dr, some middle-of-the-road voters whose ideal points are greater than
1
2
play rd,
and extreme voters vote sincerely. In Proposition 2 we prove that there exists such
a symmetric cutoff equilibrium in this game.
Proposition 2 There exists a symmetric cutoff equilibrium (z
1
, z
2
, z
3
), where z
2
=
1
2
and z
3
= 1 − z
1
.
Even though we cannot get the exact value of z
1
, by using Lemma 1 and Propo-
sition 2, we can narrow down the range where the cutpoints z
1
and z
3
are located.
Corollary 1 The symmetric cutoff equilibrium (z
1
,
1
2
, z
3
), where z
3
= 1−z
1
satisfies
that
θ
DD
+θ
DR
2
< z
1
<
1
2
and
1
2
< z
3
<
θ
RD
+θ
RR
2
.
Corollary 1 shows that moderate voters act strategically in order to balance gov-
ernment policy in equilibrium. Even though our symmetric cutoff equilibrium is
found with a specific condition that z
1
= 1 − z
3
, this equilibrium explains why some
voters split their votes persistently in on-year elections. Also, in this equilibrium, we
can find an interesting implication. In this equilibrium, voters do not use even a rel-
atively sincere voting strategy. By relatively sincere voting we mean that given four
possible policy outcomes, θ
DD
, θ
DR
, θ
RD
, and θ
RR
, voters decide whom to vote for by
the distance of the expected policy positions from their ideal points. For example,
voters whose ideal points are in (
θ
DD
+θ
DR
2
,
1
2
) should vote for D in the presidential
election and R in the legislative election because their ideal points are closer to θ
DR
than any other possible policy outcomes. However, Corollary 1 says that this is not
the case. We don’t know exactly where the cutpoint z
1
is, but we know it is to the
right of the middle point of θ
DD
and θ
DR
, which means that some voters whose ideal
points is closer to θ
DR
than θ
DD
do not split their votes and vote for D in both elec-
tions. It is because of the uncertainty of voters’ ideal points. Even though their ideal
points are closer to θ
DR
than θ
DD
, their playing dr increases not only the probability
of resulting θ
DR
but also the probability of having θ
RR
as a final outcome. Therefore,
unless voters have strong preference over θ
DR
than θ
DD
, they do not split their votes.
That is, when incorporated with uncertainty, balancing theory does not necessarily
require all of the voters who prefer a moderate policy outcome to an extreme one
use a split-ticket voting strategy.
9
Convention