition reaction toa candidate’s choice of position has not been appreciated. Indeed, Magee,Brock and Young (1989) failed entirely to account for this backlash effect intheir model of elections with campaign contributions. Correctly incorporatingthis effect in the model does affect the results in important ways. Althoughin general the introduction of campaign contributions could lead either toincreased or decreased platform convergence, example 2 showed that undercertain conditions – chiefly concavity of the interest groups’ preferences – thecandidates’ fear of opposition backlash could dominate their desire to increasethe support of their allies, resulting in increased platform convergence andposition moderation in the presence of extremist interest groups.

An interesting direction in which this work could be extended would beto examine the effects of asymmetries, either in the inherent strength of thecandidates (an incumbency effect, for example) or in the power or resourcesof the interest groups (relaxing, for instance, the assumption made in example2 that when two opposed interest groups make identical contributions there isno net effect on the election). Another obvious direction for further researchwould be to allow for more than two candidates in the election, more thantwo interest groups, and/or a multi-dimensional policy space. Whatever theconfiguration of candidates, interest groups and issues, however, the mainintuition of this paper will carry over: in choosing their positions, candidatesmust consider not just how their friends will respond, but must also look overtheir shoulders to see what response they may provoke from the enemy camp.

Notes

1. The stage game of the infinitely repeated game studied by Alesina (1988) also has thisstructure; this game also forms the basis of the experimental work done by Morton (1993).

2. A credibility problem arises in any static version of the PSVM in which candidates havepolicy preferences. For a candidate’s announced platform to be credible, there must besome mechanism that commits him to actually implementing that platform if he is elected,rather than reneging on his campaign promise and simply implementing his most preferredpolicy. Although the model developed here consists of a one-shot game, a mechanism thatwould in practice be effective in overcoming this credibility problem would be repeatedplay of the game. If candidates expect to run for office again in future elections, they wouldhave an incentive to develop a reputation for faithful adherence to campaign promises.Alesina (1988) addresses this dynamic inconsistency problem in an infinitely repeated

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version of the PSVM, and finds that (if the candidates do not discount the future tooheavily) faithful implementation of announced platforms is sustainable in a sub-gameperfect equilibrium.

3. Calvert (1985: 84, Theorem 6) shows that as the weight that the candidates’ objectivefunctions place on their expected policy payoffs (relative to the weight they place onholding office) goes to zero, the degree of platform divergence observed in equilibriumgoes to zero. This result, however, allows for some platform divergence when the candidatescare enough about policy relative to holding office. When the candidates are only policymotivated, it is easy to show that their platforms must diverge in equilibrium, and that eachchooses a platform closer to his ideal point than is his opponent’s platform. For proofs anddiscussion of this assertion, see Wittman (1983: 144, Proposition 2), Hansson and Stuart(1984: 434, Theorem 1) and Calvert (1985: 83, Theorem 5).

4. The inclusion of positive office-motivation parameters in the examples developed in thispaper would not affect the qualitative results concerning the influence of campaign con-tributions on candidates’ choices of platforms.

5. A similar probability of winning function was derived by Alesina and Cukierman (1990,Appendix 1).

6. If some voter is indifferent between the two candidates, then assume that he flips a faircoin to decide how to vote.

7. Again, ties are resolved by a coin flip.8. In other models, randomness in the outcome of the election is generated by (rational)

abstentions by voters. See Hinich, Ledyard and Ordeshook (1972, 1973), Hinich (1977)and Ledyard (1984).

9. It is easy to verify that for a and b in the policy space [0,1], the second-order conditionsfor each player’s optimization problem are satisfied. This will also be true for everyother optimization problem considered in this paper, so further reference to second-orderconditions is omitted.

10. @P(a�;b�)@a is non-positive by assumption (A2), since we are considering a solution with

0 < b� < a� < 1. Since a is greater than b, an increase in a represents a divergence ofCandidate A’s platform from Candidate B’s platform, and so decreases (or at least doesnot increase) Candidate A’s probability of winning.

11. This formulation assumes that each interest group makes a donation to only one candidate.In general, we could allow the interest groups to contribute to both candidates simultane-ously. It is easy to show, however, that in equilibrium each interest group would choose tocontribute to just one candidate. This is a result demonstrated by Magee, Brock and Young(1989, Chapter 4), who call it the “Contribution Specialization Theorem”. Restrictingattention to each interest group contributing to only one candidate therefore entails no lossof generality, and it simplifies the exposition.

12. Jacobson (1978, 1985) and Welch (1981) consider the effects that two-way causalitybetween the contributions a candidate receives and his probability of winning might haveon empirical estimates of the influence of money on election outcomes. Jacobson (1985:40) concludes that although the issue is not definitively settled, “the available, limitedanecdotal evidence supports the idea that spending money during the campaign doesindeed buy attention and support for non-incumbents”. For the case of incumbents, Greenand Krasno (1988) argue that the effects of spending on probabilities of winning are greaterthan estimated by Jacobson.

13. See Sorauf (1984: 52–55) for a discussion of the use of independent expenditures. Hedescribes them as “a genuine alternative to direct contributions” (p. 52).

14. In this case, the Contribution Specialization Theorem discussed in footnote 11 would breakdown: a single interest group might “hedge its bet” by contributing to both candidates.

15. Another point to be made here is that although the direct effect of contributions is on thecandidates’ probabilities of winning the election, they do also indirectly affect the policypositions chosen by the candidates. This point was made by Austen-Smith (1987: 124),who argued that although there is “no explicit exchange relationship between contributors

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and politicians”, it is nonetheless true that the candidates “recognize that contributions, tosome degree, will depend on their policy stance, and so they will take this into account inchoosing election strategies”.

16. Notice that this notion of “alliance” between the interest groups and the candidates has todo only with shared policy preferences, and does not entail any personal preference on thepart of the interest groups for one candidate over the other.

17. See Gibbons (1992: 71–73) for a general exposition of this method.18. This formulation of the objective functions reflects two observations. First, it relies on the

fact that, for the reasons given in footnote (3), the platforms announced by the candidatesin the first period will satisfy 0� b < a� 1. Consequently, P(a; b) = 1� a+b

2 . Second,this formulation assumes that 0� P(a; b) +

����

�+�

�� 1, so that~P(a; b; �; �) =

P(a; b) + ����

�+�

�(i.e., we can ignore themaxandminnotation in (9)). In this example,

the equilibrium campaign contributions of the two interest groups will be identical, so thiscondition will indeed be satisfied.

19. The mathematical details of these operations are available from the author.20. To find these forms of the induced game objective functions, note that, for the expressions

��(a; b) and��(a; b) given in (11), the expression��(a;b)���(a;b)

��(a;b)+��(a;b) simplifies to 1� a� b,

so that~P(a; b; ��(a; b); ��(a; b)) = 1� a+b2 + (1� a� b).

21. A similar approach is taken by Hinich and Munger (1989) and Cameron and Enelow(1992).

22. The assumption made by MBY about the objectives of the candidates is the polar oppositeof the assumption made in this paper: they assume that the candidate is purely officemotivated and is not concerned at all with the policy actually implemented. Regardlessof the assumption made about candidate motivations, however, the opposition backlasheffect that MBY omit should be included in the analysis.

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