Public Choice 89: 245-265, 1996. © 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Equilibrium and efficiency in an organized vote market

TOMAS J. P H I L I P S O N 1 & JAMES M. S N Y D E R , JR. 2

1Department of Economics, University of Chicago, Chicago, U.S.A.; 2Department of Political Science, Massaehussets Institute of Technology, Massachussets, U.S.A.

Accepted 9 November 1994

Abstract. We study an organized market for votes, in which trade is directed by a market "specialist". This market mechanism always produces an equilibrium outcome, and whenever vote buying occurs the alternative chosen is Pareto superior to the alternative that would be chosen without trade. We then characterize the equilibrium outcomes in a one-dimensional policy space, and show that if the distribution of ideal points is skewed enough, then the equilibrium with vote buying differs from the equilibrium without vote buying (the median ideal point). This difference reflects the ability of an intense minority to obtain a policy it prefers in exchange for side- payments.

I. Introduction

Vote trading and vote buying are often proposed as ways to improve majoritar- Jan collective decisions, by allowing such decisions to reflect the relative inten- sity of different voters' preferences in and across issues. Also, many observers of legislatures and other collective decision making bodies believe that vote trading and vote buying are widespread phenomena, and that something akin to vote markets exists. Two natural questions to ask about such markets are: (1) what observed collective decisions or behavior can be explained by vote buy- ing or vote trading that cannot be explained by simple majoritarian voting alone; and (2) what are the normative properties of vote markets, and in partic- ular, how efficient are the outcomes produced by such markets compared to those obtained under simple majority voting?

To answer these questions, we need plausible models that predict the likely outcomes of a vote market. As previous work has shown, however, this is a nontrivial problem: most of the existing results in the literature are negative, demonstrating the nonexistence of vote buying or vote trading equilibria (e.g., Park, 1967; Kadane, 1972; Bernholtz, 1973, 1974; Ferejohn, 1974; Schwartz, 1977, 1981; Shubik and Van der Heyden, 1978; Weiss, 1988).

One difficulty with decentralized vote markets is the following. Suppose a

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legislature must choose between one of two proposals. If legislators are able to buy and sell votes, and the price at which votes trade is strictly positive, then any legislator holding a vote that does not affect the voting outcome will want to sell his vote. This will be true, for example, of any legislator on the losing side of the issue. However, no one will want to buy the vote, precisely because it does not affect the outcome. Thus, at any positive price, there will be excess supply of votes. On the other hand, if votes are free, there will be excess demand.

In this paper, we study an organized market for votes, in which trade is directed by an intermediary, or market specialist. This mechanism overcomes the problem above, and given a choice between any pair of alternatives, always produces an equilibrium outcome. Whenever vote buying occurs in equilibri- um, the policy chosen is different from the policy that is preferred by a majori- ty without trade. Moreover, in such cases the outcome is Pareto superior to the outcome that would occur without trade. Importantly, the intermediary needs relatively little information in order to perform his tasks.

We then extend the vote buying mechanism to collective decision making over a one-dimensional policy space, in order to characterize equilibrium poli- cy outcomes. Our notion of equilibrium is a straightforward extension of the majority-rule equilibrium concept, similar to that used in previous work (e.g., Snyder, 1991). We show first that if legislators' preferences over policies are strictly concave, then there always exists an equilibrium policy. If the distribu- tion of legislators' most-preferred policies is symmetric enough, then introduc- ing vote buying has no effect; the equilibrium with vote buying is equal to the median of the most-preferred policies, and therefore coincides with the policy that would be obtained in the absence of vote buying. On the other hand, if the distribution of most-preferred policies is skewed enough, then the vote buying equilibrium is not the median. The difference between the vote buying equilibrium and the policy most preferred by the median voter reflects the abili- ty of an intense minority to obtain a policy it prefers in exchange for private payments.

Second, since the equilibrium policy with vote buying depends on the skew- ness of the distribution of most-preferred policies, as well as the median, it is possible to test for the presence and importance of vote buying in some set- tings. For example, suppose the policy space consists of levels of government spending (tax rates are fixed), and voters' preferences over spending can be written as increasing functions of their incomes. If two communities have the same median income, but one community's income distribution is more heavily skewed than the other's, then the equilibrium level of public spending with vote buying will typically be higher in the community with the more heavily skewed distribution. Without vote buying, however, the equilibrium level of spending in the two cities would be the same, as implied by the median-voter theorem.

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Later in the paper, we discuss some empirical evidence from studies of munici-

palities and school districts that supports the vote buying hypothesis. Finally, the vote buying equilibrium has the following desirable normative

property. Measuring efficiency by the sum of the legislators' willingness to pay for policies, the equilibrium policy with vote buying is always at least as effi- cient as the equilibrium without vote buying. And, whenever the two equilibri- um policies are different, the equilibrium with vote buying is strictly more effi- cient. In addition to its normative content, this result provides a reason to expect legislatures to organize a vote market, at least if the costs of maintaining it are not too great.

These results are in sharp contrast with the conclusions of much of the exist- ing literature. Although early writers such as Buchanan and Tullock (1962), Coleman (1966), Haefele (1971), and Mueller (1973) argued in favor of vote buying and vote trading, most later scholars have been rather pessimistic about the potential and realized benefits of such arrangements. This pessimism has several causes.

First, the calculus of vote buying or vote trading is considerably more difficult than Buchanan and Tullock suggested. Buchanan and Tullock ignored the complex interactions that result when many voters are trading at the same time, and the conjectures traders must make about one another 's behavior in order to calculate their optimal vote buying or vote trading strategies. Any votes bought or traded are ultimately cast in majority-rule divisions. The value of each vote therefore depends critically on whether or not the vote is pivotal

- indeed, votes that are not pivotal have no value at all - and it is therefore impossible for voters to decide what vote exchanges to accept without taking into account the likely exchanges of other voters. After surveying the literature on vote trading, Ferejohn (1974) concluded: "We really know very little theo- retically about vote trading. We cannot be sure about when it will occur, or how often, or what sorts of bargains will be made. We don ' t know if it has any desirable normative or efficiency properties" (p. 25). Twenty years later this statement is still basically true. There is still no general model of optimal and expectationally consistent vote buying or vote trading in a decentralized en- vironment, i

A second problem, closely related to the first, is that vote buying and vote trading are activities that produce externalities (see, e.g., Brams and Riker, 1973). Since voters are used to make collective decisions, when one voter buys a vote from another and thereby changes the outcome of a decision, the trade directly affects the welfare of other citizens who are not party to the trade. Of course, the externalities are positive for some voters and negative for others. However, the presence of these externalities means that the usual welfare the- orems do not apply to markets for votes, and such markets are therefore un- likely to produce Pareto optimal outcomes.

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Third, Park (1967), Kadane (1972), Bernholz (1973, 1974), and Schwartz (1977, 1981) have all shown that in cooperative situations, under rather general conditions any policy that can only be attained by vote trading can be beaten by some other policy. Thus, vote trading cannot produce equilibrium outcomes that improve upon the outcomes attainable under majority-rule without vote trading. In related work, Weiss (1988) argues that majority voting with vote buying will lead to Pareto suboptimal outcomes.

Our results provide reasons to be more optimistic, both about the possibili- ties for equilibrium in vote markets, and hence the possibilities for predicting vote buying outcomes, and also about the normative properties of such out- comes. As we show, when vote markets are organized by market specialists (possibly party leaders or committee or subcommittee chairs in legislatures), they can produce stable and Pareto improving outcomes.

The normative conclusions in our paper are similar to those of Koford (1982, 1987, 1993) and Levy and Philipson (1992), who also consider organized vote markets with market specialists. One key difference between these previous papers and our work is that we provide a complete equilibrium analysis in which both the supply and demand for votes are derived from internally consis- tent, utility maximizing, individual decisions. Equally important, the market specialist in our framework can perform his tasks with much less information about legislators' preferences than in the Koford and Levy and Philipson models.

Our efficiency results are also similar in spirit to results in Becket (1983) and Coughlin (1992), but our approach is quite different. Becker and Coughlin both assume the existence of certain aggregate or " reduced-form" political de- cision functions. Coughlin studies electoral competition for a political office where aggregate voter behavior is determined by a continuous, probabilistic vote function. Becket studies redistributive policy making given a monotonic and continuous "political influence funct ion". The functions in these models can be interpreted as the results of vote buying, but only implicitly. In contrast, we model individufil voting behavior and vote buying explicitly. Explicit con- sideration of the vote market is important because certain features of the trad- ed good - for example, the fact that only pivotal votes are valuable - mean that the existence of continuous and monotonic voting outcome functions is not obvious.

2. Equilibrium in an organized vote market

Consider a legislature with n members, which we denote by I -- { 1 ..... n}. To describe the vote buying mechanism, we first assume that the legislature's as- signment is simply to choose between two alternatives, x and y. The legislature

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makes its decisions by simple majority rule, and each legislator has exactly one vote. Later, in section 4, we address the issue of equilibrium in a larger policy space.

Each legislator i has a utility function that depends on the alternative adopt- ed by the legislature and also on the legislator's private " income . " This income may be thought of as cash, or as some political currency such as campaign ap-

pearances by party notables, private member bills, or special appropriations for the legislator's district. For analytical convenience, we assume the utility function is additively separable in policy and income. 1 Thus, if legislator i's

income is b i and alternative x is adopted, then i's utility is Ui(x,b ) ~ ui(x ) +

bi, while if y is adopted then i's utility is Ui(Y,b ) = ui(Y ) + b i. Let Ai(x,y ) -= ui(x) - ui(Y ) be the difference in i's utility if x is chosen rather than y, given a fixed level of income.

Let I x = {ilAi(x,y ) > 0] be the set of legislators who strictly prefer x to y and Iy = {ilAi(x,y ) ___ 0} be the set of legislators who weakly prefer y to x, let n x --- I Ix[ and ny ~ ] Iy], and assume that n x < ny. Thus, a minority of the legislators prefer x to y. We call I x the minority side, and Iy the majority side.

In addition to legislators, the legislature has a floor manager, who manages the market for votes. The general procedure for buying and selling votes is as follows. First, the floor manager chooses a price at which members of the minority side may buy votes. Next, he elicits vote demands from all members of the minority side, and vote supply decisions from all members of the majori- ty side. Finally, the floor manager concludes trades depending on the supply and demand choices, votes are cast, and the policy outcome (x or y) is deter- mined. As will be clear shortly, to complete his tasks the floor manager only needs to know two facts about the distribution of preferences in the legislature, and who is on each side of the issue. He does not need to know the entire distri- bution of preferences, or the preferences of individual legislators. 2

More formally, the vote buying process is described as follows. Let P =

IR + be the set of allowable prices the floor manager may announce, and let p fi P be a typical price announcement. The supply strategy of each majority member i is a function si: P -- {0,1 }, where si(P) = 1 means that i offers to sell his vote at price p, and si(P) = 0 means he does not. Let s -= <si l i~Iy> be the vector of supply decisions by all members of the majority side, and let S --- ~ s i be the aggregate supply of votes. The strategy of each m~mber j

iEIy of the minority side is a demand function dj: P -- IR + . Thus, dj(p) gives the number of votes demanded by legislator j when the market price is p. We allow legislators to buy fractions of votes, so vote demands need not be whole num- bers. This greatly simplifies the analysis, without affecting the basic intuitions. Let d = < dj I J ~Ix > be the vector of vote demands among the members of the minority side, and let D --- ~] dj be the aggregate demand for votes. A

jEI x

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strategy profile of the floor manager, the majority side, and the minority side, is a vector q --- (p,s,d).

Let N(q) be the total number of votes transferred to members of the majority from members of the minority, given the strategy vector q. The form of N is one of the rules that characterizes the market mechanism, which we specify be- low. A natural restriction is N(q) < min{D(q),S(q)}.

In the voting stage all legislators cast the votes they hold for their favorite alternative, so the side that has the most votes after trading wins. Let r = n

2 - n x denote the minimum number of votes required to yield a minority side victory. Then the voting outcome induced by the strategy vector q, which we denote v(q), is defined by

I ~ if N(q) > r (minority side victory) ' v(q) = if N(q) < r (majority side victory) (2.1)

Note that the minority side wins all ties. This is another simplifying assumption that does not effect the flavor of the results.

Legislators' payoffs depend both on the policy chosen and also on the total payments they make or receive as a result of their vote buying or selling activi- ty. Let ti(q) denote the net transfer to legislator i from trade. Then i's payoff is

O i ( q ) - - Ui(v(q)) + ti(q) (2.2)

Denote the payoff to the floor manager by UFM(q ). We assume the floor manager is compensated in such a way that he always prefers a situation with trade to one with no trade. That is, OVM(q) > OFM(q' ) whenever N(q) > 0 and N(q ' ) = 0.

Finally, we call a strategy vector q a vote buying equilibrium if it constitutes a pure-strategy, subgame-perfect Nash equilibrium in weakly undominated strategies to the game with the payoff functions above. We do not discuss mixed-strategy equilibria.

A standard model of a decentralized market equilibrium in which all buyers and sellers act as price takers would satisfy the following conditions: If the price p is such that the total supply of votes equals total demand, then N(q) = D(p), and all legislators who offer to supply their votes sell their votes for p, while all legislators who demand votes pay p per vote they demand. If the total supply of votes is not equal to total demand at price p, then no trade takes place at that price.

However, given this definition of a decentralized market, then there is never a vote buying equilibrium in which the minority side's preferred policy, x, wins. To see this, note that the minority side buys a positive number of votes only

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if x wins. Also, the minority side only buys the minimum number of votes re- quired to pass x. Thus, if x wins, then some members of the majority side do not sell their votes. However, those majority members would be better off sell- ing their votes (if p > 0), since this would not affect the outcome and they would receive a positive payment.

This observation motivates our study of a more centralized market mechan- ism. We therefore add the following three assumptions. First, we assume the floor manager rations any excess supply of votes, by randomly choosing sup- pliers. Second, we assume the floor manager distributes all revenues generated by vote selling evenly across all members of the majority side, regardless of which majority members are actually chosen to sell their votes. Third, we as- sume that trade occurs only if all members of the majority side offer to sell their votes. This means that any member of the majority side can veto a//trades, if he desires. The second assumption is especially important, because it insures that equilibria can exist with positive vote buying, at least for certain configura- tions of preferences. The third assumption insures that whenever trade occurs in equilibrium, this trade is Pareto-improving. 3 More precisely, our organized market mechanism satisfies the following:

I min{D(p),S(p)} if S(p) = ny and (i)' N(q) = 0 if S(p) ;~ ny

(ii)' ti(q) = I -Pdi(P)N(q)/D(p)

oN(q)/ny

if S(p) = ny and iEI x if S(p) = ny and iEIy, for each iEI. if S(p) ~ ny

The following result describes the conditions under which an equilibrium exists where vote buying actually occurs.

Proposition 1. If the market mechanism satisfies (i)' and (ii)', then there exists a vote buying equilibrium in which the minority side's preferred policy wins if and only if

E Ai(x,Y) > ny max IAj(x,Y)l (2.3) iEI x jEIy

Proof. All proofs are in the appendix.

The condition in proposition 1 is quite intuitive. A victory by the minority side can occur if and only if the aggregate willingness to pay for such a victory among the minority members exceeds the amount required to adequately com- pensate those members of the majority side that have the most intense prefer-

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ences for the majoritarian outcome. The reason is that any member of the majority side can veto vote buying. 4

The fact that all majority side members can veto trades implies that vote buy- ing occurs in equilibrium only if the resulting outcome is Pareto superior to the outcome obtained by majority voting without vote buying.

Comment 1. If q is a vote buying equilibrium in which the minority side's preferred policy wins, then Ui(q) _> ui(Y) for all i~I.

On the other hand, sometimes there does not exist an equilibrium with positive vote buying even though Pareto improvements are possible.

Example 1. n = 3; Al(x,y) = 5; A2(x,y ) = -1 ; A3(x,y ) = -3 . Equation (2.3) is not satisfied, since ny max IAj(x,y)[ --- 6 > 5 = ]2 Ai(x,y), so there does

jEIy iEI x not exist a vote buying equilibrium in which the policy chosen is x. However, Al(X,y ) > IAz(x,y) + A3(x,y) I, so Pareto improvements over the majority- rule outcome do exist.

If the floor manager were able to distribute the revenue generated by the minority side's vote purchases more rationally among the majority members, or, equivalently, if different members of the majority side were able to sell their votes at different prices, then efficient outcomes would occur even more often than with the single-price market mechanism described here. However, these more complicated schemes require that the floor manager know much more about legislators' preferences. In order to rationally allocate the revenue the floor manager would have to know the value of Ai(x,y) for individual majori- ty members. 5 The single-price scheme only requires that the floor leader know the aggregate of the minority side Ai(x,y) and the minimum of the Ai(x,y) (and who prefers x to y). In addition to the general desirability of designing a market-like mechanism that does not place a large burden on the market coor- dinator, the weaker informational assumptions are probably relevant when considering many real cases, such as legislatures with high turnover and voting on new or infrequently decided issues.

As noted in the introduction, one supposed advantage of vote buying and selling is that it allows collective choices to reflect the intensity of preferences, rather than purely ordinal information. As proposition 1 shows, this is clearly the case for the mechanism analyzed here. However, proposition 1 also implies that the legislators with the most intense preferences on the majority side play a special role in determining both the price of votes and the alternative chosen. This may not be exactly what proponents of vote buying have in mind, but we believe that any single-price market mechanism that produces equilibria will have a similar property.

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Proposition 1 merely establishes conditions under which a divisible surplus exists given that all votes must trade at the same price. The actual division of this surplus depends on the price chosen by the floor manager, and this choice will depend on the floor manager's payoff function. In many cases, the floor manager will be indifferent among a range of prices - for example, if his com- pensation depends only on the number of votes traded or the total surplus received by the minority and majority sides. On the other hand, if his compen- sation depends on the total dollar volume of trade, or the surplus received by the majority side only, then he may prefer a higher price.

3. Coordinating vote buying strategies and conditional trades

The minority side faces a coordination problem whenever more than one legis- lator must buy votes in order to change the policy outcome. If total demand by other legislators is too low, then no minority member will want to buy votes, since his purchases will not affect the policy outcome but he will still pay for the votes he buys. Consider the following example.

Example 2. n = 9; Ai(x,y ) = 5, i = 1, 2; A3(x,y ) = 4, Ai(x,y ) = - 1 , i = 4, .... 9. Equation (2.3) is satisfied, since ny max ]Aj(x,y)] = 6 < 14 =

jEIy

]~ Ai(x,y ), so there exist vote buying equilibria in which the policy chosen iEI x

is x. However, all of these equilibria require that two of the minority side legis- lators buy votes, since p must be at least 3. Since legislators 1, 2 and 3 make their demand decisions simultaneously, there are two types of equilibria, one with d 1 + d 2 + d 3 = 2, and one with d 1 = d 2 = d 3 = 0.

In most cases, this problem can be solved by making trade contingent on the policy outcome. Suppose legislators only pay for the votes they demand if the policy adopted by the legislature is the alternative they most prefer. Also, as- sume that legislators never choose weakly dominated strategies. Then, when- ever the price is set such that the maximum number of votes the minority side is willing to buy is equal to the number required to change the outcome, the coordination problem does not arise in equilibrium. To see this, consider the situation in section 2, but suppose that legislators must buy whole votes. Then the maximum number of votes that legislator i is willing to buy at price p is di(P) -- FLR(Ai(x,y)/P), where FLR(z) denotes the greatest integer less than or equal to z.

Comment 2. Suppose all vote sales are contingent on the policy chosen, and

suppose there exists p _> max IAj(x,y) l such that ~ di(P) - n + 1 n x and jEIy iEI x 2

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Ai(x,y ) > Pdi(P ) for all i~I x such that di(P) > 0. Then all vote buying equilibria in weakly undominated strategies have v(q) = x.

With contingent sales, legislators are not penalized for demanding votes when

total demand is too low to affect the final outcome. Moreover, if all of the minority side's demand must be filled in order to change the voting outcome,

then there is no opportunity for free riding. No minority members can reduce their own demand and hope that others will pick up the slack. Thus, for a range

of prices, any strategy with di(P) < di(P ) will be weakly dominated by one with

di(p) = di(P)- As an example, in example 2 any price p greater than 4 and less than 5 satis-

fies the conditions of comment 2. Given p E (4, 5), legislators 1 and 2 are each

willing to buy at most one vote, but legislator 3 is not willing to buy any votes. Thus, legislators 1 and 2 cannot free-ride - their weakly dominant strategies

are to buy one vote each. For some distributions of preferences, no price satisfying the conditions in

comment 2 exists. For example, if A3(x,y) = 5 in example 2, and all other A i are the same, then there is no p satisfying comment 2. More generally, the con- ditions in comment 2 cannot be satisfied whenever the number of minority

members with Aj(x,y) = max Ai(x,y ) is greater than n + 1 n x (the number iEI x 2

of votes required to change the outcome from y to x). This situation will tend to occur only when the minority and majority sides are roughly the same size,

or when Ai(x,y ) is roughly the same for all minority side members. If the

minority side is considerably smaller than the majority side, or if the Ai(x,y ) vary somewhat smoothly across the members of the minority side, then it will

generally be possible to find a price that solves the minority side's coordination

problem.

4. Equilibrium policies in one-dimensional policy spaces

We now apply the results above to the problem of choosing a policy from a one-dimensional policy space. Let the policy space be JR, and let each legisla- tor 's preferences over policies be described by a function of the form ui(x ) = u(x-zi) , where z i is legislator i's idealpoint. We assume that u is symmetric about zi, differentiable, and strictly concave. As in section 2, u describes legis-

lators' preferences over policies in units of income. Our goal is to extend the standard definition of a majority-rule equilibrium

to situations where vote buying is allowed. Thus, although we do not model the agenda process explicitly, we are considering a legislature with an "unres-

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tricted agenda", where alternatives to the status quo are routinely proposed

and any point in the policy space is a possible proposal. An equilibrium is a status quo that defeats or ties all other policies.

Recall that the vote buying game sometimes has multiple equilibria that result in different policies. We assume that whenever two policies are paired

against one another and a vote buying equilibrium exists in which the minority side's preferred policy wins, such an equilibrium occurs. That is, the members of the minority always solve their coordination problem. We therefore call

x* an equilibrium policy if and only if: (i) for all y such that y is preferred to x* by a weak majority of the legislature (i.e., ]{ilAi(x*,y)>0}] ___ I [ i lAi(x*,y)<0} l), equation (2.3) is satisfied for the pair (x*,y); and (ii) for

all y such that x* is preferred to y by a strict majority of the legislature, equa-

tion (2.3) is not satisfied for pair (y,x*).

4.1. Three legislators with quadratic preferences

We first characterize equilibrium for a simple case with three legislators. In sec-

tion 4.2, we consider a more general case. Suppose I - [ 1, 2, 3 }, and ui(x) -

- (x -z i )2 for i~I, with z 1 < z 2 _< z 3. Then the following is easily proved.

Proposition 2. There exists a unique equilibrium policy x*, which satisfies

I x R i f ~t ( X R

X* = ~t i f x R ~ ]1 ~ X L (4.1)

X L i f ~ > X L

where ~ = z2, x R ~ (2z 1 + z 3 ) / 3 , a n d x L --- (z I ÷ 2 z 3 ) / 3 .

Figure 1 shows the equilibrium policy x* as a function of ~, the median of the

ideal points, with z 1 and z 3 fixed. The intuition is as follows. As long as g is not too close to either z 1 and z3, the equilibrium policy is equal to ~t (the middle

case of (4.1)). If ~t is too close to z 1, however, then ~t is not an equilibrium policy, because z3's preference for a small rightward shift from ~t is intense enough that he is willing and able to purchase votes in order to achieve a policy to the right of g. The equilibrium policy in this case is x R (the top case of (4.1)).

The following comparative static result, which follows directly from propo- sition 2, is worth noting.

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X L

X R

X R

/

X R X L Z 3 g

Figure 1.

Comment 3. I f x* = XR, then x* is increasing in z 3, and if x* = XL, then x*

is increasing in z 1.

In section 4.2 below, we show results qualitatively similar to proposit ion 2 and

comment 3 which hold more generally. We can also say something about the efficiency of the equilibrium policy x*.

Let W(x) - ~ u(x,zi) be the sum of the legislators' dollar valuations of i~I

policy x, and let x** solve max W(x). Then x** = (z 1 + z 2 + z3)/3, the mean xEIR

of the ideal points, and the following is easily shown.

Comment 4. I f x* = XR, then x** > x* > g, and if x* = x L, then x** < x* < Ix. Therefore, W(g) <_ W(x*) _< W(x**), and if x* ~ g, then W(g) <

W(x*) < W(x**).

That is, the equilibrium policy under vote buying is always at least as efficient, in terms of the aggregate willingness to pay, as the median of the legislators' ideal points, and if the vote buying equilibrium is distinct f rom the median, then it is strictly more efficient. On the other hand, the vote buying equilibrium is not the most efficient policy.

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4.2. More general results

We now assume the legislature is composed o f a con t inuum of legislators, with ideal points distr ibuted over the interval [z,Z] according to the cumulat ive p rob- abili ty dis tr ibut ion funct ion F. Assume F has a cont inuous density, f, and

f(z) > 0 for all z E [z,Z]. Before character izing equi l ibr ium policies, we mus t character ize vote buying

equil ibria for the case o f a con t inuum of legislators. In fact , it is s t ra ightfor-

ward to p rove the fol lowing extension of p ropos i t ion 1.

Comment 5. Let x and y be any two policies, and let x be the policy prefer red by the minor i ty side. Then there exists a vote buying equi l ibr ium in which x

wins if and only if Iz~ [ u ( x - z ) - u ( y - z ) ] f ( z ) d z _ [u(Y--ZM)--U(X--ZM)]Iz f(z)dz, where Z x is the set o f legislators on the minor i ty side, Zy is the set o f legislators

on the major i ty side, and z M = _z if x > y and z N = ~ if x < y.

Recall our a s sumpt ion tha t whenever there are mult iple equil ibria to the vote buying game in which some of the equilibria involve posit ive vote buying, the equi l ibr ium that occurs is one with such vote buying. Given this assumpt ion ,

the equi l ibr ium policy can be character ized as follows.

Proposition 3. There exists a unique equi l ibr ium policy x*, which satisfies

I X R i f [.t ( x R

x* = ~t if x R ~ ~t .< x L (4.2)

x L if ~t > x L

where bt is the median of the legislators ' ideal points, x R solves

I ~ U'(XR--Z)f(z)dz + U'(XR--Z_)F(XR) = 0, and x L > X R solves x R

f~Lu'(XL--Z)f(z)dz + U'(XL--2)[1--F(XL) ] = 0.

This is similar to p ropos i t ion 2. I f the distr ibution o f ideal points is symmetr ic enough, then the equi l ibr ium policy is the median, bt (middle case of (4.2)). I f the distr ibution o f ideal points is skewed too far to the right, however , then ~t is not an equi l ibr ium, because a set o f legislators with high zi's will have suffi- ciently s t rong preferences for a small r ightward shift f r o m bt tha t they are will- ing and able to purchase votes in order to achieve a policy to the right o f bt. The equi l ibr ium policy in this case is x g ( top case). A symmetr ic a rgument holds if ideal points are skewed to the left.

The equat ions tha t implicitly define x R and x L are compl ica ted, and fur ther character izat ion is necessary. We focus on the top case, with x* = x R > bt.

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Symmetric results hold when x* = x L < Ix, and the intermediate case with x* = IX is straightforward.

The case of legislators with quadratic preferences is illuminating. In this case, u ' (g) = - 2 g for all g, and x R solves I z zf(z)dz - x R + _zF(XR) = 0. This

x R

can be rewritten as x R = [ I - F ( X R ) ] E [ z I z >-. XR] + F(XR)Z_ , where E[zlz_> XR] is the conditional mean of the ideal points greater than x R. Thus, x R is simply a weighted average of z and this conditional mean. This characterization can also

be used to show the conditions under which the equilibrium policy is x R. By proposit ion 3, x* = x R if and only if x R > Ix. It is straightforward to show that [1-F(x)]E[z[ z>_ x] + F(x)z - x is strictly decreasing in x, so x R > Ix if and

only if E[z[z>_ix]/2 + z / 2 - IX > 0, or E[z[z_>Ix]-Ix > Ix-z. Thus, x* = x R if and only if E[z[z_> IX]-IX > ~t-z. In words, this condition states that the difference between the mean of the set of ideal points greater than the median and the median is larger than the difference between the median and the smallest ideal point. This holds for any pareto distribution, many gamma dis- tributions, and other skewed distributions. 6

More generally, x* satisfies the following conditions. Given a distribution function F with support [_z,Z] and median Ix, let G be a median-preserving right- wardshift of F if and only if G(IX) = F(IX) = 1/2, G(z) < F(z) for all z > Ix, and G(_z) = 0.

Comment 6. (i) x* = x R > IX if and only if E[u'(ix-z)lz_>ix] > u'(_z-ix) (ii) if x* = x R > IX, then x* < (z_+Z)/2

(iii) if F and G are two distributions of legislator ideal points and G is a median-preserving rightward shift of F, and if

x*(F) = XR(F ) > IX, then x*(G) > x*(F).

Part (iii) is a generalization of comment 3, and is the most interesting f rom an empirical point of view, since it presents a comparat ive static that allows one to test for the presence and importance of vote buying. Specifically, it states that the skewness of the distribution of ideal points will often affect the vote buying equilibrium policy. Without vote buying, only the median ideal point matters. This is qualitatively similar to results in Coughlin (1992) and Koford (1993). We discuss this further in section 5 below.

Finally, it is straightforward to prove the followingresult about the relative efficiency of the equilibrium policy x*. Let W(x) = IZu(x-z)f(z)dz be the ag-

_z

gregate dollar valuation of policy x to the legislators, and let x** solve max x~lR

W(x). Note that x** is unique because u is strictly concave.

Comment 7. I f x* = x R, then x** > x* > Ix, and if x* = XL, then x** < x* < g. Therefore, W(IX) _< W(x*) <_ W(x**), and if x* ~ g, then W(g) < W(x*) < W(x**).

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In terms of the aggregate willingness to pay, the equilibrium policy under vote buying is always strictly preferred to the median whenever it differs from the median.

5. Some empirical evidence

Comment 6 shows how equilibrium policies with and without vote buying may differ. One way to test for vote buying is to look at public spending across a set of communities, such as states or cities. Suppose tax rates are fixed, and voters' most-preferred levels of public spending are increasing in income. Sup- pose also that decisions in the communities are made by legislative bodies whose members represent different income groups, such as state legislatures or city councils. Then the median voter theorem predicts that two communities with the same median income will have the same level of public spending, regardless of the degree to which the income distributions in the two cities are skewed. On the other hand, the vote buying equilibrium predicts that if one community 's income distribution is more skewed to the right than the other's, then spending will tend to be higher in the community with the more heavily skewed distribution. 7

We know of four empirical studies using municipal data that lend support to the vote buying hypothesis. 8 Ladd (1975) studied school expenditures in 78 Boston SMSA towns in 1970, and found that average residential wealth exerts a strong, positive influence on public school expenditures, even after controll- ing for median income, median tax-price and other factors. Feldstein (1975) found a similar effect in his study of towns in the Boston area for 1965 and 1970. Lovell (1978) found a positive effect of income skewness on school spending across a sample of Connecticut towns. Since he included no direct measure of median tax-price, however, the result may reflect a negative effect of tax-price on spending rather than vote buying. 9 Finally, in Philipson and Snyder (1992) we studied "common functions" expenditures across a sample of nearly 600 cities in 1982.1° We found that expenditures on common func- tions are increasing in the skewness of the income distribution, even controlling for median income and median tax-price. 11

While these findings are consistent with vote buying, several caveats must be mentioned. First, if the tax-price variables are poor measures of the " t r u e " tax- price perceived by the median voter, and fail to capture fully the effect of high- income citizens on the tax base, then the income skewness variables might be picking up an unmeasured tax-price effect. In communities where average household income is much larger than median household income, the tax-price of the median household will be lower because of the larger tax base. Second, not all citizens are voters, and, in particular, voters tend to have higher than average incomes. Finally, all of the studies above assume that ideal points are increasing in income. If ideal points are, say, a "backward bending" function

260

of income, then the median-income taxpayer is n o t the voter with median ideal point.

Clearly, empirical studies that treat these issues carefully are needed to better test vote buying models against the median voter model. Also, other tests might be designed by assuming that votes are bought with certain types of govern- ment spending. For example, if some spending categories can be viewed mainly as payments for votes, then spending in these categories should be positively related to the skewness of the income distribution, but not to median income. At a minimum, researchers ought to be more cautious in drawing inferences from studies that use the median voter model to estimate the demand for spending on particular programs (as is typical in the literature following Berg- strom and Goodman 1973). If the vote buying hypothesis is correct, then the estimated equations may not even be structural demand equations.

Concluding remarks

While our analysis focuses on a particular set of vote buying rules, we suspect many of the results hold more generally. In particular, we conjecture that results similar to propositions 1 -3 must hold for any "single-price" vote buy- ing mechanism. Our results also point out a necessary feature of any mechan- ism that produces even more efficient outcomes than those above, namely that it will require the floor manager to have much more information about legisla- tors' preferences (or an effective demand-revelation mechanism).

On the other hand, much work remains to be done. We consider only an en- vironment of relatively complete information, in which all legislators know at least the distribution of preferences. Relaxing these informational assumptions is an important task for future work.

Another promising direction for future research lies in studying the distribu- tion of vote payments. If we interpret the payments made to buy legislators' votes as additional government spending in the legislators' electoral districts (extra highway projects, government contracts, military bases, and so on), then a vote market similar to ours may yield interesting predictions about the geo- graphic distribution of government spending. Whenever vote buying occurs, the members of the minority side with the most intense preferences pay the most. Assuming legislators pay for votes by agreeing to eliminate or reduce spending in their own district and increase spending in other districts, spending will be lower in the districts of those who buy votes, and higher in the districts of those who sell their votes, than it would without the vote buying. If legisla- tors' ideal points are also highly correlated across issues, 12 then the intense minorities on various issues will be similar, and composed of "extremists". Vote buying on a regular basis would then result in higher government spend- ing in "modera te" districts than in more extreme districts, because legislators

261

from extreme districts would more often sacrifice spending in their districts in exchange for votes on other policy matters, while legislators from moderate spending in "mode ra t e " districts than in more extreme districts, because legis- lators from extreme districts would more often sacrifice spending in their dis- tricts in exchange for votes on other policy matters, while legislators from moderate districts would more often obtain extra spending for their districts in exchange for their votes. This is similar in spirit to results in Denzau and Munger (1986) and Snyder (1991), but the underlying logic is rather different.

A third extension is to turn the vote buying mechanism into a mechanism for organized vote trades, by considering several issues simultaneously. Whenever there are pairs of voters on opposite sides of two different issues, each of whom wishes to buy votes on just one of the issues, their purchases on the pair of is- sues can be "net ted ou t " , and thereby turned into vote trades.

Finally, more work needs to be done on the issue of free riding in vote buy- ing. Although we discussed one potential solution in section 3 - making trade conditional on the policy outcome - there are other possibilities. One is to model majority or minority leaders, as actors that may internalize the externali- ties that are inherent in vote buying on the minority side. The informational constraints on these leaders, and the "con t rac t " specifying their compensa- tion, are clear concerns. In the extreme case when each leader has complete in- formation about legislators' preferences on his side, the problem is straightfor- ward since a complete set of price discriminating trades can be executed to deliver the most efficient outcome. More interesting cases must therefore im- pose significant limits on the leaders' information.

While understudied, the potential benefits of an organized vote market have been recognized by some scholars. Koford (1982) argues that centralized vote trading is a more accurate characterization of what occurs in American legisla- tures than the decentralized vote trading models typically studied.

"F loor leaders have substantial control over the legislative agenda: what bills are to be brought up, in what order, what amendments are to be brought up, and so on.. . There is overwhelming evidence that party leaders engage in implicit and explicit vote-trading to pass bills desired by party members" (p. 248).

Although we have not attempted to find empirical counterparts to our floor manager, legislative party leaders are obvious candidates.

Appendix

Proof of Proposition 1. Suppose q is a vote buying equilibrium with v(q) = x. Then Ai(x,y ) _ Pdi(P) for all i~Ix, so 1~ Ai(x,y ) >_ pD(p). Also, pD(p)/ny

i E I x

_> [Ai(x,y)[ for all ifiIy, since otherwise i would prefer to veto all trades by

262

choosing si(P) = 0. Thus, (2.3) must hold. If (2.3) holds, then q is an equi- librium with v(q) = x as long as: p ~ [(nv/r)max IAj(x,y)], ( l / r ) ]]

j~Iy i~lx Ai(x,y)]; si(P) = 1 for i E Iy; D(p) = r; and Ai(x,y ) _> Pdi(P) for all i E I x. The floor manager chooses p in the required interval, because UrM is higher when there is trade.

Proof of Proposition 2. If x > Z2, then x is an equilibrium policy if and only if the following conditions hold: (i) for all y > x, A3(Y,X ) < -2Aa(y,x); (ii) for all y~[2Zz-X,X), A3(x,y ) _> -2Al(x,y) ; and (iii) for all y < 2z2-x, Al(Y,X ) < -2A3(y,x ). Since Ai(Y,X ) = -Ai(x,y), conditions (i) and (ii) are the same except that (i) has a strict inequality and (ii) has a weak inequality, and

can be written as: (i)' 2Ul(X)+U3(X ) > 2ul(y)+u3(y ) for all y > x ; and

(ii)' 2Ul(X) + u3(x ) _> 2Ul(Y) + u3(y ) for all y~ [2Zz-X,X ). Since u s and u 3 are both concave, xa ~ (2z 1 + z3)/3 is the unique point that

maximizes 2ul(x)+ u3(x) over IR. Suppose x R > z 2. By definition, x R satisfies conditions (i)' and (ii) ' , and satisfies (ii)' with a strict inequality. Also, x R

satisfies (iii), since for all y < 2Zz--xR, 2u3(xR)--2u3(Y ) > U3(XR)--u3(Y ) > 2ul(y)--2Ul(XI~ ) > uI(Y)--Ul(Xk). The second inequality again follows because x R maximizes 2Ul(X)+ u3(x). Thus, x R is a policy equilibrium. Also, x R is the unique policy equilibrium, since in checking (i) ' , (ii)' and (iii) we have shown that x R defeats all other points.

A symmetric argument shows that if z 2 > (z 1 + 2z3)/3, then the unique poli- cy equilibrium is x L -= (z 1 + 2z3)/3. Lastly, consider the intermediate case with

(2z 1+z3)/3 ~ z 2 _< (z 1+2z3)/3. For all x > z 2, x > z 2 _> (2z 1+z3)/3, so 2Ul(X) + u3(x) < 2Ul(Z2)+ u3(z2). Thus, z 2 beats x at the vote buying equilibri- um via the majority coalition { 1,2], so x is not a policy equilibrium (x does not satisfy (ii)). By a symmetric argument, no x < z 2 is a policy equilibrium, because z 2 defeats all such points. Finally, z 2 is a policy equilibrium, since (as just shown) it defeats all other points.

Proof of Proposition 3. The basic logic is similar to that in the proof of propo- sition 2, although the details are considerably more complicated. A complete proof is available from the authors on request.

Proof of Comment 6. Available from the authors on request.

Acknowledgement

We thank Gary Becker, James Heckman and an anonymous referee for their helpful comments on earlier drafts of this paper.

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Notes

1. This assumption is widely used in welfare economics and also in industrial organization (e.g., Groves, 1973; Salop, 1979). The assumption is not necessary for most of the qualitative results.

2. In fact, it is straightforward to design a mechanism in which legislators themselves reveal whether they are members of I x or Iy, by requiring that legislators announce who they are will- ing to buy votes from, as well as how many votes they wish to buy. In equilibrium, the members of the minority side with a positive demand for votes will reveal who is on the majority side, because they will never offer to buy votes from fellow minority side members.

3. Romer and Rosenthal (1983) show how a mechanism with vetoes can improve upon majority rule in collective choices involving pollution externalities.

4. This is what is assumed in Koford (1982, 1987, 1993) and Levy and Philipson (1992). 5. If minority members are forced to buy whole numbers of votes, then the necessary and suffi-

cient condition for a vote buying equilibrium in which the minority side's preferred policy wins

is ]2 FLR( maxrAi (x'Y)[Aj (x,y)l ) >__ r, where for any zEP,, FLR(z) is the greatest integer less iEix ny jEIy

than or equal to z. 6. Strictly speaking we must consider truncated pareto or gamma distributions, since our analysis

assumes that the support has a finite upper bound. 7. The assumption that preferences over the public good and private income are separable is not

innocuous in this context, since it implies that payments for votes do not affect voters' ideal points - that is, payments do not have a significant income effect. Relaxing the separability assumption, however, does not change the qualitative results of comment 6. Intuitively, if a voter receives a payment for his vote, then the increase in income increases his demand for pub- lic goods, which means that his vote for a given increase in the leve~ of public goods is slightly less expensive. On the other hand, the decrease in income due to expenditures on votes causes the demand for public goods by vote buyers to fall. Assuming that vote buyers take these ef- fects into account when choosing their payments, the equilibrium level of public goods still satisfies the flavor of comment 6.

8. City councils are generally small; for example, the average council size in the sample of cities studied in Philipson and Suyder (1992) is about 8. Nonetheless, they are miniature legislatures, and the literature on municipal government contains many examples of individual council members catering to special interests and engaging in apparent logrolling (e.g., Eulau and Prewitt, 1973; Goodman, 1975).

9. Lovell himself interprets the variable in terms of tax-price. 10. "Common functions" are police, fire, highways, parks and recreation, sanitation, sewerage,

financial administration, general control, and general building. Clark and Ferguson (1983) prefer this spending variable, arguing that it is comparable across cities.

11. Mueller and Murrel (1986) also provide evidence consistent with vote buying. In a study of de- veloped countries, they find that national government spending as a share of GDP is increasing in the number of interest groups. To the extent that the number of interest groups is a good proxy for the number of intense minorities that prefer high government spending on particular programs, their finding is evidence that vote buying or logrolling takes place across spending programs.

12. See Schneider (1979) and Poole and Rosenthal (1985, 1991) for evidence that this is true for the U.S. Congress.

264

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