Buying voters with uncertain instrumental preferences

Louis-Sidois, Charles; Musolff, Leon Andreas Article Buying voters with uncertain instrumental preferences Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Louis-Sidois, Charles; Musolff, Leon Andreas (2024) : Buying voters with uncertain instrumental preferences, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 3, pp. 1305-1349, https://doi.org/10.3982/TE4658 This Version is available at: https://hdl.handle.net/10419/320267 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 19 (2024), 1305–1349 1555-7561/20241305 Buying voters with uncertain instrumental preferences Charles Louis-Sidois Department of Economics, Vienna University of Economics and Business Leon Musolff The Wharton School, University of Pennsylvania We analyze a vote-buying model where the members of a committee vote on a proposal important to a vote buyer. Each member incurs a privately-drawn disutility if the proposal passes. We characterize the cheapest combination of bribes that guarantees the proposal passes in all equilibria. When members vote simultaneously, the number of bribes is at least 50% larger than the number of votes required to pass the proposal (vote threshold). The number of bribes increases with the dispersion of the disutility distribution and all members are bribed with sufficient dispersion. A proportional increase in the number of members and the vote threshold leads to a less-than-proportional increase in capture cost, and the cost may increase with the vote threshold. With sequential voting and disutility distribution U[0, 1], all members are bribed and bribes are equal. Finally, sequential voting increases capture cost in small committees and decreases it in large committees. Keywords. Vote buying, legislatures, political economy. JEL classification. D71, D72. 1. Introduction Governments often introduce bills that go against the interests of parliament members, such as a law limiting dual mandates.1To overcome members’ opposition, the government can offer rewards to those who support the bill, e.g., investments in legislative districts. Charles Louis-Sidois: [email protected] Leon Musolff: [email protected] We thank three anonymous referees for very detailed comments that greatly improved the paper. We also thank Alessandro Riboni, Ana Luiza Dutra, Andrea Mattozzi, Brendan Lucier, Christoph Rothe, Elia Sartori, Emeric Henry, Ernesto Dal Bo, Evgenii Safonov, Faruk Gul, Françoise Forges, Franz Ostrizek, Giovanni Andreottola, Hans Peter Grüner, Ian Ball, Jeanne Hagenbach, Joao Thereze, Jonas Müller-Gastell, Leaat Yariv, Margaret Meyer, Markus Mobius, Matias Iaryczower, Nicole Immorlica, Nicole Kliewer, Pietro Ortoleva, Rachel Kranton, Roland Benabou, Sergei Guriev, Stephen Morris, Thomas Romer, Thomas Troeger, Tyler Maxey, Wolfgang Pesendorfer, as well as discussants at seminars at Princeton, Science Po, Mannheim, and Yale for many insightful comments. 1A dual mandate, or double jobbing, is the practice in which elected officials serve in more than one public position simultaneously. For example, more than 80% of parliament members in France held another office before a law prohibiting dual mandates was passed in 2013. ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4658 1306 Louis-Sidois and Musolff Theoretical Economics 19 (2024) We develop a vote-buying model to analyze these situations. A committee votes on a proposal that favors the interests of a vote buyer. However, committee members prefer the proposal not to pass. To gain support, the vote buyer offers bribes to members in exchange for their votes. Our key innovation is introducing uncertainty to members’ preferences. For instance, in the case of a dual mandate prohibition, this uncertainty reflects each member’s uncertain future (re)election prospects. Our first example illustrates how a vote buyer exploits the implications of this uncertainty for pivotal probabilities. Example 1. A three-member committee votes on a proposal. The proposal passes if at least two members vote for it. In this example, members vote simultaneously. Members dislike the proposal. Crucially, each member draws his disutility viprivately at the beginning of the game: vi i.i.d. ∼U[0, 1]. A vote buyer (feminine pronoun) is interested in the proposal passing. Before the vote, she publicly commits to paying a bribe to some members if they individually vote for the proposal. We assume the value of the bribe is b≥0 and that it is the same for all bribed members. The vote buyer knows the distribution of members’ disutilities but does not observe their realizations. The proposal is important to her so she wants to guarantee that it passes with certainty in all equilibria of the voting subgame. Subject to this condition, she minimizes the cost of bribes. We compare two strategies for the vote buyer. First, suppose she bribes two members. We assume the unbribed member plays his weakly dominant strategy and votes against the proposal. The proposal passes with certainty if the two bribed members vote for regardless of their disutility. This strategy profile is clearly the unique equilibrium if b>1 because voting for is a dominant strategy for all disutilities. However, if b<1, bribed members with disutility vi>bwould deviate and the strategy profile is not an equilibrium. Moreover, as will be established, there exists an equilibrium where the proposal is rejected with a positive probability. Thus, the cheapest bribe such that the proposal passes with certainty in any equilibrium is b=1tothetwomembers,which yields a cost of 2. Instead, suppose the vote buyer bribes all three members. We will show that if b> 8 27 , there is no equilibrium where the proposal is rejected with positive probability; that is, buying a third member is cheaper for the vote buyer. For each member, voting for the proposal guarantees bribe payment. However, if the member is pivotal (i.e., if exactly one other member votes for the proposal), it also leads to the passing of the proposal. Denoting the pivotal probability by π,memberivotes for if b>v i×π. The equilibrium of the voting subgame takes a cutoff form: a member votes for the proposal if his disutility is below a threshold. For now, focus on symmetric strategies and call the common cutoff v.Thenπ(v)=2v(1−v)and an equilibrium cutoff v∈(0, 1) satisfies b=vπ(v). In Figure 1, we plot the right-hand side of this equation. For small bribes like b1,two equilibria exist with cutoffs v1and v2. Moreover, there is a third equilibrium where all members accept the bribe: committee members are not pivotal and have no incentive Theoretical Economics 19 (2024) Buying voters 1307 Figure 1. The structure of equilibrium in the voting subgame. Notes:Asthecutoffusedby other members changes, so does the value of vπ(v)(solid blue). The value of the maximum is 8 27 (reached for v=2 3). For a given bribe b1below 8 27 , there are three equilibria of the voting subgame: one with cutoff v1, one with cutoff v2,andoneinwhichallmembersvoteforthe proposal. For bribes above 8 27 , only the latter exists. to deviate. Throughout the paper, we assume committee members play the equilibrium in which the proposal is rejected with the highest probability. For instance, faced with b1, they would play v1as this lower cutoff implies the lowest probability of passing. When bis larger than the maximum of vπ(v), the third equilibrium, where the proposal passes with certainty, is the only equilibrium of the voting subgame. Here, the maximum is 8 27 . Thus, it is sufficient to pay slightly more than 8 9,whichisthecostof bribing all three members, to guarantee there is no equilibrium where the proposal is rejected with positive probability. Intuitively, bribing more members reduces members’ pivotal probabilities, forcing them to accept smaller bribes. ♦ We characterize the cheapest combination of bribes required for passing the proposal with certainty in all equilibria. We consider various factors such as the disutility distribution, the number of committee members, and the vote threshold. Specifically, we examine simultaneous voting in Section 2.First,inSection2.1, we assume symmetric strategies and equal bribes for all members bribed. Our main finding is that the cheapest capture always involves a number of bribes at least 50% higher than the vote threshold. Furthermore, the number of bribes increases with dispersion, and all members are bribed when there is enough dispersion. As for the capture cost, increasing the vote threshold and the number of members proportionally results in a less-thanproportional increase in cost (because members are less likely to be pivotal in a large committee), while increasing only the vote threshold increases the capture cost if more than half of the members must vote for to pass the proposal. In Section 2.2, members may play asymmetric strategies. Depending on the distribution, there may exist asymmetric equilibria where the proposal can be rejected when members receive the bribes of Section 2.1. This is the case when the disutility dispersion is small, but not when it is large. Section 2.3 considers unequal bribes. With large dispersion, we show with an example that unequal bribes can yield a lower capture cost. However, we establish that if vi i.i.d. ∼U[0, 1]as in Example 1, the capture cost is minimized by equal bribes. 1308 Louis-Sidois and Musolff Theoretical Economics 19 (2024) We study sequential voting with vi i.i.d. ∼U[0, 1]in Section 3. The vote buyer also exploits pivotal considerations and the cheapest capture requires offering the same bribe to all members. Finally, Section 4shows that compared to sequential voting, simultaneous voting yields a higher capture cost if the committee is large, while the opposite is true for small committees or very high or very low vote thresholds. The model has a variety of applications. Our setup primarily applies to decisionmaking in organizations. For example, a CEO may want to persuade board members to make a decision favoring his interests. If board members expect the decision to be approved regardless of their vote, the CEO can obtain their support in exchange for small favors. Alternatively, consider the application of Genicot and Ray (2006)inwhich a raider takes over a company. In such a case, the post-takeover value of nontendered shares could be diluted, harming all shareholders.2Nevertheless, if shareholders expect the takeover to happen regardless of their selling decision, shares could be bought at little cost. Finally, our model has implications for lobbying and vote-buying in committees of experts (like FDA committees) or juries. We contribute to the vote-buying literature by combining a single vote buyer with committee members who care about the vote’s outcome but are uncertain about each other’s preferences. The combination is novel, though literature on each ingredient exists. Several papers study vote-buying when members have publicly known preferences over outcomes. Dal Bo (2007) shows that a vote buyer bribes a committee at no cost by conditioning the bribes on the complete voting profile. She offers to pay an infinitesimal amount if members are not pivotal and a large bribe if votes are decisive. By contrast, we exclude any contracts based on the joint realization of votes. Moreover, the models of Rasmusen and Ramseyer (1994)andDahm and Glazer (2015) feature some equilibria unfavorable to committee members in which a supermajority accepts small bribes because no member is pivotal. Instead, we allow members to coordinate on their preferred equilibrium. Cheap capture also occurs in Genicot and Ray (2006)andChen and Zápal (2020) where the vote buyer approaches members sequentially and exploits the timing of offers. On the contrary, the vote buyer makes all offers at the same time in our model, both in simultaneous and sequential voting. We focus on the probability of a vote being decisive and do not consider information aggregation (Feddersen and Pesendorfer 1996,1997,1998). Feddersen and Pesendorfer (1998) highlight that unanimity, which in our setup maximizes capture cost, and makes information harder to aggregate. Henry (2008)andFelgenhauer and Grüner (2008)combine vote-buying and information aggregation. In Henry (2008), each committee member receives a signal about the quality of a common value proposal. Bribes determine the number of members who vote informatively, shaping members’ inferences conditional on being pivotal. Similarly, in Ekmekci and Lauermann (2019)anelectionorganizer chooses turnout to manipulate the information aggregated. These papers consider a common value proposal while we focus on private values. 2For instance, this happens in Grossman and Hart (1980) where the raider uses the dilution to force atomistic shareholders to sell, but Bagnoli and Lipman (1988) show that dilution does not necessarily happen with a finite number of shareholders. Theoretical Economics 19 (2024) Buying voters 1309 The mechanism exploited by the vote buyer in our model relies on pivotality and is not present in the literature on vote-buying with expressive preferences, e.g., in Zápal (2017), members’ responses to a bribe are uncertain, but pivotal considerations are absent because members do not take into account the effect of their vote on the outcome. Groseclose and Snyder (1996), Banks (2000), Dekel, Jackson, and Wolinsky (2008), Morgan and Várdy (2011), and Iaryczower and Oliveros (2017) also assume expressive preferences and introduce a second vote buyer. They find that the first mover bribes a large coalition to increase the cost for the follower. Our paper proposes a new explanation for the high empirical frequency of supermajorities. While early theories of coalition formation predicted minimal winning coalitions (Axelrod (1970)), some later papers predict supermajorities (Koehler (1975), Weingast (1979), Shepsle and Weingast (1981), Baron and Diermeier (2001)). The closest to us is Carrubba and Volden (2000), in which a larger-than-necessary coalition ensures no member can prevent the costly passing of other members’ bills. Supermajorities are also found in the literature on legislative bargaining (Volden and Wiseman (2007), Tsai and Yang (2010), Dahm, Dur, and Glazer (2014)); for an overview, see Eraslan and Evdokimov (2019). For instance, Norman (2002) characterizes the nonsymmetric equilibria of the classical model of Baron and Ferejohn (1989) and shows that some proposals can be unanimously approved. Chen and Eraslan (2013,2014) look at the other side of the problem and study a vote-selling model where members with uncertain preferences send messages to the vote buyer to influence the proposal. Finally, we are also related to the larger literature on unique implementation with moral hazard. In Winter (2004)andWinter (2006), agents separately perform individual tasks for a project that succeeds if all agents succeed. In case of success, the principal rewards agents who support the project. Contributions are simultaneous in Winter (2004) and sequential in Winter (2006). Our paper differs as members’ preferences are uncertain. Winter’s principal aims to prevent asymmetric equilibria where the project fails and Winter establishes that discriminatory rewards can be optimal. With sufficient uncertainty, we find that asymmetric equilibria cannot be sustained, and equal bribes may be preferred. 2. Simultaneous voting We consider a committee of nmembers voting simultaneously on a proposal. The vote threshold mis the minimum number of votes for required to pass the proposal. We exclude m=n(unanimity required to pass the proposal) and m=1 (unanimity required to reject it).3At the beginning of the game, committee members draw their disutilities from the passing of the proposal. These disutilities are drawn privately and independently from a common distribution: vi i.i.d. ∼F(·),whereviis the disutility of member i. F(·)has support [vmin,vmax]with vmin ≥0andvmax finite. We assume that the disutility distribution F(·)is continuously differentiable on (vmin,vmax), and has an increasing 3We discuss unanimous vote thresholds at the end of Section 2.1; they require technical modifications of the results but do not affect our conclusions. 1310 Louis-Sidois and Musolff Theoretical Economics 19 (2024) generalized hazard rate: ∂ ∂vvF(v) 1−F(v)≥0. Other models (e.g., Lariviere (2006)) use this assumption, which is satisfied by all Uniform and Beta distributions. Before the voting subgame, a vote buyer who favors the proposal publicly offers bribes (b1,,bn),wherebi≥0ismemberi’s payment if he votes for the proposal. We assume the proposal is important to the vote buyer, so she minimizes the capture cost, i.e., the amount spent on bribes, subject to the proposal passing with certainty. We focus on Bayesian Nash equilibria for the voting subgame. When multiple equilibria exist, we assume committee members play one of the equilibria where the proposal passes with the smallest probability. This assumption is in the spirit of Winter (2004)andGenicot and Ray (2006). First, it rules out equilibria where the proposal passes with arbitrarily small bribes because all bribed members accept and are not pivotal. Second, it follows naturally if a vote buyer to whom the proposal is important is uncertain about which equilibrium will be played. Third, it selects an equilibrium preferred by committee members. The game’s timing is as follows. First, committee members privately observe their disutility. Then the vote buyer offers bribes (b1,,bn). Members observe the bribes and simultaneously choose whether to vote for or against the proposal. Finally, the proposal passes if at least mmembers vote for it. 2.1 Symmetric voting strategies and equal bribes This subsection focuses on equal bribes: the vote buyer bribes kmembers who all receive the same bribe b. Thus, the combination of bribes is characterized by (b,k).For committee member i,astrategyσi:vi→[0, 1]is a mapping from disutility viinto a probability of voting for the proposal. We only consider members to whom the vote buyer offers a bribe; unbribed members are assumed to use their weakly dominant strategy and vote against the proposal. We focus on symmetric equilibria, i.e., equilibria in which bribed members play the same strategy. We first solve the voting subgame. Given a combination of bribes (b,k), if a member is not pivotal, the payoff difference between voting for and against is the bribe’s value. If he is pivotal, a vote for the proposal makes it pass and he incurs his disutility. We denote the pivotal probability of committee member iby πi. He accepts the bribe and votes for the proposal if b>v iπi,whereviπiis the expected cost of voting for the proposal. Moreover, he votes against if b<v iπiand can vote for with any probability if b=viπi. Thus, equilibrium strategies take a cutoff form. Since we focus on symmetric equilibria, all members vote for the proposal if their disutility is smaller than some cutoff v determined in equilibrium. Our first lemma characterizes the equilibrium of the voting subgame where the proposal passes with the smallest probability. When fewer members are bribed than the vote threshold (k<m), there exists an equilibrium of the voting subgame where the Theoretical Economics 19 (2024) Buying voters 1311 proposal is always rejected, and hence no bribe can guarantee that the proposal passes with certainty in any equilibrium. As a result, we focus on k∈{m,,n}. Lemma 1. Suppose the vote buyer offers a bribe b>0to kmembers with m≤k≤n.In the symmetric equilibrium of the voting subgame in which the proposal passes with the smallest probability: (a) If b≤maxv∈[vmin,vmax]vπk(v), bribed members vote for the proposal if their disutility is smaller than a cutoff vthat satisfies v=min{v∈[vmin,vmax]:b=vπk(v)}where πk(v)=k−1 m−1F(v)m−11−F(v)k−m. Moreover, they vote against if their disutility is larger than vand a member with vi=vcan vote for with any probability. (b) If b>maxv∈[vmin,vmax]vπk(v), all bribed members vote for the proposal regardless of their disutility: v>v max. (Proof in Appendix A.2.) First, consider the case where mmembers are bribed. vπm(v)is increasing in vand πm(v)→1asv→vmax.Forb≤vmax, Lemma 1(a) characterizes the unique equilibrium cutoff and the proposal is rejected with positive probability. For b>v max, the strategy profile described in Lemma 1(b) is the unique equilibrium, and the proposal passes with certainty. Now consider k>m. As established in Lemma A.2.1 in Appendix A.2, increasing generalized hazard rates imply that vπk(v)is single-peaked in vfor v∈[vmin,vmax]. By the intermediate value theorem, the equation vπk(v)=badmits two solutions if b<maxv∈[vmin,vmax]vπk(v), one if b=maxv∈[vmin,vmax]vπk(v), and none otherwise. Thus, equilibrium cutoffs are illustrated by Figure 1. The equilibrium where the proposal passes with the smallest probability is associated with the smallest cutoff, and this cutoff is characterized by Lemma 1(a). If we let v∗ k:=argmaxv∈[vmin,vmax]vπk(v), the smallest bribe such that the proposal passes with certainty in any symmetric equilibrium is b∗ k=v∗ kπk(v∗ k).4For Example 1, and hence in Figure 1,b∗ 3=8 27 . We now turn to the vote buyer’s problem. As just established, if the vote buyer offers kbribes, she needs to offer b∗ kto make the proposal pass with certainty. Hence, her cost c(k)is determined by the equilibrium where the cutoff is v∗ k, c(k)=k×max v∈[vmin,vmax]vπk(v)=k×v∗ kπkv∗ k=k×b∗ k. We want to determine a cost-minimizing number of bribes argmink∈{m,,n}c(k).5Intuitively, while bribing additional members requires paying more bribes, it also makes it 4More precisely, b∗ kis the “smallest number above v∗ kπk(v∗ k),” which is not defined because bribes are on a continuum, but makes sense as the limit of a grid. 5As khas to be an integer, there are distributions for which arg mink∈{m,,n}c(k)is not unique. In particular, c(k)can be minimized for two consecutive integers. 1312 Louis-Sidois and Musolff Theoretical Economics 19 (2024) harder for committee members to be pivotal with a high probability. Hence, it decreases b∗ k. Which effect dominates depends on the number of bribes, on the vote threshold, and on the disutility distribution. Our main result characterizes arg mink∈{m,,n}c(k). Proposition 1. (a) For any disutility distribution, any cost-minimizing number of bribes is at least min{3 2m−1, n}; (b) For any number of bribes k∈Nsuch that min{3 2m+1, n}≤k≤n,thereexistsa disutility distribution such that kis a cost-minimizing number of bribes. (Proof in Appendix A.2.) Proposition 1implies that the vote buyer always wants to offer a number of bribes substantially larger than the vote threshold. If she could offer any number of bribes, she would choose at least k=3 2m−1, which for mlarge represents a number of bribes 50% larger than the vote threshold. However, the number of bribes cannot exceed the number of members. As a result, when there are fewer members than 3 2m−1, this constraint binds and the vote buyer bribes all members. With more than 3 2mmembers, it can still be the case that all members are bribed, but it depends on the disutility distribution. This is true even when the number of members is arbitrarily large: for some distributions, the vote buyer’s cost is always decreasing in the number of bribes and she offers as many bribes as possible. We now show Proposition 1in three steps. First, we establish that the vote buyer bribes more members when the disutility distribution is more dispersed. Second, we show that even when dispersion is small, any cost-minimizing number of bribes is at least 3 2m−1. Finally, we demonstrate that with a sufficiently dispersed distribution, all members are bribed regardless of their number. The definition of dispersion used for this analysis is from Shaked and Shanthikumar (2007, p. 213). Definition 1. ˜ F(·)is more dispersed than F(·)if the ratio of the inverse CDFs, ˜ F−1(q)/F−1(q), is nondecreasing in qfor all q∈(0, 1).Insuchacase,wewriteF≤∗˜ F.6 An example of distributions ranked in this order are U[1 2−α,1 2+α]with α∈(0, 1 2], which become more dispersed as αincreases.7We use these uniform distributions to simulate the cost-minimizing number of bribes arg mink∈{m,,n}c(k)in Figure 2. In line with Proposition 1, the smallest cost-minimizing number of bribes is approximately 3 2m and is obtained for small dispersion (α→0). Moreover, the cost-minimizing number of bribes increases with dispersion, which is not specific to uniform distributions: for all distributions that can be ranked in our dispersion order, 6The increasing generalized hazard rate assumption implies that F(v)>0, so the CDF is strictly monotone. Hence, F−1(q)is well-defined for q∈(0, 1). 7These distributions are centered around 1 2, but some distributions with different means can also be dispersion ranked. In particular, moving the uniform support to the right on the real line decreases the variance relative to the mean, which results in less dispersion. However, note that the ≤∗order is not complete and some distributions cannot be ranked. Theoretical Economics 19 (2024) Buying voters 1319 members vote for with the same probability p(the equilibrium is formally derived in the proof of Lemma A.2.3). An equilibrium probability ¯ p∈[0, 1]solves b=1 2π(¯ p)=¯ p(1−¯ p). This expression is single-peaked and its maximum is 1 4.Aslongasb≤1 4,thereisan equilibrium with ¯ p<1 where the proposal is rejected with a positive probability. Thus, if the vote buyer offers (slightly more than) b=1 4,shepays3 4and the proposal passes with certainty in all equilibria where members play symmetric strategies. Now we allow for asymmetric strategies when the three members receive b=1 4. There is an equilibrium where one member accepts his bribe with a probability of one and the other two decline with a probability of one. Thus, the focus on symmetric strategies is not without loss. Indeed, with no dispersion, the cheapest bribes such that the proposal passes with certainty in all equilibria are b=1 2offered to two members. ♦ For sufficiently dispersed distributions, however, there is no equilibrium of the voting subgame where the proposal is rejected with a positive probability if the vote buyer offers the cost-minimizing bribes derived in Section 2.1. Proposition 3. Suppose the distribution is at least as dispersed as U[0, 1]. Offering b∗ n to nmembers, which minimizes the capture cost if members use symmetric strategies, ensures the proposal passes with certainty in any equilibrium of the voting subgame. (Proof in Appendix A.3.) Intuitively, dispersion makes the behavior of other members harder to predict, which prevents the existence of asymmetric equilibria. Formally, the proposition’s proof relies on an iterated deletion of strictly dominated strategies. Member i’s pivotal probability is maximized if others split suitably between two extreme cutoffs. In particular, if m−1 other members always accept (cutoff at vmax)andn−m always decline (cutoff at vmin), member iis pivotal with certainty. Even then, member i still votes for if vi<b ∗ n.Hence,cutoffsbelowb∗ nare not rationalizable. Once those strategies have been eliminated, member icannot anticipate being pivotal with certainty, and another set of cutoffs is not rationalizable. For distributions at least as dispersed as U[0, 1],eventually,nocutoffbelowvmax is rationalizable, and the proposal passes with certainty in all equilibria. We use U[0, 1]as a benchmark to provide a lower bound on dispersion for the proposition to be true. However, not all distributions are ranked in our dispersion order. Thus, this lower bound is sufficient but not necessary and there exist other distributions for which offering b∗ nto nmembers would also ensure that the proposal passes with certainty in any equilibrium; e.g., see Example 3. 2.3 Unequal bribes Can unequal bribes reduce the capture cost? Example 3illustrates that the vote buyer can “divide and conquer” for some distributions, and hence that unequal bribes can yield a lower capture cost. 1320 Louis-Sidois and Musolff Theoretical Economics 19 (2024) Example 3. Let (m,n)=(2, 3)and vi i.i.d. ∼Bernoulli(1 2).10 For bribed member i,astrategy consists of a probability of accepting the bribe if vi=0, and a probability of accepting if vi=1. We first characterize the cost-minimizing bribes for k=3andk=2ifthe vote buyer offers the same bribe to all bribed members. Next, we show that there exist unequal bribes that yield a lower capture cost. Suppose the vote buyer offers the same bribe b>0 to all three members. Each accepts if vi=0. Hence, the pivotal probability of member iwould be maximized if the other two members vote against if their disutility is 1. Then πi=1 2.Thus,ifb<1 2,there is an equilibrium where members vote for if their disutility is 0 and vote against if their disutility is 1. As a result, the bribe needs to be at least b=1 2. If it is (slightly more than) 1 2, an iterated deletion of dominated strategies proves that no equilibrium of the voting subgame exists where the proposal is rejected with positive probability. A member with disutility vi=0 votes “for.” Thus, no member can be pivotal with probability larger than 1 2, and all of them accept. Hence, capture costs 3 2. Now, suppose two members are bribed. The unbribed member votes “against.” Thus, a bribed member would always be pivotal if the other accepts regardless of his disutility. As a result, the vote buyer needs to offer 1 to both members to guarantee that the proposal passes with certainty in all equilibria; hence, capture costs 2. However, the unequal bribes (b1,b2,b3)=(0.51, 0.51, 0.01)yield a lower capture cost. All members vote “for” if their disutility is 0, and no pivotal probability can exceed 1 2. Thus, members 1 and 2 always accept. In turn, member 3 is not pivotal and also accepts. As a result, the proposal is accepted with certainty in all equilibria. ♦ Nevertheless, equal bribes can be preferred for other distributions. In particular, equal bribes do minimize capture cost in Example 1. Example 4. Consider (m,n)=(2, 3)with vi i.i.d. ∼U[0, 1]and allow the vote buyer to offer unequal bribes (b1,b2,b3). Denoting by vithe equilibrium cutoff of member i,an equilibrium of the voting subgame where all cutoffs are in (0, 1)satisfies:11 v1π1(v2,v3)=b1, v2π2(v1,v3)=b2,(1) v3π3(v1,v2)=b3. When bribes are large enough, an equilibrium satisfying (1) does not exist and all committee members voting for regardless of their disutility is the unique equilibrium. Thus, the vote buyer offers the cheapest (b1,b2,b3)such that (1) has no solution. To identify 10In this example, we relax the assumption that F(·)is continuously differentiable and has an increasing generalized hazard rate to provide the clearest illustration. 11Appendix A.4 considers cases where some cutoffs are 0 or 1. They do not affect our conclusion: if there exists a local perturbation of the bribes that guarantees there is no equilibrium where the proposal can be rejected, then the bribes are more expensive than bi=8 27 for all members. Theoretical Economics 19 (2024) Buying voters 1321 these bribes, it is useful to look at the Jacobian of (1): J=⎛ ⎜ ⎝ v2(1−v3)+(1−v2)v3v1(1−2v3)v1(1−2v2) v2(1−2v3)v1(1−v3)+(1−v1)v3(1−2v1)v2 (1−2v2)v3(1−2v1)v3v1(1−v2)+(1−v1)v2 ⎞ ⎟ ⎠ For given bribes (b1,b2,b3), suppose there exist (v1,v2,v3)solving (1). As long as Jis nonsingular, following any -perturbation of (b1,b2,b3)there also exists a solution to (1). By contrast, when the determinant of Jis 0, we can find an -perturbation of the bribes such that (1) has no solution within a neighborhood of (v1,v2,v3)and, potentially, no solution at all. The cost cannot be minimized if some bribes are larger than needed to pass the proposal. Therefore, the cheapest bribes such that the proposal passes with certainty in equilibrium must be arbitrarily close to a (b1,b2,b3)for which there is an -perturbation of the bribes such that (1) has no solution (the sufficient condition). This can only be the case when the determinant of Jis zero (the necessary condition). In the following, we first establish that the bribes of Example 1(bi=8 27 for all members) satisfy the necessary condition, and then show they also satisfy the sufficient condition. Finally, we argue that they are the cheapest bribes satisfying the necessary condition. To begin with, computing the determinant of Jgives 2v1v2v3(2−v1−v2−v3), i.e., the matrix is singular if v1+v2+v3=2.12 Thus, if a combination of bribes is associated with an equilibrium satisfying 3 i=1vi=2, then there exists an -perturbation of the bribes that ensures that (1) has no solution within a neighborhood of (v1,v2,v3).Inparticular, 3 i=1vi=2 is satisfied for v1=v2=v3=2 3. Plugging these values into (1)showsthat these cutoffs are an equilibrium if bi=8 27 for all members; hence, these bribes satisfy the necessary condition. Moreover, the iterated deletion of dominated strategies used for Proposition 3guarantees that if all bribes are (slightly more than) 8 27 ,nocutoffin[0, 1)is rationalizable. Hence, the -perturbation consisting in marginally increasing all bribes guarantees that (1) has no solution, and bi=8 27 for all members satisfies the sufficient condition. Finally, we establish that bi=8 27 for all members are the cheapest bribes satisfying the necessary condition. We look for (v1,v2,v3)∈[0, 1]3,3 i=1vi=2 that minimizes 3 i=1bi. Without loss of generality, suppose v1< v2< v3. We now show that decreasing the difference between cutoffs decreases 3 i=1bi. In particular, let us increase v1and decrease v3by the same amount: dv1=1, dv3=−1anddv2=0, which keeps 3 i=1vi constant. Using the Jacobian, the change in the capture cost is 3  i=1 dbi=(v1−v3)(6v2−2), which is negative because v1< v3and v2>1/3.13 As a result, under the constraint 3 i=1vi=2, v1=v2=v3=2 3, minimize 3 i=1bi. These cutoffs are an equilibrium if 12The determinant is also 0 if some cutoffs are 0. As vi≥bi,vi=0 can only be part of an equilibrium if bi=0. But then the proposal passes with probability 1 if the two other members receive a bribe of 1, which does not minimize the capture cost. 13As v1+v2+v3=2 and v3<1, we obtain v1+v2≥1. Combining with v2> v1,wemusthavev2>1/3. 1322 Louis-Sidois and Musolff Theoretical Economics 19 (2024) bi=8 27 for all members. Thus, the equal bribes identified in Example 1do minimize the capture cost. ♦ Is it generally true that the vote buyer offers the same bribes when vi i.i.d. ∼U[0, 1]?Example 4established this for (m,n)=(2, 3)by demonstrating that the Jacobian of (1)is not invertible if and only if n i=1vi=m.Wefindthesameconditionfor(m,n)=(1, 2), (1, 3),(3, 4),and(1, 4). Hence, in all these cases, equal bribes minimize the capture cost. However, we could not find a general formula for the determinant and prove the result for any (m,n). Even if Example 3showed that the restriction to equal bribes is not always without loss of generality, Example 4indicates that the main model generates economic insights going beyond the equal bribes assumption. To see this, notice that equal bribes did not prevent the vote buyer from setting k<n(i.e., offering some bribes of zero). Restricted to offering equal bribes, the vote buyer did not want to exploit this extreme form of inequality with vi i.i.d. ∼U[0, 1]and instead bribed all members. Example 4further establishes that for (m,n)=(2, 3), she never benefits from any form of inequality in the bribes. 3. Sequential voting We now consider a committee voting sequentially. The proposal passes if at least mof nmembers vote for it. Members draw their disutilities from the passing of the proposal at the beginning of the game: vi i.i.d. ∼U[0, 1]. The order of votes is known in advance and members observe previous votes, like in the US Senate where members vote in alphabetical order. The vote buyer minimizes the capture cost subject to the proposal passing with certainty. Bribes are simultaneously and publicly offered to all members before the vote begins, and a bribe is paid if a member votes for the proposal. Bribes can be unequal, but they cannot depend on the number of votes still needed to pass the proposal when the member votes.14 This section shows that the vote buyer also offers a number of bribes larger than the vote threshold to exploit pivotal considerations with sequential voting. Proposition 4. When voting is sequential and vi i.i.d. ∼U[0, 1], the vote buyer bribes all members equally, offering b=1/(n−(m−1)) to all nmembers. (Proof in text below.) Hence, with vi i.i.d. ∼U[0, 1], bribing all members equally ensures that the proposal passes with certainty in all equilibria at the lowest possible cost for both simultaneous and sequential voting. 14As in Genicot and Ray (2006), this assumption rules out bribes, which depend on the number of other members accepting. However, it has very different implications because we consider bribes offered before the vote, while members can be approached sequentially in Genicot and Ray (2006). Therefore, in their setup, bribes may depend on the number of votes still needed to pass the proposal. In our model, this would imply that the vote buyer offers 1 to all remaining members if all of their votes are required to pass the proposal, and small bribes when there are more members. If n>m, all members would vote for and receive small bribes on the equilibrium path. Theoretical Economics 19 (2024) Buying voters 1323 Table 1. Equilibrium of the voting subgame, an example. v(x,y)p(x,y) x=0x=1x=2x=0x=1x=2 y=11 b111 b10 y=21 b2 1−b1 b2 b11b1+b2b2 Note: Equilibrium cutoffs (left panel) and passing probabilities (right panel) for b2≤b1and b1+b2≤1.xis the number of votes required to pass the proposal and ythe number of members still to vote. To establish this result, we first characterize the equilibrium of the voting subgame, which itself has to be decomposed into multiple subgames. Without loss of generality, we focus on members who receive strictly positive bribes. Define S(x,y)as the subgame where xvotes are needed to pass the proposal and ymembers still have to vote. The voting subgame begins in S(m,n).IfinS(x,y)the member votes for the proposal, S(x− 1, y−1)is reached while a vote against leads to S(x,y−1). Members and bribes byare now indexed by y∈{1, ,n}, with the number of members still to vote. Members use backward induction to infer their pivotal probability as in Spenkuch, Montagnes, and Magleby (2018, July). Let v(x,y)be the cutoff played in S(x,y)and p(x,y)be the probability that the proposal passes given that S(x,y)is reached. We jointly characterize v(x,y)and p(x,y)to find the equilibrium of the voting subgame, beginning with two-member committees. Table 1gives an example for the expressions of v(x,y)in the left and p(x,y)in the right panel. When a member votes for the proposal, the subgame located North-West is reached; if he votes against, we move North. Example 5. Let (m,n)=(1, 2)and assume members receive positive bribes with b1+ b2≤1. We solve the game backward and start with the last member, y=1. If member y=2 voted “for,” y=1votesinS(0, 1). The proposal passes regardless of the vote of member y=1, who accepts with certainty. Thus, v(0, 1)=1andp(0, 1)=1. If member y=2 voted against, member y=1votesinS(1, 1), where he is pivotal. He votes for if b1>v 1and we have v(1, 1)=p(1, 1)=b1. Moving backwards, member y=2startsinS(1, 2). A vote “for” passes the proposal. Alternatively, if he votes against, S(1, 1)is reached, where the proposal passes with probability b1. Thus, member y=2 votes “for” if b2−v2>−v2b1⇐⇒ v2<b2 1−b1 , so that v(1, 2)=b2/(1−b1)and the proposal passes with probability p(1, 2)=b2 1−b1+1−b2 1−b1b1=b1+b2. Thus, bribes are substitutes from the perspective of the vote buyer: the proposal passes withcertaintyforanybribessuchthatb1+b2=1atacostof1. ♦ Using a recursive characterization of v(1, y)and p(1, y), this substitutability of bribes generalizes when one vote is needed to pass the proposal. 1324 Louis-Sidois and Musolff Theoretical Economics 19 (2024) Lemma 5. In equilibrium, p(1, y)=min{y s=1bs,1 }. (Proof in Appendix B.) With vi i.i.d. ∼U[0, 1], the vote buyer is exactly indifferent between bribing the first member to vote or a member voting later. For general distributions, the problem is not tractable, but there is still a tradeoff. On the one hand, the first member can always determine the passing of the proposal: as x=1, this member is pivotal. On the other hand, a member who votes late is less likely to be pivotal (the proposal may be already accepted), but his cutoff affects members who vote earlier: they forecast that voting against is less likely to make the proposal rejected when later members receive higher bribes. Hence, their cutoffs also increase, as we can see in Example 5where the cutoff of the first member v(1, 2)=b2/(1−b1)increases with the bribe of the second member b1. The next example shows that when more than one vote is needed to pass the proposal, bribes are not perfect substitutes. Example 6. Let (m,n)=(2, 2)and assume b1+b2≤1. We start with b2<b 1.First, consider member y=1. If member y=2 voted “for,” S(1, 1)is reached, for which we have established v(1, 1)=p(1, 1)=b1.Ifmembery=2 voted against, the proposal will be rejected and member y=1 votes “for.” Turning to member y=2, he starts in S(2, 2) and votes “for” if b2−v2b1>0⇐⇒ v2<b2 b1 so that v(2, 2)=b2/b1and the proposal passes with probability p(2, 2)=v(2, 2)p(1, 1)=b2. Bribes are not substitutes anymore: only b2, the smaller of the two bribes, affects the probability of passing. Instead, suppose b2≥b1. The strategy of member y=1isasbeforeandmember y=2 votes “for” if v2<b 2/b1.Asb2/b1>1, he always votes “for” and the proposal passes with probability p(1, 1)=b1. Again, only the smaller of the bribes affects the probability of passing. As a result, the vote buyer offers equal bribes. If not, the largest bribe does not affect the probability of passing and should be decreased. Finally, given that the probability of passing is equal to the smaller bribe, this bribe must be 1 to make the proposal pass with certainty. Hence, capture cost is minimized when b2=b1=1. ♦ The example generalizes as follows.15 Lemma 6. Let b(s) ybe the sth order statistic (i.e., the sth lowest value) among {b1,,by}. Then, for x≥1, in equilibrium p(x,y)=min{y−(x−1) s=1b(s) y,1 }. 15This lemma uses the convention that empty sums evaluate to zero. Theoretical Economics 19 (2024) Buying voters 1325 (Proof in Appendix B.) Hence, the probability of passing is the sum of the n−(m−1) smallest bribes. Intuitively, for given bribes, a member is more likely to accept if he votes early. For instance, in Example 6, when the member who receives the largest bribe votes first (b2≥b1), he accepts regardless of his disutility and free-rides on the second member, relying on him to reject the proposal. This finding generalizes: in S(x,y)with x>1, if byis one of the x−1 largest bribes among members still to vote, member y accepts regardless of his disutility. Now, suppose the first m−1membersreceivethe largest bribes. They accept regardless of their disutility and S(1, n−(m+1)) is reached with a probability of one. We have the same pattern when all bribes are equal: the m−1 first members accept and free-ride on the n−(m−1)last members, who can potentially reject the proposal. Then we have x=1 and, by Lemma 5, the probability of passing is the sum of the remaining bribes, which are the n−(m−1)lowest bribes. Instead, if the member who receives the largest bribe votes last in Example 6(b2< b1), both members decline if their disutility is large enough. In such cases, increasing the largest bribe b1has two countervailing effects on the probability of passing. On the one hand, increasing b1directly increases the probability of passing because it raises the cutoff of the second member v(1, 1)=b1. On the other hand, b1decreases the probability of passing through the first member: he forecasts that voting “for” is more likely to make the proposal pass, and hence becomes more likely to decline and make the proposal rejected. When vi i.i.d. ∼U[0, 1], the two effects cancel out and an increase in b1does not affect the probability of passing. This mechanism generalizes and the probability of passing is also the sum of the n−(m−1)lowest bribes if the members who receive the largestbribesdonotvotefirst. We can now consider the problem of the vote buyer. Given that voting starts in subgame S(m,n), the vote buyer chooses the cheapest combination {by}n y=1such that p(m,n)=1. Using Lemma 6, we can write this problem as min {by}n y=1 n  y=1 bysuch that n−(m−1)  s=1 b(s) y≥1. The m−1 largest bribes do not affect the probability of passing. Thus, the cost is minimized when the m−1 largest bribes are equal to b(n−(m−1)) y, the maximum of the n−(m−1)smallest bribes. As the sum of these bribes must be 1 to make the proposal pass with certainty, the smallest b(n−(m−1)) yis achieved when they are all equal to 1/(n− (m−1)). Therefore, the lowest bribes are 1/(n−(m−1)), which implies that all members are bribed. Furthermore, the m−1 largest bribes are also equal to the n−(m−1)smallest bribes. As a result, all bribes are equal and we obtain Proposition 4. We conclude this section with comparative statics for the capture cost. Given that thevotebuyerpaysb∗ n=1/(n−(m−1)) to nmembers, the resulting cost is Cseq(m,n)=n n−(m−1). Comparative statics are similar to Proposition 2.Ifwemultiplymand nby the same scalar λsatisfying (λm,λn)∈N2 +, the cost is multiplied by less than λ.Moreover,the effect of mis now clearly positive. Finally, the capture cost decreases with n. 1326 Louis-Sidois and Musolff Theoretical Economics 19 (2024) Figure 4. Cost comparison. Notes: Simulation of the lowest cost as a function of mand n. 4. Cost comparison We compare the capture costs under simultaneous and sequential voting for vi i.i.d. ∼ U[0, 1]. With simultaneous voting, all members are bribed and the cost is Csim(m,n)=nn−1 m−1m nm1−m nn−m . Which voting timing minimizes capture cost depends on the number of members nand on the vote threshold m. The result of the comparison, illustrated in Figure 4,isasfollows. Proposition 5. Suppose vi i.i.d. ∼U[0, 1]. (a) If it takes one or all but one votes to pass the proposal, the capture cost is lower with simultaneous voting: Csim(m,n)<Cseq(m,n)for m=1and m=n−1. (b) If unanimity is not required to pass the proposal (i.e., m<n), there is a λ∗such that Cseq(λm,λn)<Csim(λm,λn)with λ>λ ∗and (λm,λn)∈N2 +. (Proof in Appendix C.) The models with sequential and simultaneous voting differ in multiple aspects. However, the equilibrium structure of the voting subgame with sequential voting provides an intuition for the cost comparison. When all bribes are equal to b∗ n, all members accept on the equilibrium path, which shuts down some interactions and prevents a clear exposition of the underlying mechanisms. Instead, suppose members receive equal bribes slightly lower than b∗ n. Thecostofthesebribesisclose to the capture cost, but some members can vote against on the equilibrium path. As explained after Lemma 6, the group of the m−1 first members accept their bribes and rely on the group of the n−(m−1)last members to potentially reject the proposal. Intuitively, the first group free-rides on the second group and this free-riding decreases the Theoretical Economics 19 (2024) Buying voters 1327 capture cost. Moreover, free-riding is particularly pronounced if both groups are large: there should be both members who free-ride and members to free-ride on to make the capture cost lower under sequential voting. If one of the two groups is small, there is a limited effect of free-riding and we find that the capture cost is smaller with simultaneous voting. In particular, if m=1, one vote is sufficient to pass the proposal and the group of free-riders is empty. Indeed, we have Cseq(1, n)=1. Meanwhile, Csim(1, n)=(1−1/n)n−1is 1 2for n=2 and decreases in n.16 Thus, Cseq(1, n)>Csim(1, n)as stated in Proposition 5(a). Now consider a vote threshold close to unanimity m=n−1 (for m=n, pivotal considerations cannot be exploited for both sequential and simultaneous voting and the capture cost is meither way). The group of the n−(m−1)last members is empty and there is no one to free-ride on. Hence, we also find that the capture cost is lower with simultaneous voting in Proposition 5(a). Formally, Cseq(n−1, n)=n/2andCsim(n− 1, n)=n×(1−1/n)n.As(1−1/n)nis increasing in n,and1 2>e −1=limn→∞(1−1/n)n, we have Cseq(n−1, n)>Csim(n−1, n). Turning to Proposition 5(b), the result simply states that the capture cost is smaller under sequential voting if mand nare sufficiently large and the vote threshold is not one of the extreme cases already discussed. This result can also be explained with freeriding: the size of the two groups is limited when mand nare small, and Figure 4confirms that the capture cost is smaller with simultaneous voting. As we multiply both m and nby a given λsuch that (λm,λn)∈N2 +, the size of the two groups increases and the capture cost becomes eventually smaller with sequential voting because of free-riding. In the limit, we have lim λ→∞Cseq(λm,λn)=1 1−m n <∞= lim λ→∞Csim(λm,λn). Hence, free-riding even implies that the cost grows bounded with sequential voting, which is not the case with simultaneous voting. With sequential voting, the cost depends on the share of members in the two groups. The proposal is accepted with certainty if the sum of the bribes in the group of the n−(m−1)last members is one. These bribes represent a share (n−(m−1))/n of the capture cost because all members receive the same bribe, and the total cost is the inverse of this share. As λbecomes large, this share converges to (n−m)/n and free-riding implies that the cost is bounded. 5. Concluding remarks When members have uncertain preferences, the vote buyer bribes supermajorities to exploit pivotal considerations. As we considered bribes conditioned on individual voting decisions, we conclude with a discussion of other contractual environments. If bribes are conditioned on the passing of the proposal, a pivotal vote also decides the payment of the bribes. Thus, it is a weakly dominant strategy to vote against if the 16As vmin =0, the assumption m>1inSection2.1 plays no role and Csim(m,n)accurately defines the capture cost for m=1. 1328 Louis-Sidois and Musolff Theoretical Economics 19 (2024) disutility exceeds the bribe. The vote buyer cannot exploit pivotal considerations with such contracts: to make the proposal pass with certainty, she has to offer vmax to mmembers. Hence, conditioning on passing is bad for the vote buyer. Instead, suppose bribes depend on the number of votes “for.” A vote matters for the bribe even when it is not pivotal for the passing of the proposal. In a previous version of the paper (Louis-Sidois and Musolff 2023), we proposed an example where the vote buyer exploited pivotal considerations: she bribed a number of members larger than the vote threshold and paid less than when she only conditioned on passing. In an unrestricted contractual environment, bribes can be contingent on the entire vector of votes. In such a case, Dal Bo (2007) has established that capture occurs at no cost: the vote buyer promises a bribe vmax if a member is pivotal and an arbitrarily small bribe otherwise. Voting “for” is then a dominant strategy, and when more than mmembers receive such offers, the proposal always passes. If such contracts are allowed, our solution is still relevant for a budget-constrained vote buyer: even if members are never pivotal in equilibrium, the vote buyer must be able to pay the large pivotal bribes for Dal Bo’s strategy to be credible. Therefore, while our solution is more expensive (as the vote buyer actually pays the bribes), it would nevertheless be feasible for lower budget constraints. We have considered offers visible to all. If offers are privately communicated to each member, the vote buyer cannot credibly claim to have bribed more members than necessary and the number of bribes is equal to the vote threshold in equilibrium. Each bribe is equal to vmax, and capture is more expensive with private offers. To see why offering more bribes than the vote threshold is not credible, consider a voting profile where more than mmembers vote for with certainty. The vote buyer would deviate and propose exactly mbribes. This deviation cannot be detected by members who continue receiving the bribe, but in equilibrium, a bribed member cannot believe there are more than m−1 other bribes. The bribe must be equal to vmax for him to always vote “for.” Finally, our model can be reinterpreted with punishments for members who vote against instead of bribes. For the vote buyer, enforcing punishment is likely to be costly, which implies she effectively pays for members who vote against the proposal. If she has enough resources for punishment, she uses the strategy of this paper. Capture is costless because all approached members vote “for.” The capture cost we computed corresponds to the minimum resources needed to secure certain passing of the proposal. Appendix A: Proofs (simultaneous voting) A.1 Helpful facts Before proceeding to the proofs, we establish some necessary prerequisites. Recall that (·)is a continuous extension of the factorial function. In particular, (x)=(x−1)!for x∈N.Thus, k m=explog(k+1)−log(m+1)−log (k−m+1). Theoretical Economics 19 (2024) Buying voters 1335 Using the properties of the digamma function given in Appendix A.1 and noting that for any decreasing function b s=a+1g(s)<b ag(s)ds, ψ(k)−ψ(k−m+1)+1 k= m  s=1 1 k−m+s <m 0 1 k−m+sds =−logk−m k. Thus, the cost strictly decreases in kso that arg mink∈{m,,n}c(k)=n. Lemma A.2.5. For vi∼U[1 2−α,1 2+α], the cost function c(k)has a unique global real minimizer. Proof. Recall that dlogc(k) dk =1 k+ψ(k)−ψ(k−m+1)+log1−Fv∗ k. We establish that this expression crosses the horizontal axis at most once, and necessarily from below. To do so, we will show that when the FOC is satisfied (i.e., when dlogc(k) dk =0), d2logc(k) dk2>0.17 1. Consider: d2logc(k) dk2=−1 k2+ψ(k)−ψ(k−m+1)−Fv∗ k 1−Fv∗ k dv∗ k dk . The expression has the same sign as k−1 k2+ψ(k)−ψ(k−m+1)−Fv∗ k 1−Fv∗ k dv∗ k dk .(7) We now argue that for vi∼U[1 2−α,1 2+α],(7) is increasing in αso that if we want to show that it is positive, we only need to do so for α→0. 2. For vi∼U[1 2−α,1 2+α], the FOC for the choice of v∗ kreduces to 2(k−m) 2α−2v∗ k+1=2(m−1) 2α+2v∗ k−1+1 v∗ k (8) Hence, v∗ k=(4αk −8αm +4α−2k−2)2−161−4α2k−4αk +8αm −4α+2k+2 8k.(9) 17One may worry that there could be multiple local minima. However, this cannot be the case as the derivative is continuous: v∗ kis continuous, and hence so is dlog c(k) dk . This means c(k)has (at most) one local minimum (otherwise it would have to have at least one local maximum, which is ruled out by the proof). 1336 Louis-Sidois and Musolff Theoretical Economics 19 (2024) Furthermore, we can implicitly differentiate (8) to get an expression for dv∗ k dk and plug the value of v∗ kfrom (9) into F(v∗ k) 1−F(v∗ k)to find −Fv∗ k 1−Fv∗ k dv∗ k dk =m44α2−1k+−2α(k−2m+1)+k+12−(2α−1)k(m−2)+αm(4m−2)+m 2k(k−m)44α2−1k+−2α(k−2m+1)+k+12.(10) The derivative of this last expression with respect to αis d dα−Fv∗ k 1−Fv∗ k dv∗ k dk =8(m−1)(1−2α) 44α2−1k+−2α(k−2m+1)+k+123/2>0. Hence, it is increasing in α, and so is (7). 3. For α→0, (10)implies −Fv∗ k 1−Fv∗ k dv∗ k dk →m−1 (k−1)(k−m) and (7) becomes k−1 k2+ψ(k)−ψ(k−m+1)+m−1 (k−1)(k−m). We only need to show this expression is positive for ksuch that the FOC (dlog c(k) dk = 0) is satisfied. We establish a strictly stronger claim: we prove that a different expression (Zbelow) that reduces to this expression when the FOC is satisfied is positive for all k: Z=ψ(k)−ψ(k−m+1)+log1−m−1 k−1 +kψ(k)−ψ(k−m+1)+m−1 (k−1)(k−m) For α→0, F(v∗ k)→m−1 k−1and the FOC is satisfied if 1 k+ψ(k)−ψ(k−m+1)+log(1− m−1 k−1)=0. In this case, Z=(7). 4. To show Z>0, we first utilize the bounds of Qi et al. (2005) to provide a lower bound for the first line (and then for Z). To this end, note that their Corollary 8 implies 1 2x+log(x)−1 12x2≤ψ(x+1)≤1 2x+log(x), and hence ψ(k)−ψ(k−m+1)+log(k−m)−log(k−1) Theoretical Economics 19 (2024) Buying voters 1337 ≥− 1 2(k−m)−1 12(k−1)2+1 2(k−1) ≥1 k−1−1 k−m. Plugging this back into Z, it now suffices to show 1 k−1−1 k−m+kψ(k)−ψ(k−m+1)+m−1 (k−1)(k−m)≥0. This simplifies to m−1 k(k−m)+ψ(k)−ψ(k−m+1)≥0. We again utilize bounds from Corollary 8 of Qi et al. (2005), this time 1 x−1 2x2+1 6x3−1 30x5≤ψ(x+1)≤1 x−1 2x2+1 6x3. Thus, we need to show 1 m−k−1 k+1 3015 (k−m)2+5 (m−k)3−15 (k−1)2+5 (k−1)3−1 (k−1)5+30 k−1≥0. This holds for k≥m+1. As we have shown in the proof of Lemma A.2.4 that the cost decreases between k=mand k=m+1, the FOC cannot be satisfied for k∈ [m,m+1].Hence,wheneverdlogc(k) dk =0, d2logc(k) dk2>0. Proposition 1. (a) For any disutility distribution, any cost-minimizing number of bribes is at least min{3 2m−1, n}; (b) For any number of bribes k∈Nsuch that min{3 2m+1, n}≤k≤n,thereexistsa disutility distribution such that kis a cost-minimizing number of bribes. Proof.Notice that this proof does not follow the order of the text: it builds on Lemmata 2,3,and4. (a) See Lemma 3. (b) We establish the claim using uniform distributions: vi i.i.d. ∼U[1 2−α,1 2+α].Forthis proof only, let k∗:=arg mink∈[m,n]c(k)be the real (as opposed to integer) number of bribes that minimizes the cost; this number is unique by Lemma A.2.5. We first show that k∗is continuous in α. Then we establish that limα→0k∗≤ 3 2m+1 2and limα→1 2k∗=n. Hence, by the intermediate value theorem, for all k≥3 2m+1 2, there exists an α∈(0, 1 2)such that k∗=k. Finally, for any integer k∈{m,,n},k∗=kimplies k∈arg minκ∈{m,,n}c(κ).Hence,ifmis odd, 3 2m+1 2∈Nand for all k∈Nsuch that min{3 2m+1 2,n}≤k≤n, there exists an α 1338 Louis-Sidois and Musolff Theoretical Economics 19 (2024) such that k∈arg minκ∈{m,,n}c(κ).Ifmis even, 3 2m+1∈Nand for all k∈Nsuch that min{3 2m+1, n}≤k≤n, there exists an αsuch that k∈arg minκ∈{m,,n}c(κ). (i) k∗is continuous in α.To begin with, v∗ kis continuous in αas the roots of a polynomial are continuous functions of its coefficients and (2), the FOC defining v∗ k, simplifies to a polynomial when vi i.i.d. ∼U[1 2−α,1 2+α]: 4α2−1=(−2+4α−2k+4αk −8αm)v∗ k+4kv∗ k2. This, in turn, implies that c(k)is continuous in α, which implies via Berge’s maximum theorem that its minimizer k∗is continuous in α. (ii) lim α→0k∗≤(3/2)m+1/2. As vi∼U[1 2−α,1 2+α]implies F(v)=v−(1/2−α) 2αfor v∈[1 2−α,1 2+α],wehaveF−1(p)=1 2−α+2αp for p∈[0, 1]and we can express the maximization defining b∗ kin terms of p=F(v): b∗ k=max p∈[0,1]1 2−α+2αp×k−1 m−1pm−1(1−p)k−m. As α→0, we have [1 2−α+2αp]→1 2for all p∈[0, 1]: as the distribution converges to a mass point, all its quantiles converge to this point. By Berge’s maximum theorem, this convergence implies the convergence of the maximum: b∗ k→max p∈[0,1] 1 2×k−1 m−1pm−1(1−p)k−m. Hence, all b∗ kconverge to the values they have in Lemma A.2.3 (with δ=1 2) where the disutility distribution has no dispersion. By Berge’s maximum theorem, k∗thus converges to the minimizer of the cost under no dispersion as α→0. Finally, Lemma A.2.4 establishes that the cost under no dispersion increases for k≥3 2m+1 2. Hence, we also have k∗≤3 2m+1 2when α→0. (iii) lim α→1/2k∗=n.This follows from the proof of Lemma 4, which establishes that c(k)decreases with kwhen α=1 2. A.3 Other simultaneous voting proofs Proposition 2. (a) Proportional increases in vote threshold mand number of committee members n raise capture cost subproportionally: Csim(λm,λn)<λCsim(m,n)with λ∈N+. (b) Suppose only a majority can pass the proposal, i.e., n≤2m−1. Then, for any number of bribes k,b∗ k,andCsim(m,n)increase in the vote threshold m. Theoretical Economics 19 (2024) Buying voters 1339 Proof. (a) Suppose the vote buyer bribes kmembers in a committee (m,n).Asλ∈N+,she can bribe λk members in a committee (λm,λn). While λk bribes need not minimize cost in the larger committee, they give an upper bound on its minimized value. Thus, recalling p=F(v), it suffices to show that if b(λ,p)=F−1(p) p×λk −1 λm −1pm(1−p)k−mλ, then b(λ,p∗)=maxp∈[0,1]b(λ,p)decreases in λ. The log derivative of b(λ,p∗)is dlogbλ,p∗ dλ =∂logbλ,p∗ ∂λ +∂logb(λ,p) ∂p p=p∗ dp∗ dλ =(A) ∂logbλ,p∗ ∂λ =logp∗m1−p∗k−m−(k−m)ψ(λk −λm +1) +kψ(λk)−mψ(λm) =(B)logp∗m1−p∗k−m−(k−m)ψ(λk −λm +1) +kψ(λk +1)−mψ(λm +1) ≤(C)logm km1−m kk−m−(k−m)ψ(λk −λm +1) +kψ(λk +1)−mψ(λm +1) =(D)mg(m)−kg(k) <(E)0, where (A) letting p∗=arg maxp∈[0,1]b(λ,p),∂log b(λ,p) ∂p |p=p∗=0 by the envelope theorem, (B) uses xψ(λx)=x[ψ(λx +1)−1 λx ]=xψ(λx +1)−1/λ, (C) uses the fact that pm(1−p)k−m≤maxp∈[0,1]pm(1−p)k−m=(m k)m(1−m k)k−m, (D) defines g(x):=[log(x)−log(k−m)] −[ψ(λx +1)−ψ(λk −λm +1)],and (E) follows because g(·)is increasing. To see this, note g(x)=1 x−λψ(λx +1) =(i) 1 x−λψ(λx)+λ λ2x2 >(ii) 1 x−λ1 λx +1 λ2x2+λ λ2x2=0, 1340 Louis-Sidois and Musolff Theoretical Economics 19 (2024) where (i) uses ψ(u+1)=ψ(u)−1 u2and (ii) uses ψ(u)<1 u+1 u2from Guo and Qi (2010, Lemma 3, p. 107). (b) If n≤2m−1, we show that b∗ kis increasing in mfor all k∈{m,,n}.Hence, mink∈{m,,n}c(k)must also be increasing in mas c(k)=kb∗ k. By the envelope theorem, ∂logb∗ k ∂v |v=v∗ k=0and dlogb∗ k dm =∂logb∗ k ∂m +∂logb∗ k ∂v v=v∗ k dv dm =∂logb∗ k ∂m =ψ(k−m+1)−ψ(m)+logFv∗ k−log1−Fv∗ k. Lemma A.2.3 implies F(v∗ k)≥m−1 k−1.Thus, dlogb∗ k dm ≥ψ(k−m+1)−ψ(m)+logm−1 k−1−logk−m k−1 =log(m−1)−log(k−m)−ψ(m)−ψ(k−m+1). n≤2m−1impliesm−1≥k−m.Then log(m−1)−log(k−m)=m−1 s=k−m 1 sds > m−1  s=k−m+1 1 s=ψ(m)−ψ(k−m+1), where the last equality results from the property of the digamma function at the beginning of the proof section. As a result, dlogb∗ k dm >0. Proposition 3. Suppose the distribution is at least as dispersed as U[0, 1]. Offering b∗ n to nmembers, which minimizes the capture cost if members use symmetric strategies, ensures the proposal passes with certainty in any equilibrium of the voting subgame. Proof. Recall that footnote 4defines b∗ nas the smallest number above v∗ nπn(v∗ n).For this proof, we assume there exists a fixed minimum currency >0, so that b∗ n= v∗ nπn(v∗ n)+. We use a simultaneous iterated deletion of strictly dominated strategies to argue that when nmembers are bribed with b∗ n, the proposal passes with certainty in any equilibrium. We eliminate cutoffs in increasing order. Let t ibe the smallest rationalizable cutoff for member iafter iteration t.Thent+1 iis the smallest rationalizable cutoff for member iwhen no other member jplays a cutoff below t j. At each iteration, we simultaneously eliminate cutoffs for all members. We have ∀i:0 i=vmin and, as the disutility distribution is the same for all members, the same set of cutoffs is eliminated for all members at each step. Thus, t i=t∀i. Let fx(vy)be the probability that among all members but ythere are exactly xvotes for (writing vy=(v1,,vy−1,vy+1,,vn)for the vector of cutoffs of all members other Theoretical Economics 19 (2024) Buying voters 1341 than y). Without loss of generality, consider member 1. Let πmax()denote the maximal pivotal probability he can expect if every other member has a cutoff of at least : πmax():=max {(v2,,vn):∀i∈{2,,n}≤vi≤vmax}fm−1v1. Then the smallest rationalizable cutoff t+1at iteration t+1solves t+1×πmaxt=b∗ n. The remainder of the proof shows that if t<v max,then t+1−tπmaxt=b∗ n−tπmaxt≥, (11) whence t+1≥t+(as πmax ≤1) and all cutoffs smaller than vmax are eventually eliminated. We proceed by bounding tπmax(t). 1. For any i=1, we can rewrite fm−1(v1)as fm−1v1=F(vi)fm−2v1,i+1−F(vi)fm−1v1,i, where v1,i=(v2,,vi−1,vi+1,,vn)is the vector of cutoffs of all members other than 1 and i.Hence,thesignof ∂fm−1(v1) ∂viis independent of vi.Thus,thereisa solution to the maximization problem πmax(t)with vi∈{t,vmax}for all i.Inlight of this, let πn,h(t)be the value of the pivotal probability if exactly hof the n−1 other bribed members choose a cutoff of vmax and n−1−hchoose a cutoff of t; then πmaxt=max h∈{0,,n−1}πn,ht. 2. To bound tmaxhπn,h(t)from above, note tmax h∈{0,,n−1}πn,ht≤max h∈{0,,n−1},t∈[vmin,vmax]tπn,ht=max h∈{0,,n−1}ˆ bn,h, where ˆ bn,his such that when offering any amount strictly above ˆ bn,hto n−hmembers with vote threshold m−h, there is no equilibrium where members vote against the proposal with positive probability. Thus, ˆ bn,h=0forh≥mand else ˆ bn,h=max v∈[vmin,vmax]vn−h−1 m−h−1F(v)m−1−h1−F(v)n−m   πn,h(v) . 3. By definition, ˆ bn,0 =b∗ n−. When the distribution is more dispersed than U[0, 1], we now show that ˆ bn,h<b ∗ n−for all h∈{1, ,m}.Wehave ∂log ˆ bn,h ∂h =−ψ(n−h)−ψ(m−h)+logFv∗ n,h, (12) where ψis the digamma function and we define v∗ n,h:=argmaxv∈[vmin,vmax]vπn,h(v). 1342 Louis-Sidois and Musolff Theoretical Economics 19 (2024) •If vi i.i.d. ∼U[0, 1],F(v∗ n,h)=m−h n−hand (12)is −ψ(n−h)−ψ(m−h)−log(n−h)−log(m−h). Using the properties of the digamma function given in Appendix A.1 and noting that for any decreasing function b−1 s=ag(s)>b ag(s)ds, ψ(n−h)−ψ(m−h)= n−1  s=m 1 s−h >n m 1 s−hds =log(n−h)−log(m−h). Therefore, (12) is negative for U[0, 1]. •By Lemma A.2.2,F(v∗ n,h)is larger for more dispersed distributions. Thus, (12) must also be negative for distributions more dispersed than U[0, 1]. Putting these steps together,18 t×πmaxt=(1)tmax h∈{0,,n−1}πn,ht ≤(2)max h∈{0,,n−1}ˆ bn,h =(3)ˆ bn,0 =(3)b∗ n−. Plugging this into (11) indeed yields t+1−t≥. A.4 Boundary solutions in Example 4 When some members have cutoffs at the boundary, i.e., when for at least one i∈{1, 2, 3}, vi∈{0, 1}, the nonsingularity of the Jacobian is not informative about the existence of nearby equilibria for any local perturbation of the bribes: while local perturbations such that equation (1) continues to hold must exist, these local perturbations may take some vioutside of the feasible region [0, 1]. To address this concern, we consider bribes associated with an equilibrium with cutoffs at the boundary and such that a local perturbation of the bribes guarantees there is no equilibrium where the proposal can be rejected. We show that such bribes are more expensive than bi=8 27 for all members: •Cutoffs at 1. –(v1,v2,v3)=(1, 1, 1)is an equilibrium for any (b1,b2,b3), but it is irrelevant for the existence of other equilibria. –(v2,v3)=(1, 1).v1<1onlyifb1=0. Wlog, suppose b2≤b3. Then, if b2<1, (v1,v2,v3)=(0, b2,1 )is an equilibrium. Hence, the cost is at least 2 to make the proposal pass with certainty. 18The index on equalities/inequalities refers to the relevant step in the proof. Theoretical Economics 19 (2024) Buying voters 1343 –v3=1, v1≤v2<1. Then (v1,v2)satisfy v1(1−v2)=b1;v2(1−v1)=b2. (13) The vote buyer minimizes b1+b2+b3.Noticev3=1requiresb3≥π3=v1(1− v2)+v2(1−v1)=b1+b2.Thus,b1+b2+b3is at least 2(b1+b2)=2(v1+v2−2v1v2). (14) The Jacobian of (13)is J=1−v2−v1 −v21−v1. The determinant is 1 −v1−v2,whichis0ifv1+v2=1. Under this condition, the cost in (14) is minimized for v1=v2=1 2for which it is equal to 1. Thus, the cheapest bribes such that (13) has no solution are necessarily more expensive than 8 9(the capture cost with equal bribes). •Cutoffs at 0. –vi=0onlyifbi=0. With two or three cutoffs (and hence bribes) at 0, the proposal never passes. –v1=0and0< v2≤v3requires b1=0, b2>0, and b3>0. An equilibrium would have to satisfy v2v3=b2and v3v2=b3.Ifb2=b3, this system does not have a solution. Suppose b2=b3. Then, if b2<1, (v1,v2,v3)=(0, b2,1 )is an equilibrium. Hence, with one cutoff at 0, the cost is at least 2 to make the proposal pass with certainty. Appendix B: Proofs (sequential voting) Lemma 5. In equilibrium, p(1, y)=min{y s=1bs,1 }. Proof. In general, if member yvotes for, his expected utility is by−vyp(x−1, y−1) while a vote against gives −vyp(x,y−1). Therefore, in S(x,y), the member votes for of the proposal if his disutility is larger than a cutoff v(x,y)defined by v(x,y)=minby p(x−1, y−1)−p(x,y−1),1 . For all y,p(0, y−1)=1. Thus, the cutoff of member yin S(1, y)is v(1, y)=minby 1−p(1, y−1),1 . If by 1−p(1,y−1)<1, the probability of passing is p(1, y)=v(1, y)p(0, y−1)+1−v(1, y)p(1, y−1) 1344 Louis-Sidois and Musolff Theoretical Economics 19 (2024) =by 1−p(1, y−1)×1+1−by 1−p(1, y−1)×p(1, y−1) =by+p(1, y−1) We use an induction to complete the proof. Notice that the lemma holds for y=1 and assume that it holds for y−1. Then p(1, y−1)=min{y−1 s=1bs,1 }and we do have p(1, y)=min{y s=1bs,1 }, which proves the claim. Lemma 6. Let b(s) ybe the sth order statistic (i.e., the sth lowest value) among {b1,,by}. Then, for x≥1, in equilibrium p(x,y)=min{y−(x−1) s=1b(s) y,1 }. Proof. We proceed by induction on x. Lemma 5proves the base case (x=1). Suppose the result holds for x−1. We prove that it also holds for x. To do so, we use an induction on y. •Base case: y=1. If x≥2, p(x,1 )=0; if x=1, then p(x,1 )=b1as required (recall we use the convention that empty sums evaluate to zero). •Inductive step. Rearranging the equation for a member’s cutoff and then employing the inductive hypothesis, we have v(x,y)=minby p(x−1, y−1)−p(x,y−1),1  = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ minby b(y−(x−1)) y−1 ,1 if p(x−1, y−1)<1, min$by 1− y−x  s=1 b(s) y−1 ,1 %if p(x−1, y−1)=1 and p(x,y−1)<1, 1ifp(x,y−1)=1. We now verify the expression for p(x,y)following these cases: – Assume p(x−1, y−1)<1. ∗Assume by>b (y−(x−1)) y−1.Thenv(x,y)=1sothat p(x,y)=p(x−1, y−1) =min$y−(x−1)  s=1 b(s) y−1,1 % =min$y−(x−1)  s=1 b(s) y,1 %. ∗Assume by≤b(y−(x−1)) y−1.Then p(x,y)=p(x,y−1)+v(x,y)p(x−1, y−1)−p(x,y−1)