Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/interestgroupselOOsnyd
HB31
Il415
Massachusetts Institute of Technology
Department of Economics
Working Paper Series
INTEREST GROUPS & THE ELECTORAL CONTROL
OF POLITICIANS
James M. Snyder,
Jr.
Michael M. Ting
Working Paper 05-3
February 3, 2005
RoomE52-251
50 Memorial Drive
02142
Cambridge,
MA
This paper can be downloaded without charge from the
Social Science Research Network
Paper Collection
http://ssrn.com/abstract=661582
at
INTEREST GROUPS AND THE ELECTORAL CONTROL
OF POLITICIANS
1
James M. Snyder,
Departments
Jr.
of Political Science
and Economics
Massachusetts Institute of Technology
Michael M. Ting
Department
of Political Science
and SIPA
Columbia University
February
3,
2005
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
FEB
1
1
LIBRARIES
1
We
Brook
acknowledge the financial support of National Science Foundation Grant SESthank David Austen-Smith, Clare Leaver, and panel participants at the 2004 Stony
gratefully
0079035.
We
Political
Economy Conference and 2004 annual meetings
of the Midwest Political Science
Association, Society for Social Choice and Welfare, and American Political Science Association for
useful
comments and
discussion.
Abstract
Are
elections used to select
good candidate types or to reward performance? These goals
can be incompatible because the retention of good types prevents citizens from using votes
We develop a model of repeated electoral competition that examboth incentives. In each period, a randomly-drawn interest group attempts to "buy" an
incumbent's policy choice, and a voter chooses whether to replace the incumbent with an unknown challenger. The model predicts that "above average" incumbents face little discipline,
but others are disciplined increasingly and re-elected at a higher rate as the interest group
becomes more extreme. Extensions of the model consider term limits, long-lived groups, and
to discipline incumbents.
ines
—
multiple groups.
—
1.
Interest group politics
How
one of the most important topics in political economy and
However, while theorists have been analyzing formal models of interest
political science.
group politics
is
more than
for
thirty years, one aspect of the
should strategic voters vote
when they know
ways the voters do not
policies in
Introduction
like?
problem remains underdeveloped:
that interest groups are trying to skew
This issue has been overlooked because existing
models focus on the calculations and strategic interactions of interest groups and
As a result,
politicians.
these models treat voters as a black box, or at best as myopic actors that respond
1
only to the short-run campaign promises of the current election.
One obvious
tions voters
and
place to turn
politicians
approximately as
is
the work on political agency, which focuses on the calcula-
make
in a principal- agent
far as the interest
framework. That literature goes back
group literature - Barro (1973)
and has produced important insights about the
possibilities
and
vs.
Stigler (1971)
limits of using elections to
2
control the behavior of politicians and/or select "good" types of politicians.
models in
-
None
of the
this literature, however, explicitly incorporate interest groups as a strategic actor.
This paper takes a step toward combining the two literatures - to our knowledge, the
first step.
We
analyze an infinite horizon
game
in
which a representative voter
office-holder in each period. In each election, the voter chooses
a randomly-drawn challenger.
about holding
office.
The
elects
a single
between an incumbent and
voter cares about policy outcomes, while politicians care
Each period, Nature
also
draws an interest group that can
offer a
contract to the incumbent for choosing particular policies. Each challengers' value of office
(denoted w)
is
drawn from a stationary
until she achieves office.
l
An
distribution and cannot be revealed to the voter
incumbent's "price" for an interest group therefore depends on
See Peltzman (1976), Denzau and Munger (1986), Grossman and Helpman (1994, and 2001, Chapters
and Persson and Tabellini (2000, Chapter 7) for examples of the former, and Austen-Smith (1987),
7-9),
(1994), and Grossman and Helpman (1996, and 2001, Chapter 10) for examples of the latter.
sampling of the literature includes Ferejohn (1986), Rogoff and Sibert (1988), Austen-Smith and
Banks (1989), Rogoff (1990), Banks and Sundaram (1993, 1998), Harrington (1993), Fearon (1999), Barganza
(2000), Besley and Smart (2001), Hindriks and Belleflamme (2001), Le Borgne and Lockwood (2001a, 2001b),
Besley (2003), Smart and Sturm (2003, 2004).
Baron
2
A
The
w.
voter thus has incentives both to retain good incumbent types as well as to induce
good performance from them.
We
study several variants of this model. In the
first,
the voter can
stationary contracts for controlling the politician. In the second,
commit
to optimal
we drop the commitment
assumption and examine both stationary and simple (two-state) non-stationary
The model
is
then extended to incorporate
finite
replaced after serving their T-th term. Finally,
lived interest
Our
and
results reveal several
equilibria.
where incumbents must be
briefly consider the cases of
an
infinitely-
important features of the tension between inducing performance
she will re-elect an incumbent only
The
limits,
group and multiple interest groups.
selecting types. In an environment
point.
we
term
voter
may
where the voter can commit to re-election contracts,
the chosen policy
if
is
sufficiently close to her ideal
allow policy to deviate from her ideal point somewhat, however, to
prevent excessive interest group vote buying. Policies that are too far from the group's ideal
will
induce the group to "buy"
expected lifetime payoff.
An
its ideal
policy instead, at a cost equal to the incumbent's
incumbent's "price" therefore depends on her anticipation of
future re-elections.
For a given incumbent type, the voter thereby maximizes her policy
re-election in all future periods.
However, the voter
may
performance from every incumbent type. Since incumbents who value
command
by promising
office
maximum
more highly
also
higher prices, voters have an incentive to remove "low- to" incumbents. Therefore,
in the contracting equilibrium the voter will always
and always remove
retained
if
when
is
it
utility
not wish to induce
sufficiently
keep sufficiently high quality incumbents
low quality incumbents.
In between, incumbents
may be
groups are extreme, as this allows the voter to achieve some policy performance
most needed.
Somewhat
counterintuitively, our
election rates should increase as policies diverge
When we
remove the assumption of
of equilibrium assumed.
induce performance
is
In an
model thus predicts that
from the voter's
re-
ideal.
re-election contracts, the results
depend on the form
equilibrium in stationary strategies, the voter's ability to
severely constrained.
Since votes are cast after policies are chosen,
the voter's type-selection incentives are too strong and
all
incumbent types produce the same
policy results as the worst type.
When we
additionally drop the strong stationarity restriction, however, the results can
change dramatically.
We
show that by using simple non-stationary
contracting equilibrium can be restored.
We
should note that this
"trigger" strategies the
not an obvious result,
is
because we cannot use standard repeated-game punishment strategies, since the punishment
instrument eliminates players.
we consider a few extensions
Finally,
imposed, an incumbent's term of
office acts
increases, the price of her vote decreases.
formance, and becomes
As a result,
in
to this framework.
much
The
finite
term limits are
As an incumbent's experience
her w.
like
When
voter therefore loses the ability to induce per-
incumbents as they become more experienced.
less likely to re-elect
both the contracting and non-stationary, non-contracting environments, voters
cannot benefit from term
limits.
group and multiple groups.
We
also briefly consider the effect of a long-lived interest
The former weakens a
voter's electoral control
group more "buying power," while the latter strengthens
it
when groups
by giving the
are on ideologically
opposite sides of the voter.
Several recent papers in the literature on political agency argue that elections are mainly
about "selecting good types" rather than "sanctioning poor performance"
and Smart, 2001; Besley, 2003). The argument
1999; Besley
preferences,
and
if
the
game
is
if
when
it
question of type:
re-elect her,
3
See,
e.g.,
politicians
have policy
then in any subgame
it
clearly:
"although the electorate would
to a retrospective voting rule to motivate self-interested politicians optimally,
comes time
preferences?"
if
Fearon,
they care only about selecting politicians with
"good" preferences. Fearon (1999, page 57) states
commit
that
finite or if politicians are finitely-lived,
perfect equilibrium voters will behave as
like to
is
(e.g.,
If
and
to vote
it
makes sense
which candidate
is
more
for the electorate to focus completely
if
they believe the incumbent's type
Dutta (1995)
for
be principled and share the public's
likely to
voters believe the incumbent's type
on the
is
is
better than average, then they must
worse than average, then they must
a more general treatment of such repeated dynamic games.
4
replace her.
This logic might apply to a world in which politicians actually have policy preferences.
However, some theorists argue that, at least in the long run, office-motivated politicians
out policy-motivated politicians
will drive
Calvert, 1986).
(e.g.,
assuming that politicians nonetheless act as
for
if
An
alternative rationale
they are policy motivated
is
to assert
that they have "induced" policy preferences - induced by some underlying contract with a
Our
set of interest groups.
results
show that the argument
in the previous
paragraph does
not necessary hold in a world where politicians' have "policy preferences" that are induced
by payments from
interest groups.
As noted above,
in our
model the voter can employ
a voting strategy that both sanctions and selects, even with finitely-lived politicians.
that
required
is
is
that players use non-stationary strategies of a very simple type.
All
One
key difference between a world with politicians and interest groups and a world with policy-
motivated politicians
to prevent
is
that in the
first
bad policy outcomes, while
case voters
must always monitor
in the second case voters
politicians in order
who have found
a "good"
politician can simply let her act according to her preferences.
2.
Our model
We
horizon.
common
is
one of policy-making and elections in a single constituency over an
denote periods with a subscript
factor 5
In each period
interest
that
the
4
it
P
The Model
€
t
Players
t.
all
infinite
discount future payoffs by a
(0, 1).
—
1,2,... there are four players; a politician (P), a
group (G t ), and an election challenger (C
).
t
median voter (M), an
Since the incumbent
is
endogenous, note
(redundantly) denotes the incumbent politician of a given period, with Co representing
first
incumbent. Thus
if
Ci defeats Co, then
P "becomes" Ci
in period 2,
and so
on.
3, page 14) states the problem as follows: "the existence of good types makes
commit to a re-election rule as they are no longer indifferent (ex post) between
the incumbent and a randomly selected challenger. Voters would actually prefer to commit to the
Besley (2003, Chapter
impossible for voters to
voting for
is used under moral hazard. However, they cannot do so. This finding casts light on the
moral hazard model to a small variation of the model which includes politician types. This is
because the strict indifference rule that underpins incentives in that case allows the voters to commit. Once
this indifference is broken, then there is actually a constraint on optimal voting strategies."
voting rule that
fragility of the
>
P and C
period
care about holding
office,
while
M
in different
assumed that
is
G
:
3ft
is
CT
politician
w
elected in period
wT
to denote generic types.
u
but has quasilinear
+
|
:
£>(3ft+)
—
»
r,
G
G may
In each period
t
;
thus, in period
G
t 's
7
v(0)
>
ideal policy
B(X)
:
X—
=
0,
—>
all
3ft_
is
and
[0,1]
"type."
receives
t,
w
t
is
Each g
is
t
We
T.
£ X.
u M (x
w
t
t
),
where
An incumbent
receives a
office thereafter)
refer to
as a politician's "type,"
drawn
an incumbent
according to the
i.i.d.
-
drawn
CT
G
t
B =
{b
X —
:
receives:
b t (x t ).
t,
when
she receives:
b t (x t ),
continuous and single-peaked. Let gt
with support
of
moves
i.i.d.
assume that
= argmax Xt
u Gt (-) G
X
denote
according to the probability measure
for all
G
t
u Gl (x
,
t
)
=
v(x
t
—g
t)
with
in each period
t is
as follows. All actions are perfectly observable
players unless otherwise noted.
Group Draw. Nature
selects
group
Alternatively, we could assume that
their preferences
results.
P
5
so that groiip utility functions are identical up to changes in type.
The sequence
by
t
= wT +
t
:
can control
t
with support Q. She does not have policy preferences,
[0, 1]
u Gt (x )
where u Gt
We
receive non-zero utility only in period
t
M
t,
but possibly holding
u p (x t ,w T )
Finally,
G
and
a non-negative transfer or "bribe" b
utility over
measurable} offered by
b
M
9ft.
m = argmax« M (-)
per period upon each election.
probability measure
9ft
C
In each period
continuous and single-peaked and
(i.e.,
fixed benefit of
and use
X
policy in
can credibly commit to these payments.
(
Player utilities are represented as follows.
—>
The
care about policy.
t
ways; the former by voting, and the latter by offering payments in return for
policies chosen. It
uM X
G
and
an element x t from the convex, compact set
is
t
t
-
i.e.,
all
G
t
.
groups have the same ideal point
in their willingness to
pay
for favorable policy shifts.
g,
but
differ in
the intensity of
This yields qualitatively similar
G
Vote Buying.
P
Policy Choice.
a transfer schedule b & B, unobserved by M. 6
P
offers
t
C
We
(0).
t
If
M casts a vote r
period
t.
H
x
t
t
's
for the
G
{0, 1} over
revealed to
x
incumbent
That
strategy
/3t
:
as follows. Let
H
t
x
Q
>
{0, 1}
is,
after the policy choice,
maps
P
(1) or elect
a challenger
Strategies for each player
denote the set of game histories prior to
»
the history through period t—1, the
choice.
histories, types, bribes,
Xt
'
H
t
x
Finally,
ft
B~ X
x F x
*
M's strategy p t
:
and policy choices into a vote
The Contracting Game
in each period
M
we
t,
M
subgame
examine a case
game be modified
in
which
M
may
write
G
t
's
so that instead of choosing a vote
vote buying. Following convention, we
perfect equilibria (SSPE). This requires that players
and optimally when faced with
we
first
commits to a vote based on actions from that
announces p t prior to
the incumbent's type. Additionally,
£r
t
— B maps
x F
history-independent strategies. M's period
label
re-elect
or challenger.
restrict attention to stationary
we
H
and bribe schedules into a policy
X—
period. Formally, this requires that the
act identically
whether to
unobserved by M.
M.
establish a baseline for comparison,
"contracts."
is
t
perfect equilibria in pure strategies.
3.
To
t
w
and her type into a bribe schedule. P's strategy
B
x F x
is
subgame
histories, types,
Q
t
mappings defined
Then G
politician type,
w
wins,
t
will derive
are measurable
maps
C
7
chooses x t € X.
Challenger Draw. Nature selects challenger Cj, where
Election Voting.
'
t
t
identical continuation games,
and hence imply
contract can thus condition only on gt
will focus
on the
set of optimal
SSPE
for
,
x
t
,
and
M, which
.
6
The unobservability
7
This gives
intuitions hold
all
but is not required for our results.
power to the group. However, our main results and
assumptions about how the bargaining/agenda-setting power is
of the transfer schedule simplifies the analysis
of the bargaining/agenda-setting
under a variety of different
divided between the politician and the group, as discussed in note 8 below.
A One-Shot Game. Suppose
we drop time
venience,
p
is
able to
commit
to
by
£
v
and C
may
=
contract
uG
To
set of
if
p(w,g,x)
>
{p)
u G (-).
By
u G {c)
g.
M
away from m.
G(—w), then
use a type
(ii)
Note that as
G
is
that
could achieve her ideal
(i)
v
;
V=
X
{x €
p(w,
|
g, x)
=
then clear that G's optimal contract must be
It is
0}.
5
non-empty subsets;
disjoint,
choosing p
= argmaxl£ c
c
= argmaxx6 p u G (x)
u G {x) and
and zero otherwise; or
G
zero otherwise.
offers the
type
(i)
— w.
= m&xG(y) and
=
{x €
X
\
u G (x)
g(y)
=
mm G{y)-
unwilling to pay the cost of a type
can then threaten to re-elect P
>
if
y} denote the upper contour
(•),
G{y)
is
a closed interval
There are two
(ii)
and only
contract to
if
m
is
cases.
move
First,
P's policy
Second,
chosen.
if
if
M cannot induce any policy choice outside of G{—w), for otherwise G can
contract to achieve x
w
G
first
T
otherwise.
the single-peakedness and continuity of u
Let g(y)
G G{—w), then
g"
=
offer e for
(i)
choosing
for
two
into
derive M's voting contract, let G(y)
containing
tci
the result for this case holds
otherwise,
x\
=
lf
'
|
one of two types:
w+e
777.
X
{x 6
offer
(ii)
X
partition
of
offering the following contract:
k*) = {
{
Hence, we
P
p.
an optimal contract cannot be constant in
policy at negligible cost
of
thus attempts
begin by considering G's response to an arbitrary contract from M. Note
generally,
1},
M
chosen to induce the optimal policy from
is
M does not care about future policy choices,
regardless of whether she
We
i.e.,
For con-
single period.
subscripts until returning to the repeated game.
only to induce good performance;
type w. Because
game has only a
the
initially that
decreases,
=
g.
Thus the
{g{-w)
if
m
m
if
m€
g(-w)
if
m
best policy that
M
can induce
is:
< g{-w)
[g(-w) ,g(-w)]
(2)
> g{-w).
G(w) shrinks. This reduces P's
"price" for G,
and
in turn
M's
ability to discipline P.
One
contract that achieves this result
./
x
p [w,g,x)
=
is:
/
x
=
x(w,g)
1
if
(J
otherwise.
<
.
(3)
One optimal
(i)
response for
contract with
e
—
>
0).
G
P
would then be a
chooses x(w,g) and
The Repeated Game. In an
P.
is
with zero payments
re-elected.
infinite period setting,
because of multiple re-elections. As
can induce from
null contract
incumbents
(2) suggests, this
may
may
a type
expect higher payoffs
affect the set of policies that
Let l(w;{p(w,-)}) denote a type-iu incumbent's ex ante
the draw of gt ) discounted expected payoff given a set of voting contracts.
l
(i.e.,
8
(i.e.,
We
M
prior to
simply write
w when the contracts in question are generic or understood. Generalizing from the above
discussion,
it is
possible for
M to induce
t
"myopically" the following policies in period
{-w-6l w )
{gm
g t (—w — Sl w )
We
if
m <g
if
mG
[g t
if
m
gt {
>
t
(-w-5l w
(-w-6lw ),gt (-w-5lw )]
— w — 5l w ).
)
extend the previous analysis to the repeated case in two steps.
M
regardless of
w
incumbent
always re-elected under these contracts, l w
is
and gt
,
strategy profiles (which
The
strategies in £'c
offers a contract that achieves (4) in
may
not be Nash) as £'c
may be thought
Given w,
M
=
(4)
suppose that
First,
each period.
w/(l — 5).
We
Since the
refer to this set of
.
of as a "pure" ex post monitoring solution, in that
M's contract achieves the best policy possible in period
of politician retained.
t:
t
without any regard
for
the type
does better here than in the single-period case because
repetition automatically raises P's continuation value.
Second, consider whether a strategy profile in £'c can be sustained as an equilibrium.
There are two simple cases in which
it is.
If
w
t
is
constant for
all
t,
then
M
clearly has
no
8
Here we briefly sketch what happens when all bargaining/agenda-setting power lies with P rather than G.
Consider the case with
< g(—w) < g. Suppose that there is a status quo or reversion policy s < m, which
offers a contract
is enacted if P does not propose anything or if G rejects all of P's proposals. Suppose
of the form: p*(w,g,x) = 1 if x < xq and p*(w,g,x) =
otherwise, with xq < g. Given the "threat point"
G
s, P can extract at most — u (s) in payments from G, by making G a take-it-or-leave-it offer of the form:
x = g and b = —u G (s) (which G will accept). This yields P a payoff of — u G (s), since P will not be re-elected
under the proposed contract from M. Alternatively, P can choose a policy that allows her to be re-elected.
The best such policy for P is io itself, together with a take-it-or-leave-it offer to G of the form: x = Xq and
m
M
G will again accept). This yields P a payoff of w + u g (xq) — u G (s). The optimal
minimize xo s.t. w +u° (x ) — u G (s) > —u G (s), or minimize x s.t. u G (x ) > — w.
The solution is therefore x(w,g) = g(— w). Thus, M's optimal contract, and the equilibrium policy outcome,
is exactly as in the game analyzed in the paper where G has all of the bargaining/agenda-setting power.
Note also that this is true regardless of the reversion policy s (as long as s < g(—w)).
b
— u g (xq) — u G (s)
contract for
(which
M then solves:
9
incentive to select politician types. Similarly,
then
if
T
such that
is
M can achieve her ideal policy for any politician type.
an optimal
in E'c are also
More
mance
is
generally, in
equilibria, so
£c
an optimal SSPE,
=
£'
c
m
6 G(~^h)
where x*t
tracting game,
M will always replace an incumbent
re-elect P,
then
G
can
offer
=
YltL\
from a type-u> incumbent
utility
t
w
t
,
.
the policy choice under optimal play.
is
g and
In these cases, strategy profiles
expected from a replacement in the long run. Let vc {w)
w) represent M's discounted expected
for all
a contract similar to
(1),
if
better perfor-
<^~ 1 Eu
M (xj
|
wq
=
in the repeated con-
M does not
Thus, M re-elects
Clearly,
resulting in policy g t
.
if
Pif:
uM (x(w, gt )) + Sv c (w) > u M {gt )
+
5 f v c (w)uj(dw).
Note that because a type-u; incumbent's future election prospects
realizations of gt letting
,
Hence x(w, g
uM
C
t
,
for
t
)
lies
—
between the solutions
of (2)
xt
under
(4)
The
integral in (5) represents the discounted expected value
M
good
strict for all
in her election choice.
inf
{w
|
w
t
^
gt
Obviously,
.
It will
it
as a challenger.
We
is
do not require that
game
in a fictional
.
Thus, u M (x t )
>
.
from electing a challenger
provides an important point of comparison
therefore be convenient to define an "average" type
> Jn v c (w)u>(dw)}, which
v c (w)
one-time incumbent
in
the lowest type that
w
G
fi;
rather,
it
is
ex ante at least as
may be thought
which future challengers are
all
of as a
drawn according
UJ.
Our
first result
according to type:
all
It
is
concave, then
somewhat
incumbent types are never
(i.e.,
w
are re-elected.
they do not satisfy
counter-intuitively,
when groups (and hence equilibrium
of the policy
and
re-election
establishes a basic monotonicity of re-election results
incumbents with types above
not be re-elected for certain group types
uM
some
uses these observations to characterize
implications of optimal equilibria.
if
and
future
within [w, jzg].
fall
contracts in £'c
with the inequality
prior to the revelation of
may depend on
her expected payoff from re-election must
0,
(g t ),
w =
to
e
(5)
JQ
10
In the latter case,
incumbents are increasingly re-elected
policies) are farther
re-elected.
(5)).
Those below may
from m.
Finally, sufficiently
low
Proposition
In any equilibrium in
1
Sc
there exist
,
w
1
w2
and
< w < w2 < w
1
such thai
and:
if
for some
|ttt.
—
gt
\
r\
:
[w
M
if u
1
^
is
2
x T
]
{0
if
1
if
{0,1}.
77(10,
about Proposition
6C
many
it
also holds for
belong to
f2;
if
^
1
2
]
,
non- decreasing in w, and non- decreasing in
1
are worth noting. First, while the result
(but not necessarily
all)
SSPEs. Second, neither
w
1
is
nor
stated
w2
need
they are simply cutpoints characterizing behavior for any w. Third, re-election
outcomes depend greatly on
And
1
Appendix.
in the
A few observations
,
is
gt )
w <w
w € [w
w > w2
concave.
Proof. All proofs are
for
—»
r)(w,g t )
some neighborhood
may
below average types
7.
of m,
For example,
if
7
is
degenerate with gt
not contained within
is
T and
always be re-elected. Finally, the
Q,
=
3? +
,
^
m, then
w = w2
1
.
then some "slightly"
effect of S is
ambiguous. High
M can achieve her ideal
However, they also give M a greater incentive
values can enlarge Qt{-) and hence the range of group types for which
policy, holding re-election strategies constant.
to select
good future
politicians, thus
expanding the
set of politician
and group types
for
which incumbents are kicked out.
The
exercise here establishes that in an environment in which voters
tracts" for politicians,
But because
of the bluntness of the voter's contract instrument,
both problems are solved in crude fashion.
politician types (and only
The
voter
may
only select against the worst
under certain circumstances), and may only move policy a limited
are extreme.
4.
We now
write "con-
an optimal strategy in a repeated game solves both shirking and
type-selection problems.
amount when groups
may
Equilibria in Stationary Strategies
return to the case in which
sequentially rational manner.
We
M
cannot write contracts, but instead votes in a
again restrict attention to SSPE, and focus on the optimal
such equilibria for M, which we label £s
.
11
Analogously with the previous section, we define v s (w) to be M's expected payoff
in
an arbitrary SSPE, starting from a period with a type-u; incumbent. At the voting stage,
M's problem
is
By
challengers.
This implies that
equality, p^(-)
(1),
is
if
P
is
Thus,
must be constant
in
v s (w)u(dw).
always retained. Likewise,
is
types such that (6)
all
In response,
.
(6)
G
t
can
offer
ideal policy in every such period.
all
is
if
P
is
"below
not satisfied with
a contract of the form in
The
following result uses this
continuation values (up to a set of measure zero) must be equal.
=
In any SSPE, v s (w)
in equilibrium
xt
k almost everywhere for some
M's incentive to
k.
select types effectively prevents
formance. Whereas in the contracting case
policies,
/
never re-elected. As a result, for
intuition to establish that
1
>
the
if:
"above average," then she
and thus receives her
Comment
P
stationarity, she re-elects
vs (w)
average," she
random draw from among
to choose between a type-w incumbent and a
monitoring of per-
M could induce high-it; types to choose
"better"
she cannot credibly force different types to choose different actions in a stationary
equilibrium.
There are many SSPEs. As a
Gt
offers
trivial
example, consider the equilibrium set
P chooses gt and
a contract of the form in (1) in each period.
that since
G
t
always receives her ideal policy, v s (w)
equilibrium for M.
A
,
is
constant.
M
£o, in
which
elects Cj.
Note
Clearly, this is the worst
more plausible equilibrium might feature the optimal
(credible) use of
re-election incentives.
To
characterize
candidate type.
If
£s we make use
,
w =
M
if
w
Denote by
can induce any policy
can then achieve a policy of
Thus,
1.
w=
ini{w E
S7} the "lowest"
w, then (similar to the discussion of the contracting case) with the
appropriate voting strategy,
it
Comment
of
m
were the only type in
if
tt,
m
in
Gt(~^s) i n each period
t.
At
best,
€ Gt{— i^)> an d 9 (~ Jr^) or 9t(~ 1^5) otherwise.
then
t
P
is
thereby achieve the contracting result.
12
indifferent
between
all
candidates and can
The
to take the
same action
The reason
for this
t
if
M
=
types),
could induce a better
In these
+e
offering r=^
for
a
.
Proposition 2 In any equilibrium
{w,gt ,bt ,Xt)
all
outside of Qt{— yr*)-
lie
would then be willing to "buy out" type-w politicians by
policy choice of g t
P*t
straightforward:
is
must sometimes
policy in expectation, then this policy
G
can credibly induce any type of politician
as type-u; (hence creating a uniform expected value for
she cannot do any better.
cases,
M
following result establishes that while
in
£s for any type-w incumbent and
,
all t,
gt
,
and
bt :
and
1
UH?y
Xt(w,9ub
t
=
)
I
m
if
«*<&(- A)
if
m
e
[gt
{-^-5 ),gt (-^s)]
ta(-A) ifm>5
(- I^)i
As
M's strategy has elements
in the contracting case,
toring.
of type selection
and ex post moni-
However, removing commitment to re-election schedules makes the selection of types
irrelevant in equilibrium. This
happens because of
election choice occurs after P's policy choice.
remove low types)
then so strong that
is
spectively, removed).
best that
M
can do
is
if
The
all
G
M's
incentive to retain high types (respectively,
they existed, they would always be re-elected
Under these conditions
to monitor
stationarity, as well as the fact that
t
can easily buy
its ideal policy.
(re-
Thus the
types uniformly, and in a credible fashion. This causes
the performance of each type to sink to that of the "least
common
denominator," or w,
regardless of the distribution of the rest of candidate types.
5.
The preceding
welfare.
Non-Stationary Strategies
commit might play a
results suggest that the ability to
Here we demonstrate that this
is
not
so.
Instead,
central role in voter
by dropping the requirement
of
stationary equilibria and focusing instead on simple, non-stationary equilibria, the voter can
do just as well as she could in the contracting
We
case.
begin by noting a major difference between our model and other dynamic games in
which non-stationary solution concepts are applied. In the game examined here, the only
13
plausible "punishment" device
is,
not re-electing P) removes the punished player. That
(i.e.,
we do not have a repeated game.
Combined with complete information,
simplifies a potential defector's optimization problem.
punisher's problem, in that
equilibria.
M. Such
new
politicians at a later date. It
on the type of group drawn. However, our results require only that we
examine the simplest such
against
9
For example, the voter might punish
a politician by not re-electing him, and "cooperate" with
also condition
also complicates the potential
limits the promise of future "cooperative" interaction.
it
There are many kinds of non-stationary
may
It
this greatly
class of equilibria, in
equilibria are characterized
which P and
by the
triple (£
G
c ,£
t
play a "trigger" strategy
,n), the elements of
which
correspond to a cooperative phase, punishment phase, and punishment length, respectively.
Play begins in the cooperative phase at
t
=
1,
and upon any deviation by
prescribed strategies, play continues in the punishment phase for n
this phase,
G
t
offers
a contract of the form in
re-elects P. After the
P
its
During
chooses gt in each period, and
M never
game play within the punishment phase
Nash equilibrium, and that the
from
periods.
1
punishment phase, play reverts to the cooperative phase.
perfection requires that the
perfect
(1),
>
M
itself
strategies in the cooperative phase
Subgame
be a subgame
be consistent
with the incentives posed by the punishment phase.
Cooperative phase strategies
gies in
£ c that meet two
result in the
may be
may
criteria.
take
First,
forms, and for simplicity
they must be stationary. 10
we
focus on strate-
Second, they cannot
punishment phase along the equilibrium path (though the punishment phase
essential for inducing
M
to play according to
phase cannot be reached in equilibrium
least as
many
good
as g t in each period,
if
and n
£ c ).
Clearly, then, the
punishment
the cooperative phase strategies yield a policy at
is
sufficiently large.
We
call strategies satisfying
these requirements stationary trigger strategies.
Even with
its restrictions,
many
stationary trigger equilibria exist.
The simplest example
might be a "grim trigger" equilibrium (£s ,£ ,oo), where play proceeds according to the
3
See Dutta (1995) for a more general treatment of equilibria in dynamic games.
3
Note that the punishment phase strategies are stationary as
14
well.
stationary equilibrium of the previous section.
be
where
(E'c ,£ ,oo),
M
votes as she would in the "myopic" contracting game.
incentive to deviate from
its
gt forever, a result which
is
As both
weakly dominated by that under £ c Clearly, then, there
in a very
of these examples illustrate,
and does not
has no
room
is
.
rudimentary non-stationary setup.
M's minmax payoff
is
simply that implied by
Moreover, M's vote
receiving a policy at gt in every period.
i.e.,
period,
M
specified voting strategy, because doing so will result in receiving
amongst types even
for discrimination
Another, more plausible, example might
affect her payoffs in that period.
11
the last action in each
is
M
However,
Sq\
may
still
exploit
P by
replacing a "bad" incumbent after she chooses a policy in expectation of re-election. Thus,
to establish a trigger equilibrium
will
it
be important to check that
M does not have sufficient
incentive to do so.
The
following result uses these observations to characterize the optimal equilibrium for
M, which
Proposition 3 For n
rium, £ c
As
outcome £c
restores the contracting
=£
c
as the predicted result.
12
sufficiently large, in the optimal stationary trigger strategy equilib-
.
intuition might suggest, sufficiently long (and possibly infinite)
will ensure that
M
does not exploit weak incumbents.
observables, the punishment phase
is
punishment periods
Because there
no noise
is
in the
never invoked in equilibrium. Thus, the simple non-
stationary equilibrium derived in Proposition 3 produces policies that are identical in every
period to those of the contracting equilibrium of Proposition
Example
1.
M and G
t
have quadratic
Group and candidate types
f2
=
{1,5}, w(l)
It is clear
=
that
u(5)
1
2
is
5.
The discount
A
is
easily modified to
T =
= —x 2
and u Gt {x)
{1,8}, 7(1)
factor for all players
is
5
=
=
=
7(8)
—{gt
=
— x) 2
0.5,
is
=
25.
This implies that
and
its
M can induce a type-5 incumbent
no analog to "suckering" one's opponent
accommodate non-optimal stationary
15
.
0.8.
type-5 incumbent must always be re-elected, and thus
5/(1—5)
latter point implies that there
The proof
with u M (x)
are distributed as follows:
0.5.
<w <
expected lifetime payoff
'The
=
utility,
1.
in the prisoner's
trigger equilibria as well.
dilemma.
up to
to choose a policy
v c (5)
=
—4.5/(1
A
(»
is 1
t
.
=
Thus, when gt
1 (8),
—22.5. Suppose that a type-1 incumbent
an expected lifetime payoff of
this results in
the policy
— 5) =
away from g
5 units
and vc (l)
6.71),
«
M
similar exercise reveals that
=
1(0. 5)/(l— 5(0.5))
the policy
is
only re-elected
is
5/6. Thus,
t
+ 0.5[-6.7l +Sv c (l)}
0.5[-l+5{v c {l)-hJ c {5))/2]
if
when g
2
and
(3),
ta
=
gt
=
1 (8),
-68.76.
would receive —83.06 from always re-electing a type-1
incumbent, —69.17 from never re-electing a type-1 incumbent, and —91.25 from only
=
when gt
electing
1;
8;
thus the strategy of re-electing only
when
gt
=
8
is
re-
optimal in a
contracting environment.
To complete the stationary
strategy.
obvious that for any n,
It is
type-5 incumbent.
M re-elects a type-1
Finally,
M
Similarly,
M
will
will clearly not re-elect a type-1
X>E^(
>
5i )
t
=
+
5
10
=
1.
if:
(?)
n >
1.
Thus even
a mild punishment
Extensions
Term Limits
6.1
We now
T>
extend our results to the case in which legislators
terms.
The model
is
unchanged with the exception that
completes her T-th period of
Note that
at
T=
assume that a
The main
oo, the
legislator
office,
model
who
1
,
.
.
.
,
to
which
M
is
serve no
in period
t,
more than
an incumbent
automatically replaced with the challenger
identical to that in the previous sections.
intuition of our results
Thus the extent
{
is
she
if,
may
We
C
t
.
continue to
leaves office cannot return as a candidate.
not only on her value of holding
E
t
capable of sustaining "contract" behavior in this example.
is
6.
6
incumbent when g
Vc(l)+V c {5)
n
Substituting in values from above, (7) holds for any
scheme
M will follow its prescribed
always have sufficient incentive to re-elect a
incumbent when g
V C (1)
we check that
trigger equilibrium,
is
office,
that an incumbent's continuation value will depend
but also on the number of possible terms remaining.
can control politicians
T} denote the term that the
will also
depend on both
legislator is currently serving.
16
variables. Let
We
we
begin with the optimal contracting equilibria, which
resent the expected value to
Section
M
3,
M
of a type-u; legislator
label £t- Let vt(w, 9) rep-
with 9 terms of experience.
may
has two choices at the contracting stage. First, she
0+1)
a type-(tot
To
1) challenger,
,
reflect
prior to the
its
where
w
t
randomly drawn as
is
Let
draw
of gt
Next,
.
let
x(w,
re-election contract (of the
t
g
9,
t
form
+
P
performance.
{i.e.,
x(w,T,g
T).
By
t
)
gt)-
and
the notation of Section
M
can induce
in (3)) in a given period:
m
it
me{g (-w-6lWi e),g (-w-5lw>e
if
m>g
<
g t {-w-6l Wt e)
t
t
)]
down
>u M {g + 5
t)
v {w,l)u{dw).
f T
(9)
the expected payoff from an incumbent in her
form
in (1),
G
£
can then obtain
its
optimal policy
Thus,
,
T)=
M
f u {g)j{dg)
Jv
+5
f vT (w,
l)cu(dw).
(10)
Jn
Expression (10) also implies that legislators in their penultimate terms will also be
With
a single possible period of office remaining at period
performance by promising a single re-election.
t
+1
However, doing so
policy of 3t+i, followed by the election of challenger
discounted heavily, or
challenger
C
t
(8)
(-w-6l Wi e).
Since she cannot be re-elected, no voting contract can induce any
vT ( w
to control.
elect
if:
offering a contract of the
=
we extend
9,
ideal policy
if
5v T (w,9+l)
Finite term limits allow us to pin
=
its
be of type
before.
t
(-w-6l Wt e)
u M (x{w,d,gt ))
(9
buy
gt ) represent the optimal policy that
{-w-6lw>e )
Thus, the optimal contract re-elects
term
to
t
will
denote the discounted expected payoffs of a type-(w,9) incumbent
l w> e
{gm
final
M may allow G
the possible dependence of strategies on
3 as follows.
through
subsequent period. Second,
in the
in
extract the optimal
performance given P's expected lifetime and re-elect the incumbent, who
(iw,
As
G
t
is
extreme and M's
C
£
utility is concave,
to re-electing P.
17
+i.
t,
M
difficult
can extract some
will result in a period
Unless future payoffs are
M may simply prefer electing
This logic suggests that
incumbent
M
has a greater ability to use future elections to discipline
earlier in their careers.
Term
to that of high values of w.
Loosely speaking, low values of 9 have an
limits therefore constrain the
effect similar
inducements that
and reduce the long-term expected quality
offered even to the "best" incumbents,
may be
of policies
(from M's perspective). In an environment with interest groups and voting contracts, voters
would then not be able to benefit from
finite
term
limits.
13
The
following result establishes
these intuitions.
Proposition 4
(ii)
(in)
Vt{w,9)
vT (w,
9)
(i)
is
<
Combined with
vt{w,9)
is
non- decreasing in w.
non-increasing in
vc(w), VT, 9,w Eft.
(9),
an
easily established corollary is that
mance becomes more demanding
model, this implies that
to keep an
9.
incumbent
M
if u '(•)
as 9 increases or
w
decreases.
M's desire
for policy perfor-
As with the non-term
limited
concave, then progressively worse draws of g are required
is
t
in office.
Since incumbents become less desirable with seniority, our model suggests that
term limits are
limits that
finite, re-election
do not incorporate
rates should change over time.
when
Previous models of term
citizens' type-selection incentives {e.g.,
Lopez 2002) might
therefore underestimate the impact of term limits on incumbent turnover rates.
The next
result
shows that an incumbent's ex ante re-election probability, or the probaof Gj, decreases weakly over time.
14
any incumbent
is
bility re-election prior to the
draw
Proposition 5 The ex ante
re-election probability of
non-increasing in
Because type-(iu,T) incumbents have an ex ante re-election probability of
sition 5 implies that unless
incumbents are never re-elected
13
{e.g., if u;(0)
=
1,
zero,
j{m)
9.
Propo-
=
1,
or
In the model of Smaxt and Sturm (2004), term-limits can help voters, but no interest groups are present.
"Consistent with this prediction, Smart and Sturm (2004) estimate that term-limited U.S. governors are
less likely to be re-elected than governors without term limits.
18
T=
1),
there exist incumbent types for which the ex ante re-election probability
decreasing.
The
compares usefully with Proposition
result also
some types (w > w)
With
are always retained.
might be retained with certainty
w
high
for sufficiently
term
finite
and low
8,
strictly
which established that
1,
limits,
is
a type-(u>,
incumbent
6)
but her ex ante re-election
probability declines to zero eventually.
Our
to Proposition
section.
the assumption of voting contracts and
final result eliminates
therefore an analog
is
We consider the same class of stationary trigger strategies as in the previous
3.
As with the non-term
limited case, the result
is
that simple non-stationary strategies
(such as (&r, Sq, oo)) are sufficient to support strategy profiles in St-
Proposition 6 For
c
egy equilibrium, S
Example
T=
=
any re-elected
from gt
Now
re-elect,
~
sufficiently large, in the optimal stationary trigger strat-
St-
maintain
of the parameters of
all
politician will be induced
-32.5; thus, vT (5, 2)
is
by
=
G
t
Example
1,
and additionally assume
T-th term incumbents
to choose a policy of g t
v T (l, 2)
=
v2
=
-32.5
+
is
zero for
then
will receive
it
5.76 (7) for g t
As
—
8
gt
,
while by re-electing a type-(io,
M
a result,
w =
and
With
expected values of each possible contract.
these values,
it
w, and
policy
8(v T (l, l)+vr(5, l))/2.
for g t
is
=
M
does not
will
expect a
If
incumbent
1)
can induce a policy of
5 (1).
all
The expected
.
consider the set of possible contracts for newly-elected politicians.
lifetime payoff of w.
of
and n
Clearly, the expected lifetime payoff of
2.
utility
We
2.
T finite
1,
and a policy
possible to calculate the
We provide several examples here
and omit several
obviously non-optimal cases:
•
Never Re-elect: vt(w,
• Re-elect
•
iff
w =
t>r(5,l)
~
Always
Re-elect:
5:
1)
=
w; vt(w,
1)
v 2 and vT (5, 1)
w
v 2 for
v T (l, 1)
=
all
=
—162.5.
-5.76 2 /2
+
5v 2
\
~
vT (l, 1)
-139.80,
-128.45.
-133.27, uT (5,l)
v T (l,
ss
1)
ps
-7 2 /2 +
5v 2 and uT (5,
-125.38.
19
1)
«
-5.76 2 /2
+
5v 2
:
vT (l, 1)
~
• Re-elect
w=
iff
+
<J(ut(1, 1)
• Re-elect
and ur (5,
• Re-elect
w
1)
5 or g t
2
wr(5, 1)
(-5.76 2
wr(5,l)
«
-117.68.
M
would
+
+
«
1)
Sv 2 vT (l,
Sv 2 )/2
«
v 2 and u T (5, 1)
-136.85, uT (5,
1) ra
(-5.76 2
]
«
1)
-128.69, uT (5,
=
(-7 2
+
Sv 2 )/2
+
[-1 2
+
5{v T {l, 1)
[-1 2
+
+
+
+
1)
5(y r (l, l)
(
«
and
is
+ uT (5,
5(v T (l, 1)
+ vT (5,
uT (5, l))/2]/2; v T (l,
ideally write contracts that promise
new
[-1 2
+
+
l))/2]/2
-123.22.
l))/2]/2 and
1)
w
-125.57,
when
politicians re-election
groups are extreme. This strategy minimizes the damage that extreme (gt
do,
5v 2 )/2
-124.03.
= (-7 2 + dT; 2 )/2 + [-l 2 +
ur (l,l)
8:
=
uT (l, 1)
8:
v T {l, 1)
8:
w
Thus,
=
-5.76 /2
& =
iff
=
t
vT (5, l))/2]/2; u T (l,
w=
iff
and g
5
=
8)
groups can
not very costly because the policies "bought" by moderate groups (g
t
—
1)
is
relatively benign.
necessary to check that
It is finally
M
does not have an incentive to deviate from
election strategy. Carrying a calculation identical to that in (7),
if
n >
its re-
M will re-elect an incumbent
2.
6.2 Infinitely- Lived Groups
Now
analysis
fates.
consider the case of a single group, G, that
was
simplified
Under repeated
by the
fact that
is
present in
all
periods.
15
The preceding
groups did not care about incumbents' electoral
interaction, however, the
group also participates
in the retention of
incumbents.
To
see
how an
that a voter
G
is
infinitely-lived
may
affect voting
able to offer optimal contracts, as in
must accept a
politician type.
and bribing
£c Under
.
its
relatively unfavorable policy in all periods once
G
does better in periods
the policy will be her ideal point,
15
group
g.
when
It is clear,
M
strategies,
suppose
(myopic) bribing strategy,
M
finds a sufficiently
good
chooses not to re-elect the incumbent, as
then, that
G
has an incentive to "buy out"
For a treatment of dynamic interaction of long-lived interests and politicians in the regulatory context,
see Salant (1995),
Martimort (1999), and Faure-Grimaud and Martimort (2003).
20
high incumbent types in order to induce a fresh draw of politicians. In turn,
incentive to loosen
We
that
contract to prevent
5
3.
=
As
in the previous examples, let
0.8.
G's utility
at 3 (8)
under type-5 (type-1) incumbent.
= -64 +
Under the
= -125
—104.17
5(vc (l)
-
for a type-5
for a type-1
e
+
<5(-104.17
To induce
G
G
.
incumbent, and v° (1)
-
=
u M (x)
— —x 2 Q =
,
= — (8 —
x)
2
.
{1,5}, oj(1)
=
Under £c
M
;
,
w(5)
=
always
results in a policy
easily calculated that f c (5)
is
would have a continuation value of v^(5)
G
incumbent.
to choose a policy at g
-25 -
Sc
£c
= —45
and
-136.67.
ta
45)/2
strategies in
It
analog of
M.
now given by u G (x)
is
for the
and always removes type-1 incumbents. This
retains type-5 incumbents,
0.8)
from forcing such replacements.
stationary contracts that are optimal for
is,
and
v c {\)
G
with two-type example, in which we solve
illustrate this logic
Example
0.5,
its
M will have an
=
+
5(v
G (l) -
=
—(8
— 3) 2 /(l —
125)/2, implying v
G (l)
«
can do better by paying 25 + e to a type-5 incumbent
This results in a new incumbent, and an expected payoff of
8.
125)/2
«
-116.67.
not to buy out a type-5 incumbent, G's continuation value from a type-5
incumbent must solve the following system:
v
V
Solving yields v G (l)
= —50
= ^{v G (5) + v G (l))
G
= -25+ -(v G (5)+v G (l)).
5
(5)
and v G {5)
this implies that its policy choice
distant from
M's
S
G (l)
x
=
is
is
given
then to
offer
some
e
if
5(v c {l)
-
85.16)/2
As the example
«
'
~_
5
=
-75, or 8
is
always re-elected,
— \/l5, which
is
more
by a short-lived group. M's optimal contract
she chooses any policy in [— 8 + vl5, 8 — vl5]- G's
>
policy choices reduce M's expected payoff:
-64 +
by —
ideal than the policy (3) chosen
therefore re-elects a type-5 incumbent
best response
—75. Since a type-5 incumbent
for choosing policy at 8
now
t> c
(5)
=
80\/l5
—
395
— vl5. The
~
resulting
—85.16 and v c (l)
~
-163.44.
suggests,
we claim that optimal contracting
21
equilibria of
games with
larger type sets will share
M will retain
points,
type that
P and
all
allowing her to remain in
Two
The
who
features.
always retained by M,
is
policies that are closer to
6.3
two
because there
First,
and kick out
sufficiently high types,
M's
G
will
office.
ideal,
no variation
group
in
be indifferent between paying w/{\—5) to buy out
This implies that higher types
but that these policies
will
continue to choose
be closer to g than under £c
will
extension considers what happens
when two groups
are
drawn
£*(')
=
(*
l
(with positive support on
1)2).
.
in each period,
G
simultaneously offer voting contracts to P. The groups are generically labeled
ideal points g
ideal
lower types. Second, for any
Groups
Interest
final
is
all
As
in the basic
T
l
),
u°
utility functions
model, each group
1
with
,
l
(-),
and upper contour
Now
"lives" for a single period.
sets
each
group's bribes must take into account not only the cost of the politician's expected remaining
payoff from
The
offers
office,
effect of
but also any bribes from the other group as
an additional group
,
G
then
2
will
of type w, this induces
have an incentive to "buy out"
raises the possibility that
also
G
1
might
have an incentive to adjust
from
again usefully illustrated by supposing that a voter
is
optimal contracts according to £c to group
m) and an incumbent
its
well.
in turn
G
1
.
In the presence of only
some policy
P.
x'
6
[m, g
1
].
But
G
if x'
1
(with g 1
€"
£ (— jt^),
G
This upsets the bribing strategy of
be willing to pay more to buy back
contract to prevent policy from being
>
2
P.
drawn too
1
M
far
,
and
may
away
77i.
The
fully-developed model of competitive vote-buying
illustrate
two general features
realized distributions of
here.
is
we simply
quite complex, so
we show that under a
In both examples,
variety of
g\ the median voter can achieve her ideal policy with a very simple
stationary voting contract.
Example
P
Let
m
G
^(-3^) D G 2 (-jzg)
anticipates re-election in
re-elect
G
4.
2
's
P
if
and only
best response
is
if
all
for all
future periods, and that
she chooses m.
Then
if
G
1
M
offers
identical to the one-group case;
22
(g\g
i.e.,
2
)
1
2
G T x T
.
Suppose that
adopts the following contract:
P
for
any
policy,
no payment
for
any
policy.
no payment to
offer
By symmetry. G lT s
best response
Since this outcome
policy in.
is
is
M for
clearly optimal for
contract to re-elect a type-ir rncumbent
P
policy.
any draw" of
and only
if
P
if
m
chooses
therefore chooses
*g'.cr).
^77.
is
M
always
Thus, the
part of a subgame
is
equmbrium.
Example
Let u G'(-) be concave and sy
5.
m £ £r(— jz|)
2
for all (g^.g
P
wish to buy out
Suppose that
P
tract: re-elect
1
S F x
)
I"
2
mm etric,
P
if
anticipates re-election in
and only
g
r
=
1
Gl
all
G
(m)
-
u
Likewise, let
G
offer
Gl
b {x)
)
must be
strategies,
offered to
choosing g~')
:
=
-w/(l-5)
I
b (x). however.
is
G
1
otherwise.
-u G2 (m) + e
receives
P then
)
— uP'fa
1
)
and
Thus, letting
x = sf
if
x
=m
is
2
(g
)
for choosing
— vP'lg
m
1
)
for
any x
1
2
g or g
G
:
.
P
(otherwise.
1
»
clearly optimal for
is
M
for
—u G~ (g^-re
at least
— uP'lg 1 )
will strictly prefer
2u G'(m)
2
.
m. Under ^(x) and
==
G receives u//(l — 5) +
2
e — 0. 6 (x) and 6 (j) are
always re-elects P: hence P"s expected payoff
subgame
if
chooses m. For each group
no more than
Since this outcome
therefore part of a
= — uG
1
(g
to choose an}- policy other than
strictly better.
voting contract.
the contract:
otherwise.
P receives —u
so G' can receive
2
concavity)
P
offer
otherwise.
[
m. and
would al~\v;
ifx=^
if 2 = to
2
(g
G1
let
-u GV)
f
for choosing
and
the contract:
2
Under these
G
1
M adopts the con-
future periods, and that
[
2
.
.
-u GV)
u
l
m — g1 > g2 — m.
2
g
2
she chooses m. Further,
if
< m <
1
Thus, without loss of generality.
.
in the absence of a bribe from
b {x)
is
any
for
hence P's expected payoff at the beginning of each period
re-elects P;
perfect
payment
to offer no
—
e.
which
by
best responses to M's
any draw of (s1 ,^2 ),
yzj- This set of bribing
M
and voting contracts
perfect equihbrium.
The two examples may be combined
straightforwardly to
show that when group
functions are symmetric and concave, and groups are either "close" to
23
m
utility
(in the sense that
m 6 Q {— TZ5) H G (— ttx))
m for a type-io incumbent.
2
l
or
on opposite sides of m, then
This incumbent
M can always obtain a policy at
always re-elected.
is
These examples suggest that multiple groups can play a
may
policy outcomes. Extreme policy outcomes
of
77i,
and
will never
group
at least one
group
is
not close.
result
moderating
both groups are on the same side
seems safe to conjecture that the outcome
It
be more extreme than the most extreme group. However, since the most extreme
will
tend to be more extreme as the number of groups increases, the overall
adding groups on the same side of
m
is
7.
The models developed
unclear.
We
in a voter's incentives,
discipline to
improve
effect of
leave this for future work.
Discussion
here integrate strategic interest groups and strategic voters in a
general framework of policy-making and elections.
As a
if
significant role in
may be
because there
a strong short-run incentive to use electoral
and because a group can
policy,
result, the strategies of selecting
good
This combination introduces a tension
exploit a promise to retain a politician.
politician types
and rewarding good performance
are often incompatible.
The
results illustrate the effects of this tension
models generally predict that incumbents who
Incumbents who value
office less
highly can be disciplined as the interest group becomes
when adopted
policy performance
policies are
is
more extreme.
greater. This causes
We
also find that
a long-lived interest group weakens a voter's electoral control, while multiple groups
strengthen
to term-limited officials predicts that
term
limits have
an
effect similar
Thus, term limits reduce policy performance
to a reduction in the value of holding office.
from a voter's perspective. In turn, they give voters a greater incentive to replace
as their limit approaches.
term
may
it.
The extension
for
The
value office highly are difficult to discipline.
more extreme, because the short-term gain from
re-election rates to increase
under a variety of assumptions.
limits
among
Our model cannot
citizens.
While
it is
officials
explain the apparently high level of support
predicted that most voters should oppose them,
24
in survey
On
initiative voting
data voters typically support term limits by a 2 to
the other hand, one key prediction of our model
there
of
and
is
is
1
margin.
consistent with the data. Specifically,
considerable evidence that conservative and Republican voters are more supportive
term limits than
voters are probably
liberal
more
and Democratic
voters.
The
policy preferences of conservative
closely aligned with organized interest groups, since
organized interest groups have a pro-business, conservative in orientation
and Tierney, 1986). Term
limits are therefore likely to
25
move
(e.g.,
most
well-
Schlozman
policy in their direction.
Appendix
Proof of Proposition
By
bent.
stationarity,
We first
any
for
w
we
also
drop (wlog)
we use w
all
G
and
f2,
v c (w)
(ii)
=
YltZi 5
t~ 1
to denote the type of a generic incum-
time subscripts.
define an extended value function v c
coincides with v c
w
For convenience
1.
:
—
5R +
E[u M (x*
>
\
K, which
wq
=
on Q, and equals M's expected payoff
satisfies:
w)\ for any
in a fictional
incumbent and challengers drawn according to probability measure
following
lemma
will
Lemma
1 v c (w)
is
Proof.
We
be useful
w
v c (w)
(i)
g"
—
v c (w)
That
Q.
is,
vc
game with a
type-
otherwise.
The
to
for deriving the result.
non- decreasing in w.
show that
exists a voting strategy
for
<
any two incumbent types, w' and w", where w'
w", there
p that implements the same policy and re-election outcome
for
both
incumbent types.
For any optimal re-election strategy p*(w,
empty) subsets; V(w,g)
It is
0}.
c,
arg
X
{x €
p*(w,g,x)
x)
we
=
1}
|
partition
=
argmaxl€ p(KI]S)U G (x) and
max xe c(™, 5
respectively.
G
)
u G (x) and
offers the
(ii)
w =
Let x*w
jfSj,
where £
Now
{x G
X
p*(w,g,x)
this,
=
offer
contract
(i)
7 ({ 9 G T
|
V{w,g)
=
\
offer
(i)
w + 6l w + e
for
e
for
choosing
p and
if:
w-
5l w
#
(11)
.
^
0,
letting
e
—
it is
»
clear
0}).
denote the equilibrium policy with a type-u; incumbent and type-g group. There
are two cases. First,
show
=
disjoint (possibly
otherwise. These contracts result in policy choices of
type
Because P can assure herself of re-election whenever V(w,g)
l
two
into
and C(w,g)
otherwise; or
u G (p) > u a (c) -
that
X
then clear that G's optimal contract must be of one of two types:
choosing p
c =
=
g,
if
w' and g are such that x*w
,
G C(w',g), then we claim x*w
note that g G C(w\g) implies (by the definition of
suppose g G V(w',g). Then by
(11),
G
should offer a type
26
c)
(i)
x*w
,
=
,
=
g.
To
g automatically.
contract: contradiction.
To implement the
G
for all x.
then
offers
she chooses g and
Second,
if
a contract promising
is
w' and g are such that x*w
To show that
feasible for all {g
x*w
M
can choose: p(w",g,x)
policy x*w
G F
V{w',g)
\
g
,
g
f
is
that offered to a type-u/ incumbent.
Now
types
We
and
,
1 if
x
This implies that for
x^,
,
>
and p(w",g,x)
=
l(w';{p* ()}). Further,
x
all
^
G
offers a
type
P then
receives
e+w+8l(w"; {p(w",
,
and
(i)
x*w
,
,
u G (x^,
)
>
contract f3*(w',g) identical to
g,
x)}) for choosing
re-elected.
is
w =
inf
{w
vc {w)
\
>
w
G clos(Q).
which characterizes the re-election choice,
as follows.
fn v c (w)iv(dw)}. Note that
use this notation to rewrite
(5),
Lemma
M
approach in
Q(—w — 5l w )). Then
(4) (i.e.,
the optimal policy in
u M {x(w,g))
By Lemma
1
and the
fact that
The next lemma
Lemma
2 u M (x(w, g))
will
is
w", where w'
< w" By
.
be useful
i.e.,
because w'
> u M {g) +
> u M (g)
implies
given v c (w), following the
M re-elects P
of type
w
Sv c {w).
holds
if:
(12)
trivially, (12)
holds for
all
g
if
for deriving properties of (12).
u M (x(w',g)) > u M (x(w"
(4), this is
begin with two observations:
(ii)
5vc (w)
u M (x(w,g))
possible only
w'
for all g;
+
P
can induce from
1
non- decreasing in w.
Proof. Suppose otherwise;
We
=
hence l(w";{p(w",x,g)})
0};
Thus she chooses x*w
otherwise.
=
outcome
consider any optimal stationary contracting equilibrium. Define the set of "average"
as:
w.
=
Thus
otherwise.
to implement the desired
Denote by x(w,g) the optimal policy that
w>
g, x)
chosen, note that p(w",g, x) implies that re-election
u G (x) — w — 5l(w"]{p(w",g,x)}). Hence,
policy x*w
choosing policy g and
e for
G V(w', g), then
,
G V(w',g) implies that (11) holds.
,
can choose p(w",
not re-elected.
is
with a type-w" incumbent,
otherwise.
M
desired outcome with a type-u>" incumbent,
<
(i)
,
g)) for
some g and types w' and
if:
+ 5lw ,>w" + 5lw n.
by
(4), (13)
implies that
w", (13) implies that j(T w ')
27
(13)
u M (x(w',g))
> u M (x(w",g))
> 7(r„ »), where T w =
J
{g'
G
r
type w}
(12) holds for
|
the set of group types such that a type-it; incumbent
is
is re-
elected in equilibrium.
T
Partition
7(T
1
>
)
T 1 = Tw n
into
We now
0.
r2 = T
(T \ r^") and
>
T1
\
Note that by observation
.
1
T and T
consider the behavior of v c (w) over
2
(ii),
Let v c (w,g') represent
.
M's discounted expected payoff conditional on a type-w incumbent and type-g' group.
G T 1 the
First, for g'
uM (x(w',
Noting that
=
that vc (w\, eV i
come
5v c (w')
is
v c {w") lg €r
,
|
not re-elected),
any
for
/
iI )
2
G F n T w »,
g'
i
=
stationarity, v c (^')ls'er 2
we have v c (w') >
<
w2
>
may
=
M
as well.
w. Additionally, by
where
T> w n
w<
C
J? w
which
is
M
G T
clear
fl
=
=
T w »)
(F \
(i.e.,
for all x.
outcome
1 for
This con-
for a
=
x
where
type-w"
x(w",g') and
the outcome, note that x(w",g')
=
(13) implies
M can achieve this outcome in any period,
>
= j(T
This condition
Lemma
it is
v c (w")\ g e r2.
1
is
2,
+ 7(r 2 )u c (u;)| s
)v c (w)\ g er i
>
Lemma
note that by (12), for any
,
(14)
can guarantee an out-
2
also the
is
offer p(w',g',x)
Because
w, the right-hand side of this condition
for all
g'
offer p(w',g',x)
g,
may
v c (w"), contradicting
— v c (w)).
5(v c (w)
i
any w, v c (w)
Finally, note that for
(g)
Jrl
u G (x). Thus, u G (x(w"',</)) > u G (g') — w" — 5l w », which by
=1
To characterize
6vc {w").
v c (w",g)-y{dg).
For any
To show that x(w",g')
u G (x(w",g')) > u G (g')—w'—5l w
results,
M
no re-election and x
otherwise.
argmaxx p »( 10 " ig
> u M {x{w", g')) +
,
And
=
p(w',g',x)
Thus
>
identical to that with a type-ui" incumbent.
incumbent.
w <
5vc (w)
€ T 2 we show that with a type-u/ incumbent
g'
tract clearly results in
u
> u M (g') +
Jr i vc (w' ,g)i(dg)
a type- it/' incumbent
by
(12) imply:
= max{u M (x(w, g')) + 5v c (w), u M (g') + 5v c (w)},
any w, v c (w,g')
for
Second, for
+
g'))
T 1 and
definition of
,
w <
iu,
positive
the left-hand side
is
Combining
1.
P
is
not re-elected
satisfied trivially
is
6r 2.
/
and by
if
w — w
Lemma
1
u M (x(w, g))—
if:
and g
=
m. For
non-increasing in
non-negative and non-decreasing in w.
w, there exists a non-empty set of group types T>w satisfying the condition,
>
for
To characterize
w
w" >
1
,
w'.
Thus,
(12) implies
P
w2 =
is
sup{u> G
fi
|
never re-elected
28
3g G F
if
fl
T> w },
for all g:
and
u M (x(w
otherwise.
,
g))
— u M (g) <
6(v c (w)—v c (w)). Because this condition
To show
any
>
vc (w)
w =
=
hence, x(w, g)
p;
Putting
all
T> w
<
w" >
for
side
w —
l
w',
w
follow. First, note that the left-hand side of (12)
Second,
g',
(12) holds for
if
M
if u
g" such that \m
G
type-g" group; that
is,
g'
and non-decreasing in g
Proof of
Comment
[u;
=
£
Vw
M
if u
implies g"
$
Vw
it
is
all
>
<7t
,
b tl
vs
or
,
and x t
and (3t(w',gt )
v s (w')
(ii)
.
=
Given
<
vs
Vw },
Y C
for
and
at
otherwise.
0.
note that by the definition of
Two
T> w ).
comparative statics
also holds for a type-u>" incumbent.
increasing in \m
Therefore, rj(w,g)
.
Let v s
counted expected payoff from electing a challenger.
v s (w')
],
=
/
.
—
is
Thus, for any
g\.
it
also holds for a
non-decreasing in w,
concave.
is
Suppose otherwise.
1.
,^;
l(g
1
Additionally,
zero.
is
a type-^' group, then
(12) holds for
g'\, if
\
2
Clearly,
0.
w < w2
clear that
non-decreasing in w. Thus, given g and
is
u M (x(w,g)) — u M (g)
— g"\ > \m —
SI
1
1
a type-u/ incumbent, then
concave, then
is
it is
,
and the condition holds
w >w >w >
for
r\
sup{u> 6
2
of the derived inequalities together,
characterize the re-election function
w',
w2
w =
positive
is
T>w realized re-election choices are given by: r](w,g)
w" >
for
g and the left-hand side of the condition
Vw n C
Because
than that
satisfying the condition, consider
and therefore the right-hand
v c (0),
0.
To
w
existence of a
stricter
is
Suppose that
.
G
this voting strategy,
otherwise, for
some
e
>
t
JQ v s (w)LLi(dw) represent the
Then
holds.
(i)
's
=
for
By
best response
some w' € O,
=
P^(w',gt )
a contract of the form in
(i.e.,
either
(5), p*t {w' ,g t ,b t ,x t )
is:
dis-
e i£
(1)).
(i)
=
1 for
xt
=
gt,
Clearly,
= 9t- Note that retaining a type-u/ politician in each
M
u (g)j(dg). But in any period t, M can achieve at least
this induces policy choice Xt( w> 9t^t)
>
= f^ Jr
period implies v s (w')
uM
{gt )
by adopting strategy
Hence, v s
>
=
1
or
all
gt
,
bt
,
and x t thus, v s
;
^
>
M
Jr u (g)-y(dg).
v s (w'): contradiction.
Now suppose that
types. If cj(W)
(i),
p*{-)
>
0,
(ii)
holds. Let
VV
= {w
|
v s (w)
then there exists a non-empty
this set of types is
empty: contradiction.
Proof of Proposition
2.
We
first
We
<
v s } denote the set of "below average"
set of types {ii'
\
conclude that uj(W)
v s (w')
=
v s }.
By
part
0.
prove that any strategy profile in £s
29
>
is
an equilibrium,
and then prove
incumbent type w, she
optimal.
We
For M, since P's policy choice
optimality.
amongst
indifferent
is
fl
,
=
,
G
= argmaxXiCg
m
or
is
™->u M (x t ). Generally,
,
uM ^) ^ uM (^),
and
e Gt{-^r5 ) and x t
=
m
otherwise,
M
chooses the optimal policy for
t
m $ Gt(-tt)
if
I
{
if
candidates. Thus, any voting strategy p t
all
fix p*t as follows:
p*(w, gt b t xl)
where x t
constant with respect to the
is
within Gt(— 1^5)-
If
of
,
we allow
G
wi G Gt{— 335),
m; otherwise, she must choose xt which by the single-peakedness
or g t (—-j^g) (though
occur only
p£(-) requires that re-election
uM
must choose
t
either #.(— j^fj)
is
P
re-election in the out of equilibrium event that
chooses a
better policy).
Now
consider
G
least -j=g to induce
construction of
If
m
g"
t
's
P
Gt{-)->
to choose
she
is
Gt{—YZg), then for
Y^, so Gj does not
chooses x t
b t (x t )
if
>
Given
incentives.
and
^ rn
xl,
P
x t G Gt{— jt^) \ %u
e
>
0.
-^
such xt
for
^
xt
b t (x t ) for all
Letting
Next, given pi and
m
e
—
j3*
>
,
=
x t Thus
.
bt (x t )
G
optimally
t
if
=
otherwise.
Thus her strategy
payoffs for
£s
6 £?t(— j=j)
^t(— ^), then
If
To prove
m
is
P
receives
P
some gt such that
Assume without
.
xt
=
e^
offer at
— u Gt (m) <
)
,
^
=
Gt
(g t )
ui
u
Gt
-=^ by
=
0.
(m)
<
b t (x t )
—u
Gt
(x t ),
and P
offers:
xt
contract.
-^ +
receives
^+
e
for
e
for
choosing
m
and zero
choosing x £ and either zero or
-^
xl as specified.
M. Note that by Comment
m
(<) gt
must
t
optimality, suppose that there exist stationary equilibria £'s with strictly higher
there exists
>
t
G
otherwise,
v s (w)
1,
constant with respect to w, and that in
is
M receives her ideal policy whenever m € Gt{~iZg)-
Since
= argmax
Clearly, x t
.
we obtain the optimal
0,
if
otherwise.
g"
then
n °t re-elected and u
is
1
some
1^5)1
any policy other than m. But since u Gt (g
P*(w,gt )
for
^ Gt{~
unwilling to do so and thus optimally offers {3*(w, gt )
all
offer
pi
Xt(. w ,9t^t)
m
g"
Gt{—
1^5); Xti
loss of generality that
— 1^5)1 u
£t(
\Xt{ w
*
Sti^t))
30
Hence
in £'s for a
w >9t>bt) >
5ft
> m,
< — ^r6
-
type-w incumbent
9t{~\Z^) (< 9 t {~ 1^5))
so that Xt( w
i
9t->
Let Gt replace
&t)
its
>
f° r
9t(~iZ$)-
equilibrium
contract with:
^+
= —
b t (x t)
P's dominant policy choice
is
gt
x
e if
,
—
t
G
and
gt
and zero otherwise.
,
— ^=j.
receives
t
Then, letting
Thus, X*{ w >9t,b
t
e
—
>
0,
cannot be an
)
equilibrium policy: contradiction.
We
Proof of Proposition
3.
as a cooperative phase.
Suppose otherwise;
proceed in two steps. First, we establish the optimality of £c
M
that
i.e.,
attains higher discounted expected
utility under another set of cooperative phase strategy profiles £.
voting strategy under
It is clear
£ and
,
an d {Xt(
let {/?«(•)}
that to be a cooperative phase,
-
)}
{/?*(•)}
denote
the contract p
is
(-)
=
Pt{ht,
stationary because
•)
f>t{h't
,
for all
=
•)
t
)
Pt{h",
straightforward to verify that since
H
ht £
J3t (-)
phase, they are also best responses to p
t
must
be best responses each
of the contracting
H
G
,
and
(-)-
strategies, respectively.
in the cooperative phase.
for all h't h"
c
t
t
Xt(') are best
Since
J5t {-)
and
Let {p £ (-)} denote M's
and P's
and {Xt(')}
other as well as to (p t (-)}. Suppose that in each period
c
G
game
M offers
This contract strategy
in the cooperative phase.
It is
responses to p t (-) in a cooperative
Xt(')
cannot induce a punishment
phase (by assumption), they induce the same continuation games under p c (-) and
Pt(-).
c
Therefore, the policies realized under the stationary contracts {p (-)} are identical to those
under £
for all ht
6
H
t
.
This contradicts the optimality of £c in the contracting game.
Second, we show that trigger strategies of the form (£c £0l n) constitute a subgame perfect
,
equilibrium for sufficiently large n. In the punishment phase, no action can affect the duration
of the phase.
For any period
between electing and
as in (1),
/?*(•)
Now
in
and
in a
punishment phase, given Xt
re-electing, so chooses p£(-)
Xt(')
=
9t
,
=
Given
0.
(")
p£(-)
=
=
0,
9t>
M
is
indifferent
a bribing strategy
are clearly Nash.
consider the cooperative phase. Because
£c we need check only
re-elect
t
that
M
/?*(•)
and
Xt(') are best responses to {pt(:)}
casts re-election votes in accordance with
an incumbent in equilibrium
£c
.
M
will not
if:
n-l
/ v c {w)u{o\w)
Jil
Since u M (x t (w,gt ))
> u M (g
> J2 &EU
7
t )
(9t)
+
&
n
v c {w)uj{dw).
(15)
=
trivially, this
condition holds. Next,
31
M re-elects an incumbent
in equilibrium
if:
n—l
> Y,
v c (w)
Clearly,
under £c
v c (w)
,
>
$ J Vu
J
Y^JLo $ Eu
r.
M
M
{g t )
+
5
n
w
(g t ) for all
(possibly infinite) such that (16) holds for
n >
all
vc (w)u(dw).
/
6
For
n*.
Thus, there exists some n*
fi.
all
(16)
such n, £c
sustainable as the
is
cooperative phase of a trigger equilibrium.
Proof of Proposition
>
such that vt{w',9)
4.
Suppose otherwise. Then there
(i)
incumbent
x wdg denote the policy chosen
for
offers
-/
a
//
x
9,
,
gt,x)
f
=
< w")
G
an arbitrary P and
t
ideal point g t
Suppose that
.
1
if
M
x
=
x*(w',9,gt ) and p*(w',
9,
gt ,x*w
,
*
for all 9, gt
=
)
y
<
,
Let
.
and
x,
1
(17)
otherwise.
[
is,
(w'
the type-io" incumbent the following contract:
plw
That
when G* has
choosing policy x
for
t
M
fi,
Let p*(w,9,gt ,x) represent the optimal voting contract
vt{vj",9).
offered to a type- (w, 9)
w',w" €
exist
incumbent
re-elects the type- (w", 6)
and only
if
if
she chooses the equilibrium
policy of a type-(w', 9) incumbent, and the type-(u/, 9) incumbent
is
offered re-election ac-
cording to her optimal contract.
It is clear
that under
/>(),
the type-(u/', 9) incumbent
incumbent
is
not re-elected. In these cases
thus obtaining x t
=
gt
(w', 9)
Now
x w'
of
l
e,g t
w
can write a contract of the form
consider cases in which a type-(u/, 9) incumbent
=
^~ e l0^w',i)w" >
Ya=b
0,
receives zero.
G may
t
some contract
u Gt (x)—w"—5l w »
t
e
is
where T w
>
t
e
=
{g
is
\
P*i
By
re-elected.
incumbent assures
9)
group types such that a type-(w',9) incumbent
offering
t
not re-elected whenever a typein (1),
.
whenever possible, the type- (w",
n S
G
is
an expected payoff
herself of
w \8,g, x w>,e, g =
)
re-elected.
By
choosing policy
is
1}
,
induce the type-{w" 9) incumbent to deviate from x^,
,
>
b(x) satisfying b(x)
> u Gt {x*w
,
incumbent to achieve policy
e
x,
).
But then
where w
l
>
expected lifetime payoff. Since w' + 5l w
g
t
g
>
t
G
=
w"-\-5l w n
t
g for
some x
^
x*w
could have offered w'+5l w
l
YlJ=e
fi
~e
l{F w ',i)w'
< w" + 5l w" e,
t
32
is
^ ne set
not choosing x*w 6gt
>
t
e
,
g
g
,
°f
M
only by
This implies
.
to the type-(u/, 9)
a type-(w', 9) incumbent's
this implies that x*w
,
e
could not
have been chosen by a type-(u/,
9)
incumbent. Thus x*w
,
feasibly implementable
is
g
by a
type-(u/', 9) incumbent.
We
conclude that in an optimal contracting equilibrium, a type-(io",
be induced to choose policies at least as good as those chosen by a
<
Thus, vt(w',9)
vt{w,
This proof
9").
./
=
X
v c [w) for
part
all
(ii),
1
J
vt{w,9)
if
for a type-(io, 9)
X
=
=
X *w,8",,ygt
.
.
,
.
T},
but with
aild
[9'
<
9")
such that vt(w,
<
9')
<
p(-) redefined as follows:
P*{w,e",gU X*
g,,
'
y
)
=
1
(18)
'
vt(w, oo) for any w. Since vt{w,oo)
Let F Wt e
5.
incumbent
for all 9 (1
when r =
0,
{1,
(i),
=
{g
<
is
is
p*(w,9 g,xl j
)
1.
then Tw ^i
=
v c (w), vt{w,9)
<
g)
The
=
1} denote the set of group
ex ante re-election probability
claim that in an optimal equilibrium for
M,
T).
Suppose that F w j
C T w ^2
J
re-elected.
We
then j(T Wi e).
<
9
|
proceed via induction on the set {T Wt e
for all 9
IVa-i
6
otherwise.
types such that a type-(u;, 9) incumbent
We
9"
w£fl. I
Proof of Proposition
rVe C r^^a
9',
<
[
By
exist
identical to that in part
is
p(w, Q,
9 ,g t ,x)
(hi)
incumbent.
type-(io', 9)
vt(w",9): contradiction.
Suppose otherwise. Then there
(ii)
incumbent can
9)
trivially
\
T—t <
C F Wt g-i
and the
<
9
for all 9
result
is
T}.
Clearly,
and some t
(1
T w g C T w ^i
<
<
t
established. Otherwise,
by
T).
If
(9), for
any g € r w> g-i, the induction hypothesis implies:
u M (x(w,9-l,g)) + SvT (w,9)>u M (g) +
5
f vT (w, l)<j(dw\.
(19)
Jo,
To show that a
to establish that
type-(iu,
9—2) incumbent
is
re-elected
when G
t
is
of type g,
it is
sufficient
M
g)) + 5vT (w, 9-1) > u {x(w, 6-1, g)) + 5v T (w, 9). By
> vt(w, 9). Thus it is sufficient to show that u M (x(w, 9—2, g)) >
u M (x(w, 9-2,
Proposition 4(h), Vt{w, 9—1)
u M (x(w,9 — l.g)). By
(8), this is
Expanding terms,
is:
this
true
if
w + 5l Wi &-i > w + 5l w
or equivalently:
l
w ,e-\
>
l-wfi-
t-0
T-e+i
£
fi,
«5*«J7(r„,jH. fl -i)
k=o
>
£
fe=o
33
k
wMe ),
5 ury(r
(
20 )
which holds
if
7(rWi +^_1 ) >
Proof of Proposition
proof of Proposition
We
show that
^ and
F Wi g-i C Yw
hypothesis. Therefore,
<
~/(T Wi ktf) over
fc
k
< T—9.
This
is
implied by the induction
the induction hypothesis holds for r+1.
Optimality follows from an analogous argument to that in the
6.
3.
form (£t,£o,ti) constitute a subgame perfect
trigger strategies of the
equilibrium for sufficiently large n. In the punishment phase, no action can affect the duration
of the phase.
For any period
between electing and
/?*(•)
Now
=
M
9t are clearly
an incumbent
that
M
in equilibrium
/ vT (w, l)u(dw)
M (x
incumbent
t
(w, gt ))
=
> u M {g
t)
in equilibrium
Given
p*
=
(•)
t
9ti
0,
M
is
indifferent
a bribing strategy
Nash.
and
j3t {-)
Xt(') are best
responses to {pt{-)}
M will not
if:
>
y
S J Eu
M
{g t )
+
5
n
trivially, this
lWdw).
f vT (w,
(21)
Jn
j^
condition holds. Next,
M
re-elects a type-(u;,#)
if:
n—1
vT (w, 9 + 1)
>
V
5 J Eu
M
(g f)
+
5
under £?, Vt(w, 6+1)
>
Yl'jLo
^^ uM {9t)
n* (possibly infinite) such that (22) holds for
all
^
n
/
Jn
,-=o
Clearly,
0.
=
Xt(')
casts re-election votes in accordance with &r-
Jn
Since u
chooses p*t (•)
consider the cooperative phase. Because
we need check only
re-elect
punishment phase, given
in a
re-electing, so
as in (1), and Xt(')
in St,
t
for all
n >
w
n*.
vT (w, l)w(dtw).
G
Cl
For
and
all
6.
Thus, there exists some
such n, strategy profiles in
£t are sustainable as the cooperative phase of a trigger equilibrium.
34
(22)
REFERENCES
Austen-Smith, David. 1987. "Interest Groups, Campaign Contributions, and Probabilistic
Voting." Public Choice 54: 123-139.
Austen-Smith, David, and Jeffrey
S.
Banks. 1989. "Electoral Accountability and Incum-
bency." in Peter Ordeshook (ed.), Models of Strategic Choice in Politics.
Ann
Arbor:
MI, University of Michigan Press.
S., and Rangarajan Sundaram. 1993. "Adverse Selection and Moral Hazard
Repeated Elections Model." in W. Barnett et al. (eds.), Political Economy: Institutions, Information, Competition and Representation. New York: NY, Cambridge
Banks, Jeffrey
in a
University Press.
Banks, Jeffrey
lems."
S.,
and Rangarajan Sundaram. 1998. "Optimal Retention
in
Agency Prob-
Journal of Economic Theory 82: 293-323.
Barganza, Juan Carolos. 2000. "Two Roles for Elections: Disciplining the Incumbent and
Selecting a
Competent Candidate." Public Choice
105: 165-193.
Baron, David P. "Electoral Competition with Informed and Uninformed Voters." American
Political Science
Barro, Robert.
14:
Review
1973.
88: 33-47.
"The Control
of Politicians:
An Economic
Model."
Public Choice
19-42.
Besley, Timothy.
2003.
"Principled Agents? Motivation and Incentives in Government."
Unpublished manuscript.
Besley, Timothy,
and Michael Smart. 2001. "Does Tax Competition Raise Voter Welfare?"
Unpublished manuscript.
Calvert, Randall L. 1986. Models of Imperfect Information in Politics. Chur, Switzerland:
Harwood Academic
Publishers.
Denzau, Arthur T., and Michael C. Munger. 1986. "Legislators and Interest Groups: How
Unorganized Groups Get Represented." American Political Science Review 80: 89-106.
Dutta, Prajit K. 1995.
Theory
66:
"A Folk Theorem
for Stochastic
Games."
Journal of Economic
T32.
Faure-Grimaud, Antoine, and David Martimort.
2003.
"Regulatory Inertia."
RAND
Journal of Economics 34(3): 413-437.
Fearon, James D. 1999. "Electoral Accountability and the Control of Politicians: Selecting
Good Types Versus Sanctioning Poor Performance."
In Bernard Manin,
worski, Susan Stokes, (eds.), Democracy, Accountability
bridge:
Cambridge University
Press.
35
Adam
and Representation.
Prze-
Cam-
Ferejohn, John. 1986. "Incumbent Performance and Electoral Control." Public Choice 50:
5-25.
Grossman, Gene, and Elhanan Helpman. 1994. "Protection
Review 84: 833-850.
for Sale."
American Economic
Grossman, Gene, and Elhanan Helpman. 1996. "Electoral Competition and Special
est Politics." Review of Economic Studies 63: 265-286.
Grossman, Gene, and Elhanan Helpman. 2001. Special Interest
MIT
Politics.
Inter-
Cambridge,
MA:
Press.
1993.
"Economic Policy, Economic Performance, and Elections."
American Economic Review 83: 27-42.
Harrington, Joseph.
Hindriks, Jean, and Paul Bellenamme. 2001. "Yardstick Competition and Political
Problems." Queen
Mary and
Agency
Westfield College, Department of Economics Discussion
Papers, No. 444.
Eric, and Ben Lockwood. 2001a. "Do Elections Always Motivate Incumbents?"
Warwick Economic Research Paper No. 580.
Le Borgne,
Le Borgne, Eric, and Ben Lockwood. 2001b. "Candidate Entry, Screening, and the
Budget Cycle." Unpublished manuscript.
Lopez, Edward
J.
"Congressional Voting on
2002.
Term
Political
Public Choice 112(3-4):
Limits."
405-431.
Martimort, David. 1999.
"The
Life Cycle of
Regulatory Agencies."
Review of Economic
Studies 66(4): 929-947.
Peltzman, Sam. 1976. "Toward a More General Theory of Economic Regulation." Journal
of
Law and Economics
19: 211-240.
Persson, Torsten, and Guido Tabellini.
Policy.
Cambridge,
Rogoff, Kenneth.
1990.
MA: MIT
2000.
Political
Economics: Explaining Economic
Press.
"Equilibrium Political Budget Cycles."
American Economic Re-
view 80: 21-36.
Rogoff, Kenneth, and
Anne
Sibert.
Review of Economic Studies
Salant,
David
lation."
J.
55:
1988.
"Elections and Macroeconomic Policy Cycles."
1-16.
1995. "Behind the Revolving Door:
RAND
A New View
Schlozman, Kay Lehman, and John T. Tierney. 1986.
Democracy.
of Public Utility
Regu-
Journal of Economics 26(3): 362-377.
New
York: Harper and Row.
36
Organized Interests and American
2003. "Does Democracy Work? Estimating Incenand Selection Effects of U.S. Gubernatorial Elections, 1950-2000." Unpublished
Smart, Michael, and Daniel Sturm.
tive
manuscript.
Smart, Michael, and Daniel Sturm.
2004.
"Term Limits and Electoral Accountability."
Unpublished manuscript.
Stigler,
2:
George. 1971. "The Theory of Economic Regulation." Bell Journal of Economics
3-21.
37
MIT LIBRARIES
3 9080 02618 3209