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working paper
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#'.?c
•,
of economics
MODERATING ELECTIONS
Alberto Alesina
Howard Rosenthal
No.
537
October 1989
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
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91^^
MODERATING ELECTIONS
Alberto Alesina
Howard Rosenthal
No.
537
October 1989
:-,«B,sP''
DEC
3 t
law
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,
MODERATING ELECTIONS*
Alberto Alesina
Harvard University, NBER, CEPR
and
Howard Rosenthal
Carnegie Mellon University and MIT
*This work was supported by National Science Foundation grant SES8821441
Moderating Elections
Abstract
This
structure
paper
extends
the
in which policy
legislature and of
the
spatial
theory
the
executive.
In
institutional
an
model
Nash
has
and
Von
several
an
setup
stable
implications
which
in
the
("Coalition
seem
split-ticket voting;
government with different parties
majority of the legislature; and
Alberto Alesina
Harvard University
NBER and CEPR
c)
b)
To
Proof").
consistent
United States
other democracies in which the executive is directly elected.
split
which the
we apply the refinements of
Neumann-Morganstern
testable
a)
in
the
outcome depends upon how everybody else votes.
observed patterns of voting behavior
the model predicts:
institutional
each voter has incentives to
analyze equilibrium in this game between voters,
Strong
to
choices are a function of the composition of
policy outcome depends upon relative plurality,
be strategic since
voting
of
and
with
The
some
perhaps
in
For instance,
for some parameter values,
controlling
the
executive
the mid-term electoral cycle.
Howard Rosenthal
Carnegie Mellon University
and
a
the
INTRODUCTION
1.
Policy outcomes typically reflect the composition of both the executive
and
the
legislature.
United
the
In
for
States,
instance,
the
administration's influence on policy making is affected by the composition of
Congress.
Traditional
models
spatial
elections
of
ignore
this
institutional
complexity and assume that the winner of a two-party race has full control
over policy.
interaction between the
These models focus on the strategic
candidates and treat each voter as a passive player, voting for the candidate
offering the policy closest to his ideal point.
In our model the voters instead face two parties with policy-preferences
[Wittman
outcomes.
(1977,
that
1988)];
is,
preferences
party
intermediate
between
policies
ideal
the
on
policy
executives representing the
Without "checks" by the legislature,
two parties would follow distinct and polarized policies.
policies
defined
are
of
2
Voters with ideal
parties
two
the
take
advantage of the institutional structure to "moderate" the executive and thus
achieve a policy outcome closer to their ideal.
not
only depends upon the
relative plurality of
the
identity of
two
parties
feature of the model implies that,
the
in
executive but
the
the actual policy
In fact,
This
legislature.
each voter depends upon his beliefs about
other
Indeed,
one
needs
whenever policy reflects more than
a
coordination
institutionalism'
game
s"
between
a
the
concern about
the
realistic
unlike in the traditional spatial model,
the decision of
voters.
reflects
also
to
consider
strategic
single binary choice.
voters
the
in
the
interaction of
the
voting
Thus,
context
the
legislature [Shepsle (1986), Hammond and Miller (1988)].
behavior of
of
behavior
we consider
the
executive and
"new
the
Our model of "moderating behavior" has implications which are consistent
with some observed voting patterns in the United States.
the executive and the legislature are elected simultaneously,
voting
split-ticket
generally
That
occurs.
different parties in the two elections.
some
is,
Consequently,
when
For instance,
we predict that
voters
vote
for
the party winning the
election for the executive branch may not have a majority in the legislature.
In
parameter values
some
for
fact,
predicts divided government,
model
the
with different parties controlling the executive and the legislature.
Second,
United
model
the
States:
is
consistent
phenomenon
persistent
a
cycle".
the executive
the party holding
elections,
voting
"mid-term
the
with
mid-term
In
3
the
in
congressional
loses plurality relative
to
the
preceding congressional elections held simultaneously with the presidential
Even though other explanations of this strong regularity have been
election.
such as
proposed,
which
abstracts
incumbency
Rosenthal
as
well
"coattail effect",
the
from
personal
preferences,
(1989))
and
4
we offer a different
characteristics
etc.
an
In
of
empirical
presidential
study,
we
explanation
candidates,
(Alesina
and
have shown that this "moderating", model performs at least
often
better
than
traditional
empirical
voting
models
that
emphasize incumbency advantage or retrospective voting based on the state of
macroeconomic policy variables.
We
model.
ideal
investigate both a
In
the
points
former,
is
but
common
full
not
information and
in
the
knowledge.
latter,
Some
an
incomplete
information
the distribution of
of
the
results
voters'
concerning
split-ticket voting and moderating behavior emerge in both cases and can be
more easily illustrated in the certainty model,
rather
general
assumptions about
which can be developed with
utility functions,
distributions of
ideal
and
points,
the
impact of
cycle emerges only
incomplete
the
in
legislative plurality on policy.
the
information model.
mid-term
A
Explicit
solutions
for this case can be obtained under more restrictive functional forms,
are
needed
for
computational
should
assumptions
result
reasons.
small
in
Small
deviations
quantitative
from
differences
which
these
and
no
qualitative differences in the results.
[Aumann
(SN)
and
Peleg
point of view,
technical
From a
Whinston
refinements
(1987)]
games.
Since we
consider
defined
only
a
results
to
for
shows how
Coalition Proof Nash Equilibria
and
(1959)]
this paper
continuum of
a
number
finite
of
be
can
usefully
voters,
players,
and
we
develop a directly analogous refinement for
[Bernheim,
(CPN)
applied
to
voting
concept
CPN
the
use
Strong Nash
the
Greenberg' s
is
(1989)
continuum case.
the
This refinement is a Von Neuman Morgenstern (vN&M) stable set.
These
voters,
refinements
are
needed
for
two
reasons.
every strategy is Nash for the individual voter,
have any measurable
impact on the outcome.
held separately,
the
at
"mid
term"
with atomistic
First,
since he does not
When legislative elections are
of a presidential
the vNScM refinements lead to a unique equilibrium that
term,
both the SN and
is directly analogous
to the unique Nash equilibrium with a finite number of voters.
equilibrium has
the
intuitive
property
that
no
voter would
This SN-vN8iM
like
to
see
a
decline in the plurality of the party he is voting for.
Second,
with either
a
finite or an infinite number of voters,
the "all
or none" feature of the majority rule used in presidential elections requires
refinements beyond the simple Nash notion.
the situation where no voter is pivotal
In
the
We need refinements
to
address
in the presidential election.
specific context of our certainty model,
when the president
is
simultaneously
elected
legislature,
the
one with party D winning
equilibria:
identical
with
to
outcome
SN
the
are
two
possible
presidency and a legislative vote
the
when
there
party
legislative elections are held separately,
"D"
as
in
holds
executive
the
"midterm",
the
and
and another
corresponding to the outcome when party "R" holds the executive when midterm
We need refinements to indicate whether one or both of the
elections occur.
two outcomes is "stable".
We show that,
president and
always
exists
turns out that SN is too strong a refinement.
It
for some parameter values,
there is no SN equilibrium when the
legislature are elected simultaneously.
when
although,
the
parties
two
located with respect to the median voter,
the refinements appear
to match
not
too
a
vN&M
asymmetrically
both outcomes may be vN&M.
intuition.
are to the same side of the median voter,
are
contrast,
In
Again,
when both parties
For example,
there is a unique SN-vN&M where the
party closer to the median holds both the presidency and a majority in the
for symmetric utility and symmetric distributions of
Moreover,
legislature.
voter ideal points,
the outcome where the party closer to the median wins the
presidency is always vN&M.
Fiorina
has
(1988)
recently
that
model
split-ticket
of
voting
Fiorina's specification of executive-legislature interaction is such
only
four
combinations
legislature,
importantly,
Fiorina
a
There are three important differences between the
which is related to ours.
models.
proposed
of
policy
outcomes
are
holding
parties
possible,
the
executive
corresponding
and
the
while we allow for a continuum of policies.
Fiorina
considers
does
only
a
not
one
consider
period
strategic
model,
he
relationship between split-tickets and the mid-term cycle.
the
majority
Second,
voting.
does
to
in
the
and more
Third,
not
four
address
since
the
IHE MODEL WITH COMPLETE INFORMATION
2.
To facilitate intuition,
Section 3 several restrictive assumptions are relaxed.
In
2.
we first present a simple version of the model.
Assumptions and Institutional Setup
1
We consider two-party systems in which the parties
candidates) have policy preferences.
parties
and
the
voters
have
In particular,
unidimensional,
(identified with the
we assume that
continuous,
single
strictly concave utility functions over a single policy issue.
periods
utility
and
factor (3^1.
functions
the utility functions are
All
In each period
point.
objectives written
intertemporally
are
t,
=
t
=
u(x^,
e^)
has
9
9
and 9
objective
an
function
distribution of voters'
interval
=
I
and
There are two
discount
with
labelled D and
have
R,
analogous
to
R
(1)
are the two bliss points,
The population of voters is infinite,
.
= D,
j
where x represents the policy issue,
<
peaked
as:
u
9
two
identical except for the bliss
two parties,
the
1,2,
additive
the
(1),
and each voter,
with
bliss
indexed by
point
9
i,
The
.
bliss points is uniform and normalized in the closed
We also impose the unrestrictive assumption
[0,1].
with
<
<
9
D
<
9
R
1.
the
In
voters
are
information model
complete
common
control of policy,
knowledge.
it
Thus,
Alesina
would be
(19S8).
In
if
preferences of both parties and
the
follows its ideal policy
sub-game perfect equilibrium in
in
the
fact,
no
a
(9
in
or 9
office
).
has
complete
This is the only
finitely repeated electoral game as shown
other
policy
implemented by the elected party,
its ideal policy when in office.
party
Thus,
announced
prior
to
elections
given that the party can choose
rational voters would not believe in
any announcement other than the party ideal policy.
Our model of institutional structure posits that policy is determined by
between
"compromise"
a
Let
legislature.
dropped where
=
V
turnout,
1
-
is
V
V
be
to
D,R,
no possibility of
confusion.
president,
is
those
of
the
votes
of
subscripts
Time
are
Since we assume full
actual
the
and
proportion
the
elections.
When D
.
=
J
,
president
the
legislative
the
in
j
there
preferences of
define
us
party
by
obtained
the
policy,
x
is
,
as
follows:
x° = 6
+
kV
D
When R holds the presidency,
x''
the policy x
= e
kV = e
R
(2)
.
R
D
is given by:
,
-
k(l-V
(3)
)
R
R
We also impose the condition
:S
!k
<
-
'@
<4)
.
D
R
which implies that the policy outcomes are in the interval
[9 ,9
D
R
T)
X
<
any
for
x
V
6
legislature
The
is
elected
representation in a single national district.
proportional
strict
by
The president
and that
]
'R
elected by
is
majority rule.
We consider several electoral scenarios.
to mimic the United States case;
the
legislature
elected
are
The most general one is meant
in this scenario,
contemporaneously;
at
at
the executive and
t=l,
t=2,
there
is
another
legislative vote, while the president elected at t=l remains in office.
is,
president
the
serves
two
periods
while
the
That
serves
legislature
one
period.
Each
voter
moves
in
Randomization is ruled out.
DR,
or
RD,
where
the
first
both
periods.
A
voter
The first period move
letter
represents
the
is
can
vote
D
or
indicated as DD,
presidential
choice,
R.
RR,
the
second the legislative choice.
single choice between D or R describes the
A
move in the second period, when only the legislature is chosen.
period
D
1,
period 2
in
if
president,
R
R
period
in
2
if
Thus,
"RR in
president,"
D
illustrates an example of a voter strategy.
Note
strategies
that
do
reflect
not
the
conditional strategies in the second period.
is
learning
no
election at t=2.
and
shares
from
first
the
Who holds the presidency is the only information relevant for voter
period.
there
vote
on
not
entire
the
true not only because
is
legislature
up
is
for
Policy at t=2 depends only upon who holds the presidency
size
the
because
also
but
This
of
the
legislative
majority
at
t=l.
The
preceding
including the assumption that the party ideal policies are fixed,
discussion,
implies that any finite numbers of repetitions of this game can be captured
with a two period analysis.
2.2
Definitions
Equilibrium:
To focus on Nash equilibria
that
correspond
to
intuition,
we
consider
the equilibria which are in the core or the Von Neumann Morgenstern abstract
our
game
stable
set
(1989),
which we now define.
Let
of
an
be
z
for
the
individual
dominance
strategy,
relation
be
Z
i's
proposed
(nonempty
by
and
Greenberg
compact)
1
strategy set,
Z
the
Cartesian product of
where M denotes the set of players.
and
S'
denote subsets of
the
Z
and z an element
,
Let u (z) denote i's utility under
M.
Consider the abstract system (W,<):
W = {(S,z)
For
(S' ,z'
)
and (S,z) in
W,
I
S c M,
S
?=
}
of
z.
Z
,
S
(S'.z')
(S,z)
<
oScS',
u (z)
>
for all
u (z')
1
i
eS
1
i
and
•
= z
z'
8
for all
J
«£
S.
The core of the system is the set of undominated elements
B = W\A(W)
.
•
B,
where A(a)={beW|b<a) and for some subset K£W, A(K) = u{A(a)|a€K>.
is,
is an vN&M abstract
a set K
Moreover,
stable set if K = W\A(K).
K is internally and externally stable.)
dominance
This
relation
is
(That
:
particularly
because
appealing
Greenberg
shows that for a game with a finite number of players there exists a
(1989)
correspondence between the core and the set of SN equilibria and between the
vN&M stable set and the set of CPN equilibria.
More specifically,
Greenberg
establishes the following result:
For a finite set of players N,
Theorem:
(N,z)
the elements z of the pairs
(i)
that are in the abstract core B represent the set of SN equilibria to
defined by Z
the game
N
(ii)
;
(N,z)
appears as an element
of
an abstract
stable set if and only if z is in the set of CPN for the normal form of the
game defined for
Greenberg'
of part
s
N.
proof is for a finite number of players;
has not been defined for a continuum of players,
recursive
The notion of CPN
carries through to the continuum trivially.
(i)
argument.
However
the
stable
since
it
defined
set
however the proof
is
based upon a
with
Greenberg*
dominance relation extends the basic intuition of CPN to the continuum case.
That
is,
an
equilibrium
threaten to deviate from
payoff.
The
requirement
CPN
is
it
if
no
coalition
of
players
can
credibly
by changing their strategies and improving their
of
credibility,
that
is
the
requirement
sub-coalition could threaten to deviate from the original deviation,
that
no
is what
distinguishes CPN from SN.
apparent
when we
study
the
The
importance of
this distinction will
simultaneous election of
the
become
president and
the
legislature.
It
will be shown that equilibria in the stable set always yield policy
outcomes equivalent to those brought about by a subset of the equilibria that
can be described by cutpoint strategies,
Definition
element
An
:
z
of
Z
is
a
which we now define.
strategy
cutpoint
contest if there exists 9 such that voters with
6
<
an
for
9 vote D and
electoral
voters with
1
9
[The voter type
vote R in that contest.
9
>
9=9
does not influence the
1
1
analysis because of zero measure.
We denote cutpoints in presidential contests by 9 and in legislative contests
by
We will show that legislative components of strategies always follow a
9.
cutpoint rule.
Instead,
strategies are possible,
in presidential elections,
non-cutpoint equilibrium
as a result of majority rule.
an equilibrium (9,9) exists and there exists Sc[9 ,1],
Suppose,
9<9 <1,
in the presidential
all members of
to
D
for
attributed
vote
R,
given
9,
u(x ,•)
D
>
u(x ,•) for
Then the non-cutpoint strategy where members of S switch
S.
president
even
election and,
such that S is
R
'*
not pivotal
for example,
an
is
also
an
infinitesimal
equilibrium.
possibility
thus implying a cutpoint strategy.
Note
to
that
if
these
being pivotal,
Consequently,
a
they
voters
would
cutpoint for the
presidential vote is the most natural type of equilibrium even though we do
not impose this restriction.
This lack of restriction in the strategy space
makes the statement of propositions and proofs significantly more laborious.
Consequently,
in
the
main text we consider only cutpoint strategies.
This
restriction is dropped in the Appendix.
With outpoint strategies,
first period equilibrium.
ideal points below
there are only three types of voters
For instance,
Equilibrium
:
the
and 9 and RR
9
9.
Characterizations
critical
illustrate
To
in
we have DD voters who have
if 9 <
RD voters who have ideal points between
8,
voters who have ideal points above
2. 3
9
aspects
the
of
consider
we
model,
several
institutional scenarios, building up from the simplest one.
No Legislature
2. 3. a
With no
legislative
inf luence, (k=0)
policy for both periods.
elected president
the
,
determines
a SN equilibrium exists with outpoint
In this case,
implicitly defined by:
u(0
This
the
is
While if 9
,0
= u(9 ,9)
)
D
(5)
R
standard result of
theory of two candidate elections.
the
1/2 an infinity of other outpoints also are SN,
5^
9
is
unique
a
cutpoint under the assumption that no voter uses a weakly dominated strategy.
When preferences are symmetric, we obtain:
9+9
^^
6 =
As a benchmark.
Figure lA indicates whether D winning or R winning is
£ 9 < 1/2 ^ 9
the SN outcome for
2. 3.
b
Legislative Elections
Let P
party
R
12
1
-
= P u(9
u
=
A Condition on Dominance
presidency
the
objective function of voter
where P
:
be the probability,
holds
if
2
R
< 1.
R
D
i
immediately prior to period
in
period
Using
t.
t
elections,
(2)
(1),
and
(3)
that
the
at t=2 is given by:
k(l-V
R2
),9
)
+
1
2D
(1-P )u(9
=
president R is in office or P
2
2
10
+ kV
R2
if D
,9
(6i)
)
1
is
in office.
2
.t-i
«*
u
Pu(e
I
11
-k(l-V
R
1
),e)
Rt
t=l
In
11
the objective function of voter
At t=l
the complete
information model P
i
+
can be written as:
ID +kV
(l-P)u(0
1
,9
Rt
can assume only three values:
the equilibrium strategies imply an R victory in the presidential
if
imply a D victory,
they
or P =1/2
information model analyzed in Section
From
case of a
in
and the strict concavity of
(6)
In
P =1
race,
if
P =0
incomplete
the
can assume any value in [0,1].
P
4,
tie.
(6ii:
1
the
function u(*),
following
the
result follows immediately:
Lemma
1_
For fixed P
voter's indirect utility function on V
a
,
t
is
Rt
continuous
and single-peaked.
Let us define V
in
is
indirect bliss point for party R'
i's
s
vote
as the bliss point when the probability that R wins the presidency
V
and
1,
as voter
legislative election when the probability that R wins the presidency
the
is P,
IP
presidency is
t=2 and
as
V
=
Hi
bliss
Note that V
0.
that V
the
V
H2
,
when
point
1
j=D,R.
p
is
the
probability
that
R wins
the
independent of the moves to be taken at
we drop
Thus,
the
time
subscript.
We now
state our central result concerning voting in legislative elections.
Proposition
_1_
In period
t,
for any value of P
,
'
a
necessary and sufficient condition
t
for
(S.z)
period
t
to be undominated by any
legislative components
(T,y)
is:
11
where y differs from z only in the
for all measurable T c S such that leT votes R in the legislative
election,
V
(z)
p
\
^ V
R
and
R
(c;
for all measurable L c S such that ieL votes D in the legislative
election,
V
(z)
i v'^.
R
Proof
:
If
satisfied,
not
is
(C)
votes D and V
(z)
V
<
1
p
i
p
or
either
is
some LcS such that
(b)
some
TcS
such
that
leT
ieL votes R and V (z)
>
R
assume
Without loss of generality,
.
there
(a)
R
R
V
R
(a).
By continuity,
there exists a
R
measurable subset of voters
switched the period
(S,z)
does
that
T'
cT whose
utility would be
increased
legislative component of their strategy to
t
satisfy
not
condition
the
is
dominated.
So an
R.
The
they
if
proof
of
sufficiency is immediate.
Q.E.D.
This proposition implies that,
in a SN equilibrium,
there are no voters
who would like to see a reduction of the share of votes of the party they are
voting for in the legislative election.
2. 3. c
Legislative elections at t=2
We now establish a result
.
that
resembles the median voter
traditional two party electoral models,
median voter.
In our
theorem.
policy equals the bliss point of the
we have a "pivotal voter theorem" where policy
setup,
equals the bliss point of the pivotal voter in the legislative election.
us
define
as
6
elections when party
Proposition
2:
ideal
the
j
In
point
of
the
outpoint
voter
in
the
Let
legislative
holds the presidency.
Pivotal Voter Theorem
e
R
*"
The
cutpoint
strategy
^
9 =
R
:;
—
;-
1+k
represents
president R in office.
the
unique
SN
and
vN&M
for
(7)
12
e +k
The
strategy
cutpoint
——
8 =
-
represents
;
1+k
D
unique
the
and
SN
vN&M
for
president D in office.
(8)
e<e=x°<e=x"<e R
(9)
R
D
D
Proof:
For
(0,1),
g
v'-"
R
and (3) imply
(2)
9-e+k
„iR _
R
k
R
We use 9
e-e
„1D _
R ~
1
'
_1
k
as a shorthand for the cutpoint strategy with 9
By the uniform distribution of voter preferences,
1-9
and 9
for V
R
J
such that V
for 9
(9
= V
)
(10),
1
^R=
,
I^'
show that 9
satisfies
involves measurable deviations from 9
then,
(1,9
9
+
k
^0=
-W
It
is direct
(C)
for
(1,9
^''^
)
s
and that any
fails to satisfy (C).
z
*
9
,
that
(1,2)
By Proposition
does not.
Hence,
R
~
by Greenberg'
to use single-peakedness
R
belongs to the core and any (I.z),
)
R
Substituting
are given by:
R
1,
.
J
Assume president R is in office.
to
9
we obtain that the unique outpoints 9
9
1)
-
1
the cutpoint.
J
= 9
for 6
R
J
in
=
V
as
!
J
R
(Lemma
,,„,
(10)
D
theorem,
9
represents the unique SN.
The proof of (8) parallels that of (7),
substitution of
into (2) and
and 9
9
R
12
and (9) follows immediately from
(3).
D
Q.E.D.
This result establishes that the policy outcome equals the ideal point
of
the
voter at
the
legislative cutpoint.
Every voter on the left of
cutpoint would want a policy more on the left,
13
thus a reduction in
V.
K
the
and
vice versa.
This proposition also shows that voters use the legislative election to
president's
incumbent
the
pull
preferred
policy.
legislature
That
from
the
period
the
For this reason,
president.
the
away
second
the
in
is,
moderate
to
policy
incumbent's
voters
a
most
use
the
party receives
more votes in the iegislative elections when it does not hold the presidency
than when it does.
and
of
Note that Proposition 2 holds regardless of the positions
relative
e
median
the
to
voter;
the
R
D
pivotal
f
voter
is
always
J-
even if the two parties are both located to the
interior to the two parties,
same side of the median.
both
If
parties
are
same
the
to
side
of
median,
the
an
immediate
corollary to Proposition 2 is that the more extreme of the two parties can
never win control of the legislature.
are on opposite sides of the median,
legislature.
and
legislative
For expositional purposes,
the opposition can gain control of the
elections at t=l
consider the first period election in isolation.
presidential vote for strategies
M
>
1/2,
(that is,
§
.
let us consider first the case
the president and the legislature serve one period.
If:
if the two parties
"Divided government" becomes a possibility.
Presidential
2.3.d
On the other hand,
in which both
This implies that we can
Let M
be
the Republican
z.
<
1/2 with outpoint strategies) R is elected
president.
M
Rz
= 1/2,
(that is,
held,
M
<
1/2,
6 = 1/2
with outpoint strategies) a lottery is
R is president with probability 1/2,
(that is,
9 >
1/2 with outpoint strategies) D is elected
president.
14
We first establish a useful preliminary result.
Proposition 3
Any pair (I,z) is always dominated in a tied presidential election.
Proof
See Appendix
:
The basic intuition of the proof is that the voters can more effectively
moderate
election.
which
executive
the
Thus,
imply
moderation:
is
tied presidential
a
sure
winner
follows
that
a
it
there
if
with
tied
a
no
the
"optimal"
presidential
distribution
symmetric
a
about
presidential
the
race is always dominated by strategies
result holds even with complete symmetry;
preferences,
uncertainty
race
that is,
voter
of
amount
legislative
of
cannot
be
This
SN.
even with symmetric voter
ideal
points,
party
and
locations symmetric around the median voter.
If SN equilibria exist they imply either D
legislative outpoint
winning the presidency with a
and/or R winning the presidency and
9
9
.
Proposition 4
A necessary and sufficient condition for a pair (I,z) to be SN is either
1.
and
M
Hz
>l/2(Ris
;
the
legislative component of z is 6
(the following conditions
+ k
if 9
i 9
D
,
R
[l.Ai]
e
insure that 9
is
l.B
+ k <
if 9
9
D
[l.Bi]
[l.Bii]
the following
^ condition is not satisfied:
there does not exist x
,
R
=9 D
X
=
V
1
+
R
-
R
[l.Biii]
u(x*,9'
kV
+
9
9'
R
)
= u(9 ,9'
)
R
15
,
9'
~
—R -\
i "*"K
not dominated by electing D
-.
2
>
R
=
R
president)
l.A
9
"
president)
which satisfy
[l.Biv]
e'
>
-.
2
.
,
e +k
M
2.
<
Rz
1/2;
the
legislative component of z is §
—-.
—1+k
=
In addition,
;
D
z
must satisfy conditions symmetric to those given for R president.
Proof
See Appendix.
:
Corollary
There is a unique SN outcome when either 9
i 1/2.
1/2 or 9
:s
D
R
The
corollary
have
we
that
shows
unique
a
when
SN
both
legislative
outpoints are to the same side of the median, with the cutpoint closer to the
median corresponding to the stable outcome.
the same side of the median whenever
the
Of course, both outpoints are to
two party bliss points are
the
to
same side of the median.
Remarks
1.
^
(x +9 )/2
s
(x +9 )/2 = d.
D
Analogously,
r.
domination
for
of
9
The relationships
—
=
r
+
(9
+k)(l+k)
=
^
^^^^
(2-k)(l.k)
1/2,
r
and d=
9
<
parameter regions where a SN exists.
can be seen that for small
far
from
the
k
1/2,
d
>
to
the
amount
then define
9
the
D
These are shown in Figure IB for k=0.
the parties
and
"objecting"
The
of
16
moderation
2.
reasonably symmetric about
both R winning
median,
winning the presidency are SN.
relative
'
-k)(l+k) + (9 +k)(l-k)
R
median,
=
9'
(2-k)(l+k)
(9
'^
median but
have
I-
9 (1-k)
the
would
we
we obtain:
After simple algebra,
o
^
It
9'
condition [l.Biii] becomes
^
*
=
In the case of symmetric preferences,
the
party is
presidency
too
possible,
far
to
and
D
from the
upset
the
presidential victory of
with
k>0,
parties
the
For
other party.
this situation cannot occur,
0.4),
all
the
close
larger values of
as illustrated by Figure IC.
around
symmetry
full
to
(such as
k
In fact,
for
median
and
the
sufficiently close to the median, a SN does not exist.
2.
Examination of the conditions in Proposition 4 thus shows that the
Greenberg core
to
voting game
this
empty,
can be
or be characterized by two outcomes.
outcome,
characterized by one
be
that conditions
however,
Note,
for domination of an equilibrium with an R president imply that voters
[l.B]
»
with
points
ideal
below
the
policy
(x
be
to
)
implemented
if
becomes
D
president are required to vote R for Congress as domination occurs.
voters are not behaving consistently with condition
(C)
consistent behavior implies that they should vote DD.
Such
Proposition
of
(See Figure
2.
1:
Thus,
)
the absence of a core for some parameter values is not troublesome if we can
show
elimination
that
either 6
or 6
R
D
of
this
kind
of
inconsistency
always
reestablishes
policy
outcome in the vN&M abstract stable set.
as a f
J
This is
precisely the implication of the following proposition.
Proposition
5
An abstract
(I,z).
The
winning
the
stable
pair(s)
set
(I,z)
presidency
always
exists
which are
with
contains
and
in
the
certainty
and
stable
9
set
as
the
at
least
must
have
cutpoint
one
pair
party R
of
the
legislative elections and/or party D winning the presidency with certainty
and
9
Proof:
as the cutpoint of the legislative elections.
14
See Appendix.
Corollary
There is
a
unique vN&M outcome when either
17
9
^ 1/2 or 9
^ 1/2.
Remarks
A pair
1.
implying a tie in the presidential election is not in
(I,z)
the stable set.
The only pair(s)
(I.z)
included in the set
election and a policy
outcome of
presidential
'
^
^
symmetric preferences,
In the case of
2.
or 6
R
imply a certain
.
D
the analogs
to
(12)
and (13)
are:
+
9
^
,
D
(2+k)(l+k)
+k)
(9
equations
+ 9
(1+k)
(2+k)(l+k)
r'=l/2 and d'=l/2 define,
winning the presidency with outcome
members of
abstract
the
(14)
°
d'=
The
+k)(l+k)
(e
R
_
9
stable set.
R
(15)
respectively,
the
and D winning with 9
regions where R
are supported by
D
Figures
ID
least one of the two outcomes is always supported.
and
illustrate
IE
that
at
When the two parties are
reasonably symmetric about the median, both are supported.
Thus,
in summary:
DR2
If
(i)
9
<
party R wins the ^presidency with certainty,
-
:£
and
the
equilibrium outcome is unique.
(ii)
If
>
- party
i
9
D wins
the
presidency,
and
the
equilibrium
outcome is unique.
(iii)
If
9
<
D
-
<
R
the presidency with 9
9
is
as
there are two possible equilibria.
9
2
as
in
(ii),
If
the parties'
but,
in
(
i
)
In one R wins
and in the other D wins the presidency and
depending on parameters,
only one of
these
may be
vN&M.
(iv)
bliss points are symmetric about the median
(9
+
=1)
9
and preferences are symmetric,
D
R
In cases
tickets for 9
RD split
party
and
(iii)
of
the
there is always split ticket voting.
(iv),
^ and DR split
>
tickets for 9
receives
president
winning
are both equilibria.
and 9
9
D
votes
fewer
<
That
^.
the
in
We have
is,
the
legislative
elections than in the presidential election.
In the less realistic cases
legislative
knife-edge case of
2.
President
3e
President
period
serves
may be on either side of the
9
voting
ticket
Split
outpoint.
and (ii),
(i)
will
occur
except
in
the
1
the
9 = 9.
and
two
legislature
periods
elected
with
simultaneously
additional
an
in
period
legislative
;
election
in
2.
With no uncertainty the outcome of Presidential election is known before
Since no information is revealed by the first
the first period elections.
period
elections,
voters'
behavior
does
not
change
in
the
second
period.
This implies that the equilibrium cutpoints in the legislative elections in
the two periods are identical.
More precisely,
the two period case.
case
the
strategies
it
can be shown that Proposition 4 and 5 generalize to
Proposition 4 applies identically except that in this
z
of
characterized by either §
except
that
components
the
pairCs)
characterized
include
(I,z)
or
9
(I.z)
by
legislative
two
Proposition
.
in
the
either
9
5
stable
or
9
still
set
.
components
applies
have
Proofs
two
of
both
identically
legislative
these
two
Propositions for the two period case are very similar to the proofs in the
Appendix and are available upon request.
Based upon these results the following proposition is immediate:
19
Proposition 6
With
complete
information,
identical in both periods;
3.
equilibrium
in
legislative
the
vote
is
there is no mid-term congressional cycle.
GENERALIZATIONS OF THE COMPLETE INFORMATION MODEL
This section shows that our results survive,
at
least qualitatively,
if
we relax several simplyfying assumptions adopted thus far.
i)
We
Impact of the legislature on policy.
can
relax
the
assumption of
linearity and postulate
given by:
that
policies
are
'
,
x"
DDR
=
x°
=
e + g (V
e
R
.
- g (i-v
R
R
(16)
•
)
(17)
)
where g (•) and g (•) are strictly increasing in their arguments,
and twice differentiable,
=
and g (0)
g (0)
=
0.
o
R
also assume that for every value of V
In keeping
continuous
with
(4),
we
the following holds:
R
e+g(V)<e
DDR R -g(l-V)R
(18)
'^R
a)
Distribution of voter ideal points.
We now assume that this distribution is continuous and twice differentiable
on [0,1].
Consequently,
if 9 is a outpoint,
V
=
we can write,
l-H(e)
(19)
R
where H is strictly decreasing and twice differentiable.
Even
Proposition
under
1
these
general
remains unaffected:
assumptions,
one
important
feature
preferences are single-peaked in V
of
and a
necessary condition for SN equilibria in the legislative elections at t=2 is
that the equilibrium policy equals the ideal point of the pivotal voter.
20
For
suppose D holds the presidency.
instance,
x° = e
= e +g
D
Then.
(i-H(e
D
D
D
satisfies the following:
9
^^
(20)
))
An analogous result holds when R is in office:
x''
= G = e -g (H(e
R
R
R
(21)
))
R
Since g (•) and g (•) are strictly monotone,
R
the implicit functions
and
(20)
D
unique
imply
(21)
Proposition
V
,
for
x
D
and
x
R
Note
.
that
is
maintained
here.
also
is
It
immediate
proof
the
relies only on the single-peakedness of voters'
1
which
solutions
of
preferences on
establish
to
R
the
following inequalities:
9
<
The first
and
9
<
9
third inequalities follow
continuity of the distribution voters'
will
always exist
some
9
<
R
D
D
(22)
R
immediately from the assumption of
ideal points,
voters who want
which implies that there
"moderate"
to
the
president.
second inequality
follows from (18) and from the fact that V
^
indicates the legislative vote for party R when president
D
is
j
>
V
R
R
where V
,
and
legislative
elections
are
in office.
possible equilibria, both with ticket splitting voters.
wins the presidency and 9
is
the cutpoint of the
d
and e
when presidential
As before,
simultaneously,
held
\
R
R
Condition (22) also implies that the results of sections 2.3c,
also apply (qualitatively) in the general case.
The
there
are
only
two
In one equilibrium R
legislative election.
In
R
the other D wins
which,
as
before,
the presidency and 9
is
used
to
moderate
is
the cutpoint for
the
president.
the legislature,
The
argument
used
previously to rule out tied presidential elections continues to hold.
Finally the conditions under which the two possible equilibria are SN or
vNScM
are
However
in general
the
different from those derived in Propositions 4 and
basic features of these conditions should be
21
unchanged:
5.
for
instance,
more unbalanced
the
toward
candidates relative to the median voter,
the
presidency
and
with
particularly evident if g (O = g
(•).
presidents
two
can
be
degree of asymmetry in the
whether
affects
or
9
president
paribus,
R
it
is
This
moderated.
represent
of
examined
is
that R wins
the
legislative
Section
in
* g
(•)
g
(•)
influenced
by
there is
stable
outcomes.
legislature
the
with
fact,
In
•
the
than
J'
•
.
ceteris
then,
D,
say,
if,
likely that the equilibrium with R president and
is most
is
2
with respect to the median
and 9
9
together
asymmetry,
D
more
the
due to the different degree in which
location of
9
R
likely it
instead,
If,
bliss points of
the
outpoint
case
the
an additional asymmetry in the system,
the
the more
equilibrium
the
similarity
This
elections.
is
are
left
the
the
9
R
This is because the voters could more
legislative outpoint is vN&M stable.
easily achieve
easily
or'
outcomes
"moderate"
influenced
by
electing
by
legislature.
the
the
president
general,
In
mid-term
cycle
in
congressional
votes
is
ruled
out
may
it
impossible to derive these conditions analytically.
As
by
who
be
is
difficult
in Section
the
more
2,
assumption
a
of
complete information.
THE MODEL WITH UNCERTAINTY
4.
This section considers an example in which the distribution of voters'
preferences
is
perturbed by a random shock.
equilibrium with a mid-term cycle.
the
simplest
Section
2.
with
case,
all
the
We
To construct
simplifying
show
the
existence
this example,
assumptions
and
of
an
we return to
notation
of
We also assume quadratic preferences:
=
u
tj
-
)^
(l/2)(x -e
t
J
j
^
= D,R,i
(23)
The extremes of the uniform distribution of voters ideal points are now given
22
by
I
=
where a
1+a]
[a,
uniformly on the interval
random variable with zero mean distributed
a
is
with:
w]
[-w,
w<e<e<i-w
D
both
any realization of a
for
Thus,
sides
of
and
9
6
R
there are voters with bliss points
distribution
The
.
(24)
R
of
a
"common
is
on
knowledge".
D
Independent draws of a are made at
t
5
and
]_
voter types is contained in the interval
=
I
=
t_
1+w].
[-w,
equilibrium strategies are specified in terms of
The set of possible
2.
I
section,
this
In
Finally we assume that
.
w
voters cannot communicate their preferences to each other.
Consider
first
parallels Section
the
2. 3. c.
legislative
elections
at
t
=
analysis
The
2.
which now
The Republican vote for the legislature,
depends on the realization of
The expected utility of
is denoted V (a).
a,
R
voter
i,
with a bliss point
such that B
6
1
RID
9
:£
8
:£
when a president R is
in office in period 2 is given by:
IT
Eu
"
- 41
2
1
An analogous expression for Eu
-
e
kCl-V (a))
^a
2w
- 9
R
R
I
1
(25)
111
Assumption
holds if president D is in office.
Voters with bliss points
(24)
is key
1-w,
by observing their own preferences and knowing that the random variable
a
is
to
(25).
distributed uniformly,
value of
a
would update,
and,
dominant strategies for every realization of
Thus,
Denote with 9
the
presidency.
such that 9
as a
result,
<
w or 9
>
their expected
Condition (24) implies that these voters have
would not be zero.
respectively.
9
a:
they always vote RR or DD
their updating does not affect the equilibrium analysis.
the legislative cutpoint in period 2 when party
Also
let
denote
EV
i's
indirect
bliss
point
j
holds
for
R'
The
t=2
R
expected vote at t=2 when party
j
has won
23
the
presidency at
t = l.
results for the second period subgame are given in the following proposition.
Proposition 7
sufficient
necessary and
A
condition
for
(S.z)
be
to
undominated
in
the t=2 sub-game is:
for all measurable TcS such that ieT votes
R,
EV-'(z)
£
Ev'-'
for all measurable LcS such that ieL votes D,
EV-'(2)
i
Ev'"'.
Moreover,
~
Q
-,
MM
is the unique SN equilibrium when a f
president R is in office
r-
l+k
e +k
~
—D—r
=
R2
is the unique SN equilibrium when a president D is in office
:;
l+k
D2
X
—R
=
R2
R
R
e
R
—
=
-
=
——
+
j-
is the policy with a R president
ka
.-':'
e +k
X
e* = ECx'^^)
R2
is the policy with a D president
ka
+
j-
;
9*
D2
= E(x°^)
See Appendix.
Proof:
analogy with
The
the
complete
information
case
policy is equal to the ideal policy of the pivotal voter
Let us now turn to the first period election.
that
is
or §
(6
In order
to
expected
the
).
simplify the
analysis we introduce a useful assumption:
Al
at
t=l,
the second period strategy components are restricted to
the
SN strategy in the t=2 subgame.
In
period
period
2.
Al
1
is
voters'
decisions
depend
upon
the
expected
equivalent to assuming that in period
24
1
policies
in
voters expect the
policies corresponding to the SN outcome for t=2.
convenient to express
is
It
presidential context,
construct
expected
an
where
as P(e),
equilibrium
the probability of an R victory
P,
such
is
9
that
P
e
the presidential outpoint.
Let
(0,1).
EV
iP (Q)
the
in
We will
denote
i's
R
vote
bliss
point
for
legislative
the
elections
at
t=l
given
6.
Proposition (7) can be directly extended to show that:
Lemma 2
condition for
A necessary and sufficient
(S,z)
to be undominated for
a
given and fixed 6 is
for all measurable TcS such that ieT votes R in the t=l legislative
(cM
elections,
^^^^^
EV (z) s EV
and
R
R
for all measurable LcS such that i€L votes D in the t=l legislative
elections, EV (z) i EV^^^^^
R
R
Using Lemma 2 and the results developed in the certainty model,
the following
result is immediate:
Lemma 3
For
(I
,z),
given fixed
8,
the only undominated outpoint at
t=l,
8,
is
given by:
9 =
P(9)9
+
R
i-po;
Condition (26) and the uniform distribution of
25
9
(26)
D
a
imply:
e
-
:S
-
W
2
(2
+
w - e
)
K
2w
e(e) =
+ k
+ e
e
D
i-w<e<-+w
D
(27)
+ k)
(1
+ w £ 9
The next result follows immediately from (27).
Proposition 8
Given
,
!
<
P(6)
<
1,
'',.,.
<.
necessary condition for non-domination for 9 fixed
a
is that the unique legislative cutpoint 9 satisfies the following:
e
That
is,
satisfying (C
)
e < 9
<
D2
(28)
R2
with uncertain presidential elections implies a
mid-term congressional cycle:
the president always has
the party of
lower
expected plurality in mid-term legislative elections than in the legislative
elections held at the same time as the presidential elections.
The intuition of this result is straightforward.
^
voters with bliss points
such
9
that
9
9
<
i
follows:
in
identity,
these voters vote R for
the
first
period
9
9
<
9
>
the
act
as
president's
implies
This
)
then
D
1
about
legislature.
the
(and
D2
uncertainty
under
9
D2
9
<
1
say,
If
that
they
would want to "moderate" a D president despite the fact that they are taking
the risk of making a R president
less constrained,
if R wins.
D
behavior
for
Congress,
cannot
be
reducing
thus
rational,
Analogous argument holds for
9
the
>
9
.
R2
moderation
9
<
9
D2
of
cannot
Finally note,
26
the
president.
be
D
an
from (27),
in
the
they switch to
second period, when they know that a D president is in office,
voting
Then,
This
equilibrium.
that if P(9) =
1
(or
about
=
P(e)
we have 6 = 9
0),
presidential
the
(or
outcome,
=
9
R2
e
cutpoints
legislative
the
as in the complete information model,
both periods;
with no uncertainty^
Thus,
).
D2
are
the
same
in
we do not have a midterm
cycle.
Let
us
now turn to
(S,z) undominated when
presidential
the
P(-)
<
<
1
vote.
A
necessary condition for
is obviously
for all measurable TcS such that ieT votes R in the presidential
EU (R president legislative components of z)
election,
^ EU
(D
|
president legislative components of z)
I
for all measurable LcS such that i€L votes D in the presidential
EU (R president legislative components of z)
election,
|
i EU (D president legislative components of z)
I
Equating
expected
two
the
utilities
presidential election cutpoint
.k9
(C
p
ka
-
iK
9
i+k
f
k(l-9)
+
9
+
9
<
D
We
now
9
<
/3
+
R
6
we
find
the
6 + k
+ ka -
d+k]
(29)
9
2k(l+k)^9
9
2(l+k)(l+k+3)
Note that 9
Al,
is:
+ k
9
D
9(9;
using
ka - 9
1+k
D
The unique solution of (29)
+
D
ka
+
and
).
by solving the following:
9
+
in
<
D
9 < 9
(30)
R
.
R
develop
an
important
condition
components given the legislative vote.
R is president for strategies
z.
27
on
the
presidential
strategy
Let P(z) denote the probability that
Lemma
4
Given that the legislative components of strategies are fixed at
9,
9
D2
__»
9
necessary and
a
,
R2
(T,y) when
P(z)
<
<
sufficient
1
and
condition
P(y)
<
<
1
for
is
(S,z)
undominated
be
to
by
^
that the presidential component
of 2 is represented by the outpoint 9 given by (30).
Proof
Except
:
for
voter
the
type
9=9,
voters
all
~*
~
9,
election have strictly dominant strategies given
9
.
~*
and 9
R2
1.
presidential
the
in
D2
if
<
P(')
<
-
;.
.
Q.E.D.
,;
.
,
(9,9)
D
R
Since (30) is monotone on
is immediate
to show that
interior solution
A2.
9(9
)
>
R
and (27) have a solution.
(30)
if:
-
-
2
"
•
w and 9(9
)
D
Using (27) and (30),
and
(9,9)
D
R
and Q{Q) 6
<
-
is monotone,
it
There is a unique,
.
,.
.
_
+ w.
2
one can readily write the assumption explicitly in
terms of the parameters of the model.
In fact,
by doing so it can be checked
that the range of parameter values for which A2 is satisfied is quite large.
We are now ready to establish the following result.
Proposition 9
If A2 holds,
pair of the form
an abstract stable set exists.
,z).
(I
w
outpoints 9
and 9
D2
Proof:
,
and 9
,
This pair is denoted
This set contains a unique
,z
(I
)
given by the unique solution of
given
in Proposition
^
^
7.
See Appendix.
28
and corresponds to the
u
(27)
and
(30),
and 9
R2
Remarks
1.
Even when A2 is satisfied,
the
presence
incomplete
of
for some parameter values,
This
information.
can
there is no SN in
demonstrated
be
assigning fixed numerical values to the parameters of the model,
"•"
and (30) for the two t=l
search,
have
increase
an
using computer grid
outpoints and then finding,
have
expected
in
follows
it
outpoints
that
voters
all
utility
higher
Since
utility.
than
the
in
at
the
every set of parameters satisfying A2,
If and only if 9
R
election.
In
+9 D =1
interval
"
spanned
outpoints.
quadratic
have
by
Hence
(1,2
we conjecture
"'"
the
that,
)
is
for
there is no SN.
there is no split ticket voting in the first
case
this
voters
all
On the basis of our computer analysis,
dominated.
period
solving (27)
alternative "*" outpoints such that voter types at the "'" outpoints
utility,
2.
by
simple
only,
algebra
on
and
(27)
(30)
establishes that:
_•
= e
=
-.»
e
3.
.
If
-•1
9
>
^.
that
DR,
DD.
is
-"1
P(9
<
)
-•
voters:
RR,
If
9
^,
1
(32)
2
the equilibrium
implies
In addition,
-•
presidential
election
(i.e.
the
RD,
DD.
Thus the
fraction of those voting for the
a
presidential candidate more likely to win.
the
types of
1
^ the three types are RR,
<
voters who split their ticket are always
is
three
larger
|6
~
the
1
oP
less uncertain
the
larger
the
fraction of split tickets.
4.
<?(•)<
An interior solution with
uncertainty"
in
the sense
that A2
is
this occurs are shown in Figure 3A.
which A2 is violated.
and
(30).
In
this
1
occurs only if there is "enough
satisfied.
Parameter values for which
Figures 3B and 3C illustrate cases in
Figure 3B shows an unstable interior solution to (27)
case voters
can achieve an equilibrium outcome with no
29
uncertainty
the
in
presidential
The
election.
heavy
black
lines
Infinity of outpoint outcomes that correspond to the two vN&M
show
and 9
(9
R
the game of complete information.
the
that
sense
the
of
)
D
The interior intersection is unstable in
outpoints defined by the solution to
Figure
Finally,
dominated.
"credibly"
an
3C
(27)
illustrates
and
(30)
absence
the
are
an
of
The heavy black line now indicates a vN&M where the party
interior solution.
closest to the median wins the presidential election with certainty.
5.
A necessary condition for A2 is given by the relative slopes of
- 9
(9
This condition
(30).
is:
w
R
>
„
D
For
?r^—2(1 + k + p)
;
,
(27)
and
k)k
-
9
fixed,
9
,
D
R
the largest
^
w needed to satisfy this inequality occurs for some value of k intermediate
between
and
As
1.
Since there
predominant.
k
is
influence
executive
0,
->
little scope for moderation,
incentive to engage in strategic voting.
As k
->
-
9
9
legislative
influence
there
becomes
is
little
its maximum value,
,
'D
'R
the
policy
on
on policy becomes predominant.
The
legislative
vote can be used to achieve almost any desired expected policy outcome.
The
expected outcome will not be greatly influenced by the presidential outcome.
Again
there
little
is
incentive
to
eliminate
the
uncertainty
in
the
presidential election.
6.
The case in which both the president and legislature are elected every
period is obtained by setting p =
case
describe
applies
to
the
this
electoral
case
as
in (30).
equilibrium
Note
well.
The cutpoints
every
in
that
in
this
(9
Proposition
period.
case
and 9) in this
assumption
Al
9
is
irrelevant.
5.
This
paper
extends
the
CONCLUSIONS
spatial
theory
30
of
voting
to
an
institutional
structure in which policy choices are a function of the composition of
A game between the voters
legislature and of the executive.
the
considered;
is
institutional setup in which the policy outcome depends upon
in fact,
in an
relative
plurality,
voter
each
incentives
has
to
strategic
be
since
the
outcome depends upon how everybody else votes.
with
consistent
States and perhaps
some parameter values,
patterns
the model
a
voting
of
behavior
predicts:
United
the
in
where
the mid-term
and c)
the
executive
is
and
legislative
the very pervasity of
are
elected
not
the mid-term cycle
the United States which occurred even in those cases where there was
appears
who
about
far
be
to
for
b)
split government with different parties controlling
One caveat
simultaneously.
uncertainty
be
The split government prediction would also apply to nations
electoral cycle.
France
to
split-ticket voting;
a)
the executive and holding a majority in the legislature;
like
appear
democracies in which the executive is directly
in other
instance,
For
elected.
observed
some
which
implications
testable
several
has
model
The
would
richer
win
than
the
the
presidency.
standard
Thus,
this
competitive
party
two
while
in
little
model
model,
future work would hopefully lead to a more complete model.
In
First,
fact,
several
extensions
of
the
basic
model
should
be
possible.
abstentions should be considered as an additional moderating device,
for instance if supporters of the winning presidential candidate do not vote
in legislative elections.
Second,
rather
than
strategic
one could consider a party's platform as a strategic variable,
an
exogenously
variables
if
given
parties
can
ideal
point.
make
binding
These
platforms
precommittments
to
become
their
preelectoral platform or if these precommittments can be achieved if the two
31
parties select candidates in primaries with bliss points corresponding to the
desired
platforms.
election,
party
the
If
the
two candidates choose,
[See Wittman
platforms.
(1977,
would want
"moderate",
platform
their
moderating
these
to
as
in
presidential
the
in general,
strategies.
In
fact,
our
Thus,
Needless
candidates would
expected
the
the
in a model with
there would
platforms and
two
model.
before
non fully convergent
(1985).]
between the
in
policies
preferences,
1988) and Calvert
always be voters with ideal points
choosing
choose
and there is uncertainty about voters'
no legislature,
voters
credibly
can
to
take
these
say,
account
influence
of
in
of
the
legislature may create an incentive for the two parties to diverge even more,
if
they know that whichever party holds the presidency must compromise with
the legislature.
this model may have implications for the issue of entry of third
Third,
parties [Palfrey (1984), Greenberg and Shepsle (1987)].
which
the
median.
ideal
The
points
"threat"
of
of
the
two
entry
of
parties
a
are
third
on
party
Consider the case in
opposite
in
the
sides
middle
of
the
pursuing
"moderate" policies may be reduced if the voters can achieve some degree of
moderation of policy in a two party system.
Fourth,
parliamentary
the basic idea of strategic moderation could also be applied to
systems,
where
the
executive
Strategic voting would be even more essential
is
in
not
directly
elected.
1
8
multi-party parliamentary
system where the coalitions of parties which can be formed in the legislature
depend upon the distribution of seats or of vote shares.
32
ENDNOTES
For example,
see Kramer
For surveys of this literature see Enelow
(1973).
and Hinich (1984) and Ordeshook (1986).
The
existence
"polarization'
of
American
in
documented by Poole and Rosenthal (1984a,
3
politics
widely
been
has
1985).
b;
This phenomenon has occurred repeatedly in the United States and recently in
France with the "cohabitation"
of
Socialist President and a conservative
a
legislature.
4
Erikson
coattails
tests
(1988)
alternative
other
and
explanations
of
the
mid-term cycle and finds that they are not supported by the data.
Me show that moderating behavior is incompatible with an equilibrium result
of a tied Presidential election.
A more general interpretation of the parameter k would be given by writing k
as a function of 9
and 9
R
as follows:
D
k =
t/((e
R
with
<!//(•)<
1
for
:£
^
9
.e )(9 -e
R
D
1
and
(F.i)
)
D
:£
R
9
This
1.
:£
clarifies the effect of the legislature on policy outcomes as
the
ideal
policies
of
exogenous in this paper,
the
two
specification
'^
D
parties.
^
Since
and
9
a
function of
taken
are
9
as
D
R
the results would be unchanged if we used
F.
1
for
k.
A much more insightful and important role would be played by the functional
dependency
of
k
on
9
and
9
endogenously chosen as in Wittman
in
a
(1988)
point in the conclusions.
33
model
where
and Calvert
party
(1985).
positions
were
More on this
7
This institutional structure approximates not only the United States system,
generally,
but,
systems
all
legislative elections;
elections
the
Needless
and
abstract
we
say,
to
case
the
in
many
from
in particular the district system,
aspects,
direct
Presidential
and
our model can handle both the cases in which
in fact,
simultaneous
are
with
France)
(e.g.
which
are
they
important
other
staggered.
institutional
and the fact that Congress is not
completely reelected every two years.
Q
We need only consider measurable subsets since no non-measurable subset can
affect outcomes and thus individual utilities.
9
It
should be emphasized that elimination of weakly dominated strategies is
In fact even though voters with
not sufficient to select cutpoint outcomes.
bliss points
^
6^9
respectively)
the
1
and
D
0^6
1
other
have weakly
dominant strategies
'
^
R
voters
do
not
weakly dominant
have
(DD and RR
strategies
in
presidential elections.
The proposition is not restricted to cutpoint strategies.
The proposition is not restricted to cutpoint strategies.
12
~
Since
Moreover,
some
SN are
the
it
(I,z)
z
a
subset
the
of
vN&M equilibria,
(C)
5^
e
.
Assume (I,z),
z *
13
(S,y)
is
also
vN&M.
by Proposition
is undominated,
€ K.
9
Since
(I,z)
does not satisfy
R
1,
(I,z)
satisfy internal stability therefore,
1,
)
can readily be shown that there is no abstract stable set with
R
condition
(1,6
<
(S,y) where
(S,y)
i
k.
(S,y)
satisfies
But since,
(C).
To
by Proposition
external stability cannot be satisfied.
Note that in this situation sincere voting, as in Fiorina (1988), would lead
to 50-50 splits in both the presidential and legislative vote.
34
14
While we have not proved the abstract stable set is unique, Greenberg (1989)
With a finite number
shows that K is unique for a finite number of players.
of
players,
we
construct
can also
virtually
K with properties
a
(except for integer problems) to those found for the continuum.
large
finite
but
identical
properties
qualitatative
players,
of
number
would
there
those
to
be
a
identical
for a
Thus,
unique
developed
K
with
for
the
continuum.
Note
case,
that
even
the actual
though Proposition 5 holds
stable set
is
not
identically
identical.
In fact,
two
period
the element
(S,z),
in
the
Scl of the two period stable set do not fully coincide with the
(S,z),
Scl
elements of the one period stable set discussed in the Appendix.
Consider the utility function of voter
Setting the first derivative of
i.
his utility respect to policy equal to zero implies:
Imposing the equilibrium condition
IT
Suppose
instead,
V
R
>
R
V
D
.
This
9=9
implies
9
IDDD
= e
+
g (1-H(e )).
leads to (20).
that
some voters vote R for
the
R
legislature when party R controls the presidency and D when D is president.
This strategy is inconsistent with single-peaked individual preferences.
1
Austen-Smith and Banks (1988) have investigated a three party parliamentary
system with
a
strategic
voting
model.
The
process
that
links
voting
to
policy in their model follows an approach that is quite different from ours.
35
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.
"The Polarization of American
Poole, Keith T. and Howard Rosenthal (1984b).
Politics", Journal of Politics Vol. 46, 1061-1079.
,
(1985a).
and Howard Rosenthal
"A Spatial Model for
Keith T.
Legislative Roll Call Analysis", American Journal of Political Science
Poole,
.
Vol.
29.
357-84.
"The Positive Theory of Legislative Institutions:
Shepsle, Kenneth (1986).
An Enrichment of Social Choice and Spatial Models", Public Choice
50,
.
135-78.
Wittman, Donald (1977).
"Candidates with Policy Preferences:
A Dynamic
Model", Journal of Economic Theory Vol. 14, 180-189, (February).
.
"Spatial Strategies When Candidates Have Policy
Wittman, Donald (1988).
Enelow,
Melvin
Hinich,
Preferences"
in
James
and
eds.
Readings in the Spatial Theory of Elections Cambridge University Press,
forthcoming.
,
,
37
APPENDIX
Proof of Proposition 3
(The proof is not restricted to outpoint strategies.
lottery between two outcomes x
u(x",l/2)
1.
>
R
and x
D
Three cases are possible:
.
= 1/2 for the median voter.)
{Recall that 9
u(x°,l/2).
A tie generates a
)
Since all voters have identical preferences except for the location of
11
bliss
the
R
u(x ,0
under
point,
>
)
z,
must
it
D
u(x ,0
for
)
be
all
^
9
that
case
the
Since
f.
exists
there
<l/2
^
presidential
the
vote
that
tied
is
1
some of these voters must have voted D for president under
the D voters switch to R without changing their legislative vote,
X
such
If
z.
they obtain
with certainty. Hence (I,z) is dominated.
R
D
2.
u(x ,1/2)
3.
u(x",l/2) = u(x°,l/2);
Assume
A.
<
Proof symmetric to case
u(x ,1/2).
that
there
this implies that x"
are
with
voters
some
1.
1/2
>
9
x°.
>
who
1/2
>
vote
for
D
1
R
president under
Since for these voters u(x ,•)
z.
D
u(x ,•),
>
(I.z)
can be
dominated if these voters switch to R for president.
Assume,
B.
therefore,
that there are no voters with 9
1/2 who vote D
>
1
for president under
That
z.
z must
is,
by condition (C) of Proposition
1,
have a cutpoint
z must have a
cutpoint
9
=
1/2.
9 =
Moreover,
1/2.
Then there must exist (a) a measurable interval 5 = [5,5],
with 1/2 ^ 5
p
<
D,
such that, when the RR voters in 5 switch their legislative votes to
5 < X
the outcomes become x
such
[e,1/2]
R
that
D
max[u(x ,'),u(x ,•)].
change
their
r'd'
x
,
for
with x
all
Consequently,
presidential
legislative vote to
,
D.
vote
d'
<
x
voters
S
(I,z)
is
to
R
d
<
=
x
r'
r
<
x
and (b) an interval
[c,l/2]u[5],
ulx"',-]
dominated if voters
while
voters
in
5
in
switch
The construction is illustrated in Figure A-1.
A-1
[c,
>
1/2]
their
To prove Proposition
we need the following:
4,
Lemma A-1
.
.
;
..
i
,
For (S.z)
(i.e
>
S£[0,e
),
R
.
^
].
r\
i
Assume a type 9
,^
But under
e S.
= S
z,
H
1
K
1
must be worse off and
since all voters have the same utility function except for the bliss point,
^
R
k
-^
or X
9
Rl
9
R
<
e
k
9
R2
>
i
"^
>
to be better off either x
But for 9
must also be worse off.
9
<
all e
it must be the case that
But to satisfy this condition,
.
some
R
dominance is contradicted.
is worse off,
But since this 9
e S.
k
R
.,
Proof of Proposition 4
,..,
Q.E.D.
,
,;,,,,.
....
We begin by amending the statement of the proposition to permit the use
of non-cutpoint strategies
The measure of a set S is denoted
.
a-(S).
Conditions [l.Ai] and [l.Biv] become
[l.A.i]
-
1
M
+
cr({i:
i
votes
R
for
president
under
z
and
i
6
i
votes
R
for
president
under
z
and
i
€
Rz
[0,9 )}) > 1/2
R
[l.B.iv]
-M Rz
1
+cr({i:
[0,9' )})
In addition,
for 9
k
+
D
a tied
1/2.
>
9
=
is not
[to insure that 9
,
dominated by creating
R
R
<
presidential race] there does not exist 9" which solves the following
maximization problem:
9" = max
[l.v]
x° = 9
[i.vi]
+
[l.vii]
V
[l.viii]
sto
9
<
x" = 9
-
kd-v"^)
R
R
R
^
=1-9
R
•^
R
kv'^;
D
T
<
if,
+
<p,
s
(fi
s y
R
(1/2) u(x°,r)+u(x'^,r)
=
u(9
R
and which,
in addition,
satisfies.
A-2
,Tr)
[l.ix]
1
-
M
+ cr({i:
Rz
i
i
votes R for president under
and
z
[O.e")}) = 1/2.
G
Proposition
establishes
3
ties
that
be
cannot
Proposition
SN.
1
establishes that, when the presidential components of strategies result in an
R president, domination will occur unless the legislative component obeys
in the case of D president,
Similarly,
SN outcomes correspond to 9
possible
^
the proof being symmetric for 9
R
we must have 6
and 9
Therefore,
.
9
.
R
the only
We focus on the case of 9
.
D
R
,
.
First consider domination by strategies that result in a D president and
a policy x
By Lemma A-1, we need only consider domination via S £ [0,9
.
Assume
Consider a coalition S=[0,9 -?],
l.A.
R
where members of S vote DR,
elected president under
resultant policy x
also have x
<
9
>
y,
9 - ^.
and
>
simple algebra shows
that
by a
I-S.
If
implies
l.A.
For all members of S to prefer x
about
strategy y
a
holding z strategies constant for
This can be brought
.
?
to 9
strategy
).
that
,
D
is
the
we must
where some
y'
R
members of S vote DR while others vote DD.
That
be empty.)
is,
(The subset of DD voters in S may
all coalitions forming intervals
can dominate
[0,6 -^]
R
(1,9
)
Condition [l.Ai] determines whether
provided D is elected president.
R
there exists S capable of electing D president.
the case that
also
occurs
Consequently,
S.
[l.Ai]
is
necessary and
domination via shifting the presidency from R to
Assume
[l.Bii],
Sufficiency.
l.B.
gives
V
it
domination can occur via any subset of such
if
via
Moreover,
the
Let
legislative
under
obviously
domination
S,
sufficient
for
D.
S=[0,9')
vote
is
and,
y.
for
From
ieS,
(2),
y=DR.
the
From
resultant
R
policy
x
is
given
by
[l.Bi],
provided
assumptions on the utility function and
A-3
D
is
elected
[l.Biii],
president.
By
the
all members of S strictly
prefer x
to 9
[l.Biv] shows that D is elected president under
Finally,
.
Condition [l.Biv] is necessary to the election of a D president.
Necessity.
Conditions [l.Bi] and [l.Biii] are identities implied by the model.
But V
>
-
1
ruled out by the necessity for members of S to be better off under
if domination can be achieved with V
using the
previously defined,
(S.y)
<
1
- 9
R
6'
+
is
R
R
other hand,
Assume,
~
9
*
therefore that all conditions hold except [l.Bii].
occur,
y.
+ 9'
will also
it
,
On the
y.
R
with V
R
=1-9
+9'.
Thus 9
R
R
cannot be dominated by shifting the vote to D president if and only if [l.Bi]
-
[l.Biv] are not satisfied.-
Remark:
If
^
9
and
1/2
z
is
condition [l.Biv] cannot be satisfied;
presidential
a
therefore,
cutpoint
strategy,
cutpoints z with 9
all
^
R
1/2 cannot be dominated by shifting the presidency to
can be dominated by a tie that leads to a
remains to specify when 9
It
D.
H
*.4.
D
lottery over outcomes x
1
4.
<
3-
,
9
9
<
3-
This
z-
Moreover,
.
all 9.
<
X
u(x ,y)
=*
R
9.<
.
R
R
Now
R
for the unique voter type
By Lemma A-1,
lottery and 9
x
,
:£
u(9 ,r)
establishes
result
u(x
=>
R
that
3-
who is indifferent between the
y prefer the lottery to 9
,Tf)^
we
u(9 ,y) ^ u(x ,9.)
R
need
u(9 ,9.),
>
R
1
consider
only
If S could cause
subset
of
making
all
not
S
voting D
members
certainty.
Hence,
members
S
measure
of
<p
vote
of
S
1
dominating
coalitions where all members must vote D for president and where
satisfied.
.
H
[l.ix]
is
the election to end in a tie with a measurable
for
president,
better
off
S
since
could
x
throw
would
the
be
election
implemented
to
D
with
attention can be confined to coalitions S=[0,3') where all
D
for
president
and
a
(possibly empty)
subset
of
S
of
votes R for the legislature.
Conditions
[1.
v]-[l. viii]
specify a well-defined maximization problem.
A-4
The 6" and
9"
ip
which solve the problem define the dominating coalition
need be considered since
if
smaller coalition that prefers a
a
Only
S.
tie
can
dominate obviously a larger coalition that prefers a tie can also dominate.
D
If 3u(x ,r)/9(x
Remark:
R
D
-SuCx ,9')/a(x
:5
)
all symmetric utility functions,
it can be
R
a condition which holds for
),
shown that
= 0.
i^
In other words,
the dominating coalition will vote DD.
when domination must occur via a tie,
Domination via a tie need only be invoked in the knife-edge case
Remark:
where both [l.ix] holds and there are no measurable R presidential voters in
[e".e').
Q.E.D.
Proof of Proposition 5
Characterization of an abstract stable set
A
set
and
K exists
consists
of
all
the
pairs
(S.z)
SSI
that
satisfy
either:
1.
The strategy z implies R is president with vote M
component of z satisfies
>
Kz
1/2;
for members of S and implies V
(C)
the legislative
and a policy of
R
X and there does not exist
[l.Ai]
6'
,
X* = 8
kV*
+
[1.
Alii]
[l.Aiv]
R
V*= V + cr({i:
R
i€S and x' i
9
R
-
M
= u(x, 9'
)
i
9'})
)
ieS and
cr({i:
+
Rz
and
<
1
u(x',9'
1
such that:
V
,
R
D
[l.Aii]
x
[0,9' )})
€
i
votes R for president
under z
'^
1/2.
>
and that there does not exist 9" and T £ S £ [0,9") such that
[l.Biii]
u(x,9") = u(9
+
kV ,9")
R
D
[l.Biv]
1
-
M
+
cr({i:
i
e
Rz
z}) = 1/2.
A-5
T and
i
votes R for president under
2.
The strategy z implies D is president with vote M
component of
z
satisfies
(C)
strategy
The
implies
z
the legislative
1/2;
for members of S and implies a policy of x and z
satisfies conditions symmetric to those of
3.
<
HZ
legislative component of z satisfies
and a lottery between policies x
R
with
president
for
tie
a
1.
for members of Scl
(C)
and x
vote
M
,
=
Rz
the
1/2;
and implies V
•
D
and there does not exist G",
,
x
V
,
such that either:
3.
A
[3.Av]
= e
X
kv
+
R
D
[3.Avi]
= V + cr(U:
V
R
ieS and x
^
B
6"})
<
R
1
u(x*,G") = (1/2) u(x^, e")+u(x°, 6")
[3.Avii]
[3.Aviii] (r({i:
and
ieS
[0,6")})
>
votes
i
R
president
for
under
z
and
e
i
0.
or
3.B
[3.Bv]
= e
X
-
k(i-v
R
)
R
V^ = V - cr({i:
[3.Bvi]
R
ieS and x* i
0"})
>
1
[3.Bvii]
identical to [3.Avii]
[3.Bviii]
cr({i:
>
6
R
ieS and
i
votes D for president under z and i€(6",l])})
0.
Existence
If
u(e ,1/2)
u(e ,1/2),
2:
R
cutpoints e
it
is direct
to
show that
<
1/2,
e
,
satisfies (1).
(Note that,
since x
R
u(x ,1/2).)
Likewise,
1/2,
e
,
<
9
,
D
if u(e ,1/2)
uO D ,1/2),
^
R
e >
where
(I,z),
z
has
D
satisfies (2).
Hence K *
0.
D
A-6
(I,z),
u(e ,1/2)
>
R
where z has cutpoints
External Stabil ity
(C)
1.
By
not satisfied.
Proposition
legislative components for
^
is
It
1/2.
exists
there
1
such that
€ T.
i
where y differs
(T,y),
(S.z)
possible to find a dominating
(T,y)
K because cr(T)
6
in
Now consider M
(T,y).
<
only
from z
can be
sufficiently small that T is not pivotal in the presidential election.
M
=
either (T,y) satisfies (3) or,
1/2,
it does not,
if
there exists
If
T'
c
Rz
T,
presidential
where
well
as
shifted for all icT' and (S,2)
(T',y')
satisfies
as
<
legislative components of
(T'.y').
It
follows that (T',y') e K since
members of
and by construction,
(C)
strategies are
T'
prefer the outcome
with a certain presidential winner to the outcome with a tie where
satisfied
and,
moreover,
unable
are
to
shift
the
outcome
to
the
is
(C)
certain
election of the other presidential candidate.
satisfied
2.
(C)
A.
Conditions [l.Ai]
T = {i€S and i€[0,9')}.
-
[l.Aiv]
satisfied and therefore (S,z)
K.
Let
members of T in [0,x
In the dominating strategy y,
use DD and members of T in (x ,9') use DR.
«£
By the construction,
]
D is elected
president under y and members of T receive higher utility under y than under
z.
Hence,
(S,z)
<
Moreover,
(T.y).
(T,y)
trivially satisfies
(2).
Hence
-
[l.Biv]
(T,y)6K.
B.
Conditions
satisfied.
[l.Ai]
-
[l.Aiv]
satisfied,
not
[l.Biii]
For T in the construction, members of T continue to vote D in the
legislative election and thus satisfy
elect a D president.
It
(C).
is also direct
to
worse off by switching back to R president.
belongs to
but
K.
[Note
that
the
By construction,
A-7
c
T can
show that members of T would be
Hence,
lottery between
•^
no T'
(T,y)
+kV
9
D
satisfies (3) and
represents the
and 9
R
R
outcome that generates the largest "defection" from S that satisfies (C).]
A similar argument applies when D
is
elected president and
(2)
is
not
satisfied.
Finally,
either
S,
3.
it is
consider the case of a tie for president and (C) satisfied but
A or 3.B satisfied.
using an appropriate subset of voters in
Again,
possible to construct a dominating (T,y) that belongs to
K.
Internal Stability
Given that
members
other
of
construction used
K
result
that
in
members of K are obviously stable against
satisfied,
is
(C)
(3)
the
in
guarantees
that
upset by shifting to a certain winner.
same
presidential
presidential
Similary,
outcome.
ties
where the other party
wins
the construction of (1) and
presidency.
the
remains to demonstrate,
It
without loss of generality,
that a member of
K satisfying (1) cannot be dominated by a tie that satisfies
ties that satisfy [l.Biii] - [l.Biv].
tie that satisfies
(C).
(C)
Then a voter 6" is indifferent between a lottery and
in the lottery must be
to satisfy
Consequently,
.
vote D
in
the
(C)
Moreover,
(
I
,
z)
all 6
in the tie,
legislative election.
dominated by a tie that satisfies (C),
Consequently,
4,
the policy
less than x and the coalition that creates the tie
can include only voters in [0,6").
to X
other than
Let (S,z) satisfying (1) be upset by a
By arguments parallel to those used in proving Proposition
X.
to
be
guarantee that certain winners in K cannot be dominated by members of K
(2)
X
cannot
in K
The
But
<
6" strictly prefer x
all defectors must continue
then x
=
x.
So
if
(S,z)
is
[l.Biii] - [l.Biv] must be satisfied.
K is internally stable.
in the Stable Set
Finally,
if S =
I
and given a sure victory in the presidential election.
A-8
the outpoints for
the
legislative election are either
the
proof
by condition
or 6
9
(C).
complete
We
presidential outcome,
(I,z)
by
showing
must
generality,
^
9
D
T
=
^
9
conditions
The
1/2.
I,
the legislative vote
without
Assuming,
.
[3.Av]
[3.Avii]
-
satisfied because there is always some subset of voters in
X
D
(1/2) u(x
have
Now
lottery.
the
to
,
)+u(x
9
shown
thus
,
9
9"
that
A
3.
>
1/2
3.B)
(or
V>V
R
[3.Aviii]
=>
[x
R
is
where z leads to a tied presidential race is in
of
trivially
who prefer
]
*
u(x ,9
T
>
)
satisfied for S=I.
satisfied for S=I.
is
,9
D
•
loss
are
=»x>x=>
T
implies
[3.Avi]
)
tied
a
+ e
R
cutpoint
the
represents
(I,z)
if
To satisfy (C) for
« K.
e
satisfy
that
Hence,
no
We
(I,z)
K.
Q.E.D.
Proof of Proposition
The
key
7.
proof
the
to
is
that
the
realization of
only
a
affects
the
relative proportions of voters with dominant strategies of voting R and D as
a result of
(24).
It
1.
immediate
is
show
to
that
necessary
a
condition
non-domination of (S,z) is that all measurable subsets of S in [0,9
]
for
vote D
D
and all in [9,1] vote
R.
R
If
2.
<?:.(V
)
of V
R
(1.
holds,
)
(25)
can be written in terms of the derived density
as
R
^V
t2
Eu
=
l+w-9
R
RR-R
+cr
and
V
=
)
l-w-9
RR
+cr
0(V )dV
- 9
R
R
V
—R
where V
k(l-V
-
9
1
and
R
=cr{i:
cr
R
R
9
RID
>9 >9
and
i
votes R under z}.
3.
Since Eu
R
is
1
quadratic in V
,
Eu
R
A-9
R
1
can be expressed as a function of
the mean and variance of V
invariant
in
depend only on EV
shown
be
that
is
readily shown that the variance of V
preferences
between
alternative
strategies
By an argument analogous to the proof of Lemma
.
preferences
single-peaked
are
in
EV
Proposition
.
can
it
1,
follows by arguments directly parallel to the proofs of Proposition
Proof of Proposition
is
R
voter
Thus,
z.
It
.
R
then
7
and
1
2.
9.
We develop the proof using a restrictive assumption in addition to Al.
A3.
Strategies z are restricted to be cutpoint strategies.
Without this restriction,
that used for Proposition
the proof requires a construction similar
to
^
5.
,
,
Characterization of an abstract stable set.
The abstract stable set K consists of all pairs
given
either,
or,
C
z,
1
and C
p
are satisfied for all
i
(S,z),
€
any of these conditions are not satisfied for
if
SSI
such that
S
in the first
z,
S does not
period
contain
the cutpoints in z that correspond to the conditions that are violated.
addition,
.
It
*
(I
w
has z =
,z)
9,9.
the second period cutpoints are
Remark
is direct
to show that
In
D2
R2
the unique member of K of
the form
~* _*
_*
zs(9,9,e,e).
D2
R2
/^*
External Stability.
Assume that (S,z)
i
K and that there exists a cutpoint ^ contained in S
for which the appropriate C condition is not satisfied for one or more
two
or
more
contests
share
an
identical
exists a Coalition T either of the form
all
i
€
Moreover
T violate
cr(T)
cutpoint)
[C~c,C]
or
Then
contests.
[C.C + c]
(c>0)
there
such that
the C condition for one or more contests referenced by
can always be made sufficiently small
of T reverse their z votes for these contests,
A-10
such that,
leading to some
y,
(if
<^.
when members
T satisfies
appropriate
the
Hence,
conditions
C
given
T
contains
<
§
and
y
other
no
cutpoint.
e K.
(T, y)
Internal Stability
Without
loss
assume
generality,
of
9
opposite
the
,
being
case
symmetric.
is
(S.z)
(S.z)
If
1.
K and no cutpoint
in z
internally stable by A3.
(Note:
boundary,
some T
cutpoint
the
e
on
is
the
on a boundary of
in S or
is
if one boundary of S
S can
in
open but
is
defect
still
S,
another
to
cutpoint strategy y with A3 satisfied.
2.
A3,
Lemma 2 and Lemma 4 imply that (S,z) e K is internally stable if
(S,z) or its boundaries does not contain both 9 and 9.
3.
A3 and Lemma 2 and Lemma 4 further imply that if (S,z)€K and (S,z)
where
(T,y)
-'
z
=
(9 .6 ,9
9'
condition is that
* 9
D2
,9
9'
,
and
)
R2
* 9
=
y
(9'
.
9'
,
9'
D2
.
9'
)
R2
necessary
^
a
,
<
This means that y must differ from z in
.
both t=l components.
4.
Assume,
therefore,
first show that all
(S,z
)
that
the
necessary condition
internally stable.
are
must include the half-open intervals defined by 9
the stability property of A2 can be used
9(9')
(or
Hence,
both)
is
in
the direction of
5.
"
the C conditions are not satisfied on
dominated by a member of
domination
occurs
cutpoints
with
9'
z * z
K,
that both t=l cutpoints cannot
both
and
9'
We
any dominating T
and 9
9'
and
But
.
cutpoint from the
T,
implying that
or
cutpoint.
'
(S,z
is
)
not
K.
Next consider (S,z) e
Assume
By A3,
satisfied.
establish that either 9(9')
to
the
is
>
9^.
If
strictly
be
are
.
on
a
But
A-11
(S,z)
interior
boundary
then
9'
it
e K,
to
of
S.
<9^
or
is direct
to show
S.
Case
members
Assume
a.
of
S
in
a
neighborhood of
Since
T c
£
(e'.e'^)
7.
*
2.
will not be strictly better off with (T.y).
9
Moreover,
S.
at
,
one
least
C
is
satisfied
not
on S
=>
the
of
eie"^)
parallel
Case
to
strictly interior to
for
Hence,
(T,y).
Therefore,
a.
(S,z)
S,
and
e""
inequalities
therefore that domination is achieved with
is
s
C^
must
on S
be
if
(T,y)
6'
<
6
^
.
=*
b.
it
thus must be
(S,z)
e
assumption
that
z
*
case
z
.
Assume
Proof of non-domination
K and
no
t=l
outpoint
is
is internally stable.
assume the interior outpoint is 9
the
is
It
z
Finally assume one outpoint is interior and the other is not.
loss of generality,
g"".
i
eCe"")
u
that either C
Case
K.
(6^,9')
strict.
using the stability property of A2,
therefore direct to show,
or
C*"
So
that
This
0^
=
6(9^).
fact
.
Therefore,
permits
proof
of
Without
For S to satisfy C
9
5*
9(9)
from
non-domination
the
in
manner parallel to the preceeding paragraph.
Thus,
all
(S,z) 6 K are internally stable.
Q.E.D.
A-12
,
a
Strong Nash Equilibna for k =
D
Strong Nash
a:
Strong Nash Equilibria for k = 0.4
Strong Nash Equilibria for k = 0.2
1.0
2 5(Jong
0.9-
0.9
^^
D Strong
D Strong Nash/^
Naih
0.8-
\
No
R
Strong Nash
Strong Nash
0.8
a;
o
0.7-
\
R
No
\
Strong Nash
V
Strong
0.6
o
0.7
/
0.6-
^
/^Infeasible
fork=0.2
X
05-
1
0.0
f-
1
0.2
0.1
Infeasible
for k=0.4
0.3
1
0.5
1
0.4
0.0
0.5
Figure 1-B
vN&M
1.0"
0.9-
Equilibria
for
vN&M
k = 0.2
1.0
N.
0.8-
\\
D&R
0.8-
\\
o
0.7-
R vN&M
/
^y^
0.0
0.7-
0.2
Figure 1-D
RvN&M
0.6-
= 0.2
050.4
0.5
/
^^y
0.0
/
y^
y^
^
Infeasible
for k
DvN&M
x\
o
^r
/
\
a:
\\
0.6-
05-
\D&R
D vN&M
\. \
EquiUbria for k = ,04
\
0.9-
\
0.5
Figure 1-C
\
^\v
1
0.3
0.2
0.1
Infeasible
y^
fork = 0.4
0.2
Figure 1-E
0.4
0.5
Figure 2
The Strong Nash Dominating Coalition
P
This coalition dominates
6p by voting DR
<=
<^ These voters do not
satisfy (C)
they vote
These voters
not in S vote
These voters
not in S vote
D
R for the
for the
legislature
legislature
=^
when
R
for the
legislature
X'
The
<=
%
vote
e
R
vN&M Dominating Coalition
:This coalition dominates:
<= These
:p
voters must
e'
DR
^
These voters
not in S vote
These voters
not in S vote
D for the
R
legislature
legislature
Voters
DD to
satisfy (C)
x=
R
for the
Figure 3
Equilibrium with Incomplete Information
k=0.70
k=0.34
e(e)
\
.5
-P^
.45
^9(9)
V-^
V'
°
e
e„
3-B
3-A
m
k=0.05
.45
ft(0)
—
'
1
6d=-^
L_
«R=
^D
._,
'
^R
3-C
Notes:
w=
0.05 in
Heavy
all
^R
panels.
lines denote equilibria with Presidential
winner cenain.
e
'r'
e
Figure A-1
Domination of a Tie
When Median Voter is Indifferent
at Tie
DD
switch to
RD
RR
RR
voters
under
switch
z
to
RD
+
+
^
i/2 = e
1
x°'
x°
/
x^'
x^
Defecting CoaHtion
LI
^385 UIO
Date Due
MAY 05
c^'a--?^
1991
Lib-26-67
MIT LIBRARIES DIJPl
3
I
TOAD D057aT7T
^