BY
BHASKAR DUTTA, MATTHEW 0. JACKSON, AND MICHEL LE
BRETON
We study the incentives of candidates to
strategically affect the outcome of a voting procedure. We show that the
outcomes of evety nondictatorial voting procedure that satisfies unanimity will
be affected by the incentives of noncontending candidates (i.e ., who cannot
win the election) to influence the outcome by entering or exiting the election.
KEYWORDS: Strategic candidacy, voting procedures,
candidates, voting rules.
INTRODUCTION
THE DECISIONO F A
CANDIDATEto enter an election can affect its outcome even in situations where
the candidate is not the winner of the election. For instance, consider a
scenario in which three national parties A, B, and C can contest an election in
which the winner is decided by plurality rule. Although party A may have the
highest number of first-preference votes, it may still fail to win the election
if, for instance, B drops out of the race in order to let C win.2 If the voting
process is viewed as a mapping from preferences to outcomes, then strategic
behavior in the first stage can be just as important as strategic voting in the
second stage. As we shall show, this phenomenon is important to all voting procedures,
and thus spans applications ranging from political elections to committee
decisions.
To be precise, we
consider a framework in which there is a finite set of voters and potential
candidates. We allow for the possibility that some or all of the candidates may
also be voters, and consider situations where each individual (including
candidates) has preferences over the set of all candidates. We examine a
two-stage procedure where in a first stage candidates decide on whether or not
they will enter the election, and then in a second stage a voting procedure is
implemented to select from the candidates who enter. Before outlining our
analysis, let us describe in more detail the way in which we model voting
procedures. We model a voting procedure as specifying the winning candidate as
a function of the set of entering candidates and voters' preferences over the
entering candidates. The only restriction that we place on such a voting
procedure is that it satisfy unanimity. Unanimity requires that if all voters
find the same candidate most preferred out of the entering candidates, then
that candidate is selected. We focus on the following issue. Which voting
procedures are not influenced by candidates' incentives to exit an election?
More precisely, for which voting procedures is it always a Nash equilibrium for
all candidates to enter the election? We call this condition "candidate
stability." We show that if the sets of voters and candidates are
distinct, then the only voting procedures satisfying candidate stability are
dictatorial procedures. When the set of candidates and voters overlap, then
there exist nondictatorial voting procedures that satisfy candidate stability
and unanimity. However, we show that none of these voting procedures satisfy an
appealing "almost"-unanimity condition together with a very weak
monotonicity condition that is satisfied by most standard voting procedures
(e.g., tree implementable procedures, Condorcet consistent procedures, scoring
rules, etc.). This implies that most standard voting procedures fail to satisfy
candidate stability regardless of the overlap between candidates and voters. We
should mention that these results are not simple extensions of an Arrow-type
impossibility theorem, even though we invoke Arrow's theorem at one point in
the proof of the first theorem. The bulk of the proof develops the joint
implications of candidate stability and unanimity. We discuss this in more detail
in what follows. Why should we care whether a voting procedure is candidate
stable? Regardless of how one feels about candidate stability as a normative
property, the results here must be taken seriously if we are at all interested
in evaluating and comparing voting procedures. Much of what we know about
voting rules is based on comparisons of the properties of different voting
procedures when the set of candidates is taken as given.3 The results here show
that the outcome of all nondictatorial and unanimous voting procedures will be
influenced by the entry decisions of candidates. This implies that it is not
valid to treat the set of candidates as fixed for any nondictatorial voting
procedure. As most of what we know about voting rules treats the set of
candidates as fixed, our results suggest that these need to be revisited
accounting for strategic candidacy. For instance, example 5 below shows that
the Pareto property of voting by successive elimination is upset when one
allows for strategic candidacy. So, the results here show...........
