Theoretical Economics Letters, 2011, 1, 18-20
doi:10.4236/tel.[phone] Published Online August 2011 (http://www.scirp.org/journal/tel)
Elections in a Multi-Party Political System
Yeolyong Sung
Korea Institute for Industrial Economics & Trade, Seoul, South Korea
Received June 19, 2011; revised July 27, 2011; accepted August 5, 2011
Abstract
In a multi-party political system, candidates’ policy points disperse in general even with two candidates
when parties have different feasible policy sets. Also, it is shown that extremists can influence the political
outcomes with a relatively small feasible policy set.
Keywords: Election, Multi-Party, Median Voter Result, Extremist
1. Introduction
In most cases of representative democracy, individual
politicians by themselves have little power to influence
political outcomes. Rather, they collectively obtain and
wield political power through a p arty. For example, most
candidates run for an election as a member of a party not
as an individual, and actually their winn ing possibility is
larger than independent candidates’ by organizational
and financial suppo rt of the party.
Either an individual politician becomes a member of
an existing party of which policy stances contain his/her
ideal or individuals with similar political purposes form a
new party. Then individual politicians’ political actions
are governed by the political stance of the party they
belong to, which divides the spectrum of political stand-
points of the electorate. Parties pledge themselves on
their own policy stances and competition among the par-
ties with different policy stances is common in election.
A candidate affiliated with a particular party must com-
mit to his/her policy to be implemented within the par-
ty’s given policy set which is a subset of the whole pol-
icy space.
This study focuses on elections in which the competi-
tion is among office-seeking partisan candidates whose
policy set is constrained by the party they belong to. The
objective of the study is to explain dispersion in candi-
dates’ policy points and the existence of influential ex-
tremists under a multi-party system with policy con-
straints. The model is different from the citizen-candid a -
te model of Osborne and Slivinski [1] in that candidates
are partisan and their preferences are policy irrelevant.
In this note, we look over simple two- and three-party
examples and attempt to interpret the model. Also, we
discuss the modeling and its extensions.
2. The Basic Model
At first, parties decide whether or not to enter an election.
The parties which have decided to do nominate their own
candidates for the election1 and they compete with each
other for being elected by choosing a policy point to be
implemented. In the voting stage, the voters vote for one
candidate and the winner is determined by plurality rule.
The policy space is [0,1]X

and each party is exo-
genously endowed with a compact feasible policy set,
which is mutually exclusive with each other except for
the boundaries and exhausts the policy space. Formally,
party ’s feasible policy set
ii
X
X is a closed inter-
val and ii
X
X

 such that ij
X
X is an empty set
or a singleton if ij

.
Each candidate commits to a policy point in the feasi-
ble policy set. Party ’s policy point to be implemented
if elected is denoted by ii
i
x
Xi. Politicians are office
seekers, so the utility of party ’s candidate is given by




{}
1
iiw
uww c


, where is an indicator
function of the winning candidate, , on event

w
w
1A
A
and
is the running cost for the election. Thus, if a
politician is nominated as a candidate and wins the elec-
tion, then his/her utility is 1; and if the candidate
loses, then
0c
cc

. We assume that the utility of a politician
who does not enter the election is zero.
The running cost, , is assumed to be sufficiently
small relative to the expected benefit from entering the
election unless the winning probability is zero. With the
utility function specified above, it suffices to assume that
c
1We do not deal with issues of party primaries here. For them, see
Cadigan and Janeba [2] and Owen and Grofman [3].

Y. SUNG 19
c is always less than the winning probability if it is
strictly positive, which is the expected benefit from en-
tering the election. Thus, the utility function implies that
if a party expects that it will lose the election with prob-
ability 1, then it does not enter the election because no
member of the party wants to be a candidate.
The set of voters is a continuum , and each
voter’s preference on the policy space is symmetric and
single-peaked. Voters’ ideal policy points are uniformly
distributed on . We assume that each voter
votes for the most preferred candidate sincerely2. A can-
didate who obtains the most votes is elected and his/her
announced policy is implemented. If there is a tie, each
candidate with the most votes has equal probability of
winning.
[0,1]V
[0,1]
X
3. Two-Party System
Consider a two-party system with the Leftist () party
and the Rightist () party. In this case, both parties en-
ter the election only if they are balan ced in terms of their
policy stances. If the sizes of the feasible policy sets are
different, only the party with the larger set enters the elec-
tion and wins by acclamation. For example, let
L
R


ˆ
x12,1
and suppose that


ˆ
0,
L
X
x and


ˆ,1
R
Xx. Then the
Leftist can win the election by an- nouncing any policy
point


ˆˆ
,1
L
x
xx and so the Rightist does not enter
the election because then it loses with probability 1. Only
when the feasible sets of the parties equally divide the
policy space (ˆ
x12), both parties compete on the elec-
tion and the policy points are 12
LR
xx with each
party’s winning probability of 12.
4. Three-Party System
For convenience, suppose that three parties, the Leftist (),
the Moderate (L
M
) and the Rightist (), represent three
groups, respectively, corresponding to their feasible policy
sets which divide the policy space symmetrically around the
median
R
12, that is, [0, ]
L
X
x,


,1
M
X
xx
and
[1 ,1]
R
X
x for some [0,1 2]x. We have three kinds
of electoral outcome depending on the value of
x
. First, if
[0,16)x, the only candidate comes from the Moderate
and is elected by choosing

3,1

3
M
x
xx. Second, if
(16,12]x, only the two extremists enter the election
with the policy points of L
x
x and 1
R
x
x , re-
spectively. This example shows that the median voter
result does not hold in two candidate competition be-
cause the candidates’ policy choices are restricted out of
the median. Third, if 16x, all the three parties com-
pete on the election at the policies 16
L
x, 12
M
x

and 56
R
x

with equal winning probability. The lit-
erature reports that equilibrium may not exist with three
(or more) candidates (Cox [4], Osborne [5]), but the pol-
icy constraint that the median point cannot be shared by
the candidates guarantees the existence of equilibrium.
5. Conclusions
The two-party and three-party competition in the above
sections show that candidates’ policy points disperse in
general even with two candidates when parties have dif-
ferent feasible policy sets. The constraint on the policy
sets makes the parties’ entry decision strategic and leads
to the dispersio n in policies. Also, it is shown that in po-
litical systems with more than two parties, extremists can
influence the political outcomes even though their feasi-
ble policy sets are relatively small, while a moderate can
do so with a relatively large feasible set. This is because
sincere voters close to an extremist’s policy point cast
the vote for the ex tremist even thoug h their ideal lies in a
moderate party’s policy set.
In this note, only two- and three-party political sys-
tems were considered. But the analysis can be extended
to an arbitrary -party system in which the feasible
policy sets of the parties divide the policy space symmet-
rically around the median voter’s ideal point. The above
results are preserved as well in the model with par-
ties. There are a couple of things to be more studied in
this research. First of all, endogenous party formation
should be studied. In the basic model, the feasible po licy
set of a party was exogenously given. However, it is de-
termined in the course of the formation or the evolution
(expansion, contraction, mergence, or division) of the
party3. Second, if the candidate of a party was policy
motivated within the feasible set, then he/she could not
commit to a policy poin t4.
N
N
6. References
[1] M. J. Osborne and A. Slivinski, “A Model of Political
Competition with Citizen-Candidates,” Quarterly Journal
of Economics, Vol. 111, No. 1, 1996, pp. 65-96.
doi:10.2307/2946658
[2] J. Cadigan and E. Janeba, “A Citizen-Candidate Model
with Sequential Elections,” Journal of Theoretical Poli-
tics, Vol. 14, No. 4, 2002, pp. 387-407.
doi:10.1177/095162902774006804
[3] G. Owen and B. Grofman, “Two-Stage Electoral Compe-
3As an example of evolution, Owen and Grofman [3] attempt to ex-
p
lain dynamics of the split of the policy space in a two-party competi-
tion with primary election.
4For reference, Snyder and Ting [6] analyze a model of legislative
p
olicy making in which legislators have two means of communica
t
ing
their preferences to voters: party labels and roll call votes.
2If it is assumed that the set of voters is a continuum and the number o
f
the members of each party is finite, then the finite partisan voters’
influence on the election is negligible.

Y. SUNG 20
tition in Two-Party Contests: Persistent Divergence of
Party Positions,” Social Choice and Welfare, Vol. 26, No.
3, 2006, pp. 547-569. doi:10.1007/s00355-006-0087-1
[4] G. W. Cox, “Centripetal and Centrifugal Incentives in
Electoral Systems,” American Journal of Political Sci-
ence, Vol. 34, No. 4, 1990, pp. 903-935.
doi:10.2307/2111465
[5] M. J. Osborne, “Candidate Positioning and Entry in a
Political Competition,” Games and Economic Behavior,
Vol. 5, No. 1, 1993, pp. 133-151.
doi:10.1006/game.1993.1007
[6] J. M. Snyder and M. M. Ting, “Roll Calls, Party Labels,
and Elections,” Political Analysis, Vol. 11, No. 4, 2003,
pp. 419-444. doi:10.1093/pan/mpg025