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Negative Vote Buying and the Secret Ballot
John Morgan
Haas School of Business, University of California, Berkeley
Felix Várdy∗
Haas School of Business, University of California, Berkeley,
and International Monetary Fund, Washington, DC
1. Introduction
The use of voting to make collective decisions inevitably brings with it the possibility of vote buying. Thus, a crucial aspect of designing voting procedures
is to ensure that election outcomes reflect the will of the electorate, rather than
the wallets of interest groups. An important reform along these lines was the
introduction of the secret ballot. Before the introduction of the secret ballot,
vote buying was common and, often times, quite open. For instance, Seymour
(1915) reports that in English parliamentary elections of the 19th century, the
price of a vote was often posted openly outside the polling station and updated numerous times over the course of the day, much as a stock price. Some
50 years later, little had changed, at least in Texas. For instance, Caro (1982)
recounts the following vote buying scheme organized by Lyndon Johnson on
behalf of Maury Maverick’s 1934 Congressional campaign:
Johnson was sitting at a table in the center of the room—and
on the table there were stacks of five-dollar bills. “That big table
∗ Email:
[email protected]
The first author thanks the International Monetary Fund for its generous hospitality and
inspiration during the first draft of this paper. Morgan also gratefully acknowledges the financial
support of the National Science Foundation. The opinions expressed in this paper are those of the
authors and should not be attributed to the International Monetary Fund, its Executive Board, or
its Management.
The Journal of Law, Economics, & Organization, Vol. 28, No. 4
doi:10.1093/jleo/ewq016
Advance Access published November 26, 2010
c The Author 2010. Published by Oxford University Press on behalf of Yale University.
All rights reserved. For Permissions, please email: [email protected]
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We offer a model of “negative vote buying”—paying voters to abstain. Although
negative vote buying is feasible under the open ballot, it is never optimal.
In contrast, a combination of positive and negative vote buying is optimal
under the secret ballot: Lukewarm supporters are paid to show up at the
polls, whereas lukewarm opponents are paid to stay home. Paradoxically, the
imposition of the secret ballot increases the amount of vote buying—a greater
fraction of the electorate vote against their intrinsic preferences than under the
open ballot. Moreover, the secret ballot may reduce the costs of buying an
election. (JEL D71, D72, D78)
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Also in this case, the workability of the scheme depended crucially on the
observability of the votes cast. The Maverick campaign was in an enviable
position in this regard since Texas thoughtfully matched a voter’s registration
number with the number on his ballot.1
In an important study, Cox and Kousser (1981) trace the effects of the imposition of the secret ballot on vote buying in New York elections. Interestingly,
they find that the secret ballot substantially changed the form, but not necessarily the amount, of vote buying. Specifically, before the imposition of the
secret ballot, “regular” vote buying dominated. In contrast, after the imposition of the secret ballot, “negative” vote buying—that is, paying people to
abstain—increased dramatically. The rationale for this shift is nicely explained
by a turn-of-the-century New York Democratic state chairman: “Under the
new ballot law you cannot tell how a man votes when he goes into the booth,
but if he stays home you know that you have got the worth of your money.”
(Cox and Kousser: 656). In fact, if one judges the pervasiveness of vote buying
by the number of mentions in newspapers, then the study by Cox and Kousser
reveals no diminution after the imposition of the secret ballot.
The Cox and Kousser study raises a number of questions. For instance, why
was negative vote buying so rare before the introduction of the secret ballot?
Standard marginal economic analysis would suggest the optimality of using all
available vote buying tools. More fundamentally, did the secret ballot achieve
its policy objective of reducing corruption? From Cox and Kousser study, it is
unclear that vote buying was, in any way, reduced. To examine these questions,
we study a theoretical model of positive and negative vote buying with competing interest groups. More specifically, we use the Groseclose and Snyder
(1996) vote buying framework as a starting point and enrich their model in
three respects: First, we allow for abstention, thus making negative vote buying possible. Second, we study both the open and the secret ballot; under
the open ballot, interest groups can contract on actual votes, whereas under
the secret ballot, contracts can be contingent only on whether a voter shows
up at the polls. Third, we endogenize the timing of the interest groups’ vote
buying.
1. In Texas, the ballot was not truly secret until 1949.
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was just covered with money—more money than I had ever seen,”
Jones says. (. . . ) Mexican American men would come into the
room one at a time. Each would tell Johnson a number—some,
unable to speak English, would indicate the number by holding up
their fingers—and Johnson would count out that number of fivedollar bills, and hand them to him. “It was five dollars ah vote,”
Jones realized. “Lyndon was checking each name against a list
someone had furnished him with. These Latin people would come
in, and show how many eligible votes they had in the family, and
Lyndon would pay them five dollars a vote.”
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Our main results are:
The article proceeds as follows. In the remainder of this section, we review
the extant literature. In section 2, we outline the model. Section 3 characterizes
the optimal vote buying strategy under the open ballot, whereas section 4 undertakes the same exercise for the secret ballot. In section 5, we compare the
scope of vote buying and the buyability of voting bodies under the open versus
the secret ballot. Finally, section 6 concludes. All proofs are in the Appendices
A and B.
1.1 Related Literature
In their seminal paper, Groseclose and Snyder (1996) (hereafter, GS) showed
that buying a supermajority of voters is optimal when interest groups compete.
GS study the open ballot. That is, they allow contracts to be contingent on actual votes. Dal Bo (2007) shows that an even richer contract space makes vote
buying essentially free for a monopoly interest group (See, also, Morgan and
Várdy 2007). Felgenhauer and Gruener (2003) extend Dal Bo’s framework to
allow for competing interest groups and private information. They study the
effects of the open versus the secret ballot in a Condorcet type model. Stokes
(2005) offers a dynamic model of vote buying with a monopoly interest group.
Her model offers a blend between open and secret ballots—there is some probability that a voter’s action will be observed ex post. Using folk theorem type
arguments, she offers conditions in which vote buying takes place despite the
imperfectness of monitoring. Finally, Seidmann (2006) examines the effect
of ex post rewards by outsiders on votes cast by members of a committee.
The possibility of outside rewards creates a divergence between social benefits
and private incentives for committee members. The divergence is affected by
the degree of openness of the voting rule since it permits outsiders to draw
differing inferences about a committee member’s vote.
Our analysis differs from the previous literature in several respects. Abstention, and hence the possibility of negative vote buying, is absent from the extant theoretical literature. Second, none of the models endogenize the timing
2. An election is said to be bought if and only if the outcome does not reflect the intrinsic
preferences of the majority.
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1. Under the open ballot, negative vote buying is never optimal.
2. Under the secret ballot, the optimal contract entails both positive and negative vote buying—lukewarm supporters are paid to show up at the polls,
whereas lukewarm opponents are paid to stay home.
3. More voters vote against their intrinsic preferences under the secret ballot
than under the open ballot. In other words, the secret ballot increases the
amount of vote buying.
4. In close elections where a policy has many lukewarm supporters, buying
the election is cheaper under the secret ballot than under the open ballot.2
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2. The Model
Our model is essentially identical to that of GS, save for endogenizing the interest groups’ order of moves, introducing the option of abstention, and studying
both the open and the secret ballot.
There is a continuum of voters with unit mass who can choose between two
policies, labeled a and b. Voters are heterogeneous with type θ. Without loss
3. For legal and ethical considerations associated with various forms of vote buying, see, for
example, Hasen (2000) and Karlan (1994).
4. Of course, vote buying is only one possible avenue for explaining turnout suppression under
the secret ballot. The secret ballot has variously been viewed as a mechanism for marginalizing
party machines, advantaging incumbents, and disengaging voters. See, for instance, Rusk (1970),
Burnham (1970), and Converse (1972) for a lively debate about the various possible causes of
reduced turnout.
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of vote buying. Third, most of the literature (with GS as the notable exception)
focuses on small elections, where pivotality and, hence, instrumental motives
are at the fore. We focus on large elections, where pivotality incentives do
not play much of a role. Specifically, we follow GS and assume that voters
behave as if they only had expressive preferences. Morgan and Várdy (2008)
offer a formal justification for this assumption: As long as there is some uncertainty about the preferences of an arbitrarily small fraction of the electorate,
the probability of being pivotal uniformly converges to zero as the size of the
electorate grows. Hence, when voters have both instrumental and expressive
preferences, the incentives from expressive preferences completely dominate
in large elections.
In addition to the theoretical literature reviewed above, there is a considerable literature on the practice of vote buying.3 Shaffer and Schedler (2007),
for instance, offer an excellent overview of various vote buying techniques.
The seminal paper on negative vote buying is Cox and Kousser (1981). They
highlight how the imposition of the secret ballot in upstate New York affected
vote buying practices. In particular, prior to its imposition, vote buying was
almost exclusively of the “positive” (get out the vote) variety, whereas after
its imposition, negative vote buying aiming to suppress turnout became much
more prominent. Heckelman (1995) finds the same turnout suppressing effect
of the secret ballot using a panel data set of US gubernatorial elections from
1870-1910.4 Stokes (2005) as well as Nichter (2008) examine vote-buying
practices of the Peronists in Argentina. They find evidence of positive vote
buying; however, they differ in their interpretation of the data—Nichter finds
evidence to suggest that vote buying is concentrated on mobilizing lukewarm
supporters, whereas Stokes sees the data as more consistent with compensating lukewarm opponents in exchange for their votes. Finally, Vincente (2007)
uses randomized field experiments in São Tome and Principe to identify votebuying patterns. He finds evidence that vote buying disproportionately benefits well-financed challengers and that voter information campaigns can be
effective in reducing vote buying.
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v(a; θ) − v(0;
/ θ) = v(0;
/ θ) − v(b; θ).
Because v(b; θ) = 0, we have
1
v(0;
/ θ) = v(a; θ)
2
(1)
5. To see that the assumption of uniformity is without loss of generality, suppose that the underlying type, ω, is distributed according to some cumulative distribution function (cdf) F(ω). Now
remap types from ω → θ by letting θ = F(ω). Because F (∙) is a cdf, θ is uniformly distributed.
6. This result is immediate in a model with a continuum of voters, such as this one. Morgan
and Várdy (2008) show that the result extends to the discrete case, even in the presence of strategic
vote buying.
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of generality, we assume that θ is uniformly distributed on the unit interval.5
In addition, there are two interest groups, A and B, that are trying to affect
the policy choice. Group A prefers policy a, whereas group B prefers policy b.
Excluding the cost of buying votes, group B receives a payoff X > 0 when
policy b is adopted and zero when policy a is adopted. Conversely, group A receives Y > 0 when policy a is adopted and zero when policy b is adopted. Thus,
groups A and B have diametrically opposed policy preferences. We assume that
Y is sufficiently large relative to X such that group A, whose preferred policy
a does not command majority support (see below), indeed wants to buy the
election.
To induce voters to vote for their preferred policy, the interest groups can
offer (enforceable) contracts prescribing contingent transfers from the interest
group to voters. We will vary the contingencies on which these contracts can
be based.
Apart from money, voters derive utility solely from expressive motives—the
utility derived from voting according to their convictions—rather than from
instrumental motives—utility derived from the election outcome. Clearly, this
is a simplification since real-world voters likely derive utility from both expressive and instrumental motives. Instrumental motives, however, only matter to the extent that a voter is pivotal to the election outcome, and, in large
elections, this probability becomes vanishingly small. Thus, regardless of the
weight placed on instrumental motives, in large elections, voters’ optimal behavior is indistinguishable from the case where only expressive motives are
present.6
Voters’ preferences are linear in money and additively separable with respect
to the act of voting. Voting for policy a provides a voter of type θ with a utility
v (a; θ). Likewise, voting for b provides the same voter a utility v (b; θ). Finally,
expressing no view, that is, abstaining, yields a utility v (0;
/ θ). Without loss of
generality, we set v (b; θ) equal to zero for all types θ. We shall refer to voters
with types θ such that v (a; θ) > 0 as intrinsic a supporters, and to voters with
types θ such that v (a; θ) < 0 as intrinsic b supporters.
We assume that, relative to abstaining, the utility of voting for one’s preferred policy is the same as the disutility of voting for the nonpreferred policy,
that is,
Negative Vote Buying and the Secret Ballot
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1. v(θ) is continuous, strictly decreasing, and differentiable almost everywhere.
2. v 12 < 0
3. v(0) = −v(1) = ∞.
The first assumption merely ensures that preferences are “well behaved” and
that voters are ordered in a sensible way. The second assumption ensures that
the median voter strictly prefers policy b. Hence, in the absence of interest
group interference, policy b will be chosen in the election, whereas a win for a
implies that the election has been bought. The third assumption amounts to a
set of “Inada conditions” and merely rules out corner solutions.
It is useful to define θ0 to be the type who is indifferent between voting for A
and voting for B. Note that, given our assumptions on v (∙), θ0 exists, is unique
and is strictly less than 12 .
We follow GS and assume that the preferences of the voters are commonly
known to all parties, thus allowing the interest groups to write contracts specific to the preferences of each voter. Although this is clearly not literally true
in practice, it seems a reasonable approximation given the availability of observable characteristics such as race, gender, and metropolitan statistical area
that correlate with preferences in the aggregate.
The extensive form of the game is as follows. In the lead-up to the election,
which takes place at time τ = 1, interest groups may offer contracts to voters.
Time is continuous and interest groups are free to make offers at any point in
time τ ∈ [0, 1).8 Once an interest group makes its contract offers, it can make
no further offers and its move is visible to its rival. If interest groups make
offers at the same time, the order is determined by a coin toss.9 Finally, each
7. The alternative preference specification produces essentially identical results to Propositions
1–5. It differs from the main text in that the imposition of the secret ballot always increases the
cost of buying the election.
8. The openness of the time interval at τ = 1 ensures that an interest group always has the
opportunity to react to contract offers made by the opposing interest group, by making counter
offers.
9. The particular tie-breaking rule plays no role in the analysis.
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Hence, for a voter of type θ, the difference in utility from voting for a as opposed to abstaining is simply v(a; θ)/2, whereas the difference between voting
for b and abstaining is −v(a; θ)/2. Equation (1), together with the normalization of v(b; θ), implies that v(a; θ) is a sufficient statistic for voter preferences.
For ease of exposition, we use v(θ) to denote v(a; θ) in the remainder of the
paper.
In the Appendix, we show that the model is robust to other assumptions
about voter preferences. Specifically, we analyze the polar case where the disutility from abstention is the same as that from voting for the nonpreferred policy. That is, the payoffs of failing to “do the right thing” are the same regardless
of whether the voter commits a “sin of commission” (i.e., votes for his nonpreferred policy) or commits a “sin of omission” (i.e., abstains).7
We make the following regularity assumptions on the v (∙) function:
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3. Least-Cost Vote Buying under the Open Ballot
In this section, we study vote buying under the open ballot. That is, actual votes
are contractible. To develop a benchmark, we first consider the case where
abstention is not allowed. When the order of moves is specified exogenously
and A moves first, this is the model analyzed by GS. Their main result is to
show that A will optimally preempt B from making any counteroffer by paying
transfers sufficient to induce a supermajority to vote for a in the election.
Optimal vote-buying contracts are more easily characterized in terms of the
utility—or “surplus”—they provide to voters than in terms of the actual underlying transfers. As there is a one-to-one relationship between transfers and
10. Because the strategies of interest groups are functions, we cannot directly adopt Selten’s
trembling hand refinement for extensive form games.
11. The tie-breaking conventions merely serve to get rid of the usual open set problems in
studying subgame perfect equilibria. If we discretized the transfer space, then any tie-breaking
convention would produce essentially the same results as the one we have adopted.
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voter may opt for one of the contracts he has been offered and votes. The policy
receiving the majority of votes cast is adopted and payoffs are realized.
In this game, we study vote buying under the open ballot and under the secret
ballot. Under the open ballot, actual votes can be monitored and contracted
upon. Under the secret ballot, only turnout can be monitored and contracted
upon. More specifically, contracts take the following form. Under the open
ballot, interest groups can offer payments in exchange for one (and only one)
of the following contingencies: (1) a vote for a, (2) abstention, or (3) a vote
for b. Under the secret ballot, interest groups can offer payments in exchange
for: (1) showing up at the polls or (2) abstaining. To ease exposition, it is
useful to assume that if at time τ = 1 an interest group has not made an offer to
a voter, then it is assumed to have offered a null contract; that is, the promise
of a zero transfer in all contingencies.
We study subgame perfect equilibria of the game. To rule out “nuisance”
equilibria where interest groups make contract offers under the assumption
that they will not be accepted owing to a later counteroffer, we use a trembling
hand type refinement.10 Specifically, we suppose that there is an infinitesimal
possibility that no competing interest group is present. That is, with arbitrarily
small probability, an interest group is a monopolist. Moreover, since there are
a continuum of voters, any zero measure variation on a given equilibrium is
also an equilibrium. Thus, the term “unique” should be understood to mean
unique up to a measure zero change in strategies, and the term “all” should be
understood to mean “almost all.”
We adopt the following tie-breaking conventions: (1) If an interest group
moving second can do no better than to propose the null contract, we assume
that it opts for this strategy. (2) If contracts from A and B provide the same
(high) payoffs, then voters select whichever contract was proposed most recently. (3) If each side gets 50% of the votes cast, then policy b is adopted.11
Negative Vote Buying and the Secret Ballot
825
surplus, it is indeed equivalent to think in terms of surplus rather than transfers. Specifically, to provide a voter of type θ with a surplus s requires a transfer
t(θ) = s − v(θ).
GS show that the “least-cost successful contract,” that is, the cheapest possible profile of contracts through which group A can guarantee its preferred
policy, is given by:
and θ∗a is the unique solution to
arg min
Z θa
θa > 12 0
t(θ)dθ.
The proof of the Proposition 1 follows immediately from Propositions 1
and 2 on pp. 307 and 309 in GS.12
Notice that, in equilibrium, A successfully deters B from making any counteroffers. Therefore, in the proposition, we have only described A’s optimal
strategy. Indeed, interest group B’s optimal strategy is straightforward: Do
nothing if rebribing costs more than X (which is the case in equilibrium). Else,
buy a minimum winning coalition at the lowest possible cost.
Figure 1 illustrates the form of the least-cost successful contract offered by
A. All voters in group A’s coalition, [0, θ∗a ], receive surplus of at least s(θ∗a ),
and a supermajority of voters are recruited into the coalition (i.e., θ∗a > 12 ).
Now suppose that we endogenize the order of moves in the GS model. Our
next proposition shows that the extensive form GS analyzed is in fact the
unique subgame perfect equilibrium outcome of the game where the timing
of bribes is also a strategic decision. That is, group A always moves first.
To gain some intuition for why this is the case, let CM denote the cost to
group A of securing its preferred policy outcome as a monopolist, that is, in
the absence of group B. Notice that A’s optimal strategy as a monopolist is
very simple: It pays a transfer −v(θ) to voters with types θ0 6 θ 6 12 + ε in
exchange for voting for a, where ε is arbitrarily small. Let CGS denote the cost
to A of the least-cost successful contract derived in Proposition 1. It may be
readily verified that CGS > X +CM . This implies that it is never in B’s interest
12. Technically, the least-cost successful contract is only unique in terms of payoffs, not in
terms of strategies. Specifically, group A can offer a continuum of payoff-equivalent contracts that
differ only in bribe offerings that are not accepted in equilibrium. In the remainder of the article,
we use “unique” in the payoff sense.
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Proposition 1. (Groseclose and Snyder 1996) Without abstention, the unique
least-cost successful contract offered by A is as follows:
In exchange for voting for ‘a,’ all voters with type θ 6 θ∗a receive transfers
t(θ) = max(0, s(θ∗a ) − v(θ)) and earn surplus equal to max(s(θ∗a ), v(θ)).
where
X
s(θ∗a ) = ∗ 1
θa − 2
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to move first since, if it did, A could “neutralize” B’s offers at a cost of at most
X and then get its most preferred policy at an additional cost of at most CM .
Hence, by moving first, B only makes it cheaper for A to buy the election.
In fact, because with infinitesimal probability group B is a monopolist which
gets its way even if it does nothing, moving first leaves B strictly worse off.
Finally, because we have assumed that Y is sufficiently large such that group
A prefers buying the election over losing the election, and because not moving
means that policy b is adopted, group A indeed prefers moving first over neither
interest group moving at all. Notice that this intuition does not depend on the
particulars of whether voters can abstain or whether the ballot is open or secret.
Formally:
Proposition 2. In all equilibria of the vote buying game with endogenous
moves, group A moves first.
Formally, in all equilibria, group A offers a least-cost successful contract at
some time t ∈ [0, 1). As long as A has not yet offered a contract, or A has offered
a successful contract, group B does nothing. Else, at some time t 0 ∈ (t, 1), B
buys a minimum winning coalition at the lowest possible cost.
Proposition 2 shows that, generically, an equilibrium in the game takes the
following form. At some time 0 6 t < 1, group A makes its offers. If this is
a successful contract, then B optimally does nothing. If, however, the contract
is insufficient to preempt B, then B (partially) “negates” A’s contract and gets
its preferred policy adopted in the election. Notice that there is a continuum
of economically equivalent equilibria that only differ in the timing of A’s offer
and the timing of B’s potential reply.
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Figure 1. Least-cost successful contract offered by A. (Groseclose and Snyder 1996).
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Proposition 2 crucially depends on the fact that policy b is favored by the
majority of voters, thus affording B the luxury of waiting. Were policy a favored by the majority, then group A would find it optimal to wait. Furthermore,
since group B derives less benefit from seeing its preferred policy passed than
does group A, the unique subgame perfect equilibrium in this case would be a
“fair” election—neither side chooses to buy votes.
3.1 Abstention
Lemma 1. Under any least-cost successful contract offered by A under the
open ballot, there exists θa , θb ∈ (0, 1) such that
1. Voters with types θ ∈ [0, θa ] vote for a.
2. Voters with types θ ∈ (θa , θb ) abstain.
3. Voters with types θ ∈ [θb , 1] vote for b.
Lemma 1 states that, under the least-cost successful contract offered by A,
the sets of voters making each of the choices a, b, and 0/ are convex. The
intuition for the ordering of the sets is that if group A wants to recruit a given
fraction of voters, it is cheapest to recruit from those who are least hostile
toward policy a. Thus, a generic least-cost successful contract has boundary
points θa and θb as illustrated in Figure 2. Voters with types to the left of θa
are induced to vote for a, whereas those to the right of θb vote for b. Voters
with types between θa and θb are induced to abstain. (Later we will show that,
in fact, θa = θb . Hence, nobody abstains.)
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What happens when voters can abstain? Abstention provides a new and potentially useful tool to the first mover. Because Proposition 2 continues to hold,
the first mover is group A. Since intrinsic b supporters dislike abstention only
half as much as they dislike voting for a, it would seem that group A could use
this to economize on the cost of buying the election. On the other hand, abstention also provides a new tool for the opposing interest group, B, to counter A’s
offers. And since A needs to anticipate B’s response if it wants to be successful,
the possibility of abstention might, in fact, make it more expensive for A to buy
the election.
It turns out, however, that the additional tool of buying abstention is completely useless for both interest groups: In equilibrium, no voter is induced
to abstain. Indeed, we will now proceed to show that the least-cost successful
contract is identical to that in Proposition 1.
The following lemmas provide properties useful in characterizing the leastcost successful contract offered by A. First, note that in any subgame, group B
never buys more than a simple majority because it moves last. Next, note that
it is a dominated strategy for an interest group to pay its intrinsic supporters to
abstain. The reason is simple: Intrinsic supporters require more compensation
to abstain than to vote for their most preferred option, whereas an abstention
is less valuable.
Next,
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The next lemma describes the form of A’s transfers under a least-cost successful contract. As for the case where abstention was not allowed, it is helpful
to think of transfers in terms of surplus offers. To provide a voter of type θ
with surplus s from abstaining, group A has to offer a transfer t(θ) =
s − v(θ)/2. Under any least-cost successful contract offered by A under the
open ballot,
Lemma 2. All voters who receive a transfer in exchange for voting for
a are offered the same surplus, s. Similarly, all voters who receive a transfer in exchange for abstaining are offered the same surplus, s0 . Moreover,
s = 2s0 .
This lemma captures a “no arbitrage” condition resulting from competition
between A and B. Intuitively, the cost to group B of shifting the vote total
slightly in its favor should be the same irrespective of which voters it picks. If
this were not the case, then A is spending too much to “protect” some voters
from being poached by B. The second part of the lemma says that the cost to B
of poaching an a voter and turning that voter into a b voter should be exactly
twice the cost of poaching an abstaining voter and turning that voter into a
b voter. The reason is that the former changes the vote lead by two votes (a
reduction of one vote for a and an increase of one vote for b), whereas the
latter changes the vote lead by just one vote (an increase of one vote for b but
no reduction in the number of votes for a). Equalizing across voters the cost
to group B of shifting vote share requires that a voters receive twice as much
surplus as abstainers. This is illustrated graphically in Figure 2.
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Figure 2. Voters for a receive twice as much surplus as abstainers.
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We are now in a position to show that:
Proposition 3. Under the open ballot with abstention, negative vote buying
is never optimal. The contract identified in Proposition 1 remains the unique
least-cost successful contract.
Figure 3. Under the open ballot, paying for abstention is never optimal.
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Proposition 3 says that, under the open ballot, negative vote buying is never
as cost-effective as positive vote buying. Why is this? As we saw above, in any
least-cost successful vote buying scheme, group A divides voters into three
groups—those voting for a, those abstaining, and those voting for b—ordered
by their intrinsic preferences for policy a. Suppose that group A decides to
change the mix of abstainers and a voters while preserving the same vote lead
over policy b. One way it can do this is to offer additional money to the least
hostile abstainer in exchange for him voting for a, although, at the same time,
offering the null contract to the voter who until now was the most hostile abstainer, such that the latter now votes for b. This is illustrated in Figure 3.
Notice that the additional cost of “flipping” the least hostile abstainer to vote
for a is given by the pair of boxes near θ∗a in the figure. The savings from letting go of the most hostile abstainer is given by the larger rectangle near θ∗b in
the figure. Hence, such an alteration of the mix of bribes offered by A is always
profitable.
Several implications emerge from Proposition 3. First, because negative vote
buying is never optimal, one would expect to see little of it under the open ballot. This is consistent with the empirical findings of Cox and Kousser (1981).
They report that in New York elections before the introduction of the secret
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The Journal of Law, Economics, & Organization
ballot, there were very few instances of negative vote buying reported in the
popular press. It was only with the introduction of the secret ballot that negative
vote buying gained considerable prominence. Second, under the open ballot,
the imposition of compulsory voting has no effect on A’s costs of buying the
election. This may be seen by comparing the least-cost successful contract under GS, where voting is compulsory, with the result of Proposition 3, where
voting is voluntary.
Lemma 3. Under any least-cost successful contract offered by A under the
secret ballot, there exists a θb ∈ (0, 1) such that
1. All voters with types θ ∈ [0, θ0 ] vote for a.
2. All voters with types θ ∈ (θ0 , θb ] abstain.
3. All voters with types θ ∈ (θb , 1] vote for b.
Notice that compared with Lemma 1, which had two free parameters, the
partition of voter types under the secret ballot is characterized by θb only.
The reason is that an intrinsic b supporter can never be induced to vote for
a because votes cannot be monitored. Thus, a least-cost successful contract
offered by A amounts to dividing the intrinsic b supporters into abstainers and
b voters, whereas sufficiently incentivizing the intrinsic a supporters to deter
counteroffers from interest group B.
13. That vote buying was a significant concern can be seen in the very next Article, which
imposes exclusion from the political process for twenty years to life for anyone found buying or
selling votes.
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4. Least Cost Vote Buying under the Secret Ballot
Reflecting the ideals of the French Revolution, Article 31 of the Constitution
of 1795 prescribed that all elections were to be held by secret ballot.13 About
60 years later, the Anglo-Saxon world started following suit. Motivated by
Chartist principles and worried about the corruption endemic to its electoral
process, the Australian state of Victoria adopted the secret ballot in general
elections in 1856. Britain and many US states introduced the secret ballot soon
thereafter.
In this section, we analyze least-cost successful vote buying under the secret ballot. We consider the case where group A goes first, followed by group
B. This is without loss of generality since, by almost identical arguments,
Proposition 2 continues to hold for the secret ballot.
The imposition of the secret ballot limits the contracting possibilities of the
interest groups to payments contingent only on whether a voter goes to the
polls. We now offer a set of structural properties that are shared by any leastcost successful contract under the secret ballot. These properties mimic those
identified in the previous section under the open ballot.
Negative Vote Buying and the Secret Ballot
831
Under any least-cost successful contract offered by A under the secret
ballot,
Lemma 4. All voters who receive a transfer in exchange for coming to the
polls enjoy the same surplus, s, relative to abstaining. Similarly, all voters who
receive a transfer in exchange for abstaining enjoy the same surplus, s0 , relative
to voting for b. Moreover, s = s0 .
Let s(θb ) describe a surplus offer satisfying equation (1) with equality. The
problem of determining a least-cost successful contract now reduces to choosing a cost minimizing value for θb . Note that group A’s cost, C, as a function
of θb is
C(θb ) =
=
Z θb
t(θ)dθ
0
Z θb
v−1 (2s(θb ))
1
s(θb ) − v(θ) dθ
2
It may be readily verified that C(∙) is strictly convex in θb and, hence, there
exists a unique θ∗b that minimizes A’s cost. We may now conclude that:
Proposition 4. Under the secret ballot with abstention, the unique least-cost
successful contract offered by A is as follows:
Voters with type θ ∈ [0, θ0 ] are induced to vote for a; voters with type θ ∈
(θ0 , θ∗b ] abstain; voters with type θ ∈ (θ∗b , 1] vote for b.
All voters with type θ 6 θ∗b receive transfers t(θ) = max{0, s θ∗b ) − 12 v(θ }
and earn surplus equal to max{s(θ∗b ), v(θ), 12 v(θ) + s(θ∗b )}
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If B makes a counteroffer, it targets voters offered the least surplus by A.
The lemma shows how group A anticipates this and sets its transfers such that
all voters receiving payments are equally costly for B to “flip.” Finding a leastcost successful contract then consists of pinning down the surplus s offered to
abstainers and the size of the abstaining coalition sufficient to dissuade group
B from counter attacking. Intrinsic a supporters must also receive surplus s,
or more, relative to their next best option, that is, abstaining. Hence, group
A offers voters with type θ ∈ [0, θb ] a transfer of t(θ) = max{0, s − 12 v(θ)},
where s is a constant to be determined below. Voters with type θ ∈ [0, θ0 ], that
is, intrinsic a supporters, are paid for coming to the polls, whereas voters with
type θ ∈ [θ0 , θb ], that is, intrinsic b supporters, are paid to stay away.
To counter A’s offer, B would have to induce abstainers to vote for b and a
supporters to abstain. Currently, a mass θ0 of voters choose a, whereas 1 − θb
choose b. Hence, the “vote gap” B must overcome is θ0 − (1 − θb ). The cost
per voter of closing the gap is equal to s, by construction. Therefore, B’s total
cost of recruiting a minimal winning coalition is s × (θ0 − (1 − θb )). For A to
achieve deterrence, it must be that
X
(1)
s>
θ0 − (1 − θb )
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The Journal of Law, Economics, & Organization
where
s(θ∗b ) =
X
θ0 − (1 − θ∗b )
and θ∗b is the unique solution to
arg min
Z θb
θb >1−θ0 0
t(θ)dθ.
Figure 4. Least-cost successful contract under the secret ballot.
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A typical least-cost successful contract under the secret ballot is shown in
Figure 4. A key difference between this figure and Figure 1 is the upward sloping surplus for intrinsic a supporters with types just to the left of θ0 . The reason
is that these voters must receive constant surplus relative to their outside option—which is abstention—and the value of this option changes with a voter’s
type.
In contrast to the open ballot, negative vote buying takes on considerable
prominence under the secret ballot. It may take many different forms, and policy changes meant to reduce electoral fraud can sometimes have the perverse
effect of making negative vote buying a cheaper and more effective tool. In
some elections, indelible ink is used to mark the fingers of voters in order
to prevent them from voting multiple times. This makes negative vote buying
particularly easy, however: Voters in opposition districts are simply paid to dip
their fingers in ink, which prevents them from going to the polls. Similarly,
leading up to the 1998 general elections in Guyana, the government instituted
a system of voter identification cards to curtail vote fraud. Voters needed to
Negative Vote Buying and the Secret Ballot
833
5. On the Buyability of Voting Bodies
A key justification for the introduction of the secret ballot was the curtailment
of vote buying. This raises the question of how effective it is in this regard. Obviously, resolving this question empirically is difficult given that vote buying
is generally illegal. Our model offers an opportunity for a theoretical answer. It
is worthwhile to distinguish between two separate metrics of vote buying. The
first measure is the number of people paid in exchange for not voting according
to their intrinsic preferences. The second measure is how much it costs group A
to buy the election. It might seem apparent that the secret ballot unambiguously
improves the situation on both counts. After all, it would seem that depriving
interest group A of a key tool—the ability to contract on individual votes—
would make it harder to bribe voters and, thereby, induce fewer people to vote
against their intrinsic preferences and raise the cost of influencing the election.
However, this ignores the competition between interest groups. The imposition
of the secret ballot deprives rival interest group B of the same key tool. Could it
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present these cards at the polling stations to vote. Ironically, these cards, combined with readily observable racial identifiers, served to make the process
of negative vote buying cheap and easy for the ruling party. The ruling party,
which was favored by most Indo-Guyanans, set about buying the identification
cards of Afro-Guyanans, who were the main opposition. See Shaffer (2002).
Finally, in the Philippines, negative vote buying and demobilization campaigns
often take the form of offering opposition supporters coach trips to interesting
resorts with lots of booze on the day of the election (Quimpo 2002).
Our model predicts voter turnout to decrease when the secret ballot is introduced. This is consistent with Heckelman (1995) who found that the introduction of the secret ballot accounted for a seven percentage point drop in turnout
in US gubernatorial elections during 1870–1910. In addition to negative vote
buying, the optimal contract under the secret ballot also entails positive vote
buying—lukewarm intrinsic a supporters are compensated in exchange for
coming to the polls. “Get out the vote” campaigns are indeed commonplace
in modern US politics. In practical terms, this might take the form of ward
heelers offering transportation or other amenities to increase voter participation. For example, in 1999, Oakland Democratic campaign workers gave away
almost 4000 coupons for free chicken dinners in targeted neighborhoods with
traditionally low turnout (Matier and Ross 1999). The practice of handing out
“street money” to party operatives in advance of election day is also well entrenched. Indeed, during his presidential primary run in 2008, Barack Obama
made headlines and infuriated Philadelphia ward bosses by refusing to pay
(Nicholas 2008). The predicted changes in vote buying under the secret ballot
also resemble the empirical findings of Cox and Kousser (1981). As mentioned
above, there was a lot of positive vote buying but little negative vote buying before the introduction of the secret ballot. After its introduction, the amount of
negative vote buying went up dramatically, but parties continued to engage in
positive vote buying as well.
834
The Journal of Law, Economics, & Organization
be that, as a result of having its rival “disarmed” in this fashion, interest group
A can actually more cheaply exert influence under the secret ballot?
The Pervasiveness of Bribery
C=
Z θa
v−1 (s)
(s − v(t))dt
subject to s > X 1 . We may then think of the problem in price theory terms
θa − 2
and construct an iso-cost curve for group A. The slope of this iso-cost curve,
which we shall refer to as its marginal rate of substitution, is MRSopen =
s−v(θa )
ds
dθa = − θ −v−1 (s) . Similarly, the slope of the iso-cost curve under the secret
a
s− 1 v(θ )
b
ballot is MRSsecret = − θ −v2 −1 (2s)
. Under the open ballot, the cost of buying
b
the marginal voter consists of compensating him for voting against his intrinsic preference. Under the secret ballot, the cost of buying the marginal voter
consists of compensating him for abstaining. Since abstaining is less abhorrent
to him than voting against his preferred policy, the required change in surplus
needed to maintain cost neutrality is lower. Hence, it follows that:
Lemma 5. For all (θ, s), |MRSopen | > |MRSsecret |.
Recall that an optimal contract minimizes costs subject to meeting the deterrence constraint. Under the open ballot, the deterrence constraint is given by
X
s = X 1 , whereas under the secret ballot it is given by s = θ −(1−θ
. Like the
θa − 2
0
b)
marginal rates of substitution, we can also order the slopes of the deterrence
constraints.
Lemma 6. For all θ, the slope of the deterrence constraint is flatter under the
X
open ballot than under the secret ballot. Formally, − X1 2 > − (θ −(1−θ))
2
0
(θ− 2 )
for all θ.
The proof follows immediately from the fact that θ0 < 12 . Intuitively, group
A’s savings from exceeding a minimal winning coalition decrease as the size
of the coalition grows. The slope of the deterrence constraint simply expresses
the speed of this decline. In the case of the open ballot, the size by which a
coalition exceeds the minimal winning coalition is θ − 12 , whereas under the
secret ballot the size is given by θ0 − (1 − θ) = θ − (1 − θ0 ), where
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We first study how pervasive bribery is under the secret ballot as compared to
the open ballot; that is, we compare the number of people voting against their
intrinsic preferences under the two regimes. To do so, we take advantage of the
fact that the structure of least-cost successful vote buying contracts derived in
Propositions 3 and 4 allow us to characterize optimal contracts only in terms
of s and θ . Specifically, under the open ballot, the strategy of interest group A
amounts to choosing a surplus level s and a fraction of voters θa to induce to
vote for policy a, subject to the constraint that s and θa are jointly sufficient to
deter B. That is, group A chooses s and θa to minimize
Negative Vote Buying and the Secret Ballot
835
1 − θ0 > 12 . Thus, for each value of θ, the marginal savings from expanding the
supermajority are smaller under the open ballot than under the secret ballot.
Finally, a least-cost successful contract occurs at a tangency point of the
iso-cost curve and the deterrence constraint. Under the open ballot, such a
tangency point (θ∗a , s∗a ), satisfies
MRSopen (θa , sa ) = −
X
θa − 12
(1)
Under the secret ballot, a tangency point, (θ∗b , s∗b ), satisfies
MRSsecret (θb , sb ) = −
X
.
(θ0 − (1 − θb ))2
(2)
Together, the orderings of the marginal rates of substitution and the deterrence
constraints allow us to make an unambiguous statement about the fraction of
voters paid to change their voting behavior under the open ballot as compared
to the secret ballot.
Proposition 5. The introduction of the secret ballot raises the fraction of
voters not voting according to their intrinsic preferences. Formally, θ∗a 6 θ∗b .
The proposition shows that the secret ballot leads group A to bribe more
pervasively than under the open ballot. There are two economic forces driving
the result. First, the marginal benefit of increasing θ is higher under the secret
ballot since the effective supermajority is smaller and, hence, the feasible surplus reduction greater. Second, the marginal cost of increasing θ is lower under
the secret ballot since the marginal voter, who is an intrinsic b supporter, need
only be compensated for abstaining as opposed to voting for a under the open
ballot. Both forces push the optimal contract in the direction of buying more
voters under the secret ballot.
The Cost of Buying an Election
Up until now, we have simply assumed that group A values its preferred policy
“sufficiently highly” relative to B that it is indeed willing to engage in vote
buying. In this section, we examine the absolute costs of buying the election,
thus obtaining conditions on group A’s valuation Y relative to group B’s valuation X such that vote buying indeed takes place. Second, we make relative
comparisons between the cost of vote buying under the open versus the secret
ballot and show that buying the election may, in fact, be cheaper under the
secret ballot.
To calculate absolute costs of buying the election requires that we place
additional structure on the model. Suppose, for instance, that the preferences
of voters are a linear function of their type.14 Specifically, let v(θ) = β(θ0 − θ)
14. Strictly speaking, linear v functions do not satisfy the “Inada conditions,” that is, v(0) =
−v(1) = ∞. However, as long as we restrict attention to interior solutions, this is irrelevant.
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2
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The Journal of Law, Economics, & Organization
where β > 0. In the absence of competition from B, group
A’s “monopoly”
open
cost under the open ballot is CM = 12 β 14 − θ0 (1 − θ0 ) , whereas the cost
under the secret ballot is exactly twice this amount. In fact, also for general
v (θ), monopoly vote buying is always more expensive under the secret ballot
than under the open ballot.15 Thus, for a fixed valuation Y for group A, we may
conclude:
Remark 1. Absent competition, vote buying is less likely under the secret
ballot than under the open ballot.
1
Copen = (θ∗a − v−1 (s∗open ))(s∗open − v(θ∗a ))
2
p
open
= CM + 2X + βX(1 − 2θ0 ),
whereas A’s cost under the secret ballot is
1
1
Csecret = (θ∗b − v−1 (2s∗secret ))(s∗secret − v(θ∗b ))
2
2
p
√
secret
= CM
+ 2X + 2 βX(1 − 2θ0 ).
(3)
(4)
Equations (3) and (4) reveal that group A must value its preferred policy
more
√ than twice as much as group B to engage in vote buying. The terms 2X +
βX(1−2θ0 ) represent the costs of preempting group B from counter bribing.
These costs are obviously at least X, the value to B of getting its preferred
policy. However, the fact that A must anticipate and defend against a variety of
possible counter-bribing strategies by B raises A’s costs even higher. Defence
costs increase with B’s valuation, X, and the intensity of voter preferences, β,
and decrease with the fraction of intrinsic a supporters, θ0 . If Y 6 Copen (Csecret ,
respectively), then there
√ is no vote buying under the open (secret) ballot. Since
secret > Copen and 2 > 1, it follows that, for linear v(θ), vote buying is more
CM
M
expensive under the secret ballot than under the open ballot. That is,
Proposition 6. When preferences are linear and interior solutions obtain,
the imposition of the secret ballot raises the costs of vote buying. Hence, vote
buying is less likely under the secret ballot than under the open ballot.
Although Remark 1 holds generally, Proposition 6 is special to the linear
case, which induces a type of symmetry to the strength of preferences held
15. The general result follows from the fact that the monopoly contract under the secret ballot
is also feasible under the open ballot, but never optimal; hence, costs under the open ballot are
lower.
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Now consider the case where there is competition. Calculating the least-cost
successful contracts using equations (1) and (2), it may be readily verified that,
for linear v(θ), the cost top A under the open ballot is:
Negative Vote Buying and the Secret Ballot
837
by intrinsic a supporters and intrinsic b supporters. If we relax this symmetry
using the preference specification16
v (θ) =
(
(1 − 2θ)ρ
β 12 − θ
θ6
θ>
1
2
1
2
(5)
where ρ > 0, then we can show that:
1. If ρ < 1, Copen < Csecret
2. If ρ > 1, Copen > Csecret
3. If ρ = 1, Copen = Csecret
Proposition 7 illustrates that the effect of the imposition of the secret ballot
on the buyability of an election crucially depends on the structure of preferences. Roughly, the proposition says that, in close elections where much of the
intrinsic support for policy a is lukewarm, the imposition of the secret ballot
makes it easier for group A to achieve its desired policy. The reason is that,
owing to the inability of B to contract on votes directly, group A is able to
economize on payments to lukewarm supporters of its preferred policy, while
still deterring B from making any counter offers. In short, despite the common intuition that the imposition of the secret ballot offers an antidote to vote
buying, the theory suggests this may not necessarily be true.
6. Conclusion
We have analyzed vote buying when voters have the option to abstain. First, we
derived the optimal vote buying strategy under the open ballot, that is, when
interest groups and voters can contract on votes directly. Here, we found that,
although negative vote buying was feasible, it was never optimal. Indeed, we
showed that the option of abstention has no effect at all on vote buying under
the open ballot.
Next, we studied optimal vote buying under the secret ballot. In this case,
interest groups and voters can only contract on whether to show up at the
polls—not on actual votes. We showed that this changes optimal vote buying
significantly: Interest groups make extensive use of negative vote buying to
induce lukewarm opponents of their preferred policy to stay home on election
day. Positive vote buying is also used: Interest groups optimally pay lukewarm
16. Strictly speaking, this specification does not satisfy θ0 < 12 . However, in Proposition 7
below, we derive strict inequalities between the costs in the two regimes. It then follows from
continuity that for θ0 strictly smaller than, but close to, 12 the same ordering applies.
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Proposition 7. For the class of preference functions in equation (5), the imposition of the secret ballot reduces the cost of buying the election if and only
if the preferences of intrinsic a supporters are convex in θ. Formally,
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The Journal of Law, Economics, & Organization
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supporters of their preferred policy to show up at the polls. Our results are consistent with the empirical findings of Cox and Kousser (1981) who observed
little evidence of negative vote buying before the imposition of the secret ballot
in New York, but considerable evidence of the practice thereafter.
We then compared both the scope and the cost of successful vote buying
under the secret versus the open ballot. We found that the imposition of the
secret ballot always increases the scope of vote buying—more people vote
against their intrinsic preferences under the secret ballot than under the open
ballot. We also found circumstances where, paradoxically, the imposition of
the secret ballot makes it easier for interest groups to wield influence. In particular, for close elections where the bulk of the supporters for an interest
group’s desired policy are lukewarm in their support, it is cheaper for the interest group to buy the election under the secret ballot than under the open
ballot. Taken together, these results suggest that the common intuition about
the effectiveness of the secret ballot as a deterrent to electoral corruption is not
robust.
Our analysis implies that a combination of policy reforms may be needed
to reduce electoral corruption. In our model, combining the secret ballot with
mandatory voting would, in principle, remove all scope for vote buying. That
this is not merely a theoretical possibility is suggested by the case of the aborigines in Australia. Unlike the rest of the country, voting was not mandatory
for this group from the time they got the vote in 1962 until 1984. In those years,
free-flowing alcohol was used extensively and successfully to lure aborigines
away from the polls (see Orr 2004).
However, even seemingly robust policy combinations need not put an end
to vote buying. Indeed, a key insight from the model is that the form of vote
buying adapts to the policy environment, sometimes with unintended consequences. A recent example occurred in Thailand. In 1997, the Thai constitution put in place numerous safeguards to deter vote buying, including the
imposition of compulsory voting and the creation of an independent election
commission. In response, during the 2000 and 2001 elections, it became commonplace to pay salaries to voters for joining a political party. Voters were also
hired as canvassers in hopes of winning their support. Indeed, the net effect of
the reforms seems to have been to increase payments to voters compared to
past elections (Hicken 2002). On the one hand, this is, perhaps, evidence that
the measures were successful—vote buying is now more expensive than it was
before. On the other hand, it also attests to the fact that the policies did not
succeed in eliminating vote buying. Rather, they shifted the practice toward
more discrete and indirect channels.
Our results also have implications for transparency in legislative settings.
The usual rationale for roll call voting in legislatures is that legislators will be
held to greater account by constituents when votes are transparent. However, a
potential cost associated with this benefit is that the individual voting behavior
of legislators—that is, whether and how they vote—is also observable and,
hence, contractible, by interest groups. The possibility of such an underlying
cost indeed merits attention.
Negative Vote Buying and the Secret Ballot
839
Although our results focus on the case where there are two competing interest groups, in some settings, this is easily generalized: As long as the
policy space remains binary—and ignoring the familiar coordination/free-rider
problems—our model can accommodate any number of interest groups if we
think of each group as a coalition of subgroups favoring a particular policy.
Like much of the voting literature, extending the model outside of binary policy spaces introduces a host of strategic voting issues that have proved difficult
to handle, even absent interest group competition. Our results may be thought
of as a key first step toward this more ambitious goal.
Proofs
Proof of Proposition 2. Suppose, to the contrary, that neither interest group
offers contracts. In that case, A can profitably deviate by offering the monopoly
contract and obtaining its preferred policy at a cost CM < Y .
Next, suppose that B moves before A. Define U to be the set of voters that
accept B’s contract if A stays out. Since B is a monopolist with infinitesimal
probability, payments made to voters in U must be less than or equal to X.
Suppose that A responds as follows: It “negates” all of B’s offers to voters in
U and, in addition, offers the monopoly contract. Negation consists of offering
the same amount as B in exchange for an a vote. Adding the monopoly contract consists of offering voters with type θ ∈ [θ0 , 12 + ε ] where ε is arbitrarily
small, an additional amount −v(θ) in exchange for an a vote. Since the cost
of negation is, at most, X, whereas the cost of adding the monopoly contract
is CM , A’s total expenditure on vote buying following a move by B is at most
X + CM < CGS < Y . Therefore, this is a profitable response by A. As a result,
B can do no better than to wait for A to move first. In fact, because with infinitesimal probability group B is a monopolist that gets its way even if it does
nothing, moving first leaves B strictly worse off.
Finally, suppose that B moves at the same time as A. When the coin flip
determines that B moves first, then, by the same argument as above, B is strictly
better off moving second. When the coin flip determines that A moves first, B is
just as well off as when it offers contracts after A has already moved. Therefore,
B is strictly better off moving second.
Proof of Lemma 1. To establish the lemma, we show that, under the leastcost successful contract, (1) the set of a voters is convex and equal to [0, θa )
and (2) the set of b voters is convex and equal to (θb , 1].
To establish (1), first note that since v (0) = ∞, voters with types [0, ε ) vote
for a, for ε sufficiently small. Next, suppose that there is a set O of voters who
do something other than vote for a and a set M of voters all of whom vote for
a, such that for all θ ∈ O and θ0 ∈ M, it is the case that < θ0 . It is without
loss of generality to assume that O and M have equal mass. Now consider the
following deviation on the part of A: voters in O are paid to vote for a, whereas
voters in M are paid to do whatever the former O voters did. This has no effect
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Appendix A
840
The Journal of Law, Economics, & Organization
on the cost to B of switching these voters to b votes. However, A’s costs are
reduced by an amount at least
Z
Z
1
v (θ) dθ −
v (θ) dθ > 0.
ΔC =
2 θ∈O
θ∈M
Proof of Lemma 2. To prove the lemma, it is useful to first establish several
properties of optimal vote buying. The first of these properties simplifies the
set of B’s counter moves we need to consider.
Lemma 7. In any subgame, group B never rebribes voters to abstain.
Proof. Because interest groups do not buy abstention from intrinsic supporters,
group B rebribes intrinsic b supporters to vote for b. Hence, it only remains to
consider rebribing by B of intrinsic a supporters, who were bribed by A to vote
for a.
To see that it is never optimal for B to rebribe an intrinsic a supporter to
abstain, notice that a voter who has been offered a transfer tA > 0 by A to vote
for a enjoys a surplus v(θ) + tA . To get this voter to vote for b instead, B must
offer him a payment tB such that
tB = v(θ) + tA .
To gain an equivalent increase in vote share, B would have to rebribe two
voters to abstain. It costs tB0 to induce a voter to switch from a to abstention,
where tB0 solves
1
tB0 + v(θ) = v(θ) + tA .
2
Thus, the cost of achieving an equivalent increase in vote share through
abstention is
2tB0 = v (θ) + 2tA
which is clearly more expensive.
Lemma 8. Any successful contract offered by group A entails:
(1) bribes to a measure of voters strictly more than 12 − θ0 ; and
(2) bribes to a positive measure of voters with types θ > 12 .
Proof. Suppose, to the contrary, that A bribes less than 12 − θ0 voters or does
not bribe a positive measure of voters with types θ > 12 . Then, policy b receives
at least half of the votes cast. Hence, A’s contract is not successful.
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But this contradicts the notion that A’s original contract was cost minimizing.
This completes the proof of (1).
The proof of (2) proceeds analogously to the proof of (1), starting out with
the observation that since v(1) = −∞, voters with types (1 − ε , 1] vote for b,
for ε sufficiently small.
Negative Vote Buying and the Secret Ballot
841
Next, it is helpful to define V to be the set of types group A pays to vote for
a and V 0 to be the set of types A pays to abstain.
Lemma 9. In any subgame, group B only makes counters offers to a strict
subset of voters in V ∪V 0 .
1
− μ(V ∪V 0 ) + m = θ0
2
or, equivalently,
μ(V ∪V 0 ) − m =
1
− θ0 > 0
2
where the inequality follows from the fact that, by Lemma 8, the measure of
voters in V ∪V 0 > 12 − θ0 . Hence, a positive measure of voters in V ∪V 0 are not
rebribed.
Now we are in a position to proceed with the proof of Lemma 2. We will
prove the lemma by verifying a series of claims. It is convenient to verify
these claims for the following closely related game: Group A’s objective is to
minimize its costs subject to the constraint that it costs B at least X to assemble
a winning coalition. Group B’s objective is to obtain a winning coalition at
minimum cost. Obviously, any least-cost successful contract of the original
game must also satisfy the properties of A’s optimal contract in this modified
game.
Claim 1: If B rebribes none of the voters in V (V 0 ), then A must offer all
voters in V (V 0 ) the same surplus.
If not, then A can reduce its costs by offering every voter in V (V 0 ) as little as the infimum of the surpluses of voters in V (V 0 ). Group B will not alter
its behavior following such a deviation since, by revealed preference, it was
not cost effective to recruit voters enjoying the infimum of the surpluses in
V (V 0 ).
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Proof. In essence, the lemma follows from the fact that B never buys more
than a minimum winning coalition. A formal proof
is
as follows.
Recall that Lemma 8 implies that (V ∪V 0 ) ∩ 12 , 1 is nonempty.
Case 1: Suppose that (V ∪V 0 ) ∩ [0, 12 ] is nonempty. Then either group B does
not rebribe all voters in the set (V ∪ V 0 ) ∩ [ 12 , 1], in which case the statement
of the lemma is true; or group B does rebribe all voters in the set (V ∪ V 0 ) ∩
[ 12 , 1]. In the latter case, by Lemma 7, these rebribes induce voters in this set to
cast b votes rather than to abstain. Thus, when B rebribes all voters in the set
(V ∪V 0 ) ∩ [ 12 , 1], it obtains a minimum winning
coalition. Therefore, it will not
rebribe any voters in the set (V ∪V0 ) ∩ 0, 12 .
/ Since no voters with types θ 6 12
Case 2: Suppose that (V ∪V 0 ) ∩ 0, 12 = 0.
receive bribes from group A, then a measure 12 − θ0 will cast b votes. Hence,
group B need only induce a measure θ0 of types θ > 12 to cast b votes. Let m
denote the measure of types rebribed by B. Then it must be that m solves
842
The Journal of Law, Economics, & Organization
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Claim 2: If B rebribes a positive measure of voters in V (V 0 ), then the infimum of the surpluses offered by A to voters in V (V 0 ) who are not rebribed,
must be greater than or equal to the supremum of the surpluses offered by A to
voters in V (V 0 ) who are rebribed.
If not, then B could reduce its costs by reallocating its bribes from voters
with the highest surplus offers from A to previously not rebribed voters receiving the lowest surplus offers from A.
Claim 3: If B rebribes a positive measure of voters in V (V 0 ), then the supremum of surpluses offered by A to voters in V (V 0 ) who are not rebribed must
be less than or equal to the supremum of surpluses offered by A to voters in V
(V 0 ) who are rebribed.
If this were not the case, then A could pay strictly less to voters in V (V 0 )
who are not rebribed.
Claims 1–3 imply:
Claim 4: Group A offers all voters in V (V 0 ) who are not rebribed by B the
same surplus. Furthermore, that surplus equals the supremum of the surpluses
offered by A to those who are rebribed, if any.
Claim 5: If B does not rebribe all of the voters in V (V 0 ), then A offers all
voters in V (V 0 ) the same surplus.
Claim 4 already established that A offers the same surplus to all voters who
are not rebribed by B. Hence, it suffices to show that if B does not rebribe all
voters in V (V 0 ), then A offers the same surplus to all voters who are rebribed
by B.
Suppose, to the contrary, that A does not offer the same surplus to all voters
who are rebribed. In that case, A can reallocate its bribes to these voters without
affecting B’s costs of rebribing, by giving all of them their average surplus.
Furthermore, by Claim 2, this average surplus must be strictly smaller than
the surplus offered by A to voters who are not rebribed by B. But then, A can
reduce the surpluses paid to the nonrebribed voters.
Claim 6: Suppose that B does not rebribe all of the voters in V nor all of
the voters in V 0 , then A’s surplus offer to voters in V is equal to two times its
surplus offer to voters in V 0 .
Note that by claim 5, A offers the same surplus to all voters in V , and the
same surplus to all voters in V 0 . Now suppose, to the contrary, that the surplus
of voters in V is strictly greater than two times the surplus of voters in V 0 . If
group B is rebribing voters in V , then it can reduce its costs by reallocating its
bribes from voters in V to voters in V 0 . Thus, it must be the case that no voters
in V are rebribed. In that case, A can reduce the surplus paid to voters in V .
A similar argument holds if the surplus in V is strictly less than two times the
surplus in V 0 .
Claim 7: Suppose that B does rebribe all of the voters in V (V 0 ) but not all
of the voters in V 0 (V ), then A’s surplus offer to voters in V is equal to two
times its surplus offer to voters in V 0 .
We show that the claim holds for the case that B rebribes all of the voters in
V but not all the voters in V 0 . The proof of the other case is analogous.
Negative Vote Buying and the Secret Ballot
843
Proof of Proposition 3. Suppose that, contrary to the proposition, the leastcost successful contract entails buying a mass of abstentions, that is, θb > θa
in Lemma 1. Let the surplus of the abstainers be s.
ˉ From Lemma 2, we know
that these abstainers must be offered surplus
2
s
ˉ
to
vote
i for a.
h
θa +θb
are paid to vote for a,
Consider a deviation whereby voters θa , 2
whereas the remaining voters are not paid at all. Note that B remains deterred,
whereas the change in the cost to A is
1
ΔC =
2
Z θb
θa +θb
2
v(θ)dθ −
Z
θa +θb
2
θa
v(θ)dθ
!
< 0.
Hence, this is a profitable deviation.
It now follows immediately that the contract identified in Proposition 1 remains the unique least-cost successful contract.
Proof of Lemma 3. Note that only types θ ∈ [0, θ0 ], that is, intrinsic a supporters, will ever vote for a, because ballots are secret. Furthermore, it costs
group A more to induce these voters to abstain or vote for b than to induce
them to vote for a, and either alternative is clearly worse for A. Hence, in any
least-cost successful contract, A induces all voters with types [0, θ0 ]—and only
these—to vote for a.
Thus, we need only consider the interval [θ0 , 1] of intrinsic b supporters.
Under the secret ballot, these voters can only be induced to abstain or to vote
for b, as they will never vote for a under any contract.
Suppose, contrary to part 2 of the lemma, that there exists a set O containing a positive measure of voters voting for b (and hence not paid by A) and
a set M containing a positive measure of voters paid by A to abstain, such
that, for all θ ∈ O and θ0 ∈ M, it is the case that θ < θ0 . It is without loss of
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First, notice that if the supremum of surpluses offered by A to voters in V is
strictly greater than two times the surplus offered by A to voters in V 0 , then B
is not optimizing. Hence, the supremum is smaller than or equal to two times
the surplus of voters in V 0 .
Next, notice that if the infimum of surpluses offered by A to voters in V is
strictly smaller than two times the surplus offered by A to voters in V 0 , then
A has the following profitable deviation. First, it can reallocate its bribes to
voters in V without affecting B’s costs of rebribing, by giving all of them their
average surplus. Note that this average must be strictly smaller than two times
the surplus of voters in V 0 because the supremum was smaller or equal to,
whereas the infimum was strictly smaller than, two times the surplus of voters
in V 0 . Next, it can slightly reduce the surplus it pays to all voters in V 0 , passing
on the savings from the rebribed in V 0 to the voters in V and pocketing the
rest.
This completes the proof of Lemma 2.
844
The Journal of Law, Economics, & Organization
generality to assume that O and M have equal mass. Consider the following
deviation: Voters in O are paid to abstain, whereas voters in M are not paid
at all. This deviation has no effect on the cost to B of achieving a minimum
winning coalition. However, A costs are reduced because
1
ΔC =
2
Z
θ∈O
v(θ)dθ −
Z
θ∈M
v(θ)dθ < 0,
Proof of Lemma 4. It is helpful to define W to be the set of types whom A
pays to come to the polls and W 0 to be the set of types whom A pays to abstain.
Also, define w to be the set of voters in W who receive a counteroffer from B,
whereas w0 is the set of voters in W 0 who receive a counteroffer.
Again, it is convenient to prove the lemma for the closely related game
where group A’s objective is to minimize its costs subject to the constraint
that it costs B at least X to assemble a winning coalition. Group B’s objective
is to obtain a winning coalition at minimum cost. Obviously, any least-cost
successful contract of the original game must also satisfy the properties of A’s
optimal contract in this modified game.
Claim 1: Group A offers the same surplus s0 to all voters in W 0 , relative to
their outside option, which is to vote for b.
First, notice that for the same reasons as under the open ballot, in any
subgame, B always buys a minimal winning coalition. Hence, any best response by B entails making counteroffers to only a subset of voters in
W 0 = (θ0 , θb ].
Next, suppose, contrary to Claim 1, that A offers a positive measure of voters
in W 0 different surpluses. We have to consider two cases.
Case 1: Some voters in W 0 receive a counteroffer from B, that is, w0 is
nonempty.
First, if A does not offer the same surplus to all voters in W 0 \w0 —that is,
to voters in W 0 who do not receive a counter—then A can lower the surplus
of voters in W 0 \w0 getting surpluses strictly greater than the infimum, without
affecting B’s incentives.
Next, suppose that A does offer all voters in W 0 \w0 the same surplus. Then,
by assumption, not all voters in w0 receive the same surplus offer from A. Moreover, A must have offered some voters in w0 surpluses strictly smaller than the
surplus of voters in w0 \W 0 , whereas nobody in w0 was offered a surplus strictly
larger. Else, B would not have optimized its counteroffers. This implies that A
could instead pay all voters in w0 the average of their surpluses without affecting B’s incentives. Because this average is strictly smaller than the infimum of
the surpluses of voters in w0 \W 0 , the surpluses of all voters in w0 \W 0 can then
be lowered without affecting B’s incentives.
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since v (∙) is strictly decreasing and strictly negative. This contradicts the
notion that the original contract was least cost.
Finally, part 3 of the lemma follows immediately from parts 1 and 2.
Negative Vote Buying and the Secret Ballot
845
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Case 2: Nobody in W 0 receives a counteroffer from B, that is, w0 is empty.
Then either all voters in W 0 get the same surplus, or, for a positive measure
of voters receiving the highest surplus, group A can lower their transfers while
not affecting B’s incentives.
Claim 2: A offers the same surplus, s, to all voters in W , relative to their
outside option, which is to abstain.
We have to consider three cases.
Case 1: Nobody in W receives a counter offer; i.e., w is empty.
Then either everybody in W gets the same surplus from A, in which case we
are done, or we can reduce the surplus of those getting strictly more than the
infimum without affecting B’s incentives.
Case 2: Everybody in W receives a counteroffer, that is, w = W .
Then, either A offers the same surplus to all voters in W , in which case we
are done, or all voters in W 0 receive a surplus offer of at least the supremum
of what voters in W are offered. (Else, rebribing everybody in W would not
have been a best response by B.) In that case, A can redistribute bribes in W
in a budget-neutral fashion such that all voters in W are offered the average
surplus, without affecting B’s incentives. This makes the surplus of all voters
in W strictly smaller than the surplus of all voters in W 0 . If w0 is empty—that
is, nobody in W 0 gets rebribed—then A can marginally reduce the surplus of
everybody in W 0 without affecting B’s incentives and pocket the money saved.
If w0 is nonempty—that is, some voters in W 0 do get rebribed—then A can
marginally reduce the surplus of voters in w0 and redistribute it to voters in W .
This is budget neutral for A and does not affect B’s incentives since it costs
B exactly the same as before to rebribe voters in W ∪ w0 , whereas no other
coalition of the same measure is cheaper. Note, however, that voters in W 0 no
longer receive the same surplus offers from A. And, by Claim 1, we know that
this is suboptimal.
Case 3: Some but not all voters in W receive a counteroffer, that is, both w
and W \w are nonempty.
Then the claim follows from arguments identical to those in Case 1 of
Claim 1.
Claim 3: s = s0 .
Suppose not. Then either s < s0 , or s > s0 . Suppose that s < s0 . In that case,
a best response for B is to rebribe as many voters in W as are needed to form
a winning coalition. If there are insufficient voters in W , then B should also
rebribe some voters in W 0 . Regardless, a positive measure of voters in W 0 will
be left without a counteroffer, that is, W 0 \w0 is nonempty. Consider the following deviation by A. Suppose that A marginally reduces the surplus offered to
all voters in w0 and transfers it to voters in W . This is budget neutral for A and
has no effect on the incentives of B since it costs B exactly the same as before
to rebribe voters in W ∪ w0 , although no other coalition of the same measure is
cheaper. Note, however, that voters in W 0 no longer receive the same surplus
offers from A. And, by Claim 1, we know that this is suboptimal.
The proof for the case where s > s0 is analogous.
846
The Journal of Law, Economics, & Organization
Proof of Lemma 5. Recall that
MRSopen = −
and
s − v(θ)
θ − v−1 (s)
MRSsecret = −
s − 12 v(θ)
θ − v−1 (2s)
Proof of Proposition 5. Suppose, to the contrary, that θ∗a > θ∗b . Consider a
solution to
X
MRSsecret = −
2 .
θ − 12
That is, find a point where the marginal rate of substitution under the secret
ballot is tangent to the deterrence line under the open ballot. Call this point θ0b .
By Lemma 5, it follows immediately that θ0b > θ∗a . Next, notice that evaluated
at θ0b ,
MRSsecret > −
X
(θ0 − (1 − θ0b ))2
by Lemma 6. Furthermore, since the absolute value of the slope of the deterrence constraint is decreasing in θ, it then follows that θ∗b > θ0b . But this is a
contradiction.
Proof of Proposition 7. Under the open ballot, the costs are
Z 1
2
1
1
1
Copen =
(s − v(θ))dθ + θ −
s+
θ−
(−v(θ))
2
2
2
v−1 (s)
=
Z
1
2
1
2
1
1−s ρ
(s − (1 − 2t)ρ )dt + X
+
β
1 2
θ−
2
2
2ρ
1+ρ
1 3ρ+1
1
ρ
2 3ρ+1
+
+X
=
(βX )
2
2 1+ρ
while under the secret ballot,
Z 1
2
1
1
1
1
1
Csecret =
s − v(t) dt + θ −
s+
θ−
− v(θ)
2
2
2
2
2
v−1 (2s)
3ρ−1
1+ρ 1
1 3ρ +1
ρ
βX 2 3ρ+1
+
+ X.
=
2
2 1+ρ
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Now note that s − v(θ) > s − 12 v(θ), whereas θ − v−1 (s) < θ − v−1 (2s). Hence,
|MRSopen | > |MRSsecret |.
Negative Vote Buying and the Secret Ballot
847
Comparing the vote buying costs under the two regimes reveals:
!
2ρ ρ−1
1+ρ
1 3ρ+1 1
ρ
1 3ρ+1
2 3ρ+1
−1
βX
+
Csecret −Copen =
2
2 1+ρ
2
and this expression is positive iff ρ < 1, negative iff ρ > 1, and equals zero iff
ρ = 1.
In this section, we assume that for all θ 6 θ0 , u(b; θ) = 0, whereas for θ >
θ0 , u(a; θ) = 0. We show that for this preference specification, essentially the
same results hold as in the model presented in the main text. However, now
that buying an intrinsic opponent’s abstention is equally expensive as buying
his vote, unsurprisingly, the secret ballot always raises the cost of buying the
election.
Proposition 8. Under the open ballot, negative vote buying is always feasible
but never optimal.
Proof. Suppose, to the contrary, that there exists a least-cost vote buying
scheme in which a positive measure of voters are paid to abstain. In that case,
group A can deter B by deviating and offering a fraction 1 − ε of these voters
the same payment in exchange for voting for a while offering the remaining
fraction of voters the null contract. This strictly reduces A’s costs and, hence,
contradicts the notion that the original scheme was least cost.
Proposition 9. Under the secret ballot with abstention, the unique least-cost
successful contract offered by A is as follows:
Voters with type θ ∈ [0, θ0 ] are induced to vote for ‘a’; voters with type
θ ∈ (θ0 , θ∗b ] abstain; voters with type θ ∈ (θ∗b , 1] vote for ‘b’.
All voters with type θ 6 θ∗b receive transfers t(θ) = max{0, s(θ∗b ) − v(θ)}
and earn surplus equal to max{s(θ∗b ), v(θ)}
where
W
s(θ∗b ) =
θ0 − (1 − θ∗b )
and θ∗b is the unique solution to
arg min
Z θb
θb >1−θ0 0
t(θ)dθ
Proof. The argument is identical to that leading up to Proposition 4.
Proposition 10. The introduction of the secret ballot raises the fraction of
voters not voting sincerely. Formally, θ∗a < θ∗b .
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Appendix B
Alternative Preference Specifications
848
The Journal of Law, Economics, & Organization
Proof. Under these preferences, it is easily verified that for all (s, θ),
|MRSopen | = |MRSsecret |. Next, notice that the deterrence constraints are independent of voter preferences over abstention. It then follows that the ordering
given in Lemma 6 is unchanged. Together, these two orderings orderings imply
that θ∗a < θ∗b .
Proposition 11. The introduction of the secret ballot raises the cost of buying
the election.
Next, notice that, if we substitute for s using the deterrence constraint, the costs
of buying the election are only a function of θ. Moreover, since sopen < ssecret ,
it then immediately follows that for all θ,
Copen (θ) < Csecret (θ).
Finally, since θ∗a minimizes costs under the open ballot, it then follows that
Copen (θ∗a ) < Copen (θ∗b ) < Csecret (θ∗b ),
which establishes the result.
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