pay-to-play politics: informational lobbying and contribution limits when money buys access

Journal of Public Economics 96 (2012) 369–386

Contents lists available at SciVerse ScienceDirect

Journal of Public Economics

j ourna l homepage: www.e lsev ie r .com/ locate / jpube

Pay-to-play politics: Informational lobbying and contribution limits when moneybuys access☆

Christopher Cotton ⁎Department of Economics, University of Miami, Coral Gables, FL 33146, United States

☆ I appreciate comments from Ralph BoleslavskyKaushik Basu, Arnaud Dellis, David Kelly, and seminar psity. Any errors are my own.⁎ Tel.: +1 321 426 8866.

E-mail address: [email protected] This quote appears in Schram (1995).

0047-2727/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.jpubeco.2011.11.005

a b s t r a c t
a r t i c l e i n f o
Article history:Received 8 June 2010Received in revised form 9 November 2011Accepted 23 November 2011Available online 8 December 2011

Keywords:Political contributionsContribution limitsAccessCostly information disclosure

We develop a game theoretic model of informational lobbying between two interest groups and a politician,in which the politician can require political contributions in exchange for access. The analysis considers threeclaims: (1) the rich have better access to politicians than less-wealthy groups, (2) this access advantagemakes the rich better off and skews policy in their favor, and (3) contribution limits can reduce the richgroup advantage and result in less-skewed policy. We show that the rich do have better access, with thepolitician always offering access to the rich groups and only sometimes offering access to the less-wealthy group. This does not, however, mean that the rich group is better off or that policy is biased in itsfavor. The politician sets access fees to extract the greatest amount of rent from the political process.When only the rich group has access, its expected benefit from gaining access is fully offset by its paymentto the politician. In this case, the less-wealthy interest group who is not targeted by the politician is betteroff. Contribution limits decrease the politician's ability to extract rent, which improves the payoffs of richinterests and decreases politician payoffs. Finally, the paper presents a novel benefit of contribution limits:they can encourage the formation of lobby groups or the search for evidence, which results in moreevidence disclosure and better policy.

© 2011 Elsevier B.V. All rights reserved.

I really appreciate your additional contribution of $100,000… I hopeyour meeting with Trent Lott was productive.

Jim Nicholson, then Chairman of the Republican National Com-mittee, in a letter to businessman Phil Anschutz, October 23, 1998

Money doesn't buy…a position. But it will definitely buy you some
access so you can make your case.
Thomas Downey, former US Congressman1

2 For example, in Herndon (1982, p1000), an anonymous interest group representa-tive stated: “About all you get [in exchange for your contribution] is a chance to talk tothem… If you have a good case you can win them over. But you have to be able to talkto them.” In Schram (1995), former US Senator Dennis DeConcini said “What they got

1. Introduction

Special interests often provide political contributions in an effortto gain access to decision makers. Access allows one to present infor-mation or arguments in favor of a preferred policy. Contributions aretypically not provided in a quid pro quo exchange for policy favors.These statements not only summarize the views of interest groups

, Steve Coate, David Easley,articipants at Cornell Univer-

rights reserved.

and policy makers (e.g., Herndon, 1982; Schram, 1995),2 they arestrongly supported by the empirical evidence (e.g., Langbein, 1986;Hall and Wayman, 1990; Milyo et al., 2000; Ansolabehere et al.,2002; Clawson et al., 1992; Wright, 1990). Even advocates of cam-paign finance reform argue that the current system of money inpolitics is unfair because the rich have better access to politiciansthan the poor (Makinson, 2003).3 Although there is substantial evi-dence that political contributions help secure access to politicians,the majority of the theoretical literature on lobbying does not incor-porate access, instead assuming that money directly or indirectlybuys policy favors.

We develop a game theoretic model of informational lobbyingbetween two interest groups and a politician, in which the politiciancan require political contributions in exchange for access. Access

out of me for that contribution is access to come in … and to tell me why … it's good‘for America.’”

3 We encourage anyone skeptical of the contributions-for-access story to readthrough the assortment of letters, memos, and fundraising call sheets that have beenmade public in recent years during court battles over campaign finance reform legisla-tion. A substantial collection of these documents may be found on the News Now onPBS website, http://www.pbs.org/now/politics/cfmemos.html. The first quote aboveis taken from a letter available on this site.

http://dx.doi.org/10.1016/j.jpubeco.2011.11.005

mailto:[email protected]

http://www.pbs.org/now/politics/cfmemos.html

http://dx.doi.org/10.1016/j.jpubeco.2011.11.005

http://www.sciencedirect.com/science/journal/00472727

370 C. Cotton / Journal of Public Economics 96 (2012) 369–386

allows an interest group to present verifiable evidence to the politi-cian about the benefits of its preferred policy.4 Within this frame-work, our analysis focuses on the impact of interest group wealthdifferences. We consider three claims: (1) the rich have better accessto politicians than a less-wealthy group, (2) this access advantagemakes the rich better off and skews policy in their favor, and (3) con-tribution limits can reduce the rich group advantage and result inless-skewed policy.

The politician awards access in a way that maximizes the amountof rent she can extract from the political process.5 In equilibrium, thepolitician always prefers to offer access to the rich interest group.Only when the probability that the relatively poor group has favor-able evidence is sufficiently large will the politician also offer itaccess. This means that the politician often excludes the poor groupand never excludes the rich group from the political process,supporting the claim that rich groups tend to have better access topoliticians.

Payments collected from the rich interest group are maximizedwhen the politician implements the rich group's policy if and only ifthe rich group presents favorable evidence. In this case, the politiciangives the “burden of proof” to the rich group and the policy choice iscompletely independent of any action taken by the other group.Although the rich group is included in and the poor group excludedfrom the political process, the rich group tends to be worse off thanthe poor group in equilibrium. This is because the politician sets theaccess fee so high as to fully extract the rich group's expected rentfrom gaining access. The poor group, who is not targeted by the poli-tician's efforts to extract payments, tends to be better off.

Including both groups in the political process decreases theexpected payment from the rich interest group, and increases theexpected payment from the poor group. The politician only prefersto include both groups when the poor has a sufficiently large proba-bility of favorable evidence (and thus being willing to pay for access).When this is the case, the politician prefers to give the “benefit of thedoubt” to the rich interest group, implementing the rich group's pre-ferred policy unless only the poor group provides favorable evidence.When the poor group is included in the political process, it becomes atarget of the politician's efforts to extract payments, has a lower prob-ability of its policy being implemented, and tends to be worse off thanif it was excluded. The rich group, on the other hand, is better offwhen it is not the only group included in the political process.

The analysis illustrates a flaw in common intuition about the ben-efits of political access. Onemay believe that an interest groupwill bebetter off when the politician offers it access. When the politiciansets access fees to extract the maximum payments from interestgroups with access, this is not the case. The rich group tends to bebetter off and the poor group worse off when the politician offersaccess to both groups rather than just the rich group. Additionally,we find no theoretical support for the claim that better poor groupaccess makes it less likely that policy is biased in favor of the rich in-terest group. In fact, the opposite tends to be true, with the politicianmore likely to implement the rich group policy when she includes

4 Fundamental to our model is an assumption that the politician has control overwhich agents are allowed to present evidence. That is, the politician can avoid or ignoreevidence held by an interest group without access. Without this assumption, there areno benefits to gaining access to the politician, and the game will resemble a standardgame of strategic evidence disclosure (e.g., Milgrom and Roberts, 1986).

5 Our discussion assumes that political contributions are monetary payments frominterest groups to the politician. Other interpretations of contributions are also consis-tent with our analysis. For example, they may be in-kind contributions such as the in-terest groups providing staffing for fundraising or get-out-the-vote campaigns, or theymay be business decisions that benefit the politician's district (e.g., choosing the loca-tion of a new manufacturing facility, purchasing intermediate goods from a firm in thepolitician's district). In these cases, wealth differences between the interest groupsmay be interpreted as differences in the cost of undertaking activities that benefitthe politician. The analysis concerning contribution limits, however, is less applicableunder these alternative interpretations.

both groups in the political process (in which case, she gives the ben-efit of the doubt to the rich group) than when she excludes the poorgroup (and assigns the burden of proof to the rich group).

Next, we consider the impact of political contribution limits.Limits decrease the wellbeing of the politician not because theyresult in less lobbying or worse policy, but rather because theylimit politician's ability to extract payments from special interests.For the same reason, limits tend to benefit rich interest groups. Theanalysis illustrates how, contrary to popular belief, contributionlimits can make poor special interests worse off. This is because acontribution limit reduces the expected difference between richand poor group contributions, and can result in the politician includ-ing both groups in the political process and setting fees to fullyextract any benefit the poor group expects from access. We alsoshow, however, that there are situations in which sufficiently lowcontribution limits benefit both special interest groups. In such set-tings, the politician is worse off because she can capture less rent;however, her ability to identify and implement good policies isunchanged.

Campaign finance reform advocates often argue that contributionlimits help level the playing field between rich and poor special in-terests, and reduce the rich-interest bias in the political process.Our analysis shows that limits can increase the probability thatpoor groups receive access. But, improved access does not implythat the poor group is better off, as a limit may cause the politicianto focus her rent seeking efforts on extracting the maximum pay-ment from the poor group. Furthermore, we show that whenmoney buys access (rather than policy favors) a limit does not in-crease the likelihood that the politician implements good policy. Ifa contribution limit makes less-wealthy interest groups better offor decreases political bias in favor of rich groups, then it is due to fac-tors outside of our model, not by decreasing the rich group's accessadvantage. To illustrate how a limit may improve the quality of thepolicy outcome, we consider an extension in which interest groupsdecide whether to form a lobby presence prior to the politician an-nouncing access fees.

One may interpret an interest group's decision about whether toform a lobbying presence as a choice of whether to establish an officein Washington, or whether to search for evidence. In the game withendogenous lobby formation, the politician is able to offer access toan interest group only if it forms. In the absence of a contributionlimit, neither interest group will form since they recognize that thepolitician will react to their formation by setting access fees to extractthe maximum payment through the lobbying process. If an interestgroup forms, it becomes the target of the politician rent seeking ef-forts, and in equilibrium will be worse off than if it did not form inthe first place. By committing to a contribution limit, the politician ef-fectively commits to constrain her ability to extract payments lateron. Only when the politician commits to a low-enough contributionlimit do interest groups find it worthwhile to form. In the absenceof a contribution limit, the politician collects no payments andchooses policy without observing any evidence from the interestgroups. A contribution limit, by encouraging lobby formation, in-creases expected payments, results in a more informed politicianand improves policy outcomes. This is the first paper we are awareof to highlight this benefit of contribution limits.6

Our analysis considers an aspect of political contributions and lob-bying that has largely been overlooked in the literature. We do notsuggest that gaining access to lobby a politician is the only reasonfor making contributions, or that contributing is the only way to

6 In a bargaining game between interest groups and a politician, Drazen et al. (2007)show that contribution limits can increase interest group bargaining power and thusencourage interest group formation. In their model, however, this can result in morepressure on the politician to deviate from the socially optional policy, rather than a mo-re informed politician.

9 Other “access” models, including Austen-Smith (1995) and Lohmann (1995), as-sume that evidence is completely unverifiable. Therefore, the presentation of evidence

371C. Cotton / Journal of Public Economics 96 (2012) 369–386

gain access to a politician. In many cases, e.g., on issues where a poli-tician's policy preferences are known, interest group contributionsmay be primarily intended to support the election of a politician al-ready predisposed in favor of the group's policy. In this case, the pol-itician may still give interest groups access, but in order to use thegroup's expertise when drafting legislation or lining up support be-hind policy (similar to the lobbying as legislative subsidy model inHall and Deardorff, 2006) rather than to learn about the issue. Wesuspect that lobbying to present evidence and lobbying as legislativesubsidy both play important roles in US politics, with the more-appropriate description of access depending on the issue and wheth-er a legislator already has a firm preference over the policy outcome.In other settings, access itself may be unnecessary for conveying evi-dence to a politician. When evidence is straightforward and easilyconveyed (e.g., when evidence is simply made up of the most recentpoll numbers about constituent support for a policy), a politicianmay be unable to prevent interest groups without access from con-veying their evidence. Our analysis is most-applicable to evidencecomplicated enough that it takes a conscious effort to verify its sourceor understand what it means about the best policy.7 For these rea-sons, our framework is less applicable to hot issues such as abortionon which people have well established positions, and more applicableto technical issues such as industry specific regulatory, tax, or tradereforms.

In the next section, we review the relevant literature. Section 3describes the model in detail, and Section 4 solves the game.Section 5 allows for various extensions of the model, starting withthe incorporation of a contribution limit into the lobbying frame-work (Section 5.1) and then considering whether contribution limitscan encourage lobby groups to form (Section 5.2). Section 6 con-cludes the paper with a discussion of the results, policy implications,and potential extensions of the model.

2. Literature review

Stratmann (2005) reviews the literature linking political contribu-tions and policy outcomes. Much of the evidence, such as the findingthat political contributions peak around the time of a major legislativeinitiative (Stratmann, 1998) or that the likelihood of a legislator vot-ing for a bill increases in political contributions from those who favorthe bill (Stratmann, 2002), is consistent with both a contribution-for-access and contributions-for-favors story. The papers that considerwhether contributions buy favors or access find significant supportin favor of the access story. For example, Langbein (1986) finds alink between contributions and the amount of time the politicianspends meeting with constituents and interest groups. Ansolabehereet al. (2002) present evidence that an organization's spending onlobbyists is increasing in its political contributions, suggesting thatinfluencing policy requires both contributions to politicians (tosecure access) and the presentation of information (which requireslobbyists). Herndon (1982) surveys interest groups to determinetheir reason for contributing and finds that all business leaders inhis survey emphasize access.8 Schram (1995) and Makinson (2003)interview retired politicians and interest group representatives, find-ing substantial support for the idea that contributions are often givento gain access to politicians. Our results are also consistent with the

7 Even if the politician cannot guarantee that she ignores a group without evidence,she will still be able to extract some (but not as much) payments from the interestgroups as long as a group without access has a lower probability of being able to con-vey its evidence to the politician than a group with access. That is, as long as there is apositive probability that the politician does not observe evidence transmitted by agroup without access, groups with favorable evidence will still be willing to pay a(now lower) fee to gain access.

8 Other interest groups, including labor unions, say their main reason for contribut-ing is to help their preferred candidates win election.

evidence presented in Baumgartner et al. (2009), which provides adetailed account of the lobbying efforts surrounding 98 randomly se-lected policy issues. The analysis emphasizes the role of access andshows that business groups have much greater access to high-leveldecision makers than relatively poor citizen groups. Consistent withour analysis, it also finds little evidence that policy decisions favorthe more-wealthy side of an issue.

Despite the evidence supporting the contributions-for-accessframework, the theoretical literature tends to assume that contribu-tions are given in the quid pro quo exchange for policy favors. Foran overview of this literature, see Grossman and Helpman (2002).Such models involve awarding a policy favor through an all-pay orwinner-pay auction (Che and Gale, 1998; Baye et al., 1993), a lottery(Tullock, 1980), or a menu auction (Bernheim and Whinston, 1986;Grossman and Helpman, 1994, 1996). The access motivation for con-tributing has received far less attention in the literature.

To our knowledge the only other model of access fees is Austen-Smith (1998), where paying an access fee allows an interest groupto conduct an experiment, the results of which are observed by thepolitician. Interest groups differ in terms of their policy preferencesrelative to the politician. The politician can only offer access to oneof the interest groups, and the analysis focuses on which group is of-fered access. We take an alternative approach, relying on a more-simple evidentiary structure and fixing interest group policy prefer-ences. This allows us to endogenize the number of groups to receiveaccess, and to consider in more detail the impact of interest groupwealth differences. Our analysis also puts more emphasis on the im-pact of contribution limits.

Similar to our analysis, Cotton (2009) assumes that an interestgroup needs access before presenting verifiable evidence. In thatmodel, a politician chooses between selling a policy favor or sellingaccess through an all-pay auction. The advantage of using an all-payauction to model contributions-for-access is that the framework isrelatively straightforward to incorporate with a standard model ofcontributions-for-policy favor (e.g., Gavious et al. (2002)), and it re-sults in sharp predictions about full evidence revelation in equilibri-um. The disadvantage is that such a model may not match therealities of the lobbying process; there is little evidence that interestgroups view their efforts to gain attention as a contest. Rather, we ex-pect that interest groups have an expectation about gaining access ifthey give a certain amount (this is consistent with the evidence pre-sented in Herndon (1982), Schram (1995) and Makinson (2003)).9

Various papers consider contribution limits. In Cotton (2009), Prat(2002a,b), and Coate (2004a), contribution limits decrease the incen-tives politicians have to sell policy favors, and increase the likelihoodthat a politician chooses the policies preferred by his constituents. Inthis way, limits can result in better policy decisions. In Austen-Smith(1998) a limit can have a similar effect, causing the politician to grantaccess to a more-informative interest group rather than a group witha higher willingness to pay for access. Other papers suggest that con-tribution limits may harm constituent welfare. Coate (2004b)

by itself can have no affect on the politician's beliefs, and the impact that any piece ofinformation has on the politician depends on who provides it and how much moneythey attach to the evidence. This paper, as well as Austen-Smith (1998) and Cotton(2009), make the alternative assumption that evidence can have an impact on the poli-tician's beliefs independent of who provides it, or the size of the contribution attachedto it. In other words, interest groups have verifiable evidence. The novel aspect of ourframework compared to other models of verifiable evidence is that we assume that thedecision maker (i.e., politician) controls which agents (i.e., interest groups) can presenttheir evidence. Typically in the verifiable information literature, an agent with privateinformation can disclose its information whenever it chooses to do so (e.g., Milgromand Roberts, 1986; Bennedsen and Feldmann, 2002, 2006; Bull and Watson, 2004,2007).


incorporates limits into an election model, where contributions fundadvertising. There limits may result in lower revenue for the politi-cian, less advertising, and less informed voters. In Riezman andWilson (1997), a politician may sell additional policy favors in orderto compensate for lost revenue due to a limit. Drazen et al. (2007)shows how a contribution limit can result in the formation of morelobbying groups, and worse policy from the perspective of constitu-ents. We also show that a limit can result in the formation of morelobby groups; however, unlike in Drazen et al. (2007), more lobbygroups does not imply worse policy. When we allow for endogenouslobby formation, contribution limits can increase the number of spe-cial interests that lobby, which can result in more information disclo-sure, a better-informed politician, and better policy choices.

Finally, it is worth pointing out that the underlying evidentiaryframework is related to models of judicial decision making. In thelegal framework, two agents (i.e., a plaintiff and defendant) maypresent a judge with evidence in support of or against conviction.10

In such settings, whether a defendant is assumed innocent until prov-en guilty or guilty until proven innocent significantly affects the in-centives that the plaintiff and defendant have for collecting andpresenting evidence (Shin, 1994; Hay and Spier, 1997; Demouginand Fluet, 2008). These concepts are related to our concepts of burdenof proof and benefit of the doubt. Sobel (1985) considers burden ofproof rules in a game theoretic model of evidence disclosure whereevidence is verifiable (as in this paper), and the costs of evidence pro-duction is fixed. There, the judge prefers to assign the burden of proofto one of the agents and the agents prefer for the burden of proof tobe assigned to the other agent rather than themselves. Our analysisin Section 4 produces a similar result: the politician may prefer toassign the burden of proof to the rich interest group, and the interestgroup without the burden of proof is better off than the one with theburden of proof.

13 The analysis assumes that e1 and e2 are independently distributed according totheir respective Bernoulli distribution. We could add (less-than-perfect) correlationbetween the values without changing the qualitative results.14 For this reason, our model better applies to more-technical evidence about policychoices for which a politician must make a conscious effort to process evidence, verify

3. Preliminaries

3.1. Model

Wemodel an informational lobbying process with political contri-butions and access. There are two advocate interest groups (hence-forth IGs) representing alternative policies, and a politician whomust decide which of the two policies to implement.11 We usei∈ {1,2} to denote both a policy and its IG. Each IG may have private,verifiable evidence in favor of its preferred policy, which it can dis-close to the politician only if the politician grants it “access;” that is,if the politician or her staff meet with the interest group and reviewsthe merits of its case. Then, after observing revealed evidence, thepolitician chooses a policy, i*∈{1,2}.

The evidentiary structure is one of verifiable information in whichIGs have private evidence in favor of their policy.12 To keep the modeltractable and focus on developing intuition, we assume a discrete ev-idence space where IGs either have favorable evidence or they do not.Group i's evidence is denoted ei∈{0,1}, with ei=1 if i has favorableevidence. IGs know their own evidence (or lack thereof), but not

10 See for example evidentiary models of the legal process, including Cooter andRubinfeld (1994), Shin (1994) and Posner (1999).11 The framework is consistent with Baumgartner et al. (2009), which provides ex-tensive evidence that the majority of issues considered by US Congress are definedby two policy alternatives (a status quo and a known reform), and groups of individ-uals and organizations that work together in support of or against the reform.12 Formally, one may think of the evidence as favorable research reports, estimates ofjob creation, or other information that can be verified. For a more thorough discussionof verifiable evidence, see Bull and Watson (2004, 2007).

the evidence of the other group. πi is the ex ante probability thatei=1, and 1−πi is the ex ante probability that ei=0.13

Whether an IG can reveal its evidence depends on whether it hasaccess to the politician. Unlike other models of verifiable informationdisclosure, we do not assume that all IGs have unlimited, free accessto the politician. Rather, the politician chooses which IGs can discloseevidence. Implicit in this is the assumption that reviewing an IG's ev-idence requires a conscious effort by the politician. It is not enoughfor an IG to disclose its evidence if the politician chooses not to payattention.14 The model is one of contributions-for-access, where IGsprovide political contributions, and receive access if their contribu-tions are large enough. Variable ci≥0 is the value of any political con-tribution paid by IG i, with c=(c1,c2). Prices p=(p1,p2) represent thepolitician's access strategy, where the politician awards access to IG iif and only if ci≥pi. We say that IG i “lobbies” when ci≥pi.

After IGs contribute, the politician awards access according to p.Then, the IGs choose whether to reveal their evidence. It is well estab-lished in the literature that an IG with access reveals any favorableevidence.15 To simplify presentation, the remainder of the papertakes this revelation result as given. This means the politician learnsei when ci≥pi. Denote revealed evidence by r=(r1, r2). Therefore,ri=1 if and only if i pays for access and has favorable evidence, andri=0 otherwise.

At the conclusion of the game, the politician implements policyi*∈{1,2}. We define her policy strategy by function θ, where θ(r) isthe probability that the politician chooses policy 1 given revealedevidence r.

3.1.1. Strategies and beliefsIG i's strategy is given by its contribution ci. The politician's strat-

egy profile is given by (p,θ), which represents an access decision foreach combination of c, and a policy choice for each c and r.16

The state of the world is fully defined by (e1,e2). Ex ante beliefsabout the state are captured by π=(π1,π2). The politician's updatedbeliefs at later stages of the game are represented by ~π . At the timethe politician awards access, her beliefs depend on contributions. Atthe time the politician chooses policy, her beliefs depend on both con-tributions and revealed evidence.

3.1.2. Price and policy mechanismAt the beginning of the game, before the IGs choose contributions,

the politician announces her strategy profile (p,θ). We assume thatthe politician has partial ability to commit to her strategy in that byannouncing p=(p1,p2), she commits to provide access to any groupthat pays their respective pi.17 Otherwise, we limit the politician'sannouncement to strategies that are sequentially rational at the

its source, or understand what it means about the best policy.15 Failure to reveal evidence once a group receives access results in the politician be-lieving that the interest group must have below-average evidence. Therefore, onlythose with below-average evidence would ever not reveal their evidence. There is“unraveling” and in equilibrium, only those with the lowest possible evidence wouldever consider not revealing their evidence after being granted access. This result wasfirst established in Milgrom (1981).16 To simplify notation, we write θ(r) as a function of r, not c. Of course, θ(r) could de-pend on c as well. But, in equilibrium c affects the policy outcome through its influenceon beliefs, rather than through its direct integration into θ.17 This assumption plays no role in our analysis, as we assume that the politiciandoes not find granting access costly. However, allowing the politician to commit togrant access to anyone who pays a large enough political contribution is requiredfor the results to continue to hold if costs of providing access are incorporated intothe model.

19 In equilibrium, the politician can correctly infer the evidence of an IG who is in-cluded in the political process after observing whether or not the IG paid the accessfee. As we show in the analysis, the politician always implements a good policy whenone exists. Because of this, the politician never has an incentive to review evidencefrom agents who do not pay their fee. This means there is no need for the politicianto commit to avoid or ignore evidence of the agents who do not pay their fees; such be-havior is consistent with equilibrium. However, the politician also does not have an in-


later stages of the game. That is, the politician must never want toaward access to an interest group that contributes cibpi. Similarly, θmust be a sequentially rational strategy for choosing policy afterobserving contributions and revealed evidence. (This rules out thepolitician's ability to commit to choose policy in favor of the highestbidder, for example.) This is consistent with the idea that the politi-cian is the monopoly seller of access, and can strategically choosehow to award access and pick policy in an effort to extract rentfrom the political process. However, she is limited to choosingrules that she does not have an incentive to deviate from in laterperiods.18

3.1.3. PayoffsThe politician earns policy payoff 1 from implementing a policy

with favorable evidence, and 0 from choosing a policy with no favor-able evidence. Her benefit from choosing policy i instead of policy − itherefore equals ei−e− i. The politician also benefits from collectingpolitical contributions, where ϕ>0 denotes the relative weight thepolitician puts on contributions relative to policy. Her final payofffrom implementing i* equals

W ¼ ei� þ c1 þ c2ð Þϕ:

Each IG is an advocate for its respective policy. IG payoffs dependon the implemented policy and any contribution, but are indepen-dent of the evidence. i's payoffs may be written U i ¼ 1 i ¼ i�½ �vi−ω ici, where vi > 0 denotes the value of the policy and ω i > 0 denotesthe relative value IG i puts on contributions. Define Ui≡U i=vi, andωi≡ω i=vi. Because preferences are unchanged by positive affinetransformations of the utility function, we may rewrite IG i's payofffunction as

Ui ¼ 1 i ¼ i��

−ωici:

Higher wealth (i.e., a lower ω i) and higher policy valuation (i.e., ahigher vi) have similar effects on IG preferences and the optimalchoice of political contribution. The policy-valuation effect and thewillingness-to-pay effect are indistinguishable in the model. Wecombine these into a single parameter ωi. Because the policy debatefocuses on the advantages that rich interests have over poor interests,our discussion assumes that v1 ¼ v2 and that differences in ω1 and ω2

correspond to differences in IG wealth. We assume that the IGs havewealth asymmetries. Without loss of generality, let 0bω1bω2. That is,1 is the “rich” group and 2 is the “poor” group.

3.1.4. Order of playThe game takes place as follows:

1. The politician announces price and policy mechanism (p,θ).2. Each IG simultaneously and independently chooses its contribu-

tion, ci≥0.3. The politician awards access to any IG that pays ci≥pi. An IG with

access reveals any favorable evidence.4. The politician implements policy i* according to θ.

18 If the politician cannot announce p and θ up front, then there may exist multipleequilibrium. For example, there will exist an equilibrium in which an indifferent politi-cian chooses policy in favor of IG 1, one in which an indifferent politician chooses policyin favor of IG 2, and one in which an indifferent politician flips a coin to choose policy.An equilibrium in the game when the politician announces a sequentially-rationalstrategy up front is also an equilibrium in a game without the upfront announcement.

3.2. Equilibrium concept

We solve for the Perfect Bayesian Equilibrium (PBE) of the game.Although we allow the politician to commit to provide access to any-one who pays at least pi, we otherwise require that (p,θ) are sequen-tially rational at the later periods of the game.19 The analysis alsomakes an assumption about the IG strategy.

A1. An IG who is indifferent between paying for access and revealingevidence and not paying for access or revealing evidence will pay andreveal if and only if revealing its evidence has the potential to affectpolicy.

IG i may be indifferent between revealing and not revealing itsevidence because the politician's policy strategy θ is independent ofri, or because pi is high enough to fully offset the expected benefit ofrevealing ei. A1 implies that i reveals its evidence if its indifferenceis caused by high access fees, but not if its indifference is caused bythe politician not accounting for its revealed evidence when choosingpolicy. This is consistent with the politician's ability to charge a pricejust under, but infinitely close to pi such that IG i strictly prefers to payfor access when it has positive evidence. It is also consistent withevidence revelation not being cost free. Although we do not modelcosts of evidence revelation, positive costs (even as they approach0) would result in IGs choosing not to reveal evidence when revealingevidence has no possibility of affecting policy. A1 simplifies the anal-ysis and description of equilibrium without weakening our concept ofequilibrium. Any equilibrium found under A1 will also be an equilib-rium in the absence of A1.

We call PBE under these assumptions an “Access Equilibrium.” Afull description of equilibrium must define a strategy profile foreach IG and the politician, and the politician's beliefs when sheawards access and chooses policy. Each player's strategy must be asequentially-rational best response to the strategies of the otherplayers, given the player's beliefs. The politician's beliefs must be con-sistent with Bayes' Rule given the ex ante distribution of evidence andIG strategies.

4. Analysis

Here, we begin with some initial results and terminology that willbe helpful when describing the equilibrium of the model.

4.1. Buying access

In equilibrium, an IG will contribute either ci=0 or ci=pi. Giventhat an IG gets access if and only if ci≥pi, paying more than pi is strict-ly dominated by playing ci=pi, and paying a positive amount less

centive to review agents who pay their fee, even though he must review these agentsin order to incentivize the agents to pay their fees in the first place. Because of this weassume that the politician can commit to give access for the price. The ability to com-mit to honoring the price may come from concerns about maintaining a positive repu-tation for dealing with interest groups on other issues in the future. These concerns,and future interactions with other interest groups, are not formally included in themodel. This assumption pays a more important roll if the politician found granting ac-cess costly; in which case it is required for the politician not to withdraw access afterinferring an agent's type. In our setting, where it is costless for the politician to grantaccess, it is not required to maintain the same behavior in equilibrium.


than pi is strictly dominated by paying ci=0 when the politicianbelieves that only a group without evidence does not pay at least pi.20

An IG with no favorable evidence does not pay to reveal its evi-dence. If ei=0, then ci=0. This is because paying pi>0 is costly andhas no benefit when ei=0. When an IG does have favorable evidence,it will pay for access only if the price is not too high. An IG's expectedbenefit from revealing its positive evidence to the politician is non-negative and constant. Its cost of doing so is strictly increasing in pi.Therefore, the sequentially rational IG strategy is captured by a cutvalue pi such that IG i pays pi if and only if pi≤pi.

4.2. Policy choice

In the final stage of the game, the politician implements policyaccording to strategy θ, where θ(r1, r2) is the probability that the pol-itician chooses policy 1 given r1 and r2. In equilibrium, θ must repre-sent a sequentially rational strategy. Therefore, it must implementthe policy the politician believes is more likely to have favorable evi-dence. If ~π i > ~π−i, then θ must implement policy i with probability 1.

If the politician observes ri=1, then ~π i ¼ 1. If ri=0 and if q∈ [0,1]is the equilibrium probability that IG i with ei=1 pays pi, then~π i ¼ πi 1−qð Þ= πi 1−qð Þ þ 1−πið Þb1. Therefore, if only one of the IGsreveals evidence, the politician must choose policy in favor of thatgroup. That is, ri=1 and r− i=0 implies that ~π i ¼ 1 > ~π−i. Thus, se-quential rationality implies that θ(1,0)=1 and θ(0,1)=0.

On the other hand, the politician will be indifferent between thetwo policies when r1=r2=1, and she may be indifferent (dependingon the type of equilibrium) when r1=r2=0. This means that the pol-itician is free to choose any θ(1,1)∈ [0,1], and may be free to chooseany θ(0,0)∈ [0,1].

There are two feasible pure strategies involving θ(1,1) and θ(0,0):

• Benefit of the Doubt: Policy strategy θ gives the “benefit of thedoubt” to IG i if it implements policy i whenever r1=r2.

• Burden of Proof: Policy strategy θ assigns the “burden of proof” to IGi if in equilibrium it implements policy i if and only if ri=1.

The policy strategy gives the benefit of the doubt to IG 1 whenθ(1,1)=θ(0,0)=1, and to IG 2 when θ(1,1)=θ(0,0)=0. The policystrategy assigns the burden of proof to IG 1 when θ(1,1)=1 andθ(0,0)=0, and to IG 2 when θ(1,1)=0 and θ(0,0)=1. When bothIGs lobby in equilibrium when they have favorable evidence, theseare the only values of θ that constitute a burden of proof strategy. If,however, only one of the IGs lobby when they have positive evidence,then r1=r2=1 is an off equilibrium path event and any θ(1,1) maybe part of a burden of proof strategy since it does not directly affectpolicy. (In all of these cases, θ(1,0)=1 and θ(0,1)=0 are requiredby sequential rationality.)

In some instances, the politician is indifferent between a range ofvalues for θ(1,1) or θ(0,0). Here, we provide two alternative assump-tions about the politician's choice of θ in the event that she is indiffer-ent between multiple values.

A2.1. When the politician is indifferent betweenmultiple values of θ,she chooses the rule that maximizes the probability of choosing pol-icy 1.

A2.2. When the politician is indifferent between multiple values of θ,she chooses the rule that maximizes the probability of choosing poli-cy 2.

20 Let ~π i cið Þ denote the politician's updated beliefs about ei given contribution ci wheni does not receive access. We focus on equilibria in which ~π i cið Þ ¼ 0 for all cibpi. How-ever, given that any ci∈(0,pi) is an out of equilibrium payment, any beliefs for such ciwill be consistent with Perfect Bayesian Equilibrium. Therefore, technically, ~π i cið Þ cantake on any value sufficiently low that expect at least as great a payoff from not con-tributing anything as from contributing some other cibpi.

A2.1 and A2.2 may be interpreted as alternative assumptionsabout politician bias. Under A2.1, the politician has a bias in favor ofIG 1, which affects her decisions when she is otherwise indifferentbetween different strategies, but is sufficiently weak as to not domi-nate her other preferences. Under A2.2, the politician has such biasin favor of IG 2. For the majority of the analysis, these assumptionsserve only to simplify exposition at the knife-edge case of parametervalues where the politician is indifferent between assigning theburden of proof or the benefit of doubt to IG 1. In Section 5 theyplay a more-significant role when we consider the impact of contri-bution limits.

4.3. Inclusion in the political process

We now introduce the concepts of political inclusion and exclu-sion. An IG that is excluded from the political process can have no in-fluence over policy in equilibrium. Otherwise it is included. We definethe concept formally below.

Definition 4.1. IG i is “excluded” from the political process if in equilib-rium either

1. pi > pi, or2. θ is independent of ri.

An IG that does not meet either of these conditions is “included” in thepolitical process.

The first exclusion condition means that pi is sufficiently high thati never pays to disclose its evidence in equilibrium, and can thereforehave no impact on policy. Similarly, when θ is independent of ri, anyevidence revealed by i has no impact on policy, regardless of theother group's revealed evidence. This is the case when the politicianassigns the burden of proof to one's opponent. When the burden ofproof is put on − i, the politician follows a strategy of choosing policyindependent of any action taken by IG i; thus, excluding i from the po-litical process.

The first lemma relates the concept of exclusion back to the con-cepts of burden of proof and benefit of the doubt that we introducedearlier.

Lemma 4.1. In any possible access equilibrium,

• if θ assigns the burden of proof to IG i, then IG i is included and IG− i isexcluded from the political process;

• if θ assigns the benefit of the doubt to IG i, then both IGs are included inthe political process.

All formal proofs are in Appendix A. The first part of the Lemmafollows directly from the definition of exclusion in terms of θ. The sec-ond implies that the politician never prefers to exclude one of the IGsby setting a high price of access in the benefit of the doubt case.

At this point, it is also helpful to rule out the possibility that thedecision maker excludes both IGs.

Lemma 4.2. There does not exist an access equilibrium in which bothIGs are excluded from the political process.

If the politician's strategy excludes both IGs from the politicalprocess, then she collects no contributions and learns nothing abouteither IG's evidence. She has an incentive to deviate from such a strat-egy to include at least one IG in the political process. Doing so assuresthat the politician learns at least one IG's evidence, increasing herexpected payoffs.

Given the discrete evidentiary structure, granting access to at leastone group enables the politician to maximize her policy utility. Thereis no additional policy benefit to granting access to a second IG. Thatis, the burden of proof strategy and the benefit of the doubt strategyboth guarantee that a good policy is implemented whenever oneexists. This means that the politician expects the same policy utility


from including only the rich IG, including only the poor IG, or includ-ing both IGs in the political process. Her choice between these possi-bilities will be done to maximize political contributions.21 Weconsider this choice further below.

4.4. Price of access

An IG's threshold strategy pi depends on the policy strategy θ andwhether the other IG pays to lobby when it has favorable evidence.For example, IG i may be willing to pay less for access when − i po-tentially lobbies, than when− i never participates in the political pro-cess. We calculate the equilibrium values of pi below. We firstdetermine the politician's choice of pi given IG cut values pi.

If the politician plays a strategy that includes IG i in the politicalprocess, then she sets the price of access that extracts the greatestexpected payment from the group. That is, she sets pi ¼ pi. If, on theother hand, the politician's strategy excludes i from the political pro-cess, then she sets pi > pi.

22

Now, we can calculate p1 and p2. Sequential rationality assuresthat θ(1,0)=1−θ(0,1)=1. On the other hand, any valuesθ(1,1),θ(0,0)∈ [0,1] are sequentially rational. Let ψi∈ {0,1} indi-cate whether pi≤pi. If the politician allocates prizes according toθ, then IG 1 contributes if e2=1 and π2ψ2θ(1,1)+(1−π2ψ2)−p1ω1≥(1−π2ψ2)θ(0,0), which implies

p1 ¼ π2ψ2θ 1;1ð Þ þ 1−π2ψ2ð Þ 1−θ 0;0ð Þð Þω1

: ð1Þ

Similarly, IG 2 contributes if e2=1 and π1ψ1(1−θ(1,1))+(1−π1ψ1)−p2ω2≥(1−π1ψ1)(1−θ(0,0)), which implies

p2 ¼ π1ψ1 1−θ 1;1ð Þð Þ þ 1−π1ψ1ð Þθ 0;0ð Þω2

: ð2Þ

4.5. Equivalence of exclusion methods

At this point it is helpful to show that the analysis can focus on thesituation where pi≤pi for both IGs. That is, we do not need to explic-itly consider the setting where the politician excludes one of the IGsby setting their pi > pi. This is because payoffs are independent ofwhether the politician excludes an IG by setting access fees abovetheir willingness to pay (i.e., pi > pi) or by assigning the burden ofproof to the other IG even if pi≤pi. Therefore, if (when restricting at-tention to the case when pi≤pi) we find that the politician prefers toexclude i through her choice of θ, then we can simply conclude thatthe politician prefers to exclude i through either method.

If the politician excludes IG 2, then p1 ¼ 1ω1

and p2 ¼ 0. This is true

when the politician setsp2≤p2 but excludes IG 2 by assigning the bur-den of proof to IG 1, in which case θ(1,1)=1−θ(0,0)=1 and thevalues for p1 and p2 follow immediately from Eqs. (1) and (2). It isalso true when IG 2 is excluded due to p2 > p2 and therefore ϕ2=0.

Here, Eq. (1) simplifies to p1 ¼ 1−θ 0;0ð Þω1

, and with IG 2 excluded

the politician chooses θ(0,0)=0 to maximize p1 and potential pay-ments from IG 1. Similarly, if IG 1 is excluded, then p1 ¼ 0 and

p2 ¼ 1ω2

. Regardless of how an IG is excluded from the political pro-

cess, the included IG pays the same access fee pi ¼ pi when it has

21 Being able to focus on maximizing political contributions without worrying aboutpolicy utility makes the analysis of the politician's problem more tractable. We discussthis further in the conclusion.22 She may exclude i by setting pi ¼ pi when her policy strategy θ is independent of ri.

favorable evidence, and the politician implements the same policyalong the path of play (since θ(0,0) is the same under both types ofexclusion). Therefore, all payoffs are independent of the method ofexclusion.

It is helpful for the discussion to recognize that whenever the pol-itician assigns the burden of proof to IG i, she excludes IG− i from thepolitical process.

Lemma 4.3. In equilibrium, the politician assigns the burden of proof toIG i if and only if she excludes − i from the political process.

If IG − i is excluded and p−i≤ ¼ p−i, then by definition of exclu-sion the politician must choose policy independent of IG − i's evi-dence, and therefore must assign the burden of proof to IG i. If IG− i is excluded andp−i > p−i, then we have established that θ(0,0)will award policy to group − i given r1=r2=0 (doing so maximizespi and the expected payment from IG iwho is included in the politicalprocess). Therefore θ satisfies the definition of assigning the burden ofproof to i, given that in equilibrium the politician chooses policy infavor of i if and only if ri=1.

These results allow the analysis to assumepi ¼ pi (and thus ψi=1)for both i=1,2, and to let the politician choose θ(1,1) and θ(0,0). Ifthe politician prefers a choice of θ which assigns the burden of proofto IG i (i.e., θ(1,1)=1−θ(0,0)∈{0,1}), then we can say that the pol-itician prefers to assign i the burden of proof and exclude− i from thepolitical process, regardless of the method she chooses for doing so.

4.6. Maximizing political contributions

Above, we establish that the politician will always include at leastone of the IGs in order to maximize policy utility. Her choice of whichIG to include, or whether to include both IGs is made to maximizeexpected contributions.

If two groups each with favorable evidence differ only in terms oftheir wealth (parameter ω), then the relatively rich IG is willing topay more than the poorer group to disclose evidence. Because ofthis, the politician prefers to interact with the rich IG relative to thepoor IG when they have similar probabilities of having favorable evi-dence (and, thus, making a contribution). Only when the poor IG issufficiently more likely than the rich IG to make a contribution inequilibrium does the politician prefer to include the poor IG in the po-litical process. Even when this is the case, however, we show that thepolitician prefers to give access to both IGs rather than only to thepoor group. This implies the following result.

Proposition 4.4. The politician always includes the rich IG in thepolitical process. The politician excludes the poor IG unless

π2≥π1ω2

1−π1ð Þω1 þ π1ω2.

Excluding the poor IG from the political process increases the richgroup's willingness to pay to disclose favorable evidence. This is be-cause an IG is willing to pay more to disclose favorable evidencewhen it has the burden of proof (and exclusive access) than when itis being given the benefit of the doubt (and the other group mayalso reveal evidence). Therefore, the politician prefers to excludethe poor IG from the process unless the probability, π1, that the richIG has favorable evidence is sufficiently low compared to the proba-bility, π2, that the poor IG has favorable evidence. Only then doesthe politician prefer to include both IGs in the political process, asthe expected contribution from the poor group exceeds the expecteddecrease in political contributions from the rich group.

4.7. Equilibrium

Proposition 4.5 describes the equilibria of the game.


Proposition 4.5. If π2 >π1ω2

1−π1ð Þω1 þ π1ω2, then the only access equi-

librium involves the politician giving the benefit of the doubt to IG 1. Inthis case:

• IG i pays for access if and only if ei=1 and pi≤pi, where p1 ¼ π2

ω1and

p2 ¼ 1−π1

ω2.

• The politician includes both IGs in the political process, offering each IGaccess at price pi ¼ pi.

If π2bπ1ω2

1−π1ð Þω1 þ π1ω2, then the only access equilibria involve the

politician assigning the burden of proof to IG 1. In this case:

• IG 1 pays for access if and only if e1=1 andp1≤p1, wherep1 ¼ 1ω1

andIG 2 never pays for access.

• The politician includes IG 1 in the political process offering access atprice p1 ¼ p1. IG 2 is excluded from the political process.

If π2 ¼ π1ω2

1−π1ð Þω1 þ π1ω2, then under A2.1 the only access equilibri-

um is the benefit of the doubt equilibrium, and under A2.2 the only accessequilibrium is the burden of proof equilibrium described above.

Notice that the required condition for the politician to includeboth IGs in the political process is only feasible if π2>π1. Only thendoes there exist possible values of ω1 andω2 such that π2 is sufficient-ly large compared to π1. If π2≤π1, then the politician always prefers toexclude the poor group from political participation. Because thisrange includes equality, this implies that when the interest groupsonly differ in terms of their wealth and realized evidence (i.e.,π1=π2), the politician strictly prefers to include the rich and excludethe poor IG from political participation.

The result is consistentwith the arguments fromcampaignfinance re-form advocates that the rich have better access to politicians than thepoor (e.g., Makinson, 2003). However, the result does not directly implythat equilibrium policy is biased in favor of rich groups, or that the richare necessarily better off than the poor through their involvement inthe political process. We consider these aspects of equilibrium below.

4.8. Does the policy choice favor the rich IG?

If in equilibrium the politician only includes IG 1 in the political pro-cess, then the politician chooses policy 1 with probability π1. That is, thepolitician chooses policy in favor of the rich IG if and only if the richgroup discloses favorable evidence. Alternatively, if both IGs are includ-ed in the political process, then the politician prefers to give the benefitof the doubt to the rich IG in equilibrium. In this case, she chooses policy1 when only IG 1 reveals favorable evidence or if r1=r2, which happenswith total probability π1+(1−π1)(1−π2). This means that the proba-bility of choosing policy is dependent upon the probabilities that IGshave and are able to disclose favorable evidence; not on their relativewealth parameters.

Suppose that πi>π− i. In this case, policy i is more likely a good choicethan policy − i, and the politician would implement policy i in the ab-sence of any lobbying game. In the access equilibrium of the lobbyinggame, the policy choice depends on the revelation of evidence (or lackthereof) by at least one of the IGs. With lobbying, the probability thatthe politician implements policy i is a function of π, and is always lessthan one. Compared to the absence of any lobbying game, where policyi is always implemented, the lobbying game strictly decreases the prob-ability that policy i is implemented and strictly increases the probabilitythat policy− i is implemented in equilibrium.

Lemma 4.6. Compared to no lobbying, lobbying strictly decreases theprobability that the politician implements policy i if π− ibπi.

Lemma 4.6 holds independent of the other parameter values, andtherefore applies within both burden of proof and benefit of the

doubt equilibria. Whether the lobbying process increases or decreasesthe probability that the politician chooses different policiesdepends only of the distribution of evidence, and not on IG wealthdifferences.

It is feasible that the lobbying process increases the likelihood thatpolicy favors the poor IG. Consider, for example, a case whereπ2bπ1b

12. Here, from an ex ante perspective, it is more likely that the

rich group has favorable evidence compared to the poor group. With-out any lobbying game, the politician would always choose policy 1based on her priors alone. When we introduce lobbying, the politicianchooses policy in favor of the rich IG with probability less than one.Here the lobbying game decreases the probability of policy 1 andincreases the probability of policy 2.

Continuing the example in which π2bπ1b12, we can also show that

in equilibrium the politician chooses to exclude IG 2 from the politicalprocess. Despite this, the politician implements policy 2 more thanhalf the time. When π1>π2 the only equilibrium involves the politi-cian assigning the burden of proof to IG 1, excluding IG 2. In this equi-librium, the politician implements policy 1 with ex ante probability

π1b12, and policy 2 with probability 1−π1 >

12.

Next, consider a case when π2>π1, which means that either theburden of proof and benefit of the doubt equilibria are possible,depending on the relative wealth differences between the IGs. Takingπ1 and π2 as given, there exists ω1 and ω2 such that the politician ex-cludes the poor IG from the political process, and such that the politi-cian includes both IGs. When the poor IG is excluded, the burden ofproof is given to the rich group and the politician chooses the poorgroup's policy whenever the rich group does not reveal favorable ev-idence. When the poor IG is included in the process, the politicianprefers a policy strategy that chooses the poor group's policy ifand only if the rich group does not reveal and the poor group doesreveal favorable evidence. This means that it less likely that the pol-itician chooses policy in favor of the poor group when both groupsare involved in the political process, than when the poor group isexcluded.

Lobbying does not necessarily bias policy in favor of the rich IG,even when the process excludes the poor IG.

4.9. Is the rich IG better off than the poor IG?

In equilibrium, the politician either assigns the burden of proof orthe benefit of the doubt to the rich IG, depending on parametervalues. If the game results in a benefit of the doubt equilibrium, IG 1earns ex ante expected utility EU1=1−π2 and IG 2 earns EU2=0,net policy utility and contributions. Alternatively, if the game resultsin a burden of proof equilibrium, IG 1 earns ex ante expected utilityEU1=0 and IG 2 expects EU2=1−π1.

Proposition 4.7. In the access equilibrium,

• if the politician gives the benefit of the doubt to IG 1 (which includesboth IGs in the political process), then EU1>EU2=0, and

• if the politician assigns the burden of proof to IG 1 (which excludes IG 2from the political process), then EU2>EU1=0.

The Proposition shows that IG 1 expects a higher payoff whenboth IGs are included in the political process than when IG 1 is theonly group to be included. IG 2, on the other hand, expects a higherpayoff when it is excluded from the process than when it is includedin the process. These findings are in contrast to the standard intuitionsuggesting that groups with access are better off than those withoutaccess. We illustrate a flaw in this intuition. When the politician stra-tegically sets access fees, and updates her beliefs about evidencewhen a group does not present favorable evidence, being offered ac-cess does not necessarily benefit an interest group. When the param-eters are such that the politician includes both IGs in the political


process, she sets access prices to extract rent from the poor IG. Forother ranges of parameters, the politician finds it more profitable toexclude the poor IG from the political process, only including therich IG. When this is the case, the politician finds it most profitableto set access prices that extract the maximum possible paymentsfrom the rich group.

We can consider once again the case where IGs only differ in termsof wealth and realized evidence. We summarize these findings inCorollary 4.8.

Corollary 4.8. When π1=π2, the politician always gives the burden ofproof to the rich IG and excludes the poor IG from the political process.In this case, 0=EU1bEU2; the poor IG is better off compared to therich IG.

In this case, the rich IG may not have an initial policy advantage,and it is made worse off by being rich. The politician sets access feesto extract the policy rents from the interest group that is likely topay the most. This means that although the rich group can buy access,the price of access offsets any benefit achieved from having its policyimplemented. The poor group, on the other hand, is not the target ofaccess fees and benefits whenever the rich group does not reveal fa-vorable evidence. As Corollary 4.8 shows, a poor group is actually bet-ter off than a rich group in equilibrium.

5. Extensions

We consider a series of refinements to the model for the casewhen interest groups only differ in terms of wealth, ω2>ω1>0. Be-cause π1=π2=π, any differences in the treatment of the two IGsare driven solely by their wealth differences, not by differences inthe probability of producing favorable evidence. This allows us tofocus the analysis and discussion on one of the central claims fromthe policy debate, that rich groups have an advantage due to theirability to outspend less wealthy groups.

From Proposition 4.5 and Corollary 4.8 we know that without re-finements, the politician assigns the burden of proof to the rich IG,

setting p1 ¼ 1ω1

and excluding the poor IG from the political process.

In equilibrium, EU1=0 and EU2=1−π.

5.1. Contribution limit

Suppose there exists an exogenous contribution limit, cmax≥0. In-terest groups cannot pay and the politician cannot charge more thancmax for access.

Proposition 5.1 describes the equilibrium for the game with limitcmax.

Proposition 5.1.

• If cmax≥1ω1

, then the only access equilibria involve the politician

assigning the burden of proof to IG 1, and setting p1 ¼ 1ω1

. IG 2 is

excluded from the political process.

• If1

ω1 þω2bcb

1ω1


choosing θ(1,1) and θ(0,0) such that p1 ¼ cmax and p2 ¼ 1−ω1cmax

ω2,

and setting p1 ¼ p1 and p2 ¼ p2. Both IGs are included in the politicalprocess.

• If c≤ 1ω1 þω2


choosing θ(1,1) and θ(0,0) such that p1≥cmax and p2≥cmax, and set-ting p1=p2=cmax. Both IGs are included in the political process.

Whencmax≥1ω1

the limit is not binding.When cmaxb1ω1

, the politician

will be indifferent between multiple values of θ which are payoff

equivalent. Proposition 5.1 makes no assumptions about the choice of θwhen the politician is indifferent. The multiplicity of equilibrium doesnot result in a coordination problem, as the politician announces θ beforethe IGs choose payments. For the welfare analysis, we may apply as-sumptions A2.1 or A2.2 to identify a unique choice of θ that favors eitherthe implementation of policy 1 or policy 2. In Appendix A, we fully char-acterize the equilibrium under both assumptions.

Without a limit, the politician finds it optimal to exclude the poorIG from the political process. Under a binding limit, the politician in-cludes both IGs in the process. Doing so does not improve her abilityto choose good policy. Rather, under a limit it is no longer optimal totarget her rent extraction efforts at the rich group alone. She is unableto choose the access rules that maximize expected payments from therich IG, and instead she finds it optimal to target both groups.

The contribution limit cmax affects the expected payoffs. Thechanges to utility are summarized in Proposition 5.2.

Proposition 5.2. In equilibrium, compared to the case of no limit, any

binding contribution limit cmaxb1ω1

• strictly decreases politician expected utility EW,• either increases or does not change rich IG expected utility EU1, and• either increases, decreases or does not change poor IG expected utilityEU2.

Without a limit the politician chooses p and θ to maximize total IGpayments. With a limit, the politician is unable to achieve the samemaximum payments and does not improve her ability to identifygood policy. Therefore, the limit makes the politician strictly worseoff. Whether the limit changes the IGs' expected utility depends onthe parameter values and the politician's choice of θ in the eventthat she is indifferent between multiple values.

Appendix A fully characterizes equilibrium and the affect that acontribution limit has on IG utility for each combination of parametervalues. Here, we describe some of the more interesting features ofequilibrium.

When cmax is high enough, the politician sets the access fee for IG 1at p1=cmax and a lower access fee for IG 2. This is because the politi-cian benefits from extracting the maximum feasible payment for therich IG, even when that requires extracting a lower than feasibleamount from the poor IG. As the contribution limit decreases, thepoor group access fee p2 increases until cmax is low enough that

both p1=p2=cmax. This is the case for all cmax≤1

ω1 þω2. This

means that p1=cmax always, and p2=cmax only when the limit issufficiently low.

When1

ω1 þω2bcmaxb

1ω1

, the politician sets access fees for each IG

such that the price of buying access equals the expected benefit ofpresenting favorable evidence. This means that in equilibrium overthis range of cmax both IGs are indifferent between paying for accesswhen they have favorable evidence and not paying for access. Thisdoes not mean, however, that their expected payoff is zero; rather,EUi equals the expected payoff from not revealing favorable evidence.That is, EU1=(1−π)θ(0,0) and EU2=(1−π)(1−θ(0,0)), the proba-bility that the other IG does not reveal evidence times the probabilityof receiving a favorable policy choice in the event that neither IGreveals evidence. In this case, the minimum value of EU1 and themaximum value of EU2 are equal to the IG payoffs in the absence ofa contribution limit. At worse IG 1 can be just as well off, and atbest IG 2 can be no worse off than without a limit.

When cmaxb1

ω1 þω2, at least one of the IGs is willing to pay more

than cmax to reveal favorable evidence. The more an IG is willing topay for access relative to cmax, the better off the IG expects to be inequilibrium. Only a limit in this lower range has the potential tomake IG 2 better off compared to the case without a limit; although

23 The formal advantage of assuming that the formation decision takes place beforethe realization of e is that the analysis does not have to consider one's choice of lobbyformation or signals to the politician about one's type. However, when formation is de-cided after observing one's own type, there still exists an equilibrium with noformation.


this is not always the case. IG 1 continues to be no worse off (and inthe majority of cases, strictly better off) in this range of cmax.

Finally, we compare IG payoffs under different assumptions aboutpolitician behavior in the event that she is indifferent over alternativevalues of θ. Under A2.1, the politician chooses θ which, conditional onmaximizing EW, maximizes the probability of implementing policy 1.Under A2.2, the politician chooses θwhich, again conditional on max-imizing EW, maximizes the probability of implementing policy 2.

Corollary 5.3. If1

ω1 þω2bcmaxb

1ω1

, then

• IG 1 is better off under A2.2than under A2.1, and• IG 2 is better off under A2.1than under A2.2.

If cmaxb1

ω1 þω2, then

• IG 1 is better off under A2.1 than under A2.2, and• IG 2 is better off under A2.2 than under A2.1.

One may interpret A2.1 as the politician having a weak bias infavor of IG 1, and A2.2 as a weak bias in favor of IG 2. This meansthat under a high-enough contribution limit, IGs prefer the politicianto be biased in favor of the other IG rather than their own group.

When there is a contribution limit, the politician may no longer beable to set prices high enough to capture all of the policy rent earnedby a rich group with positive evidence and access. This means thateven a rich groupwith evidencemay now earn positive expected utilityin equilibrium. The same factor that makes the rich group better off alsoworks tomake the politicianworse off in equilibrium. Under the contri-bution limit, the politician is limited in her ability to extract rent frominterest groups. She therefore expects to collect lower total contribu-tions. At the same time, her ability to implement policy is unchangedsince she continues to offer access to IGs, just at a lower price.

5.1.1. Reconsidering the popular argument in favor of limitsA popular argument in favor of contribution limits by campaign fi-

nance reform advocates goes as follows. Rich interest groups have betteraccess to policymakers. Therefore policywill be biased in favor of rich in-terests. Contribution limits can help level the playing field between richand poor special interests, and will therefore result in better policy.

Our first critique of this argument relies on the fact that offeringaccess to an interest group does not commit a policy maker to act infavor of the group. In our framework, the policy maker rationally up-dates her beliefs about the best policy, and then chooses her preferredpolicy regardless of whether that interest group bought access. In oursetting, a contribution limit does not result in a better informed poli-tician and “better” policy. Our second critique of the standard pro-limit argument involves interest group expected payoffs. At theheart of the argument is the idea that rich interest groups have an ad-vantage over poor groups, and that contribution limits eliminate thisadvantage. This reasoning is only partially correct. Without a limit,rich groups receive access and poor groups do not, but poor groupshave higher expected utility than rich groups because they are notthe target of politician rent seeking. Contribution limits offer a greaterbenefit to rich interest groups than poor IGs.

5.2. Endogenous lobby formation

The previous section showshow contribution limits can benefit inter-est groups, but alwaysmake the policymakerworse off.Why thenwoulda politician ever support the implementation of a contribution limit? Onepossibility is that voters believe the argument put forth by campaign re-form advocates and the politician gives in to voter pressure. Or, the pol-itician herself may believe the arguments and chooses the reform in aneffort to achieve better policy decisions. This section considers anotherexplanation: to encourage the formation of lobby groups.

Herewe consider a gamewhich is identical to the game in Section 3,except for two initial stages of play. First, the politician commits to a

contribution limit, cmax. Second, the IGs simultaneously decide whetheror not to form a lobbying presence. Forming costs μ>0. After this, thegame precedes identically to the earlier sections, except that the politi-cian can only offer access to a group that formed a lobbying presence.We assume that IGs decidewhether to establish a lobbying presence be-fore they observe their evidence; although similar results can be foundif they first observe their own e.23 In this sense, we assume that estab-lishing a lobbying presence is a long-term commitment.

First, consider the game with no contribution limit. Here, not form-ing a lobbying presence is a dominant strategy for both IGs, and theunique equilibrium involves neither IG forming. If no interest groupforms, then the politician chooses either policy 1 under A2.1, or policy2 under A2.2. Suppose that the politician implements policy i in theevent that neither IG forms. Then, EUi=1 and EU− i=0. If only one IGforms, the politician sets access fees tomaximize the expected paymentfrom the formed group. That is, she assigns the burden of proof to thegroup that forms, extracting the maximum payment from the IG. Fur-thermore the formed group already paid the sunk cost of formation,and will thus earn EUj=−μ. Neither IG has an incentive to formwhen the politician can require political contributions so large thatthey fully offset the benefits of lobbying.

The politician benefits if either one or both IGs lobby. Even in theabsence of political contributions, there are informational benefits toproviding access to IGs. However, IGs have no incentive to form whenthey expect the politician to target themwith her rent seeking efforts.IGs will only form if the politician credibly commits to less rent seek-ing behavior in the later stages of the game.

The politician can effectively commit to limiting her rent seeking be-havior in the lobbying game by implementing a contribution limit. Shecannot extract more rent from an interest group than the contributionlimit allows. This can encourage lobby formation, which increases evi-dence revelation and political contributions in equilibrium.

Proposition 5.4 describes the IG formation decision given limitcmax. We provide a formal statement of equilibriumstrategies, includingmixing probabilities, in Appendix A.

Proposition 5.4. Under A2.1, there exists equilibria of the subgame(taking as given cmax) in which

• neither group forms a lobbying presence when

π−μπω2

≤ cmax

• only IG 2 forms a lobbying presence when

minπω1

;1

ω1 þω2

� �≤ cmax ≤

π−μπω2

; and

• both IG play a mixed strategy, forming with positive probability, when

cmaxbminπω1

;1

ω1 þω2;π−μπω2

� �:

24 There are many ways one may interpret the results in terms of overall welfare. Onepossibility is to assume that the politician and interest groups represent a very smallportion of the overall population, and therefore social welfare depends only on thequality of the policy outcome. In this case, social welfare is independent of whetherthe politician includes or excludes the poor group in the political process, and a contri-bution limit improves welfare only if it encourages lobby formation. A more interestingview of social welfare, and one that is more consistent with the policy debate, assumesthat conditional on implementing a good policy welfare is increasing in the ratio ofpoor group to rich group payoffs. Under this view of welfare, our analysis shows howwelfare may be higher when the politician excludes the poor group from the politicalprocess. In this case, contribution limits may decrease welfare by as they can make therich group better off relative to the poor group. The conclusions are less drastic if weassume that welfare is increasing in policy quality and the payoffs of both IGs, and in-dependent in the politician's payoff. In that case, a contribution limit can improve IGpayoffs without resulting in worse policy, thereby improving welfare. A low enoughlimit, under A2.2, for example, improves the payoffs of both IGs and will certainly havethis effect.


Under A2.2, there exists equilibria of the subgame (taking as givencmax) in which

• neither group forms a lobbying presence when

π−μπω1

≤ cmax

• only IG 1 forms a lobbying presence when

minπω2

;1

ω1 þω2

� �≤ cmax ≤

π−μπω1

; and

• both IG play a mixed strategy, forming with positive probability, when

cmax b minπω2

;1


� �:

Interest groups form lobbies only when the politician is sufficient-ly constrained in her ability to charge access fees in the next stage.Too high of a contribution limit does not sufficiently constrain thepolitician's rent seeking ability, and neither IG forms a lobby in equi-librium. For intermediate levels of contribution limits, only the IG thatthe politician is biased against (given A2.1 or A2.2) finds formationworthwhile. When the contribution limit is low enough, both IGsform lobbies with positive probability. The politician finds any limitthat results in lobby formation beneficial compared to the casewhen there is no limit and no IG forms. Even banning contributions(i.e., cmax=0) benefits the politician as it encourages lobby formationand the resulting disclosure of evidence; this in turn results in amore-informed policy choice.

From Proposition 5.5, we see that IGs only form if cmax is sufficiently

low. Under A2.1, IGs only form if cmax≤π−μπω2

, and under A2.2, IGs only

form if cmax≤π−μπω1

. When the politician commits to cmax at the onset

of the game, she is able to select a limit that results in lobby formationif and only if the right hand side of these inequalities are non negative;that is, when μ≤π. This brings us to the main result for this section.

Proposition 5.5. If μ≤π, then the politician can set a contribution limitunder which at least one IG forms a lobbying presence in equilibrium.Under such a limit, the politician is better off than without a contributionlimit.

This says that as long as the cost of forming is sufficiently low, thepolitician can encourage lobby formation by committing to a contri-bution limit. Appendix A provides details of the equilibrium strate-gies, including the politician's optimal choice of cmax. There, weshow that when feasible, the politician prefers to set cmax such thatonly one IG forms a lobbying presence in equilibrium. Only whenparameters are such that the politician cannot entice one IG to formwithout also enticing the other group to form does the politicianselect a cmax under which both IGs form with positive probability.

Finally, it is worth pointing out that the contribution limit does notin itself benefit IGs. When the limit entices only one IG to form, thepolitician will strategically set the limit such that the group is justindifferent between forming and not forming. The group that formsexpects payoff of 0, and the group that does not form expects payoff1−π, which is strictly lower than its payoff of 1 in the absence of acontribution limit. When both IGs form with positive probability,

than the IGs will continue to be indifferent between forming andnot forming (a requirement of playing a mixed strategy in equilibri-um). Here, the IG the politician is biased in favor of (IG 1 underA2.1 and IG 2 under A2.2) is made strictly worse off and the IG thepolitician is biased against is made strictly better off by the limit.

6. Conclusion

This paper makes two primary contributions. First, it presents aninnovative model of money in politics that is both tractable and con-sistent with the story that political contributions help secure access topoliticians. Although the money-for-access assumption is consistentwith claims by both politicians and interest groups (e.g., Herndon(1982), Schram (1995) and Makinson (2003)), it has generally beenoverlooked by theoretical considerations of the political process. Theanalysis shows that much of the common intuition about the advan-tages that rich interest groups have over poor interest groups in thelobbying process are not supported by a theoretical model.

Second, the paper contributes to the debate on campaign financereform by considering the effect of contribution limits. Much of thepopular support for contribution limits centers on the idea that richspecial interests have better access to decision makers than do poorspecial interests. Contribution limits, it is said, help eliminate therich group advantage, which will result in better representation forpoor interests, and better policy. Our analysis shows that rich interestgroups do tend to have better access to politicians than poor interestgroups. However, we also show that this rich group access advantagedoes not imply a policy or payoff advantage. What is missing from thepopular argument (but not from our analysis) is the recognition that apolitician will be strategic when offering access to interest groups andwhen choosing policy. In equilibrium, the politician offers access tothe rich interest group, observes any evidence the rich group presentsin favor of its policy, then chooses whichever policy she believes isbest. Because the politician is not committed to acting in favor ofthe interest group with access, the likelihood of implementing agood or bad reform does not depend on the identity of the groupwith access. Furthermore, the interest group with access is the targetof politician rent seeking, which eliminates any possible advantagethe group gains from disclosing positive evidence. We show thatpoor interest groups (who are not offered access, and are thus notthe target of politician rent seeking) may have higher expected pay-offs than their rich counterparts. In the basic framework, contributionlimits tend to benefit rich interest groups and make the politicianworse off.24

Although the basic model allows for a critical analysis of the pop-ular arguments in favor of contribution limits, we find the setting toosimple to put much faith in the conclusion that contribution limits arenever beneficial for policy or the politician. To address this concern,we allow for endogenous lobby formation by the interest groups. Indoing so, we present a novel argument in favor of contribution limits:


that limits can encourage lobby formation, which increases informa-tional lobbying, and results in a more-informed politician and betterpolicy decisions. Interest groups recognize that forming a lobbyingpresence will result in them being the target of politician rent seek-ing. In equilibrium when there is no contribution limit, interestgroups do not form a lobbying presence and the politician collectsneither evidence nor contributions. By committing to a contributionlimit, the politician effectively constrains her rent seeking ability, en-couraging lobby formation and a better-informed politician.

Our results are most intuitive when we assume a discrete evidencespace, where IGs either have or do not have favorable evidence. Here,the politician is just as likely to make a good policy decision when shelearns only one IG's evidence as when she learns both IGs' evidence.This means that, conditional on the fees being low enough that at leastone group is willing to lobby, the politician prefers to set access fees tomaximize expected payments from the IGs. The simple structure keepsthe analysis tractable, and allows for us to focus the analysis on the poli-tician's rent seeking efforts. This is helpful when discussing the mainpoint of the paper: an IG who is offered access is the target of politicianrent seeking and is therefore not necessarily better off compared to anIGwho is not given access, andwe do not expect that such complicationsto the framework would change the qualitative results.25

There are other arguments in favor of and against contribution limitsthat we do not address in the analysis. For example, in an election, con-tribution limits maymake it more difficult for challengers to mount a vi-able campaign against a sitting incumbent. Limits may also result insmaller budgets to run campaigns,whichmay result in fewer informativeadvertisements, and less-informed voters (e.g., Coate, 2004b). Futurework may incorporate the access model into an election game, allowingvoters to rationally respond to politicians funding campaigns by sellingaccess. Furthermore, our analysis makes the conservative assumptionthat interest group policy preferences and wealth are independent ofthe politician's preferred policy or the preferences of the electorate.Relaxing this assumption, however, may strengthen our results as it be-comes more important (from a standpoint of constituent welfare) tolimit the politician's ability to extract rents from the interest groups.

Finally, we conclude with a brief discussion of the January 2010 U.S.SupremeCourt decision inCitizens United v. the Federal Elections Commis-sion. The ruling eliminated limitations on private spending in support ofpolitical campaigns.26 To the extent that campaign contributions andprivate ads run in support of campaigns are substitutes, the CitizensUnited ruling is equivalent to an elimination of contribution limits inour model. Our results suggest that the ruling may not strictly benefitrich interests, at least on issues on which contributions are intended to

25 The downside of the structure is that it means that the politician is no better in-formed when he meets with both IGs than when he meets with only one IG. Allowingfor a more complex evidence structure would eliminate this concern. For example, aproject's quality and evidence may be continuously distributed on (0,1), and the poli-tician may prefer to implement the policy with the highest evidence value. In this case,even if the politician gives IG i access and observes ei close to 1, there remains a posi-tive probability that the other IG has even stronger evidence. The politician would beable to observe that the other policy is marginally more beneficial only if she offers ac-cess to both groups. In this environment, strategies will be more complex, with IGspaying for access only if their evidence is above some threshold value, and the politi-cian needing to take this into account after an IG does not pay an access fee. Even inthis setting, however, we expect our conclusions to continue to hold as long as the pol-itician cares enough about revenue relative to changes in policy outcome. The politi-cian's access strategy will be designed to capture rent from the IGs, and an IG thatreceives access will not necessarily be better off than one that does not. Contributionlimits will still constrain politician rent seeking ability, playing the same role as inthe discrete evidence game.26 Although campaign contributions (i.e., money given to political campaigns for thecampaign to use at its own discretion) were unaffected by the ruling, the ruling allowsfor the formation of “super PACs,” political action committees which can collect andspend unlimited money on behalf of a candidate's campaign. This means that an inter-est group can make an unlimited payment to a candidate's super PAC at any time, rath-er than to the candidate herself.

gain access. In the standard model without endogenous lobby forma-tion, eliminating a limit can make rich interest groups worse off, as thepolitician expects larger payments in exchange for access. This meansthat the politician will be less likely to give access to poorer interestgroups after Citizens United, but does not necessarily make these poorgroups worse off. When our model allows interest groups to decidewhether to form a lobbying presence or search for favorable informa-tion, the model suggests a more-pessimistic effect of Citizens United.In that case, the Citizens United ruling may discourage the formationof lobby groups or the search for evidence in the first place, as interestgroups attempt to avoid the politician's now less-constrained efforts toextract rent from interest groups with favorable evidence. Future workmay empirically test these implications.

Appendix A

Throughout this section, let BoP denote “burden of proof” and BoDdenote “benefit of the doubt.”

a.1. Proofs from analysis in Section 4

Proof of Lemma 4.1. If θ assigns the BoP to i, then the policy choicedepends on ri but not r− i. Thus − i is excluded. For i to be included,it must also be that the politician sets pi≤pi, which she will sinceany pi > pi results in the politician remaining fully uninformed andcollecting no payments.

If θ assigns the BoD to i, then the policy choice depends on ri andr− i. Conditional on θ being a burden of proof strategy, the politicianprefers to set both pi and p− i no greater than their respective pi andp−i. Setting a higher price guarantees no payment or evidence fromthe group, making the politician worse off in expectation. Thus, sheincludes both i and − i in the political process. ■

Proof of Lemma 4.2. Follows immediately from the discussion in thebody of the paper. ■

Proof of Lemma 4.3. Follows immediately from the discussion in thebody of the paper. ■

Proof of Prop 4.4. The politician expects payoff

EW θð Þ ¼ π1 þ π2−π1π2 þ π1

π2θ 1;1ð Þ þ 1−π2ð Þ 1−θ 0;0ð Þð Þω1

þπ2π1 1−θ 1;1ð Þð Þ þ 1−π1ð Þθ 0;0ð Þ

ω2

!ϕ:

Expected politician utility is strictly increasing in θ(1,1) sinceω1bω2 is assumed by the model. That is,

∂EW θð Þ∂θ 1;1ð Þ ¼

ω2−ω1

ω1ω2π1π2ϕ > 0:

Whether EW is increasing in θ(0,0) depends on the model parame-ters.

∂EW θð Þ∂θ 0;0ð Þ ¼ π2

1−π1ð Þω2

−π11−π2ð Þω1

� �ϕ:

This expression is strictly negative when

π2bπ1ω2

1−π1ð Þω1 þ π1ω2;


in which case the politician strictly prefers the pure strategy θ that as-signs the BoP to 1, excluding 2 from the political process (i.e., θ(1,1)=1 and θ(0,0)=0). The inclusion of 2 in the process is guaranteedwhen the politician strictly prefers assigning the BoD to 1, which isthe case when

π2 >π1ω2

1−π1ð Þω1 þ π1ω2:

It is also possible when she is indifferent between assigning theBoP or BoD to 1, which happens when

π2 ¼ π1ω2

1−π1ð Þω1 þ π1ω2:

In the event of indifferent, the politician's choice of θ depends onwhether we use assumption A2.1 or A2.2. As we establish in thetext, assigning the BoD (as opposed to BoP or some mixed strategychoice of θ) to IG 1 results in a higher probability of choosing policy1. Thus, A2.1, which requires θ to maximize the probability of imple-menting policy 1 in the event that the politician is indifferent, meansthe politician chooses the BoD strategy. In a similar way, A2.2 resultsin a BoP equilibrium when the politician is indifferent. ■

Proof of Prop. 4.5. The proof to Prop. 4.4 shows that the politician

will either assign the BoP or BoD to 1. If π2≤π1ω2

1−π1ð Þω1 þ π1ω2, then

the politician gives the BoP to IG 1, setting θ(1,1)=1 and θ(0,0)=0.

In this case, Eq. (1) simplifies to p1 ¼ 1ω1

, where 1 pays any p1≤p1,

the politician sets p1 ¼ p1, and 2 is excluded from the political process(in this case, her choice of p2 does not matter since 2 is excluded fromthe political process independent of p2).

If π2 >π1ω2

1−π1ð Þω1 þ π1ω2, then the politician gives the BoD to IG 1,

setting θ(1,1)=1 and θ(0,0)=1. In this case, Eqs. (1) and (2) simpli-

fy to p1 ¼ π2

ω1and p2 ¼ 1−π1

ω2, where i pays any pi≤pi, the politician

sets pi ¼ pi for both i. Establishing that both IG's strategies and thepolitician's choice of θ are sequentially rational follows Lemmas 4.1and 4.2, the analysis in the proof to Prop. 4.4, and the discussion inthe body of the paper.

It remains for us to establish that the politician's choice of pi issequentially rational in later stages. In equilibrium, the politician mustnot have an incentive to provide access to any IG that provides a pay-ment less than its assigned access fee. If group i pays less than pi ¼ pi

and the other group is included in the political process, then the politi-cian has no incentive to review agent i's evidence; she is already becom-ing as informed as she needs to be through her interaction with theother interest group (which in equilibrium pays for access wheneverit has favorable evidence). If group i pays a positive amount less thanpi ¼ pi and the other group is excluded from the political process,then the politician updates her beliefs about ei, denoted ~π i cið Þ. Theseare off equilibrium path beliefs, and as such any beliefs such that thepolitician does not want to deviate are consistent with the PerfectBayesian Equilibrium of the game. This will be the case when ~π i cið Þ issufficiently low for all ci∈(0,pi). Such a ~π i is always feasible, giventhat the condition is satisfied when ~π i ¼ 0. ■

Proof of Lemma 4.6. In the absence of lobbying, the politicianchooses the policy with the highest probability of being good. Shechooses i with probability 1 if πi>π− i, and with probability 0 ifπibπ− i. In the equilibrium of the game with lobbying, the ex anteprobability of policy i is (i) πi if the BoP is given to i, (ii) 1−π− i ifthe BoP is given to − i, (iii) πi+(1−πi)(1−π− i) if the BoD is givento i, and (iv) (1−π− i)πi if the BoP is given to − i. All four of theseprobabilities are strictly greater than 0 and strictly less than 1. ■

Proof of Prop. 4.7. If the politician assigns the BoD to IG 1, then p1 ¼π2

ω1and p2 ¼ 1−π1

ω2. In this case 1 has EU1=1−p1ω1=1−π2 when

e1=1 and EU0=0 when e1=0, and 2 has EU2=(1−π1)−p2ω2=0when e2=1 and EU2=0 when e2=0. From an ex ante perspective,EU1=π1(1−π2)>0=EU2.

Similarly, if the politician assigns the BoP to IG 1, then p1 ¼ 1ω1

and

2 is excluded from the political process. In this case 1 has EU1=1−p1ω1=0 when e1=1 and EU1=0 when e0=0, and 2 has EU2=1−π1 regardless of e2. From an ex ante perspective, EU1=0b1−π1=EU2. ■

Proof of Corollary 4.8. As Proposition 4.5 shows, the access equilib-rium involves the politician assigning the BoP to IG 1 whenever

π2bπ1ω2

1−π1ð Þω1 þ π1ω2. If π1=π2, this condition simplifies to ω1bω2,

which is assumed by the structure of the game. Therefore, whenπ1=π2, the poor group is excluded from the political process. Therest of the proof follows from Propositions 4.5 and 4.7. ■

a.2. Detailed analysis with contribution limit

The following analysis is provided in place of individual proofs toProposition 5.1, Proposition 5.2, and Corollary 5.3.

a.2.1. EquilibriumThe politician chooses p and θ to maximizeW subject to the various

constraints on p and θ. The only difference between the politician'sproblemhere, and her problem in Proposition 4.5 is that here the choiceof pi must also be less than the limit cmax. As before, we can solve for p1and p2 as given by Eqs. (1) and (2), and sequential rationality requiresthat θ(0,1)=0 and θ(1,0)=1. Given π1=π2, we can also apply a posi-tive affine transformation to W to simplify the politician's objectivefunction. The politician's maximization problem simplifies to

p1; p2; θ 1;1ð Þ; θ 0;0ð Þmax

p1 þ p2

s:t: 0 ≤ θ 1;1ð Þ ≤ 1; 0 ≤ θ 0;0ð Þ ≤ 1

p1≤ cmax; p1 ¼ πθ 1;1ð Þ þ 1−πð Þ 1−θ 0;0ð Þð Þω1

� �;

p2≤ cmax; p2 ¼ π 1−θ 1;1ð Þð Þ þ 1−πð Þθ 0;0ð Þω2

� �:

The first intermediate Lemma follows directly from the politician'smaximization problem:

Lemma 7.1. In equilibrium, the politician sets pi ¼ min cmax; pif g.

Proof. If not, then she could increase expected payoffs by increasingpi. A contradiction. ■

The next intermediate lemma establishes that the contribution

limit may only impact player strategies when cmaxb1ω1

, the optimal

access fee in the absence of limits.

Lemma 7.2. If cmax≥1ω1

, then the only access equilibria involve the pol-

itician assigning the burden of proof to IG 1, setting p1 ¼ 1ω1

, and exclud-ing IG 2 from the political process.

Proof. In the absence of a contribution limit, the politician sets

θ(1,1)=1 and θ(0,0)=0, such that p1 ¼ 1ω1

and p2 ¼ 0, and sets ac-

cess fees pi ¼ pi. This strategy is still feasible when cmax≥1ω1

. ■

a


We next consider the four potential equilibrium cases given

cmaxb1ω1

:

1. when p1 ¼ cmax ≤ p1 and p2 ¼ p2 b cmax

2. when p1 ¼ p1 b cmax and p2 ¼ cmax ≤ p23. when p1 ¼ cmax ≤ p1 and p2 ¼ cmax ≤ p24. when p1 ¼ p1 b cmax and p2 ¼ p2 b cmax

For the following analysis, it is helpful to establish another inter-mediate Lemma.

Lemma 7.3. If pi > cmax, then p1=p2=cmax.

Proof. Establishing pi=cmax follows immediately from Lemma 7.1given pi > cmax. To establish that p− i=cmax, suppose instead the al-ternative, p−i ¼ p−ibcmax. The politician is able to marginally changeθ(1,1) or θ(0,0) to both decrease pi and increase p−i. Doing so doesnot impact pi=cmax as long as the change is small enough thatpi≥cmax. It does however increase p−i ¼ p−i, thereby increasing thepolitician's expected payoffs. Therefore, the politician will neverchoose θ such that pi > cmax unless she is already setting the maxi-mum possible access fee for both IGs. ■

This means that p1 ¼ cmax in case (1), and p2 ¼ cmax in case (2).Consider now the politician's strategy given that it is consistentwith case (1). She sets p1 ¼ cmax ¼ p1 and p2 ¼ p2, and choosesθ(1,1) and θ(0,0) to maximize p2. That is, she solves

θ 1;1ð Þ; θ 0;0ð Þmax

�p2 ¼ π 1−θ 1;1ð Þð Þ þ 1−πð Þθ 0;0ð Þω2

s:t: 0 ≤ θ 1;1ð Þ ≤ 1; 0 ≤ θ 0;0ð Þ ≤ 1; �p2 b cmax;

nd cmax ¼πθ 1;1ð Þ þ 1−πð Þ 1−θ 0;0ð Þð Þ

ω1:

Given that π1=π2, the politician is indifferent between any com-bination of θ(1,1) and θ(0,0) such that the constraints in the abovemaximization problem hold. Therefore, without making any assump-tions about politician behavior when indifferent, there exists an equi-librium for each of these θ combinations. In each of these equilibria,the politician expects total contributions equal to π cmax þ p2ð Þ ¼πω2

1þ cmax ω2−ω1ð Þð Þ.This case is only feasible if there exists a combination of feasible θ

such that p2bcmax. That is if

π 1−θ 1;1ð Þð Þ þ 1−πð Þθ 0;0ð Þω2

bcmax

and cmax ¼πθ 1;1ð Þ þ 1−πð Þ 1−θ 0;0ð Þð Þ

ω1:

This simplifies to the condition that

cmax >1

ω1 þω2:

Case (2) is symmetric to (1). The politician chooses any θ(1,1) andθ(0,0) such that p2 ¼ cmax. In each of the corresponding equilibria, the

politician expects payoffπω1

1þ cmax ω1−ω2ð Þð Þ. This is a possibility

only when cmax >1

ω1 þω2. Notice that the expected total payment is

strictly lower under case (2) than under case (1) given that

cmax >1

ω1 þω2. Therefore the politician prefers equilibria consistent

with case (1) to equilibria consistent with case (2) since both result inthe same policy payoff and (1) results in higher expected payments.

In case (3), the politician sets p1≥cmax and p2≥cmax, and expectstotal payments 2πcmax. Any combination of θ that results in p1 andp2 greater than cmax results in the same payoffs. This is only feasible

if cmax≤1

ω1 þω2.

Case (4) is similar to the initial politician maximization problemwith the added constraint that pi≤cmax. Notice that if the optimalpi ¼ cmax, then the solution is consistent with one of the other casesrather than case (4). We will show this to be the case. Here, the pol-itician will choose pi ¼ pi for both agents, according to Lemma 7.1.Therefore, we must determine the value of θ(1,1) and θ(0,0) thatmaximize π p1 þ p2ð Þ. As determined previously, this value is strictlyincreasing in θ(1,1) and strictly decreasing in θ(0,0) whenever

cmaxb1ω1

. Thus, the politician prefers to increase θ(1,1) or decrease

θ(0,0) until p1 ¼ cmax, at which point case (4) is violated. Thismeans that the politician always prefers one of the other cases to

case (4) when cmaxb1ω1

. (Notice that if cmax≥1ω1

, then case (4) is the

only case consistent with equilibrium behavior, and the politician

chooses θ such that p1 ¼ 1ω1

and p2 ¼ 0).

Therefore, if1

ω1 þω2bcb

1ω1

, then the only access equilibria in-

volve the politician choosing θ(1,1) and θ(0,0) such that p1 ¼ cmax

and p2 ¼ 1−ω1cmax

ω2, and setting p1 ¼ p1 and p2 ¼ p2. Alternatively,

if c≤ 1ω1 þω2


choosing θ(1,1) and θ(0,0) such that p1≥cmax and p2≥cmax, and set-ting p1=p2=cmax. In both cases, the politician includes both groupsin the political process since pi > 0.

In the above analysis, we make no assumption about the choice ofθwhen the politician is indifferent between multiple values. Next, wedetermine equilibrium behavior under alternative assumptions, A2.1and A2.2.

Under A2.1, the politician chooses the rule (of those she is indiffer-ent between) that maximizes

π2θ 1;1ð Þ þ π 1−πð Þ þ 1−πð Þ2θ 0;0ð Þ;

which is the probability the politician implements policy 1 in equilib-

rium. When1

ω1 þω2bcmaxb

1ω1

(case (1) from above), the politician's

choice of θ(1,1) and θ(0,0) must satisfy the condition that p1 ¼ cmax.

Alternatively, when cmax≤1

ω1 þω2(case (3) from above), her choice

of θ must maintain p1≥cmax and p2≥cmax.Alternatively, under A2.2 the politician chooses the rule (of those

she is indifferent between) that maximizes

π2 1−θ 1;1ð Þð Þ þ π 1−πð Þ þ 1−πð Þ2 1−θ 0;0ð Þð Þ;

which is the probability that the politician implements policy 2 inequilibrium. This maximization problem is subject to the same con-straints as the maximization problem under A2.1.

Here we describe the politician's choice of θ(1,1) and θ(0,0), butleave the step-by-step derivation to the reader.

27 If p1≤cmax , then p1 ¼ p1 and by definition of p1, IG 1 is indifferent between paying p1and not paying p1 when it has favorable evidence. Not paying p1 results in EU1=(1−π2)θ(0,0), which is the probability that IG 2 reveals no evidence times the probability thatthe politician chooses policy 1 in the event that neither reveals favorable evidence. Indif-ference implies IG 1 must also expect this EU1 when it pays p1 ¼ p1.


Under A2.1:

• If1

ω1 þω2bcmaxb

1ω1

, then

θ 1;1ð Þ; θ 0;0ð Þð Þ ¼cmaxω1

π;0

� when cmax ≤

πω1

1;ω1cmax−π

1−π

� when

πω1

b cmax ≤1ω1

:

8>><>>:

• If cmax ≤1

ω1 þω2,

θ 1;1ð Þ; θ 0;0ð Þð Þ

¼

1;1−cmaxω1

1−π

� �when π ≤ ω1

ω1 þω2and

πω1

≤ cmax

1−cmaxω2

π;1

� �when π ≥ ω1

ω1 þω2and

1−πω2

≤ cmax

1;1ð Þ when cmax b minπω1

;1−πω2

� �:

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

Under A2.2:

• If1

ω1 þω2bcmax b

1ω1

, then

θ 1;1ð Þ; θ 0;0ð Þð Þ

¼0;

cmaxω1

1−π

� when cmax ≤

1−πω1

ω1cmax− 1−πð Þπ

;1� �

when1−πω1

b cmax ≤1ω1

:

8>><>>:

• If cmax≤ 1ω1þω2

,

θ 1;1ð Þ; θ 0; 0ð Þð Þ

¼

0;cmaxω2−π

1−π

� when π ≤ ω2

ω1 þω2and

πω2

≤ cmax

cmaxω1− 1−πð Þπ

;0� �

when π ≥ ω2

ω1 þω2and

1−πω1

≤ cmax

0;0ð Þ when cmax b minπω2

;1−πω1

� �:

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

Wewill use these results when calculating expected payoffs in thenext section.

a.2.2. PayoffsIn deriving the equilibrium above, we determined the expected

total payments collected by the politician when there is a contribu-tion limit. Since all possible equilibrium outcomes produce the sameexpected policy payoff for the politician, it is sufficient to comparetotal expected payments in each case to determine when the politi-cian is made better off. A contribution limit cmaxb

1ω1

strictly decreasespolitician expected payoffs. This follows immediately from the factthat the politician's payoff has a unique maximum in which p1 ¼ 1

ω1

in the unconstrained game, and the contribution limit prevents thepolitician from playing her payoff maximizing strategy.

Determining whether the contribution limit benefits IGs requiresadditional analysis. Here,

EU1 ¼ πθ 1;1ð Þ þ 1−πð Þ−ω1p1ð Þπ þ 1−πð Þ2θ 0;0ð Þ; andEU2 ¼ π 1−θ 1;1ð Þð Þ þ 1−πð Þ−ω2p2ð Þπ þ 1−πð Þ2 1−θ 0;0ð Þð Þ:

The first set of parentheses in each equation denotes the expectedpayoff in the event that the IG draws favorable evidence. This is addedto the expected payoff in the event that the IG does not draw favorableevidence. Notice that if p1≤cmax, then EU1 simplifies to EU1=(1−π)θ(0,0).27 Similarly, if p2≤cmax, then EU2=(1−π1)(1−θ(0,0)).Given this, we calculate the following expected payoffs.

Under A2.1:

• If1

ω1 þω2b cmax b

1ω1

, then

EU1 ¼0 when cmax ≤

πω1

case a½ �

ω1cmax−π whenπω1

b cmax ≤1ω1

case b½ �

8>><>>:

EU2 ¼1−π when cmax ≤

πω1

½case a�

1−ω1cmax whenπω1

b cmax ≤1ω1

½case b�

8>><>>:

• If cmax ≤1

ω1 þω2,

EU1 ¼

1−cmaxω1 when π≤ ω1

ω1 þω2and

πω1

≤cmax ½case c�

1−cmaxπ ω1 þω2ð Þ when π≥ ω1

ω1 þω2and

1−πω2

≤cmax ½case d�

1−π 1−π þ cmaxω1ð Þ when cmaxbminπω1

;1−πω2

� �½case e�

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

EU2 ¼

cmax ω1−π ω1 þω2ð Þð Þ when π≤ ω1

ω1 þω2and

πω1

≤ cmax ½case c�

0 when π≥ ω1

ω1 þω2and

1−πω2

≤ cmax ½case d�

π 1−π−cmaxω2ð Þ when cmaxbminπω1

;1−πω2

� �½case e�

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

Under A2.2:

• If 1ω1þω2

b cmaxb1ω1, then

EU1 ¼cmaxω1 when cmax ≤

1−πω1

½case f �

1−π when1−πω1

b cmax ≤1ω1

½case g�

8>><>>:

EU2 ¼1−π−cmaxω1 when cmax ≤

1−πω1

½case f �

0 when1−πω1

b cmax ≤1ω1

½case g�

8>><>>:

• If cmax ≤1

ω1 þω2,


EU1 ¼

cmax ω2−π ω1 þω2ð Þð Þ when π≤ ω2

ω1 þω2and

πω2

≤ cmax ½case h�

0 when π≥ ω2

ω1 þω2and

1−πω1

≤ cmax ½case i�

π 1−π−cmaxω1ð Þ when cmaxbminπω2

;1−πω1

� �½case j�

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

EU2 ¼

1−cmaxω2 when π≤ ω2

ω1 þω2and

πω2

≤cmax ½case h�

1−cmaxπ ω1 þω2ð Þ when π≥ ω2

ω1 þω2and

1−πω1

≤cmax ½case i�

1−π 1−π þ cmaxω2ð Þ when cmaxbminπω2

;1−πω1

� �½case j�

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

We now compare these payoffs with the IG payoffs under no con-tribution limit, EU1=0 and EU2=1−π. A comparison of the payoffsproduces the following results:

Lemma 7.4.

1. In [case a] and [case i], IG 1 is just as well off under the contributionlimit as under no limit. In all other cases, IG 1 is strictly better offunder the limit.

2. In [case a], IG 2 is just as well off under the contribution limit as underno limit. In [case i] and [case j], IG 2 is strictly better off under thelimit. In all other cases, IG 2 is strictly worse off under the limit.

Proof. Follows immediately from a comparison of EU1 and EU2 underno limit to their values in the 10 cases with a limit. ■

We also compare the IG's payoffs under the two assumptionsabout politician behavior in the event that she is indifferent: A2.1

and A2.2. Consider first the situation when1

ω1 þω2bcmaxb

1ω1

. Here,

given the range of cmax, one can see that EU1 is strictly lower andEU2 is strictly higher in both [case a] and [case b] compared to [casef] and to [case g]. In this range of cmax, IG 1 strictly prefers that the pol-itician act according to A2.2 and IG 2 strictly prefers A2.1. When

cmaxb1

ω1 þω2, a comparison of payoffs in cases c, d and e with the

payoffs in cases h, i and j yields the opposite result: EU1 is strictlyhigher under A2.1 and EU2 is strictly higher under A2.2.

a.3. Detailed analysis with endogenous lobby formation

The following analysis is provided in place of individual proofs toPropositions 5.4 and 5.5.

The body of the paper describes the equilibrium of the game whenthere is no contribution limit: neither IG forms a lobbying presence,and the politician either implements policy 1 (under A2.1) or policy2 (under A2.2).

Here we determine the equilibrium with endogenous lobby forma-tion under contribution limit cmax. In the analysis below, it is helpfulto remember that μ is a sunk cost by the time the politician announcesaccess fees, and thus plays only a limited role in the following analysis.The analysis limits attention only to the equilibrium inwhich the politi-cian updates negatively her beliefs about πi in the event that i forms alobbying presence but does not pay an access feepi≤pi; this is a reason-able assumption concerning beliefs off of the path of play, and guaran-tees that the politician does not choose a policy supported by an IGthat is the only group to form and then does not reveal favorableevidence. Additionally, the analysis assumes that the politician onlycommits to cmax before the IGs decide whether to form, and that sheannounces θ and p after observing which IGs form; alternative assump-tions about the order of the politician decisions will not change the re-sult that a contribution limit can encourage lobby formation andimprove politician payoffs.

Subgame equilibrium taking as given the IG formation decision.There are four possible cases to consider:

Case I: Neither group forms a lobbying presence. Payoffs are thesame as in the game with no contribution limit.

EW ¼ πEU1 ¼ θ ∅;∅ð ÞEU2 ¼ 1−θ ∅;∅ð Þ:

Under A2.1, θ(∅,∅)=1 and thus EU1=1 and EU2=0.Under A2.2, EU1=0 and EU2=1.

Case II: Only IG 1 forms a lobbying presence. In this case, the politi-cian chooses a price and access strategy to maximize the ac-cess fee p1 while satisfying IG 1's individual rationalityconstraint; thus p1 ¼ min cmax;p1f g. When IG 1 has favor-able evidence, it is indifferent between paying p1 and notpaying the access fee. That is p1 solves 1−p1ω1 ¼ 0, wherethe left hand side denotes EU1 conditional on paying thefee and having favorable evidence, and the right hand sidedenotes EU1 conditional on not paying the access fee, inwhich case the politician believes that policy 2 is more likely

good. Therefore, p1 ¼ 1ω1

.

Under any binding contribution limit (i.e., cmaxb1ω1), p1≥cmax.

Given the IG 1 alone forms, the politician maximizes herpayoff by setting p1=cmax.

EW ¼ π þ 1−πð Þπ þ ϕπcmaxEU1 ¼ π 1−ω1cmaxð Þ−μEU2 ¼ 1−π:

Case III: Only IG 2 forms a lobbying presence. This case is similar tothe previous one. Here, the politician extracts the maximumpayment from IG 2, setting p2 ¼ min cmax;p2f g where

p2 ¼ 1ω2

. For any binding contribution limit (i.e., cmaxb1ω2

),

the politician maximizes payments by setting p2=cmax.

EW ¼ π þ 1−πð Þπ þ ϕπcmaxEU1 ¼ 1−πEU2 ¼ π 1−ω2cmaxð Þ−μ:

Case IV: Both IGs form lobbying presences. In this case, the equilibri-um of the game and the payoffs are identical to those inAppendix Section A.2 for the case of the limit, except thatboth IGs also experience the formation costs of μ.

From this point forward, we present the analysis under A2.1. A sym-metric analysis may be conducted to derive the results under A2.2.

a.3.1. IG formation decision

Consider the possibility that neither group forms a lobbying pres-ence in equilibrium. For this to be the case, IG 1 must prefer not form-ing and earning 1 to forming and earning π(1−ωcmax)=μ, which willalways be true. IG 2 must prefer not forming and earning 0 to formingand earning π(1−ω2cmax)−μ, which will be the case only if

μ≥π 1−ω2cmaxð Þ⇔cmax≥π−μπω2

:

When this condition is satisfied, there exists an equilibrium in

which neither IG forms. It will certainly be satisfied if cmax≥1ω2

. For

lower cmax, it is satisfied if and only if μ is sufficiently large.Next, consider the possibility that only one IG forms. If only IG 1

forms, it must prefer and earn expected payoff of π(1−ω1cmax)−μ


than to not form. This will never be the case under A2.1 since thepolitician chooses policy in favor of IG 1 when neither group forms.Instead, consider the possibility that only IG 2 forms a lobbying pres-ence in equilibrium. For this to be the case, IG 2 must prefer formingand earning π(1−ω2cmax)−μ to not forming and earning 0. This willbe the case as long as neither cmax nor μ are too large. That is if

μ≤π 1−ω2cmaxð Þ⇔cmax≤π−μπω2

:

Additionally, IG 1 must prefer not forming and earning 1−π toforming and earning the payoffs derived in previous section. Of thefive parameter cases associated with A2.1 ([case a] through [casee]), IG 1 only prefers to deviate in [case d]and [case e]; that is when

both cmax≤1

ω1 þω2and cmax≤

πω1

. For all other parameter cases, IG 1

has no incentive to deviate. Therefore, such an equilibrium existswhen

minπω1

;1

ω1 þω2

� �≤ cmax ≤

π−μπω2

; ð3Þ

which is only feasible when

μ b π: ð4Þ

Finally, we consider the possibility that both IGs form in equilibri-um under A2.1. It is possible to show that there does not exist a purestrategy equilibrium in which both IGs form. Rather the equilibrium isin mixed strategies. Such an equilibrium only exists when cmax is suf-ficiently low that there does not exist the pure strategy equilibria inwhich neither or one IG forms a lobbying presence. That is, such anequilibrium exists when

cmax≤minπω1

;1


� �:

Note that this condition is feasible only when μbπ(1−cmaxω2)

and cmaxb1ω2

.

Here, we describe the equilibrium mixed strategies, where ρidenotes the probability IG i forms:

28 Given cmax ¼ π−μπω2

, there exist two subgame equilibria in which either (i) neither group

forms, or (ii) only IG 2 forms. However, only the second of these subgame equilibria has the

potential to be on the equilibrium path of play for the overall game. To see this, suppose in-

stead that neither group formswhen cmax ¼ π−μπω2

; then the politicianwill never choose a con-

tribution limit equal to this value, preferring instead a limit that ismarginally below this value

and which guarantees that at least one IG forms in equilibrium.

• When1−πω2

≤π≤ 1ω1 þω2

(e.g., [case d] from the previous section),

ρ1 ¼ π−μ−cmaxω2π1−cmaxω1 1−πð Þ ; and ρ2 ¼ 1−π þ μ þ cmaxω1π

1−cmaxω2π:

• When π≤minπω1

;1−πω2

� �(e.g., [case e] from the previous section),

ρ1 ¼ π−μ−cmaxω2π1−π þ π2 ; and ρ2 ¼ 1−π þ μ þ cmaxω1π

1−π þ π2 :

a.3.2. Politician choice of cmax

If neither group forms in equilibrium, then the politician earnsexpected payoff EW=π. If only IG 2 forms in equilibrium, then thepolitician earns expected payoff

EW ¼ 2π−π2 þ ϕπcmax:

This achieves its maximum value when the politician sets thehighest cmax such that such an equilibrium exists; that is when

cmax ¼ π−μπω2

. If both IGs form in equilibrium, then the politician

earns expected payoff

EW ¼ ρ1ρ2 2π−π2 þ 2ϕπcmax

� þ ρ1 1−ρ2ð Þ þ ρ2 1−ρ1ð Þð Þ 2π−π2 þ ϕπcmax

� þ 1−ρ1ð Þ 1−ρ2ð Þπ: ð5Þ

Notice that EW is strictly higher when either one group forms orboth groups formwith positive probability compared to when neithergroup forms in equilibrium. If the politician is able to choose a cmax

that results in an equilibrium in which at least one group forms, shewill be better off than if she imposes no contribution limit. Any

cmax≤π−μπω2

achieves this outcome, improving the politician's expected

payoff.28 When μ≤π this is possible; otherwise, a contribution limitcannot improve politician payoffs.

Next, assuming μ≤π, we determine the optimal value of

cmax≤π−μπω2

for the politician. Conditional on only IG 2 forming a lob-

bying presence, the politician's expected payoff is strictly increasing

in cmax. Therefore, if cmax ¼ π−μπω2

results in only IG 2 forming in equi-

librium, then the politician prefers this maximum value of cmax to anylower value that also results in only IG 2 forming. By comparing EWfor the case when only IG 2 forms to maximize payoffs in the casewhen both IGs form, one can also show that the politician prefers

such a cmax ¼ π−μπω2

to any contribution limit that is low enough to en-

tice both IGs to form with positive probability. Such an equilibriumexists only if

minπω1

;1

ω1 þω2

� �≤π−μ

πω2:

If this condition does not hold, the politician chooses cmax to max-

imize Eq. (5) conditional on cmax≤π−μπω2

. In doing so, remember that

the values of ρ1 and ρ2 depend on cmax. For our results, it is sufficientto recognize for which parameter values the politician sets a contri-bution limit under which only IG 2 forms and under which both IGsform with positive probability.

A symmetric analysis may be conducted to find the range of valuesunder A2.2.

References

Ansolabehere, Stephen, Snyder Jr., James M., Micky, Tripathi, 2002. Are PAC contributionsand lobbying linked?New evidence from the 1995 LobbyDisclosure Act. Business andPolitics 4 (2), 131–155.

Austen-Smith, David, 1995. Campaign contributions and access. American Political Sci-ence Review 89 (3), 566–581.

Austen-Smith, David, 1998. Allocating access for information and contributions. Journal ofLaw, Economics, and Organization 14 (2), 277–303 Fall.

Baumgartner, R. Frank, Berry, Jeffrey M., Hojnacki, Marie, Kimball, David C., Leech, Beth L.,2009. Lobbying and Policy Change:WhoWins,Who Loses, andWhy. TheUniversity ofChicago Press, Chicago.

Baye, Michael R., Kovenock, Dan, de Vries, Casper G., 1993. Rigging the lobbying process:an application of the all-pay auction. American Economic Review 83 (1), 289–294.

Bennedsen, Morten, Feldmann, Sven E., 2002. Lobbying legislatures. Journal of PoliticalEconomy 110 (4), 919–946.

Bennedsen, Morten, Feldmann, Sven E., 2006. Informational lobbying and political con-tributions. Journal of Public Economics 90 (4/5), 631–656.


Bernheim, B. Douglas, Whinston, Michael D., 1986. Menu auctions, resource allocation,and economic influence. Quarterly Journal of Economics 101 (1), 1–32 February.

Bull, Jesse, Watson, Joel, 2004. Evidence disclosure and verifiability. Journal of EconomicTheory 118, 1–31.

Bull, Jesse,Watson, Joel, 2007. Hard evidence andmechanismdesign. Games and Econom-ic Behavior 58, 75–93.

Che, Yeon-Koo, Gale, Ian L., 1998. Caps on political lobbying. American Economic Review88 (3), 643–651.

Clawson, Dan, Neustadtl, Alan, Scott, Denise, Talks, Money, 1992. Corporate PACs andPolitical Influence. Basic Books, New York.

Coate, Stephen, 2004a. Pareto improving campaign finance policy. American EconomicReview 94 (3), 628–655.

Coate, Stephen, 2004b. Political competitionwith campaign contributions and informativeadvertising. Journal of the European Economic Association 2 (5), 772–804.

Cooter, Robert D., Rubinfeld, Daniel L., 1994. An economic model of legal discovery. TheJournal of Legal Studies 23, 435–463.

Cotton, Christopher, 2009. Should we tax or cap political contributions? A lobbyingmodel with policy favors and access. Journal of Public Economics 93, 831–842.

Demougin, Dominique, Fluet, Claude, 2008. Rules of proof, courts, and incentives. TheRAND Journal of Economics 39 (1), 20–40.

Drazen, Allan, Limao, Nuno, Stratmann, Thomas, 2007. Political contribution caps andlobby formation: theory and evidence. Journal of Public Economics 91, 723–754.

Gavious, Arieh, Moldovanu, Benny, Sela, Aner, 2002. Bid costs and endogenous bidcaps. The RAND Journal of Economics 33 (4), 709–722.

Grossman, Gene M., Helpman, Elhanan, 1994. Protection for sale. American EconomicReview 84 (4), 833–850.

Grossman, Gene M., Helpman, Elhanan, 1996. Electoral competition and special interestpolitics. Review of Economic Studies 63 (2), 265–286.

Grossman, Gene M., Helpman, Elhanan, 2002. Special Interest Politics. The MIT Press,Cambridge, MA.

Hall, Richard L., Deardorff, Alan V., 2006. Lobbying and legislative subsidy. AmericanPolitical Science Review 100 (1), 69–84.

Hall, Richard L., Wayman, FrankW., 1990. Buying time: moneyed interests and the mobi-lization of bias in congressional committees. American Political Science Review84 (3),797–820.

Hay, Bruce L., Spier, Kathryn E., 1997. Burdens of proof in civil litigation: an economicperspective. The Journal of Legal Studies 26, 413–431.

Herndon, James F., 1982. Access, record, and competition as influences on interest groupcontributions to congressional campaigns. Journal of Politics 90 (2), 280–294.

Langbein, Laura I., 1986. Money and access: some empirical evidence. Journal of Politics48 (4), 1052–1062.

Lohmann, Susanne, 1995. Information, access, and contributions: a signalling model oflobbying. Public Choice 85 (3/4), 267–284.

Makinson, Larry, 2003. Speaking Freely:Washington Insiders Talk aboutMoney in Politics.Center for Responsive Politics, Washington D.C.

Milgrom, Paul, 1981. Good news and bad news: representation theorems and applica-tions. Bell Journal of Economics 12, 380–391.

Milgrom, Paul, Roberts, John, 1986. Relying on the information of interested parties.The RAND Journal of Economics 17 (1), 18–32.

Milyo, Jeffrey, Primo, David, Groseclose, Timothy, 2000. Corporate PAC campaign con-tributions in perspective. Business and Politics 2 (1), 75–88.

Posner, Richard A., 1999. An economic approach to the law of evidence. Stanford LawReview 51, 1477–1546.

Prat, Andrea, 2002a. Campaign advertising and voter welfare. Review of EconomicStudies 69 (4), 997–1017.

Prat, Andre, 2002b. Campaign spending with office-seeking politicians, rational voters,and multiple lobbies. Journal of Economic Theory 103, 162–189.

Riezman, Raymond, Wilson, John Douglas, 1997. Political reform and trade policy. Jour-nal of International Economics 42, 67–90.

Schram, Martin, 1995. Speaking Freely: Former Members of Congress Talk aboutMoney in Politics. Center for Responsive Politics, Washington D.C.

Shin, Hyun Song, 1994. The burden of proof in a game of persuasion. Journal of Eco-nomic Theory 64, 253–264.

Sobel, Joel, 1985. Disclosure of evidence and resolution of disputes: who should bearthe burden of proof? In: Roth, Alvin E. (Ed.), Game Theoretic Models of Bargaining.Cambridge University Press, Cambridge, UK, pp. 341–361. Chapter 16.

Stratmann, Thomas, 1998. The market for congressional votes: is timing of contributionseverything. Journal of Law and Economics 41, 85–114.

Stratmann, Thomas, 2002. Can special interests buy congressional votes? Evidencefrom financial services legislation. Journal of Law and Economics 45, 345–374.

Stratmann, Thomas, 2005. Some talk: money in politics. A (partial) review of the litera-ture. Public Choice 124, 135–156.

Tullock, Gordon, 1980. Efficient rent seeking. In: Buchanan, James M., Tollison, RobertD., Gordon, Tullock (Eds.), Towards a Theory of the Rent Seeking Society. TexasA&M University Press, College Station.

Wright, John, 1990. Contributions, lobbying, and committee voting in the U.S. House ofRepresentatives. American Political Science Review 84, 417–438.

pay-to-play politics: informational lobbying and contribution limits when money buys access

Documents