Voting Rules in a Changing Environment
Antoine Loepery
Wioletta Dziuda
September 2014
DRAFT
Abstract
We identify a novel distortive e¤ect of supermajoritarian voting rules such as quali…ed majority, …libuster or unanimity. Under such rules, di¤erent voters become pivotal under di¤erent
status quos. If policies are continuing in nature and a changing environment creates the need for
renegotiation, legislators distort their voting behavior in favor of alternatives that make them
pivotal in the future. As a result, players disagree more often, and the policy is less responsive
to the environment. We show that the voting distortions are bigger, the greater the supermajoritarian requirement. The distortions generated by supermajoritarian requirements are larger
when the environment is volatile.
In the last decades, multiple U.S. states have passed constitutional amendmends that require
a quali…ed majority to increase taxes. Arguably, these amendments are designed to reduce
spending and tax levels. Our results imply that such amendments can severely limit the states’
ability to adjust taxation throughout the business cycle. We also show that they can even fail
to achieve their primary goal of keeping taxes low.
JEL Classi…cation Numbers: C73, D72, D78
Keywords: Dynamic voting, supermajoritarian voting rules, quali…ed majority, …libuster,
checks and balances, continuing policies, polarization, policy inertia.
y
Kellogg School of Management, Northwestern University. Email: [email protected]
Universidad Carlos III de Madrid. Email: [email protected]
1
1
Introduction
The …libuster has come lately under heavy criticism. Commentators and pundits have argued
that with a highly polarized Congress, supermajority requirements lead to policy inertia and legislative gridlock. Such inertia prevents Congress from reacting in a timely fashion to a volatile
environment.1
Even though supermajority requirements frustrate proponents of reforms, it is not clear whether
their costs outweigh their bene…ts. They prevent e¢ cient reforms, but they also prevent ine¢ cient
ones, and the total e¤ect is ambiguous. Economists and political scientists have predominantly
found that supermajority requirements and veto power have desirable properties. They reduce the
likelihood of irrational collective decisions (Condorcet 1785), can mitigate dynamic commitment
problems (Gradstein, 1999; Messner and Polborn, 2004), and most importantly, prevent policy
makers from implementing laws that bene…t only a bare majority, and force them instead to internalize the interests of a greater share of the population (Buchanan and Tullock, 1962; Riker,
1962).
In this paper, we provide a novel strategic rationale for the suboptimality of supermajority
requirements and veto power. We consider policies that are continuing in nature but require
periodic revisions in response to exogenous shocks. In such dynamic settings, supermajoritarian
rules implicitly tie the future bargaining power to the current policy, creating incentives to distort
voting behavior. We show that these distortions make legislators appear more polarized than they
are, and hence lead to excessive policy inertia. In other words, it is not polarization that renders
…libuster ine¢ cient; it is the …libuster that leads to a polarized behavior.
To see the intuition for this rationale, consider legislators that are ordered on the ideological
spectrum. When voting, each legislator takes into account not only how adequate the policy is to the
current environment, but also how it a¤ects the identity of the pivotal player in the next periods.
Under simple majority, the same median legislator is pivotal independent of the current status
quo. Hence, when voting, the median legislator considers only the policy’s adequacy to the current
environment, because she expects to be pivotal in the next periods irrespective of the outcome of
1
A …libuster is a parliamentary procedure where one or more members can delay or entirely prevent a vote on
a given proposal by extending the debate. In the current political environment, the …libuster e¤ectively means that
most major legislation requires a 60% vote to pass.
2
the vote today. Under a supermajoritarian rule, the identity of the pivotal player depends on the
policy change under consideration: If the status quo is to the left of the proposed policy change,
the pivotal player is more leftist than the median legislator; if the status quo is to the right of the
proposed policy change, the pivotal player is more rightist than the median legislator. Therefore,
in order to increase the probability of being pivotal tomorrow, relatively leftist (rightist) legislators
prefer leftist (rightist) status quos. These incentives induce legislators to vote in a polarized way,
and hence lead to more disagreement and status quo inertia.
We show that the resulting polarization is bigger, the greater the supermajoritarian requirement
is. Hence, increasing the required supermajority leads to more political inaction not only because
policy reforms require the approval of a greater number of legislators, but also because all legislators
behave as if they were more polarized.
Understanding the e¤ects of quali…ed majorities on policymaking is of great importance as many
decision bodies operate under such voting rules, and consider or have recently implemented changes
to these rules. For example, the EU requires unanimity or quali…ed majority for decision making
in the European Council, and generally requires an agreement of the Council and the Parliament.
Since the Single European Act, the scope of the voting procedure for unanimity has become more
restricted. The Lisbon Treaty further increases the number of areas where quali…ed majority voting
in the Council will apply. Our results provide yet another rationale for such changes.
In the last decades, several U.S. states have passed constitutional amendments that require a
quali…ed majority to increase taxes.2 These amendments were designed to reduce spending and tax
levels.3 Between 1996 and 1999, the US Congress has voted every year on an amendment to require
two-thirds supermajority in order to increase taxes.4 Our model implies that such amendments, by
leading to excessive polarization, can severely limit the states’ability to adjust taxation throughout
the business cycle. More surprisingly, we show that they can even fail to achieve their primary goal
of keeping taxes low: In our dynamic setting, the policy change that requires higher majority may
actually become more prevalent.
We show that the distortions created by the supermajoritarian requirements are larger when the
2
See Table 1 in Section 6.
If the preferences of the society coincide with the median voter, one should not expect the society to introduce
measures to curb taxes. Gradstein (1998), Messer and Polborn (2004) argue that quali…ed majority can be adopted
by a median voter as a commitment device not to raise taxes in the future.
4
These attempts failed.
3
3
environment is more volatile. Intuitively, in a volatile environment, the current policy is likely to require a revision next period; hence, securing pivotality for the next period becomes more important
than implementing an adequate policy given the current circumstances. This result provides a new
rationale for endowing the governments with power to introduce state of emergency during which
the usual legislative procedures are suspended and the decision power becomes concentrated. This
is also in line with the rules of the EU. The EU requires unanimity for membership of the Union,
harmonization in the …eld of social security and social protection, the common foreign and security
policy, the granting of new rights to European citizens, anti-discrimination measures, and certain
institutional issues such as the electoral system and composition of the Parliament and committees.
Arguably, the stance of the member states on these issues is unlikely to change rapidly. On the
other hand, the EU requires smaller quali…ed majority for other regulatory areas that are arguably
more a¤ected by recurring shocks, such as immigration, crime prevention, transport, structural and
cohesion funds, the EU budget.
In purely democratic systems, the decision process is based on the simple majority rule. However, in modern democracies, bills and policy reforms have to pass several additional institutional
hurdles to be enacted. These checks and balances can take many forms: presidential vetoes, concentrated power to set agenda, committee approval, bicameralism, the possibility of public initiatives,
or judicial review by a constitutional court. Their original intent is to limit the concentration
of power by increasing the set of individuals whose approval is necessary to enact policies. Our
analysis highlights a detrimental, unintended consequences of checks and balances: by changing the
identity of future pivotal players, they induce decision makers to behave in a more polarized way,
and thus prevent them from implementing Pareto improving reforms. In Section 5.4 we discuss
institutional arrangements that can mitigate the polarizing nature of checks and balances.
The existing literature on voting rules considers mostly static environment, and hence identi…es
their static properties. In contrast, we focus on a dynamic environment in which decision makers’
preferences are a¤ected by shocks. This is arguably relevant, as the preferences of the legislators
are stable in few policy areas. For example, legislators’ preferences over …scal policies re‡ect
heterogeneous ideologies and constituencies, but they also depend on the business cycle, changes
in the country’s credit rating, demographic trends, or the vagaries of public opinion. Similarly,
citizens’and legislators’preferences over civil liberties and privacies are a¤ected by shocks such as
4
terrorist attacks and national security threats.
The paper is organized as follows. Section 2 discusses the related literature. Section 3 describes
the basic model. Section 4 characterizes the general properties of all equilibria. Section 5 shows
how these properties vary as we vary voting rules and volatility of the environment. Section 5.4
concludes. All proofs are in the appendix.
2
Related literature
Studying the properties of various voting rules has been a subject of Public Choice for a long
time. The literature has found that various rules may be optimal depending on the considered
environment as well as the assumed optimality criterion. Quali…ed majorities seem to dominate,
however, in a broad class of problems.
Since Condorcet (1785), the social choice literature has argued that majority rule may note
aggregate heterogeneous preferences in a rational way (Plott 1967). Caplin and Nalebu¤ (1991)
show that for a broad class of preferences distribution, a quali…ed majority can solve the Condorcet
paradox (see Enelow 1997 for a survey of that literature).
Voting rules have been also analyzed with respect to their information aggregation ability (see
Grofman and Owen, 1986, for a summary). Austen-Smith and Banks (1996) show that strategic
voters may not vote sincerely. As a consequence, Feddersen and Pesendorfer (1998) show that
unanimity rule can have perverse incentives. Coughlan (2000) argues, however, that unanimity can
be optimal if communication among jurors is allowed and jurors have similar preferences.
Another strand of literature focused on the properties of the implemented policy changes. Wicksel (1896) has advocated that since governments are formed to bene…t the citizens, unanimity should
be used to assure that all changes are Pareto improving. Similarly, Buchanan and Tullock (1962)
single out unanimity as the only rule that guarantees Pareto improvements. Rae (1975) points out
that under unanimity the societies may be stuck with an ine¢ cient policy as long as such policy
bene…ts one player. Using a similar argument, McGann (2004) argues that simple majority o¤ers
the best protection to minorities as it makes it the easiest to overturn minority-harming policies.
Guttman (1998) points out that under unanimity, projects that give large bene…ts to all but a few
will be vetoed (see also Rae 1969), which is socially detrimental. In response, Tullock (1998) argues
5
that in the presence of transfers, all socially bene…cial changes will be undertaken even under unanimity. Arrow (1998) counters that this result holds only under the assumption that distributive
policies can violate the generality of treatment of taxpayers. However, Buchanan (1998) argues that
rent-seeking costs would again most likely tilt the verdict in favor of more inclusive decision rules,
as such rules minimize the resources wasted on assuring one’s position in the winning coalition.
Supermajoritarian requirements have been advocated also on the basis of protecting minority
from "the tyranny of majority" (a view associated with James Madison and Hayek 1960). When
distributive policies are considered, majority rules create incentives to particularize the bene…ts
and collectivize the costs, and the extent of that is higher the lower the required majority. As
a result, ine¢ cient distributive policies can be adopted (Buchanan and Tullock 1962, and Riker
1962). Ferejohn, Fiorina, and McKelvey (1987) and Baron (1991) …rst formalized this prediction
in models of legislative bargaining. Aghion, Alesina and Trebbi (2004) analyze a model in which
higher supermajority means that any reforms: bene…cial as well as harmful are more easily passed.
They show that a higher quali…ed majority is better when the probability of expropriation and the
cost of it are higher, the bene…t from reforms is small, the distribution of bene…ts is less polarized,
and players are more risk-averse. Aghion and Bolton (2003) show that when optimal contracts are
possible, unanimity is the best rule. When they are not and when transfers are ine¢ cient, smaller
majority may be optimal. The reason for this is that a smaller number of loser-voters need to be
compensated in exchange for their vote, which restricts the deadweight loss from transfers.
May (1952) shows that the majority rule is the only pairwise decisive and responsive voting
rule that satis…es anonymity (all voters treated equally) and neutrality (all alternatives treated
equally). With any supermajority rule, the status quo is privileged, and since the status quo is
favorable to some voters, these voters are privileged by a supermajority (Rae 1975). The literature
has long recognized that under higher supermajority requirements fewer changes can be adopted
(see Tsabellis 2002) as any change requires an approval of a larger set of voters. That does not
mean, however, that supermajority is suboptimal; the higher the supermajority is, the more e¢ cient
the adopted policies are, and under all voting rules Pareto e¢ cient changes are implemented. In
contrast, we show that policy inertia of the supermajoritarian rules is exacerbated by players’
dynamic concerns, and such voting rules will result in Pareto e¢ cient policies not being adopted.
Persico (2004) and Hastard (2005) compare voting rules in environments in which voters make
6
costly investments in the pre-voting stage (in information and preparedness for reforms respectively). Hastard (2005) shows that higher supermajority requirements lead to more underinvestment; and hence, fewer changes being undertaken. The rationale for this is that higher majority
requirement makes each party’s approval more likely to be needed for the change. Hence, parties
that have invested heavily will be willing and forced to compensate those who did not for the vote
in favor of the change. This decreases the incentives to invest. Persico (2004) shows that a large
quali…ed majority can be optimal only if the information available to each committee member is
su¢ ciently accurate. When individual information is noisy, however, such rules discourage the
committee members’from acquiring information: When information is noisy, it is unlikely that all
remaining members of the committee vote exactly in the same way; and hence, that the vote of the
voter in question is pivotal.
Our paper analyzes a di¤erent environment than the aforementioned literature. In our paper,
the preference domain is restricted so that all considered voting rules are decisive. We consider
common interest policies with no transfers; hence, the question of tyranny of majority is mute. All
information is public, and there are no investments. Instead, we focus on a dynamic environment,
where an implemented change may need to be revised later in response to an exogenous shock.
There is a strand of literature which recognizes that a quali…ed majority can serve as a commitment tool. In Gradstein (1999), by adopting a quali…ed majority, the society de facto delegates the
decision about taxes to voters that prefer lower taxes than the median voter does, herby committing
itself to smaller future tax increases. Messner and Polborn (2004) assume that older voters prefer
lower taxes than the younger ones. In such a setting, a median voter may prefer to adopt a quali…ed
majority as the decision rule for the future, as such a voting rule assures that she is pivotal when
she becomes older. In contrast to that, in our paper legislators would like to decrease the quali…ed
majority. Moreover, contrary to the aforementioned literature, in our model, the voting rule a¤ects
not only who is pivotal, but also how voters vote. In fact, the reason why simple majority rule is
better is entirely due to this second e¤ect.
Our paper builds upon our previous work. In Dziuda and Loeper (2014), we analyze a twoplayer model with unanimity. We show that players become polarized, interpret this polarization as
partisanship, and we focus on the magnitude of the resulting status quo inertia and its robustness.
In this paper we focus solely on the relationship between voting rules and voting distortions. Hence,
7
we work with a multi-player model. We show that the e¤ect identi…ed in our previous work does
not extend to quasi-dictatorial voting rules. We also tie the magnitude of the distortions generated
by non quasi-dictatorial rules to the volatility of the environment.
Another related paper is Eraslan and Merlo (2002), who analyze a divide the dollar game in
which the size of the size of the surplus to be allocated varies stochastically over time. They
show that majorities lower than unanimity may lead to ine¢ cient agreements: a current winning
coalition may prefer to divide the current surplus even though waiting for a bigger surplus would
be bene…cial because they fear that they lose their place in the winning coalition next period.
Our paper is also related to Roberts (1999), which studies a dynamic model of club formation.
Even if they prefer a larger club, the current club members may vote against enlarging it out of fear
that they will lose the decision power over subsequent enlargements (See also Acemoglu, Egorov,
and Sonin, 2014). In our paper, the legislators veto Pareto improving policies our of fear that they
lose the decision power over subsequent changes.
3
The model
A set of legislators N = f1; :::; n; :::; N g are in a relationship that lasts for in…nitely many periods.
In each period, the legislators must decide which of the two alternatives L and R to implement.
We will interpret L as a liberal policy such as a high tax rate or high government spending, and R
as its conservative counterpart.
The utility of legislator n 2 N from implementing alternative R in some period t 2 N is
n (t) ;
and the utility from alternative L is normalized to 0; without loss of generality. Hence, if
n (t)
is positive (negative), legislator n prefers the conservative alternative R (the liberal alternative
L) in period t: We refer to
(
1 (t) ; :::; N
n (t)
as legislator k’s current preference in period t, and to
(t) =
(t)) as the state of nature in period t:
We assume that in each period, (t) = (t
1) with probability p; and with probability 1
p;
(t) is drawn from a distribution P ( ) : Parameter p measures the persistence of the environment:
the higher p; the larger the persistence, or equivalently, the smaller the volatility. We assume that
has a …nite expectation.
8
Assumption 1 For all t 2 N;
1 (t)
legislators n; m 2 N ,
m (t)
n (t)
and
< :: <
N
(t) with probability one, and for any two distinct
are of opposite sign with positive probability.
Assumption 1 has a natural interpretation in political economy: legislators can be unambiguously ranked on the ideological spectrum. Speci…cally, legislators with lower indices are more leftist
than legislators with higher indices. This means that whenever a more leftist legislator disagrees
with a more rightist legislator, then the former prefers L and the latter prefers R: Note, however,
that this assumption imposes no restriction on the preference distribution of a single legislator nor
on the severity of the con‡ict of interest between legislators: all legislators might prefer the same
policy arbitrarily often.
In each period t; a status quo q (t) 2 fL; Rg is in place. First,
(t) is realized and observed by
all legislators.5 Then legislators choose which alternative to implement in that period using some
voting rule
; which we describe later. The implemented policy determines the legislators’payo¤s
in this period and becomes the status quo for the next period q (t + 1). Each legislator maximizes
her expected discounted sum of payo¤s, and
game by
> 0 is the discount factor. We denote the above
( ).
Voting rules
A voting rule
is characterized by a pair of collections of winning coalitions (
L;
R) ;
which
determine the outcome as follows: if the status quo is L (R) in a given period, then it is replaced
by R (L) if and only if the set of legislators who vote for R (L) in this period is an element of
(
R ).
L
We impose the following conditions on the voting rule.
De…nition 1 A pair of collections of winning coalition
if for all q 2 fL; Rg,
q
=(
L;
R)
2N
2N is a voting rule
satis…es the following:
(i) Monotonicity: if C 2
q
(ii) Nonemptyness: N 2
q;
and C
(iii) Properness: for q 0 6= q, if C 2
C 0 , then C 0 2
q,
then N n C 2
=
q,
q0 :
Conditions (i) to (iii) are standard in the voting literature (see, e.g., Austen-Smith and Banks
2000). Monotonicity ensures that if a coalition can impose R; then so can a larger coalition.
5
own
This assumtion is without loss of generality: in equilibrium, each legislator conditions her action only on her
k ; hence, whether she observes other legislators’preferences is inconsequential.
9
Nonemptyness ensures that the voting rule is Paretian. The properness condition means that if a
coalition can change the status quo, then the legislators outside this coalition cannot reverse this
change.
These conditions characterize a large class of voting rules such as majoritarian voting rules, but
also other nonanonymous, and nonneutral, voting rules. An example of a nonanonymous voting
rule is the combination of a simple majority and a veto legislator v 2 N :
R
=
L
=
C
N : jCj >
N
and v 2 C :
2
Such nonanonymous voting rules are the de-facto rules for many democracies in which the legislative
body uses simple majority but is subject to a veto of the president, or the agenda control of
committees.
An example of a nonneutral voting rule is the simple majority rule when q = R and a quali…ed
majority rule with threshold M >
R
=
C
N
2
when q = L:
N : jCj >
N
2
and
L
= fC
N : jCj > M g :
(1)
Such rules are used by many US states (see Section 6) to legislate taxation: while simple majority
is needed to lower taxes, a quali…ed majority is required to raise them.
Equilibrium
We look for Markov perfect equilibria in stage-undominated strategies (henceforth, equilibria)
as de…ned in Baron and Kalai (1993). A Markov strategy for legislator k in
( ) maps in each
period t the current state (t) and the current status quo q (t) into a probability distribution over
votes. Stage undomination requires that in each period, each legislator votes for the alternative that
gives her the greater continuation payo¤. It rules out pathological equilibria such as all legislators
always voting for the status quo.6 Without loss of generality, we assume that when indi¤erent, a
legislator votes for R:
Comments
Two comments on the modeling assumptions are in order. First, the model used in this paper
6
Stage-undominated Markov perfect equilibrium is the dominant concept in the dynamic voting literature.
10
builds on the two-player model analyzed in Dziuda and Loeper (2014). We extend it to more than
two players to analyze the impact of the voting rule on the equilibrium outcome, which is the main
focus of this paper. We simplify it in other dimensions. Speci…cally, the policy space consist of
two alternatives, and we assume a speci…c form of correlation of players’ preferences over time.
The latter assumption is only for expositional simplicity. The former assumption implies that the
policy making process within each period depends on the voting rule but not on the allocation of
other procedural rights, e.g., amendments rights or proposal rights. Therefore, this simpli…cation
allows us to isolate the impact of the voting rule, independently of the details of the bargaining
procedure.7
Second, our model applies only to continuing policies: policies that do not have an expiration
date and remain in e¤ect until a new agreement is reached. Many legislative decisions have this
feature. For example, about two-thirds of the U.S. federal budget— called mandatory spending—
continues year after year by default. Outside of the …scal sphere, many ideologically charged issues
such as taxes, immigration, …nancial regulation, minimum wage, civil liberties, and national security
are typically regulated by permanent legislation.
4
Equilibrium
4.1
Pivotal voters and the dispersion of power
Since most of the literature on voting is cast in an inherently static setting, we will use such a
setting as a benchmark. Consider hence the game in which legislators play a single period of
( ). In the unique undominated equilibrium of this game, legislators vote according to their
current preferences. That is, each legislator n votes for R whenever
n
0 and for L if
n
< 0;
irrespective of the status quo. Under such behavior, Assumption 1 implies that if a legislator n
votes for R, then so do all more rightist legislators k
n: Therefore, by monotonicity of
; if
the status quo is L; policy R is implemented whenever there exists n 2 N such that
n
0 and
fn; :::; N g 2
n
< 0; and
L:
Conversely, status quo L stays in place if there exists n 2 N such that
7
One can use the technics developed in Dziuda and Loeper (2014) to extend the analysis to a one dimensional
space of alternatives under some reasonable conditions on the bargaining procedure. However, such an extension
would come at a signi…cant cost in terms of technical and expositional complexity, and the main results would be
qualitatively unchanged.
11
fn + 1; :::; N g 2
=
L:
Hence, if legislator n is such that fn; :::; N g 2
L
and fn + 1; :::; N g 2
=
L,
this legislator is pivotal when q = L in the sense that the outcome of the vote coincides with her
preferences. Condition (i) and (ii) in De…nition 1 guarantee that such a legislator exists and is
unique. This leads to the following de…nition.
De…nition 2 For a given voting rule
; the pivotal legislator under status quo L is denoted by
l ( ) and is characterized by:
fl; :::; N g 2
L
and fl + 1; :::; N g 2
=
L:
The pivotal legislator under status quo R is denoted by r ( ) and characterized by
f1; :::; rg 2
R
and f1; :::; r
1g 2
=
R:
For instance, under the unanimity rule, r = N and l = 1, while under the simple majority rule,
when N is odd, r = l =
then f1; :::; l
1g 62
N +1
2 .
R:
Note that condition (iii) in De…nition 1 implies that if fl; :::; N g 2
Together with condition (i) ; this implies that l
L;
r : the legislator pivotal
under status quo L is more leftist than the legislator pivotal under status quo R:8
When
is such that l = r; we shall say that
is quasi-dictatorial. In that case, the equilibrium
outcome of the static game is the same as it would be if the unique pivotal voter were the dictator.
Quasi-dictatorial rules include rules in which one player is a dictator, but also simple majority
when the number of voting players is odd. Similarly, majority voting with the median voter as the
agenda setter is quasi-dictatorial. On the other hand, systems in which the legislature operates
under simple majority but is subject to the presidential veto are not quasi-dictatorial.
The following partial order on the voting rules will be useful in the subsequent analysis.
De…nition 3 Let
under
if
L
0
L
and
0
and
R
be two voting rules. The dispersion of power is greater under
0
than
0 :
R
In words, the dispersion of power increases when changing the status quo requires the approval
of a greater set of legislators. When restricted to majoritarian rules, a voting rule has greater
8
Our de…nition of pivotal legislators follows Krehbiel (1999). Note that l and r are also the lowest and the highest
quasi-median voter of Acemoglu, Egorov and Sonin (2014).
12
dispersion of power if it requires a higher threshold to approve the change. In particular, the
dispersion of power under the unanimity rule is greater than under the simple majority rule, as
under the latter, a bare majority is needed to change the status quo, while under the former, an
agreement of all legislators is required. It is easy to see that when the dispersion of power under
0
is greater than under
, then the pivotal voters are more extreme under
l(
Note that when
0
)
l( )
r( )
r(
0
0
:
):
(2)
is quasi-dictatorial, the dispersion of power under
is minimal because l ( ) =
r ( ), so from (2), any further decrease in the dispersion of power must leave the unique pivotal
voter, and thus the equilibrium, unchanged.
4.2
Equilibrium characterization
Let us turn now to the dynamic game
of
( ) : The following proposition characterizes the equilibria
( ):
Proposition 1 Suppose p < 1: In all equilibria of
( ), legislators use cuto¤ strategies: there
exists c 2 RN such that for all n 2 N ; legislator n 2 N votes for R if
If
n
is quasi-dictatorial, then c = (0; :::; 0) is the unique equilibrium. If
cn and for L if
n
< cn .
is not quasi-dictatorial,
in any equilibrium c;
(i) c1 > :: > cN ;
(ii) cl(
)
> 0 > cr( ) :
Proposition 1 states that each legislator votes for R if and only if her current payo¤ from R is
higher than a certain threshold. The threshold is 0— and hence, there is no distortion in the voting
behavior as compared to the static game— if and only if the voting rule is quasi-dictatorial.
For any other voting rule, each legislator n votes as if there was a single period, and her current
preferences were given by
n
cn instead of
n:
The term
n
captures legislator n’s preferences
over the alternative implemented in a given period, while the term
cn re‡ects her preferences over
the next status quo, given the continuation play prescribed by the equilibrium. Therefore, one can
view the term
n
cn as legislator n0 s intertemporal preferences.
13
When there is some dispersion of power, part (i) states that legislators’intertemporal preferences
are more polarized than their current preferences in the sense that the di¤erence between the
intertemporal preferences of any two legislators is always bigger than the di¤erence between their
current preferences: For all n > m;
(
n
cn )
(
cm ) = (
m
m)
n
+ (cm
cn ) >
n
m:
Therefore, the dispersion of power exacerbates the initial polarization of the legislators. They
strategically vote as if their ideological positions where more spread out than they are.
To see how this polarizing e¤ect of the dispersion of power a¤ects the equilibrium outcome, note
…rst that from part (i),
c satis…es Assumption 1. Therefore, the same reasoning as for the static
game shows that the statically pivotal legislators l ( ) and r ( ) are also pivotal in the dynamic game
( ) when the status quo is L and R; respectively. Part (ii) further states that the intertemporal
preferences of these legislators are biased in favor of the status quo under which they are pivotal.
More formally, in any period t in which the status quo is L and
l( ) (t)
2 0; cl(
)
; legislator l ( )
prefers to stay at the status quo L even though policy R would give her a greater payo¤ in that
period. Similarly, in any period t in which the status quo is R and
r( ) (t)
2 cr( ) ; 0 ; legislator
r ( ) prefers to stay at the status quo R even though policy L would give her a greater payo¤ in
that period. Hence, the dispersion of power creates excessive status quo inertia. The magnitude of
the strategic polarization and the excessive status quo inertia is captured by the magnitude of the
voting thresholds cl(
)
and cr(
)
:
The intuition for Proposition 1 is as follows. Legislator l is pivotal as long as the status quo
stays at L: Therefore, to keep control of the policy in the next period, she is willing to sacri…ce her
current payo¤ and cling onto L even if L gives her a lower payo¤ than R. How much of her current
payo¤ is she willing to sacri…ce to remain pivotal in the next period? Being pivotal matters only if
it leads to a di¤erent outcome than not being pivotal, and this happens only in the states in which
l ( ) and r ( ) disagree. Consequently, as one can see from the proof of Proposition 1, the voting
threshold of legislator n satis…es
(1
cn =
1
p)
p
Z
(cn
f 2RN : l <cl and
14
r
cr g
n ) dP
( ):
From Assumption 1, distinct pivotal legislators disagree with positive probability in every period, which explains why the voting distortions are nonzero whenever the voting rule is not quasidictatorial.
We end this section on a technical note. Polarization in this model feeds on itself: the more r
distorts her behavior, the more likely it is that status quo R stays in place, so the more important
it is for l to defend status quo L; and thus the greater the incentives for l to behave in a polarized
way. As a result, there may be multiple equilibria, which makes it hard to derive comparative
statics. The proposition below, however, states that there exists an equilibrium in which the voting
distortions are the smallest. This result allows us to perform comparative statics on the least
polarized equilibrium in the subsequent sections.
Proposition 2 There exists an equilibrium c 2 RN such that for any other equilibrium c0 2 RN ;
c0r
cr and cl
c0l :
In the results below, for any voting rule
we call it simply the equilibrium.
5
; c ( ) refers to the least polarized equilibrium, and
9
The determinants of polarization
5.1
Dispersion of power
The following proposition shows that polarization, and hence status quo inertia, increase with the
dispersion of power.
0
Proposition 3 If the dispersion of power is greater under
cr( 0 )
If
n
0
cr( 0 ) ( )
cr(
)(
)
0
cl(
)(
)
than under
, then
cl( 0 ) ( )
cl( 0 )
0
:
(3)
is not quasi-dictatorial, if (l ( ) ; r ( )) 6= (l ( 0 ) ; r ( 0 )) ; and if for each n; the distribution of
has full support,10 then all of the above inequalities are strict.
9
There exists also a most polarized equilibrium c such that for any other equilibrium c0 ; cr c0r and c0l
comparative statics for this equilibrium would be the same as for the least polarized one.
10
More precisely, what is needed is that for all a < b; the probability that n 2 (a; b) is strictly positive.
15
cl : The
Proposition 3 states that if a voting rule
0
is replaced by a voting rule
with a greater
dispersion of power, the polarization of the pivotal players increases for two reasons. First, from
(2), more extreme legislators become pivotal. From Proposition 1 part (i) and (ii), we know that if
we …x the voting rule
, legislators that are more extreme than l ( ) and r ( ) are more polarized.
This e¤ect explains the four inner inequalities in (3). Second, the two outer inequalities in (3)
further say that the polarization of a given legislator increases as the dispersion of power increases.
The intuition for the second e¤ect is as follows. As argued in Section 4.2, legislators distort
their behavior in order to remain pivotal in future decisions. Pivotality, however, has value only
in states in which the pivotal legislators disagree. Since the pivotal legislators under
ideologically extreme than the pivotal legislators under
0
are more
; they are more likely to disagree. Hence,
0:
the incentive to distort behavior is higher under
Proposition 3 has important implications for policy inertia. From Proposition 1, the status quo
stays in place independent of its identity if
l( ) (t)
cl(
)(
)
0<
r( ) (t)
cr(
)(
). So as the
dispersion of power increases, status quo inertia increases not only because a reform requires the
approval of policy makers that are farther apart on the ideological spectrum— that is,
r( ) (t)
l( ) (t)
and
move farther apart— but also because policy makers on either side of the political spectrum
are less likely to agree— that is, cl(
)(
) and cr(
)
also move farther apart.
Few pundits and academics dispute that the political polarization in Congress have grown
dramatically in the recent decades (see Table 2). At the same time, a …libuster— once an infrequently used tool reserved for the most important legislation— became one of the central features
of American politics (see Table 3). Political scientists have come forward with various explanations
for the increase in polarization (polarized electorate, gerrymandering, primary elections, economic
inequality, money in politics, media, majority-party agenda control, and party pressures) and argued that an increased polarization increases the use of the …libuster, at the same time making
the consequences of the …libuster more severe (Barber and McCarty 2013, Koger 2010). Proposition 1 suggests that there may also exist a reverse relationship. Since it has became easier to
…libuster— just the threat of the …libuster is su¢ cient to block a bill— legislators understand that
reversing policy changes in the future became harder. As a result, they distort their behavior. As
a consequence, they appear more polarized.
16
Bafumi and Herron (2010) indeed showed that legislators seem to be taking positions that
are even more extreme than their constituency. This observation is consistent with a theory that
there was no change in the ideological position of the legislators, but their behavior became more
polarized in response to the easier use of the …libuster.
The following example demonstrates numerically the e¤ect of increasing dispersion of power.
5.1.1
Example: Increasing the supermajoritarian requirement
Suppose p = 0; that is, preferences are redrawn every period. For each n 2 N ; let
" (t) ; where " (t)
n
N (0; 1). The parameter
n
n (t)
=
is interpreted as the ideological position of the
legislator. That is, the ideological di¤erences between legislators are constant across periods, and
their preferences are a¤ected by the common shock. The shock " can be interpreted for example as
the bene…ts from public spending: when they are high, legislators become more favorably inclined
to increase taxes (implement L), and when they are low, they are more inclined to lower them
(implement R). The bene…t from public spending ‡uctuates in response to the state of the economy,
wars, natural disasters, investors’sentiment, etc.
Suppose that the median legislator’s position is
distributed around the median legislator. Let
m
= 0; and legislators are symmetrically
= 0:9: In this setting, increasing the quali…ed
majority means that the ideological positions of the pivotal legislators become more polarized
r
=
l
$ . Panel A of Figure 1 below demonstrates the voting distortions of the pivotal player l
in the least polarized equilibrium, cl ; as increases (the distortions for player r are symmetric). Note
that as
increases, the polarization of the voting behavior, as measured by
becomes 10 times larger than the polarization of the current preferences
17
r
cl
r
l
:
l
cr
;
Panel B plots the probability that the current policy (be it L or R) stays in place. The dashed
curve depicts this probability for the static game, and the solid curve depicts it for the dynamic
game
( ) : As the supermajoritarian requirement increases, the status quo stays in place more
often even in the static game. However, it stays in place much more often in the dynamic game. For
r
above 0:45; the probability of status quo inertia is almost 1: Hence, we have complete gridlock
in which the initial status quo is replaced only with negligible probability.
5.2
Nonneutral voting rules
Some democratic institutions use di¤erent voting rules depending on the policy change under consideration. For example, in 16 U.S. states, a bill that increases taxes requires the approval of a
quali…ed majority in each house (plus the governor’s signature), while tax decreases can be approved
using simple majority. Similarly, in the U.S. budget process the Byrd Rule requires a …llibusterproof majority to pass bills that raise the de…cit and a simple majority to pass bills that lower
it.11
Arguably, the explicit goal of these voting arrangements was to limit the size of the public
sector and the growth of the public debt. The analysis below shows, however, that they may fail
to achieve their primary objectives.
If we interpret policies L and R as high and low taxes, respectively, then introducing the
nonneutral voting rules described above can be modelled as follows: a simple majority rule
replaced by the voting rule
quo L— i.e.,
0
L
=
L—
0
is
de…ned in (1): It entails the same dispersion of power under status
but a greater dispersion of power under status quo R— i.e.,
Hence, the median player m is always pivotal under
; but under
0;
0
R
R:
she is only pivotal when taxes
are high (q = L) : When taxes are low (q = R) ; the pivotal legislator r is more rightist than the
median legislator.
Clearly, if we …x the legislators’voting strategy, increasing the hurdle to raise taxes has a direct
e¤ect of increasing the likelihood that status quo R stays in place. However, this reallocation of
voting power has a strategic e¤ect as well, as it a¤ects legislators’equilibrium behavior. Proposition
11
The U.S. federal budget process is governed by the Congressional Budget Act of 1974, which prohibits the use of
the …libuster against budget resolutions. This act was amended in 1985 (and later in 1990) by the Byrd Rule to allow
the use of a …libuster against any provision that increases the de…cit beyond the years covered by the reconciliation
measure.
18
3 implies that after the nonneutral rule is introduced, the median legislator will distort her behavior
in favor of higher taxes L: In other words, a …scally conservative voting rule makes the median
legislator less willing to reduce taxes, because she realizes that the the bias in the voting rule makes
it more di¢ cult to increase them in the future. Hence, whether the bias in the voting rule makes
policy R more likely on the equilibrium path depends on the relative strength of the direct and
strategic e¤ect. As the example below illustrates, the strategic e¤ect might dominate. That is, a
…scally conservative voting rule might in fact generate higher taxes.
5.2.1
Example: Increasing the supermajority for tax increases
The distribution of payo¤s is like in Section 5.1.1. Under
, the median legislator m is always
pivotal, and from Proposition 1, legislators vote according to their current preferences. Therefore,
R replaces L when
m
increase is equal to 1
Under
0,
" (t)
0 and L replaces R when
m
(
and that they decrease is
(
m );
" (t) < 0. The probability that taxes
m ):
the median legislator m is still pivotal when taxes are high (q = L) ; so l = m:
However, when taxes are low (q = R), a more rightist legislator r > m is pivotal. First, note that
if legislators continued voting the same way as they do under
; then
0
would indeed increase
the probability that R is implemented in a given period. High taxes L would be still replaced by
low taxes R whenever the median voter preferred lower taxes, but low taxes would be replaced by
higher one less often: with probability 1
( r) < 1
(
From Proposition 1 we know that in the dynamic game
voting behavior. Legislator n will vote for L if
will replace L) when
m
" (t)
n
" (t)
m ):
( 0 ) ; however, legislators distort their
cn : In this case, taxes will decrease (R
cm ; and taxes will increase (L will replace R) when
r
" (t) < cr :
Proposition 1 implies further that cr < 0 < cm : Hence, the probability that taxes increase is
1
(
r
cr ); which is lower than under simple majority 1
(
m ):
However, the probability that
taxes decrease is also lower: it is equal to the probability that the median voter votes for R; which
is
(
m
cm ) < (
m ):
Hence, one can see that compared to
; the probability of tax increases goes down, but the
probability of tax decreases goes down as well. So when starting from a low tax level, the constitutional amendment will keep taxes low for longer. However, once they increase, it will be harder
to lower them when the need arises. If the second e¤ect is strong enough, the average tax rate in
19
the long run can go down.
Figure 2 below depicts the invariant probability of L (probability that in a distant future the
policy is L) as we vary
r:
The dashed line is the probability of L under simple majority, the dotted
curve demonstrates how this probability would change under
0
The solid curve demonstrates what happens in equilibrium under
and Panel B for
m
=
if players played a static game.
0:
Panel A is drawn for
m
=0
0:5. When the median player is on average indi¤erent between low and
high taxes (i.e., she is indi¤erent when " (t) = 0), then increasing the majoritarian requirement to
raise taxes indeed makes high taxes less prevalent. In fact, it makes them less prevalent than what
a static model would imply. However, when the median player on average prefers high taxes, then
increasing the majoritarian requirement may result in higher average taxes in the long run.
The empirical evidence on the e¤ect of supermajoritarian rules on taxes is relatively scant and
relatively inconclusive. For the period 1980-2008, the average tax level in the states in which
tax increases require a supermajority approval is nearly identical as in the states without such
supermajority requirements. The average state with no supermajority requirement taxes as a share
of personal income have been between 9.7 percent and 10.9 percent , while in the seven states with
strict supermajority requirements they have been between 9.7 percent to 10.8 percent (Leachman,
Johnson, Grundman, 2012, and Jordan and Ho¤man, 2009).12 On the other hand, using …xed e¤ects
models, Knight (2000) and Besley and Case (2003) …nd that supermajoritarian requirements reduce
taxes by about $50 per capita.
12
These calculations are based on Arizona, California, Delaware, Mississippi, Nevada, Louisiana, and Oregon.
Knight (2000) reports that in 1995, among the continental states, states with supermajoritarian requirements had
identical average e¤ective tax rates of 7.13% as states without such requirements.
20
5.3
Volatility
The following proposition shows that the polarizing e¤ect of the dispersion of power is greater in
more volatile environments.
Proposition 4 Let
be a voting rule that is not quasi-dictatorial. Let p; p0 2 R be such that
p0 < p < 1; and let c and c0 be the equilibrium thresholds with persistence p and p0 respectively.
0
Then
c0r < cr < 0 < cl < c0l :
When the environment is static, that is, when p = 1; then c = 0:
The intuition for Proposition 4 is the following. As the volatility of the environment increases—
that is, as p decreases— the policy implemented today is more likely to require a revision tomorrow;
hence, remaining pivotal tomorrow becomes more important relative to implementing the right
policy today. In the extreme case when p = 1; preferences never change; hence, a policy adopted
today will never be revised. As a result, all players vote according to their current preferences.
This result implies that the inertial e¤ect of the dispersion of power is greater in more volatile
environments. This suggest that simple majority rule should be used in policy domains which
require frequent adjustments, such as …scal or monetary policies, and requiring a quali…ed majority
is less costly for policies over which preferences are more stable, such as constitutional laws.
5.4
Discussion and Conclusions
When binary decisions are taken via quali…ed majority rule, two sorts of mistakes can be made.
First, no quali…ed majority wants to move away from the status quo though changing the status
quo would be socially optimal (type 1 mistakes). Second, a quali…ed majority wants to move away
from the status quo, but it would be socially optimal to stay at it (type 2 mistake). By increasing
the quali…ed majority requirement in a static setting, we increase the likelihood of type 1 mistakes,
but decrease the likelihood of type 2 mistakes. Whether this is welfare improving depends on the
expected cost of each type of mistake (see Gutmann 1998). Our results show that in dynamic
settings and volatile environments, increasing the quali…ed majority may increase the likelihood
21
of type 1 mistake quite substantially. Hence, the optimal quali…ed majority is lower than the one
implied by the static game.
Propositions 3 and 3 have important consequences for constitutional design. As we discussed
in Section 1, in most modern democracies there exists a system of check and balances. Admittedly,
these checks and balances are not designed to smooth the decision process. Rather, their role is to
limit agency costs and abuses of power by any government branch. Our model shows, however, that
when checks and balances are introduced in a decision process, they tend to make legislators more
polarized, which can greatly exacerbate their inherently inertial e¤ect. Hence, a system of checks
and balances should be complemented with solutions that mitigate polarization. One option is to
allow or require a more frequent use of sunsets, especially during volatile times. Sunset provisions
are clauses attached to a legislation that determine its expiration date. They break the dynamic
linkage between today’s decision and tomorrow’s pivotality; hence, mitigate the voting distortions
identi…ed in this paper.
Legal scholars and politicians, starting from Thomas Je¤erson, have long encouraged the use
of sunsets.13 The modern conception of sunset provisions was spelled out in Lowi (1969). The
main rationale was to ensure that obsolete policies and agencies do not last solely on the basis of
legislative inertia. Our paper provides an explanation for the source of this inertia.
Over the past three decades, Congress has increasingly employed sunset provisions to tax legislation. In 2001, Congress enacted the largest tax cut in twenty years, the Economic Growth and
Tax Relief Reconciliation Act (EGTRRA), and two years later, it passed the third largest tax cut in
history, the Jobs and Growth Tax Relief Reconciliation Act (JGTRRA). Both acts included sunset
provisions. Sunset provisions have been recently used also to legislate gun ownership (1994 Anti
Crime Act which banned the manufacture, sale, and possession of certain type of assault weapons)
and civil liberties (2001 USA PATRIOT Act). The more frequent use of sunset provisions for major
legislations coincided with higher polarization and gridlock in the US Congress. If this polarization
re‡ected a true change in ideological positions of the legislators, the sunsets should not facilitate a
bill passage more than they did when polarization were small. If on the other hand, the increased
13
“[T]he power of repeal is not an equivalent [to mandatory expiration]. It might indeed be if [...] the will of the
majority could always be obtained fairly and without impediment. But this is true of no form. [...] Various checks
are opposed to every legislative Proposition [...] and other impediments arise so as to prove to every practical man
that a law of limited duration is much more manageable than one which needs a repeal.” (See Woods 2009, p. 93).
22
polarization of the US Congress is an endogenous phenomenon driven by the recent changes in
voting rules and in norms governing the use of the …libuster, then sunsets should become more
attractive.
In this paper, we abstracted from the ability of legislators to use sunsets, as we intended to
focus solely on the distorting e¤ect of di¤erent voting rules. However, a formal analysis of the
interaction of voting rules and the frequency (and impact) of sunset provisions is an interesting
avenue for future research.
6
Tables
Table 1. Source: National Conference of State
Legislators.
23
Table 2. Source: Barber and McCarty (2013)
Table 3. Source: US Senate.
7
Appendix
Throughout the appendix, we use the following notations:
24
Notation 1 For all p 2 [0; 1] ; l; r 2 N ; and c 2 RN ;
(1
Hn (p; l; r; c) =
1
p)
p
Z
(cn
f 2RN : l <cl and
r
n ) dP
( ):
(4)
cr g
We omit p; l; and r in the argument of H when these parameters are clear from the context.
The next remark and lemmas derive some important properties of H.
Remark 1 Note that (Hl ; Hr ) (c) depends on c only though (cl ; cr ) : Moreover, the …xed points of
c ! H (c) are in bijection with the …xed points of (cl ; cr ) ! (Hl ; Hr ) (cl ; cr ) in the following sense:
if c is a …xed point of H; then (cl ; cr ) is a …xed points of (Hl ; Hr ), and reciprocally, if (cl ; cr ) is
a …xed points of (Hl ; Hr ) ; then there exists a unique c0 that is a …xed point of H and such that
(c0l ; c0r ) = (cl ; cr ).
cl and c0r
Notation 2 De…ne the order ( ; ) as follows: (c0l ; c0r ) ( ; ) (cl ; cr ) if c0l
cr :
We say that (Hl ; Hr ) (cl ; cr ) is isotone in the order ( ; ) if whenever c0 ( ; ) c then (Hl ; Hr ) (c0l ; c0r ) ( ; ) (Hl ; Hr )
Lemma 1 The mapping (cl ; cr ) ! (Hl ; Hr ) (cl ; cr ) is isotone in the order ( ; ) ; and for n 2 fl; rg
Hn (cl ; cr ) is
(1
p) Lipshitz continuous in cn :
Proof. For all cl ; c0l ; cr ; c0r such that cl
Hl (c0l ; c0r ) Hl (cl ; cr )
(1
(1 p)
p) =
c0r ;
c0l and cr
Z
f l <cl and
+
Z
f
r
0
l <cl and
cr g
r
c0l
cl dP ( )
c0r gnf l <cl and
r
cr g
c0l
l
dP ( )
The integrand of the two integrals on the right-hand side of the above equation are positive on their
domains of integration, so Hl (c0l ; c0r )
Hl (cl ; cr ) : Therefore, Hl (cl ; cr ) is increasing in (cl ; cr ) in
the order ( ; ) : A similar proof shows that Hr (cl ; cr ) is decreasing in (cl ; cr ) in the order ( ; ) ;
so (Hl ; Hr ) is isotone in (cl ; cr ) in the order ( ; ) :
When cr = c0r ; the above equality implies
25
Hl (c0l ; cr ) Hl (cl ; cr )
(1
(1 p)
p) =
Z
f l <cl and
+
Z
f cl
For all
l
2 [cl ; c0l ] ; cl
l
c0l
(1
cr g
r
0
l <cl and
cl dP ( )
cr g
r
c0l
l
dP ( )
cl ; so the above equality implies in turn that
Hl (c0l ; cr ) Hl (cl ; cr )
(1
(1 p)
so Hl (cl ; cr ) is
c0l
p)
c0l
cl P
l
< c0l and
r
c0l
cr
cl ;
p) Lipshitz continuous in cl : The proof for Hr (cl ; cr ) proceeds in the same
way.
Lemma 2 There exists a < 1 such that (Hl ; Hr ) [ a; a]2
[ a; a]2 and all the …xed points of
(Hl ; Hr ) are in [ a; a]2 :
R
:
Proof. Let k k = maxn2N j
n j dP
:
( ), and kck = maxn2fl;rg jcn j. From (4), we see that for all
(cl ; cr ) 2 R2 and all n 2 fl; rg ; jHn (cl ; cr )j is bounded by
(1 p)
1 p
point of (Hl ; Hr ), then kck
(1 p)
1 p
(k k + kck). So if (cl ; cr ) is a …xed
(k k + kck), which in turn implies that kck
The following lemma shows that the equilibria of the game
(1 p)
1
k k
a.
( ) can be characterized as the
…xed points of the mapping H:
Lemma 3 If
is an equilibrium, then
is a cuto¤ strategy pro…le for some pro…le of cuto¤ s c
such that cn is decreasing in n. A pro…le of cuto¤ s c is an equilibrium if and only if it is a …xed
point of the mapping c ! H (c) ; and
cn =
Proof. Let
1
p
(1
(1 p)
p) Pr (f l < cl and
r
cr g)
Z
f l <cl and
n dP
r
( ):
(5)
cr g
be a pro…le of Markov strategies, and for all q 2 fL; Rg ; let Wn (q) be the
continuation value of the game for player n at the beginning of some period t; conditional on the
status quo being q; on the state being redrawn at the beginning of the period, and on players
playing
. Since players play Markov strategy, and since the state is redrawn, Wn (q) does not
26
depend on previous states.
Since we can assume without loss of generality that players revise the policy only when the
state
is redrawn, the di¤erence in continuation value for player n from implementing policy R
versus L in a period with state
n
can be expressed as follows:
+ (1
p) Wn (R) + p (
0 + (1
=
1
1
n
p
n
+ (1
p) Wn (L) + p (0 + (1
+
(1
1
p)
(Wn (R)
p
p) Wn (R) + p (:::))
(6)
p) Wn (L) + p (:::))
Wn (L)) :
In any period with status quo q (t) 2 fL; Rg and state
(t) 2 Rn ; if (6) is nonnegative (negative),
then R (L) is the only stage-undominated action for player n: Manipulating (6), this implies that
n
prescribes stage-undominated actions if and only if it prescribes n to vote for R if and only if
n (t)
(1
:
Wn (R)) = cn :
p) (Wn (L)
(7)
Since a stage-undominated strategy must prescribe stage-undominated actions for all possible realizations of
n (t)
(even those that may arise with probability 0), this shows that a stage-undominated
strategy must be a cuto¤ strategy, and for any strategy pro…le ; the unique stage-undominated
cuto¤ is given by the above condition.14
Let D ( ) denote the set of realizations of
such that no winning coalition of players vote for
the alternative di¤erent than the status quo when the state is : Note that the status quo in period
t + 1 a¤ects the policy outcome in t + 1 only when (t + 1) 2 D ( ) : In such states, the status quo
stays in place. Hence, Wn (R)
Wn (R)
Wn (L) =
Wn (L) is simply the expectation of (6) over all
Z
1
2D( )
1
p
n
which delivers
Wn (R)
Wn (L) =
1
(1
1
+
p
14
R
p)
(Wn (R)
p
2D( ) n dP
(1
p)
R
2 D ( ), so
Wn (L)) dP ( ) ;
(8)
( )
2D( ) dP
( )
:
Technically, the strategy does not have to specify the behavior for that are not in the support of P ( ) ; so if
the support of is not connected, the thresholds may not be uniquely determined. We ignore this issue for simplicity,
as the resulting multiplicity does not reverse any of our …ndings and only complicates the exposition.
27
Substituting the above expression for Wn (R) Wn (L) into (7), we obtain that if c is an equilibrium
cuto¤ pro…le, cn is decreasing in n: Substituting (7) into (8), we obtain
(1
cn =
1
We have shown that if
p)
p
Z
(cn
n ) dP
( ):
(9)
2D( )
is an equilibrium,
is a cuto¤ strategy pro…le for some pro…le of cuto¤
c such that cn is decreasing in n. Therefore, voters vote as if there is a single period and their
current preferences is given by
n
cn instead of
n;
and
c satis…es Assumption 1. Therefore,
the same reasoning as in Section 4.1 shows that players l and r are pivotal in all periods in which
the status quo is L and R; respectively (see De…nition 2). So, up to a zero measure set, D ( ) can
be rewritten as a function of the preference realization of the pivotal legislators only:
2 RN :
D( ) =
l
< cl and
cr :
r
(10)
Substituting the above expression for D ( ) into (9), we obtain that c must be a …xed point of H;
and (5) follow immediately from (4).
Conversely, let c be a …xed point of H; and consider the corresponding strategy pro…le : Then
(4) must hold, and (4) implies that cn is decreasing in n; and thus that (10) holds. Therefore,
Wn (L) must still satisfy (8), and viewed as an equation in
(9) also holds. Note that Wn (R)
Wn (R)
Wn (L), (8) has a unique solution. Manipulating (9), we obtain that
the same equation, and is thus equal to Wn (R)
is an equilibrium.
By Lemma 3, in any equilibrium
, players use status-quo-
independent cuto¤ strategies, and the pro…le of cuto¤s c is such that c = H (c) and c1
If
H (c) = (0; :::; 0) : Suppose now that
Hl (c)
2 RN :
is quasi-dictatorial, then l = r so
0
2 RN :
Hr (c) ; and thus that cl
l
< cl and
r
satis…es
Wn (L). Therefore, (7) holds, which implies that
the cuto¤ strategy cn is stage undominated for player n; so
Proof of Proposition 1.
1+ p
(1 p) cn
l
< cl and
r
cr
:::
cN :
= ;; and therefore, c =
is not quasi-dictatorial. One can easily see from (4) that
0
cr . Since l 6= r; Assumption 1 implies then that
> cr has strictly positive probability, and thus that Hl (c) > 0 > Hr (c),
which proves property (ii) :
28
Since
2 RN :
l
< cl and
r
> cr has strictly positive probability, Assumption 1 implies that
the integral on the right-hand side of (5) is strictly increasing in n; which proves property (i) :
Proof of Proposition 2. Using Lemma 3 and Remark 1, to prove Proposition 2, it su¢ ces
to prove that the …xed point of (Hl ; Hr ) are a complete chain for the order ( ; ) :
From Lemma 2, the …xed points of (Hl ; Hr ) lie in [ a; a]2 which is a complete lattice for the
order ( ; ) ; and (Hl ; Hr ) [ a; a]2 is included in [ a; a]2 : Lemma 1 together with Tarski’s …xed
point theorem imply that the set of …xed points of the restriction of (Hl ; Hr ) on [ a; a]2 (and hence
the set of …xed points of (Hl ; Hr ) on R2 ) is a complete lattice in the order ( ; ) :
Let (cl ; cr ) and (c0l ; c0r ) be two …xed points of (Hl ; Hr ) ; and suppose w.l.o.g. that c0r
c0l < cl ; then since from Lemma 1, Hl is (1
cr : If
p) Lipshitz continuous in cl ; we obtain
cl = Hl (cl ; cr ) < Hl c0l ; cr + cl
c0l
Hl c0l ; c0r + cl
c0l = cl ;
where the last equality follows from Hl (c0l ; c0r ) = c0l : Hence, we have a contradiction. Therefore,
c0r
cr implies that c0l
cl ; which shows that the set of …xed points of (Hl ; Hr ) is completely
ordered by ( ; ) ; and is thus a chain.
0
Lemma 4 If the dispersion of power under
cr(
0)
0
is greater than under
cr( ) ( )
0
cr(
0
cl( ) ( )
cl(
0)
; then
0
:
(11)
Proof. Let
D =
D0 =
r( )
r(
0)
cr(
0)
0)
and
0
29
and
l( )
l(
< cl(
0)
< cl(
0)
0)
0
;
0
:
Assumption 1 together with (2) imply that modulo a zero probability event, D
H (l ( 0 ) ; r ( 0 ) ; c ( 0 )) ; and since cl(
cl(
0
0)
=
0)
0)
0)
l(
Z
p)
c
p D0 l(
Z
p)
c
p D l(
Z
p)
c
p D l(
(1
1
(1
1
(1
1
= Hl(
A similar argument shows that cr(
( 0)
0
( 0)
Hr(
0
l(
0)
0
l( )
)
dP ( )
0)
l(
0)
l ( ) ; r ( ) ; cl(
)
2 D0 ; we have that
is positive for all
0)
dP ( )
0)
0
dP ( )
; cr(
0)
l ( ) ; r ( ) ; cl(
0)
0)
with Lemma 1, this implies that for all (cL ; cR ) 2 R2 such that cL
D0 : Since c ( 0 ) =
cl(
0
:
( 0 ) ; cr(
0)
0)
( 0 ) : Together
( 0 ) and cR
cr(
0)
( 0) ;
(Hl ; Hr ) (l ( ) ; r ( ) ; cL ; cR )
( ; ) (Hl ; Hr ) l ( ) ; r ( ) ; cl(
( ; ) cl(
0)
Using the notations of Lemma 2, if C =
implies that Hl( ) ; Hr(
)
0
; cr(
a; cl(
(l ( ) ; r ( ) ; C)
0)
0)
0
cl(
0)
)(
( 0 ) ; cr(
) ; cr(
0)
)(
; cr(
0)
0
:
( 0)
cr(
0)
( 0 ) ; a , the previous two inequality
C: Therefore, (cL ; cR ) ! Hl( ) ; Hr(
admits a …xed point in C which is smaller than cl(
Since cl(
0
0)
0)
( 0 ) ; cr(
0)
)
(l ( ) ; r ( ) ; cL ; cR )
( 0 ) for the order ( ; ) :
) is the smallest …xed point of this mapping, it must be smaller than
( 0) :
Proof of Proposition 3. We …rst show that four inner inequalities in (3):
cr(
0)
( )
cr(
)(
)
0
cl(
)(
)
cl(
0)
( ):
When
is quasi-dictatorial, Proposition 1 implies that these inequalities in (3) hold with equality.
When
is non dictatorial, Proposition 1 part (i) and (ii) imply that they hold strictly.
30
We now show the two outer inequalities in (3): cl(
0)
( )
cl(
( 0 ) and cr(
0)
0)
( 0)
cr(
0)
( ).
Let
D =
cr(
r( )
D0 =
r(
)(
cr(
0)
) and
0
0)
l( )
and
< cl(
0)
l(
)(
) ;
< cl(
0
0)
:
From (5),
cl(
0
0)
cl(
0)
=
( ) =
1
p
1
p
Z
(1 p)
(1 p) Pr (D0 ) D0
Z
(1 p)
(1 p) Pr (D) D
l(
dP ( ) ;
0)
l(
dP ( ) :
0)
D0 ; and for almost all
Assumption 1 together with (11), modulo a zero probability event, D
2 D0 ;
l(
0)
> 0; so the above equations show that cl(
shows that cr(
0)
( 0)
cr(
0)
0)
( 0)
(12)
cl(
0)
( ) : A similar reasoning
( ). It remains to show that these inequalities are strict when
(l ( ) ; r ( )) < (l ( 0 ) ; r ( 0 )) : Suppose to …x ideas that r ( ) < r ( 0 ) ; the proof in the case
l ( ) > l ( 0 ). By de…nition of D and D0 ;
< cr(
)(
) and
From Proposition 1, cr(
)(
)
r( )
1,
r( )
< cr(
)(
) implies
l(
cr(
0)
0 and cl(
0)
r(
If r ( ) < r ( 0 ) and
r(
< cl(
0)
0)
0)
0
0)
( 0)
and
l(
0)
< cl(
0
0)
D0 n D
0, so Assumption 1 implies that with probability
( 0 ) : Therefore, the above inclusion implies that
2 cr(
0)
0
; cr(
)(
D0 n D
)
(13)
is not quasi-dictatorial, from what precedes, we have that
cr(
0)
0
cr(
0)
( ) < cr(
)(
):
The above inequalities and the full support assumption imply that the probability of the event on
the left-hand side of (13) is strictly positive. Therefore, P (D0 ) > P (D) : Moreover, cl(
Substituting the latter inequalities in (12), we obtain cl(
31
0)
( 0 ) > c l(
0)
0)
( 0 ) > 0.
( ) : A similar reasoning
shows that cr(
0)
( 0 ) < cr(
0)
( ).
Proof of Proposition 4.
Note …rst that when
is quasi-dictatorial, Proposition 4 follows
trivially from Proposition 1.
Suppose now that
is not quasi-dictatorial. Using the notations of Proposition 4, Lemma 3
and Remark 1 imply that (c0l ; c0r ) is a …xed point of (cL ; cR ) ! (Hl ; Hr ) (p0 ; cL ; cR ) : Let (cL ; cR ) be
such that
c0r
cR and cL
c0l :
(14)
Applying H to the above inequality, and using Lemma 1 and
@Hn
@p
=
(1 )
(1 p)(1 p) Hn ;
: we have
that
c0l = Hl p0 ; c0l ; c0r < Hl p; c0l ; c0r
and 0
If we denote C = [0; c0l ]
Hr (p; cL ; cR )
Hl (p; cL ; cR )
Hr p; c0l ; c0r < Hr p0 ; c0l ; c0r
0;
c0r :
[c0r ; 0] ; the above inequalities show that (Hl ; Hr ) (p; C)
C: Since C is a
complete lattice, Tarski’s theorem implies that (cL ; cR ) ! (Hl ; Hr ) (p; cL ; cR ) admits a …xed point
in C: Moreover, the above inequalities imply that for all such …xed point, the inequalities in (14)
hold strictly. From Lemma 3 and Remark 1, (cl ; cr ) is one of these …xed points.
References
[1] Acemoglu, D., G. Egorov, and K. Sonin, 2014. Political Economy in a Changing World. Journal
of Political Economy, forthcoming.
[2] Aghion. P., A. Alesina, and F. Trebbi, 2004. Endogenous Political Institutions, The Quarterly
Journal of Economics, 119, 565–611.
[3] Aghion P. and P. Bolton, 2003. Incomplete Social Contracts. Journal of the European Economic
Association. 1, 38–67.
[4] Arrow K. J., 1998. The External Costs of Voting Rules: A Note on Guttman, Buchanan, and
Tullock. European Journal of Political Economy, 14, 219–222.
32
[5] Austen-Smith, D. and J. Banks, 1996. Information Aggregation, Rationality and the Condorcet
Jury Theorem. American Political Science Review, 90, 34–45.
[6] Bafumi, J. and M. C. Herron, 2010. Leapfrog Representation and Extremism: A Study of
American Voters and Their Members in Congress. American Political Science Review 104,
519–542.
[7] Barber M. and N. McCarty, 2013. Causes and Consequences of Polarization. In Negotiating Agreement in Politics, eds. J. Mansbridge and C. J. Martin. American Political Science
Association.
[8] Baron, D., 1991. Majoritarian Incentives, Pork Barrel Programs, and Procedural Control.
American Journal of Political Science, 35, 57–90.
[9] Besely, T. and A. Case, 2003. Political Institutions and Policy Choices: Evidence from the
United States. Journal of Economic Literature,41, 7–73.
[10] Buchanan, J. M.,1998. Agreement and E¢ ciency: Response to Guttman. European Journal
of Political Economy, 14, 209–213
[11] Buchanan, J. M. and G. Tullock, 1962. The Calculus of Consent. Ann Arbor, MI, University
of Michigan Press.
[12] Caplin, A., and B. Nalebu¤, 1988. On 64%-majority rule. Econometrica, 56, 787–814.
[13] Condorcet, N., 1785. Essai sur l’application de l’analyse a‘la probabilite´ des de´cisions rendues a‘la pluralite´ des voix. Paris: Imprimerie Royale.
[14] Coughlan, P. J., 2000. In Defence of Unanimous Jury Verdict: Mistrials, Communication and
Strategic Voting. American Political Science Review, 94, 375–393.
[15] Dziuda, W. and A. Loeper, 2014. Dynamic Collective Choice with Endogenous Status Quo.
Northwestern University, mimeo.
[16] Enelow J., 1997. Cycling and Majority rule. In D. Mueller, ed., Perspective on Public Choice:
A Handbook, Cambridge: Cambridge University Press, 149–62.
33
[17] Eraslan, H. and A. Merlo, 2002. Majority Rule in a Stochastic Model of Bargaining. Journal
of Economic Theory, 103, 31–48.
[18] Gradstein, M., 1999. Optimal Taxation and Fiscal Constitution, Journal of Public Economics,
72, 471–485.
[19] Feddersen, T. and W. Pesendorfer, 1998. Convicting the Innocent: The Inferiority of Unanimous Jury Verdicts under Strategic Voting. American Political Science Review, 92, 23–35.
[20] Ferejohn, J. A., M. P. Fiorina, and R. D. McKelvey, 1987. Sophisticated Voting and Agenda
Independence in the Distributive Politics Setting. American Journal of Political Science, 31(1),
169-93.
[21] Gradstein, M. 1999. Optimal Taxation and Fiscal Constitution. Journal of Public Economics,
72, 471–485.
[22] Grofman, B. N. and G. Owen, 1986. Information Pooling and Group Decision-Making. JAI
Press, Greenwich, CT.
[23] Guttman, J., 1998. Unanimity and Majority Rule: The Calculus of Consent Reconsidered.
European Journal of Political Economy, 14, 189–207.
[24] Hastard, B., 2005. Majority Rules and Incentives. Quarterly Journal of Economics, 120, 535–
568.
[25] Hayek, F. A., 1960. The Constitution of Liberty. University of Chicago Press.
[26] Jordan, M. M., and K. U. Ho¤man, 2009. The Revenue Impact of State Legislative Supermajority Voting Requirements. Midsouth Political Science Review, 2009, 10, 1-15.
[27] Knight, B., 2000. Supermajority Voting Requirements for Tax Increases: Evidence from the
States. Journal of Public Economics, 76, 41–67.
[28] Krehbiel, K. 1999. Pivotal Politics: A Re…nement of Nonmarket Analysis for Voting Institutions, Business and Politics, 1, 63–81.
[29] Leachman, M., N. Johnson, and D. Grundman, 2003. Six Reasons Why Supermajority Requirements to Raise Taxes Are a Bad Idea. Report for Center on Budget and Policy Priorities.
34
[30] Lowi, T.J., 1969. The End of Liberalism: Ideology, Policy, and the Crisis of Public Authority.
New York, NY, Norton.
[31] May, K. O., 1952. A Set of Independent, Necessary and Su¢ cient Conditions for Simple Majority Decision. Econometrica 20, 680–684.
[32] McGann, A. J., 2004. The Tyranny of the Supermajority: How Majority Rule Protects Minorities. Journal of Theoretical Politics, 16, 53–77.
[33] Messner, M. and M. Polborn, 2004. Voting on Majority Rules. Review of Economics Studies,
71, 115–132.
[34] Persico, N., 2003. Committee Design With Endogenous Information. Review of Economic
Studies, 70, 1–27.
[35] Plott, C. R., 1967. A Notion of Equilibrium and Its Possibility Under Majority Rule. American
Economic Review, 57, 787–803.
[36] Rae, D. W., 1969. Decision-Rules and Individual Values in Constitutional Choice. The American Political Science Review, 63, 40–56.
[37] Rae, D. W., 1975. The Limits of Consensual Decision. The American Political Science Review,
69, 1270–1294.
[38] Riker, W. H., 1962. The Theory of Political Coalitions. New Haven: Yale University Press.
[39] Roberts, K., 1999. Dynamic Voting in Clubs. Unpublished manuscript.
[40] Tullock, G. 1998. Reply to Guttman. European Journal of Political Economy, 14, 215–219.
[41] Tsabellis, G. Veto Players, 2002, Princeton: Princeton University Press.
[42] Wicksel, K., 1896.A New Principle of Just Taxation. Reprinted in R. Musgrave and A. Peacock,
eds., Classics in the Theory of Public Finance, New York: St. Martin’s Press, 1967.
[43] Brett F. Woods. 2009. Thomas Je¤erson: Thoughts on War and Revolution. Algora Publishing,
New York.
35