Preliminary Draft September 2005
Socially Optimal Districting: An Empirical Investigation∗
Abstract
This paper provides an empirical exploration of the potential gains from socially optimal districting.
Districtings, or the allocation of voters of differing ideologies across legislative districts, matter for welfare
because they influence the partisan composition of the legislature and hence policy outcomes. Under a
socially optimal districting, the ideological composition of the legislature is welfare maximizing for any
realization of voter preferences. We develop a methodology for analyzing how actual districting plans
differ from optimal districting plans and for measuring the potential welfare gains that would emerge
from implementing optimal districting. We apply this method to U.S. state legislative elections using
data from the 1990s. We find that the districtings used by the states in our data set lead to electoral
systems that are overly responsive to changes in voters’ preferences. While there is significant variation
across states, the potential welfare gains from optimal districting are on average small relative to the overall
surplus generated by legislatures. This appears to be because state districting plans are reasonably close
to optimal rather than because there are little aggregate gains to be had from varying districting plans.
Interestingly, implementing proportional representation would lead to welfare gains in all states and would
produce welfare levels quite close to those achieved with socially optimal districting.
Stephen Coate
Department of Economics
Cornell University
Ithaca NY 14853
[email protected]
Brian Knight
Department of Economics
Brown University
Providence RI 02906
brian [email protected]
∗ We
are greatly indebted to Jim Snyder for providing the data on state legislative elections used in this study.
1
Introduction
The decennial redrawing of district lines used in electing candidates to federal, state, and local
legislatures in the United States has generated intense interest among voters, politicians, and
parties alike. The intensity of this interest in redistricting, as witnessed through both political
and legal battles, should come as no surprise. Democrats and Republicans are often sharply
divided in their preferences over the choice of public policies. These policies are determined in
part by the representation of political parties in legislatures, which in turn depend upon the
allocation of voters across legislative districts.
This public interest in redistricting has helped to spurn an academic literature analyzing redistricting plans and the redistricting process. In this literature, redistricting plans are typically
characterized by the implied relationship between seats in the legislature and support for parties
among voters. In particular, the seat-vote curve relates the fraction of seats won by the Democratic
party to its support among voters across all districts. Commonly-studied properties of seat-vote
curves include responsiveness, defined as the increase in seats for the Democrats associated with
an increase in voter support across all districts for the Democrats, and partisan bias, defined as
the advantage or disadvantage afforded the Democratic party in terms of seats, relative to the
Republican party, when voters are equally divided between the two parties. Researchers have
developed statistical methods to estimate seat-vote curves and to measure the associated responsiveness and partisan bias parameters. Key questions are then how responsiveness and bias are
altered following the redrawing of district lines and also how different institutional arrangements
for redistricting influence these changes.
While this literature is certainly interesting from a positive perspective, the normative lessons
to be drawn from its findings are unclear. How should voters be allocated across districts? Is
partisan bias necessarily a bad thing from the perspective of voters? What is the optimal degree of
responsiveness? In Coate and Knight (2005), a companion paper, we develop a welfare economics
approach to address these questions from a theoretical perspective. Following the literature, in our
framework districting matters because it determines the equilibrium seat-vote curve. The fraction
of seats held by Democrats determines the ideological make up of the legislature and thereby policy
outcomes and citizen welfare. To understand optimal districting, we first derive the optimal seatvote curve, which describes the welfare-maximizing relationship between Democratic seats and
1
votes. This optimal seat-vote curve is linear with a slope, or responsiveness, that depends on the
degree of variation in the preferences of Independent voters. Furthermore, it is biased in favor
of the party with the largest base. We then show that, under seemingly reasonable conditions,
there exists districtings under which the equilibrium and optimal seat-vote curves coincide. These
districtings are socially optimal districtings because they generate the ideal relationship between
seats and votes.
These theoretical findings raise several natural empirical questions. How close are the equilibrium seat-vote curves associated with real world legislatures to the optimal seat-vote curves
identified via this social welfare approach? Are the conditions under which there exist districtings
that make equilibrium and optimal seat-vote curves coincide satisfied in practice? If so, how large
are the potential welfare gains associated with implementing optimal seat-vote curves? What are
the sources of these welfare gains?
In this companion empirical paper, we begin the development of a methodology that can be
used to answer these and related questions. We first develop an empirical methodology for using
district-specific electoral returns in order to estimate equilibrium seat-vote curves. This approach,
while allowing for a general functional form for the equilibrium seat-vote curve, has micro foundations, and the parameters of the equilibrium seat-vote curve can be related to the parameters
of the theoretical model. Turning to the normative questions, we then develop expressions for the
optimal seat-vote curve and its associated optimal bias and responsiveness parameters; this optimal seat-vote curve can also be identified using data on electoral returns along with information
regarding the fraction of Independent voters across all districts. We then address whether or not
the optimal seat-vote curve can be implemented in practice by developing a sufficient condition
that depends upon the fraction of Independent voters and showing that this condition can be
verified empirically. Finally, we develop expressions for the welfare gains from socially optimal
districting and show that these depend on squared distance between the equilibrium and optimal
seat-vote curves.
We apply this method to U.S. state legislative elections using data from the 1990s. One key
finding is that equilibrium seat-vote curves tend to be overly responsive, relative to optimal seatvote curves, suggesting that, if anything, elections in the U.S. are overly competitive and that there
are too few safe seats from a social welfare perspective. In addition, we find that optimal seatvote curve can be implemented in practice in the sense that there exists districtings that make the
2
equilibrium and optimal seat-vote curves coincide. In general, while there is significant variation
across states, our measures of the potential welfare gains from socially optimal districting are quite
small, at least relative to the overall surplus generated by state legislatures. We argue that this
is because state districting plans are reasonably close to optimal rather than because there are
little aggregate gains to be had from varying districting plans. We also find that implementing
proportional representation would lead to welfare gains in all states and would produce welfare
levels quite close to those achieved with socially optimal districting.
The paper proceeds as follows. Section 2 describes the formal model underlying the analysis
and reviews the key theoretical results. Section 3 presents our empirical methodology for comparing equilibrium and optimal seat-vote curves and evaluating the potential gains from socially
optimal districting. In Section 4, this empirical methodology is applied to data from U.S. state legislative elections during the 1990s. Section 5 compares our approach and results with the existing
districting literature, and Section 6 concludes.
2
Theoretical background1
We consider a state divided into n equally sized districts indexed by i ∈ {1/n, 2/n, ...., 1}. Policymaking is done by a legislature consisting of a representative from each district. Each district
chooses its representative in an election. The policy outcomes chosen by the legislature depend
upon the average ideology of the elected representatives, where ideology is measured on a 0 to 1
scale. In terms of ideologies, citizens can be divided into three groups - Democrats, Republicans,
and Independents. Democrats and Republicans have ideologies 0 and 1, respectively. Independents
have ideologies that are uniformly distributed on the interval [m − τ, m + τ ]. Reflecting the fluid
nature of these voters’ attitudes, the ideology of the median Independent is ex-ante uncertain.
Specifically, m is the realization of a random variable uniformly distributed on [1/2 − ε, 1/2 + ε],
where ε < τ and ε + τ ≤ 1/2. The former assumption guarantees that both candidates receive
support from Independents, while the latter assumption guarantees that the ideologies of the
Independents are always between zero and one. The fraction of voters in district i who are
i
i
, πR
and πIi . Let πD , πR and πI
Democrats, Republicans, and Independents are, respectively, πD
1 This section will be brief, and readers are referred to Coate and Knight (2005) for additional interpretation
and discussion of alternative modeling assumptions.
3
denote, respectively, the fraction of voters in the whole state who are Democrats, Republicans,
and Independents.
Each district must elect a representative. Candidates are put forward by two political parties:
the Democrats and Republicans. Following the citizen-candidate approach, candidates are citizens
and are characterized by their ideologies. Each party must select from the ranks of its membership,
so that the Democrat Party always selects a Democrat and the Republican Party a Republican.
Elections are held simultaneously in each of the n districts and the candidate with the most votes
wins.
The surplus a citizen with ideology x obtains from having a legislature with average ideology
x0 is given by β − γ(x − x0 )2 . The parameter β is the surplus a citizen would obtain from having a
legislature that is perfectly congruent with his own ideology. The parameter γ measures the rate at
which this surplus is dissipated as the ideology of the legislature diverges from the ideology of the
citizen. The ratio γ/β will play an important role in the welfare analysis and can be interpreted
as the fraction of the surplus a partisan (i.e., a Democrat or Republican) obtains from having
a perfectly congruent legislature that is dissipated by having a legislature composed entirely of
the opposition party. We assume throughout that this ratio is bounded between zero and one
(γ/β ∈ [0, 1]).
In each district, every citizen votes sincerely for the representative whose ideology is closest to
his own. If the median Independent has ideology m, the fraction of voters in district i voting for
the Democrat candidate is given by:
i
+ πIi [
Vi (m) = πD
1/2 − (m − τ )
].
2τ
(1)
This consists of all the Democrats and the Independents whose ideologies are less than 1/2.
Similarly, the statewide fraction of voters voting for Democrats is
V (m) = πD + πI [
1/2 − (m − τ )
].
2τ
(2)
Let V and V denote, respectively, the average maximum and minimum Democratic vote shares;
i.e., V = V (1/2 − ε) and V = V (1/2 + ε).
4
2.1
The equilibrium seat-vote curve
We can use the model to derive the equilibrium relationship between seats and average votes.
First, for all V ∈ [V , V ], let m(V ) denote the ideology of the median Independent that would
generate the vote share V ; i.e., m(V ) = V −1 (V ). It is straightforward to verify that
m(V ) =
1
πI + 2πD − 2V
].
+ τ[
2
πI
(3)
Substituting this into the district-specific support, we obtain
i
+ πIi [
Vi (m(V )) = πD
V − πD
].
πI
(4)
Thus, district i elects a Democrat if and only if Vi (m(V )) ≥ 1/2, or, equivalently, if and only if:
V ≥ Vi∗ = πD + πI [
i
1/2 − πD
].
πIi
(5)
Here Vi∗ is the average vote threshold above which district i elects a Democrat candidate. District
i is a safe Democrat (safe Republican) seat if Vi∗ ≤ V (Vi∗ ≥ V ). A seat which is neither a safe
Democrat or Republican seat is competitive.
∗
∗
≤ V2/n
≤ .... ≤ V1∗ . Then, the fraction
Relabelling as necessary, order the districts so that V1/n
of seats the Democrats receive when they have aggregate vote share V is:
S(V ) = max{i : Vi∗ ≤ V }.
(6)
This is the equilibrium seat-vote curve. Its range is the interval [V , V ] and its shape is determined
by the pattern of threshold vote levels (Vi∗ ). These threshold vote levels are in turn determined
by the districting; that is, the allocation of citizen types across districts.
2.2
The optimal seat-vote curve
Consider now the problem of a planner deciding on the number of seats S that should be allocated
to the Democrats when their vote share is V . Aggregate surplus when the median Independent
has ideology m and the Democrats have seat share S is given by:
W (S; m) = β − γ[πD (1 − S)2 + πR S 2 + πI
Z
m+τ
m−τ
(1 − S − x)2
dx
].
2τ
(7)
To avoid tedious integer concerns, assume that the number of districts is very large, so that
we can interpret S as the fraction of seats held by the Democrats and treat the choice set in
5
the optimization problem as the unit interval [0, 1]. It is then straightforward to establish the
following result:
Proposition 1: The optimal seat-vote curve S o : [V , V ] → [0, 1] is given by
S o (V ) = 1/2 + (πD − πR )(1/2 − τ ) + 2τ (V − 1/2).
(8)
Proof: See Appendix.
Thus, the optimal seat-vote curve is linear. Its partisan bias - defined as the advantage or
disadvantage accorded to the Democrats when the two parties split the vote - is (πD −πR )(1/2−τ ).
The system is therefore optimally biased towards the party with the largest partisan base. This
optimal bias reflects the fact that when the legislature is equally divided between Democrats
and Republicans, a marginal change in the composition of the legislature has a greater impact
on partisans’ surplus than on Independents. This in turn reflects the assumption that the loss
experienced by citizens from having a legislature that diverges from their own ideology is a convex
(indeed, quadratic) function of the extent of this divergence.
The responsiveness of the optimal seat—vote curve - defined as the rate at which Democrat
seats increase as their vote share increases - is 2τ . Thus, as the policy preferences of Independent
voters become more diverse, the Democrats’ seat share becomes more responsive to changes in
their aggregate vote. Notice, however, that since τ < 1/2, optimal responsiveness is always less
than 1 meaning that if the Democratic vote share goes up by ∆V , the Democrats share of seats
in the legislature should optimally go up by less than ∆V . To understand this result, note first
that the Democratic seat share will optimally be such as to make average legislator ideology equal
to average ideology in the population at large. Then observe that changes in vote share always
over-estimate changes in average population ideology; that is, if the Democratic vote share goes up
by ∆V , the decrease in average population ideology will be less than ∆V . This is because changes
in vote share are driven by changes in the voting behavior of Independents and such changes can
arise from fairly small shifts in preferences. To illustrate, consider the case in which τ is very
small so that Independents all have basically the same ideology - namely m. Now suppose that m
changes from 0.51 to 0.49 so that all the Independents switch from voting Republican to voting
Democrat. Then there is a massive change in the Democratic vote share but virtually no change
in the average ideology of the population and hence virtually no change in the optimal share of
Democratic seats in the legislature.
6
2.3
The conditions for implementation
The optimal seat-vote curve describes the ideal relationship between seats and votes. It is by no
means obvious that there exist districtings that would actually make the equilibrium and optimal
seat-vote curves coincide. When there do, we say that the optimal seat-vote curve is implementable
and refer to the underlying districtings as socially optimal districtings.
Whether the optimal seat-vote curve is implementable depends upon the fraction of Independents. Two special cases provide intuition for this. First, if there were no Independents, then the
optimal seat-vote curve would be a single point [S o (V ) = πD ] and could be implemented by creating a fraction πR of majority Republican districts and a fraction πD of majority Democrat districts.
On the other hand, if the entire population were Independents, then all districts would necessarily
be identical and the equilibrium seat-vote curve would jump from zero to one at V = 1/2. The
optimal seat-vote curve, by contrast, is linear in this case [S o (V ) = 1/2 + 2τ (V − 1/2)] and is thus
clearly not implementable. In the companion theoretical paper, we establish:
Proposition 2: The optimal seat-vote curve is implementable if and only if
πI [
ε
ε
+ ε − (τ + ε) ln(1 + )] ≤ min(πD , πR ).
2τ
τ
(9)
Proof: See Coate and Knight (2005).
Thus, the fraction of independents must be sufficiently small. The condition does not appear
particularly restrictive since it is necessarily satisfied whenever πI ≤ 2 min(πD , πR ). Moreover,
as shown in Coate and Knight (2005), when the condition is satisfied, there will typically exist a
large class of districtings under which the equilibrium coincides with the optimal seat-vote curve.
This class of optimal districtings typically includes districtings that look quite “straightforward”
in the sense that they do not require patterns of concentration of voter types that would seem
likely to infeasible when account is taken of the geographic constraints that real world districters
face. This nurtures the hope that the optimal seat-vote curve may be an attainable benchmark.
2.4
The welfare gains to socially optimal districting
Assuming that the optimal seat-vote curve is implementable, what would be the welfare gains
from moving to a socially optimal districting? We address this question next. While aggregate
surplus has previously been expressed conditional on the aggregate vote share V, which can be
7
interpreted as welfare at a single point on the seat-vote curve, we require a measure of expected
welfare over the entire seat-vote curve in order to rank alternative curves from a social welfare
perspective. This expected welfare is defined by:
E[W (S(V ))] = β−γ
Z
V
[πD (1−S(V ))2 +πR S(V )2 +πI
V
Z
m(V )+τ
(1−S(V )−x)2
m(V )−τ
dV
dx
](
). (10)
2τ V − V
The welfare gain from socially optimal districting is therefore given by
G = E[W (S o (V ))] − E[W (S(V ))].
(11)
This welfare gain turns out to be proportional to the squared distance between the optimal and
equilibrium seat-vote curves. To see this, note first that expected citizen welfare can be expressed
as a function of the equilibrium and optimal seat-vote curves:
Lemma: It is the case that
E[W (S(V ))] = β − γ{c + E[S(V )2 ] − 2E[S(V )S o (V )]}
(12)
where c is a constant given by c = πD + πI [1/4 + ε2 /3 + τ 2 /3].
Proof: See Appendix.
Using this formula to compute the welfare gain immediately establishes:
Proposition 3: The welfare gain from socially optimal districting can be written as
G = γE[(S o (V ) − S(V ))2 ].
(13)
Intuitively, the smaller the degree of similarity between the optimal and equilibrium seat-vote
curves, the larger are the welfare gains associated with socially optimal districting.
3
Empirical methodology
We can now describe our methodology for comparing actual and optimal districting plans and
evaluating the welfare gains from socially optimal districting. We develop empirical counterparts to
each of the four theoretical concepts described above: the equilibrium seat-vote curve, the optimal
seat-vote curve, the conditions for implementing the optimal seat-vote curve, and the welfare gains
to socially optimal districting. In developing this methodology, we consider a researcher with
8
access to data on district-specific electoral returns, as well as the state-wide fraction of voters who
identify as Independents.2
3.1
Estimating the equilibrium seat-vote curve
As explained in Section 2.1, the equilibrium seat-vote curve is determined by the range of possible
aggregate Democratic vote shares [V , V ] and the pattern of district-specific threshold vote levels
(Vi∗ ). We will show that both the range of vote levels and the vote thresholds can be expressed
as a function of the first two moments of the district-specific and statewide voting distributions.
This then permits the estimation of the equilibrium seat-vote curve by simply estimating these
moments.
The first step in establishing this is to provide expressions for the moments of the voting
distribution. Beginning with the district-specific moments, recall from the theoretical model that
the fraction of votes received by the Democrat candidate in district i is given by:
i
+ πIi [
Vi = πD
1/2 − (m − τ )
].
2τ
(14)
Given that m is uniformly distributed on [1/2 − ε, 1/2 + ε], the mean and standard deviation of
votes in district i are
i
i
i
+ 12 πIi = 12 [1 + πD
− πR
]
µi = E(Vi ) = πD
q
p
πIi2 var(m)
πi ε
σi = V ar(Vi ) =
= 2√I3τ
4τ 2
where the second expression uses the fact that var(m) = ε2 /3.3
(15)
Observe that the district
expected vote share (µi ) reflects a partisan advantage for the Democratic candidate (µi > 1/2) if
i
i
i
i
> πR
but for the Republican candidate (µi < 1/2) if πD
< πR
. The standard deviation of vote
πD
shares is proportional to the fraction of Independents. It is increasing in the degree to which the
median Independent shifts support between the two candidates from election to election (ε) but
is decreasing in the diversity of preferences among Independents (τ ). This latter property reflects
the fact that a small swing in preferences of the median Independent towards the Democrats
2 Data on voter identification are typically available from a variety of sources, such as survey data or, in some
states, voter registration rolls.
3 More generally, a random variable X distributed uniformly on the interval [a, b] has a mean equal to (a + b)/2
and a variance equal to (b − a)2 /12.
9
translates into a large shift in support for the candidate if the distribution of the preferences of
Independents is tight but a small shift if this distribution is disperse.
Turning to the statewide moments, we take cross-district averages of the district-specific means
and standard deviations to obtain:
µ = E(V ) = πD + 12 πI = 12 [1 + πD − πR ]
.
q 2
p
πI var(m)
π
Iε
√
σ = V ar(V ) =
=
4τ 2
2 3τ
(16)
The statewide expected Democratic vote is independent of the fraction of Independents, reflecting
the fact that these voters favor neither party ex-ante. The variance is again proportional to the
fraction of Independents, reflecting the fact that there is no uncertainty in how partisans vote.
Using the state-wide moments and equation (2), we can now write the maximum and minimum
aggregate Democratic vote shares as
V = V (1/2 + ε) = µ −
√
3σ
(17)
√
3σ.
(18)
and
V = V (1/2 − ε) = µ +
Moreover, using the district-specific and statewide moments and the definition of Vi∗ in equation
(5), we can write the vote threshold for electing a Democratic candidate in district i as follows:
Vi∗ = µ +
σ
(1/2 − µi ).
σi
(19)
These results permit the researcher to estimate the equilibrium seat-vote curve by estimating
the means and standard deviations of the Democratic vote share in each district and the associated
bi )1i=1/n
statewide mean and standard deviation. Suppose that these estimates are given by (b
µi , σ
and (b
µ, σ
b). Relabelling the districts so that
σ
b
σ
b
σ
b
(1/2 − µ
b1/n ) ≤ µ
b+
(1/2 − µ
b2/n ) ≤ .... ≤ µ
b + (1/2 − µ
b1 ),
σ
b1/n
σ
b2/n
σ
b1
√
√
the estimated equilibrium seat-vote curve is the function defined on the interval [b
µ − 3b
σ, µ
b + 3b
σ]
µ
b+
given by:
S(V ) = max{i : µ
b+
σ
b
(1/2 − µ
bi ) ≤ V }.
σ
bi
10
(20)
3.2
Estimating the optimal seat-vote curve
Turning to the normative component of the empirical analysis, we next develop measures of the
optimal seat-vote curve, thereby allowing for comparison with the equilibrium seat-vote curve.
Using the fact that πD − πR = 2µ − 1, we can re-write the optimal seat-vote curve described in
Proposition 1 as follows:
S o (V ) = 1/2 + (2µ − 1)(1/2 − τ ) + 2τ (V − 1/2).
(21)
Observe that the optimal seat-vote curve and, in particular, its responsiveness and partisan
bias parameters depend critically upon the diversity of preferences among Independents (τ ). Unfortunately, this preference parameter is not directly identified in the voting data. To see this,
recall that the statewide standard deviation of votes for the Democratic candidate can be written
as follows:
πI ε
σ= √ .
2 3τ
(22)
Thus, with only information on σ, we cannot separately identify the key underlying parameters
πI , ε, and τ . Intuitively, large swings in the Democratic vote share within a state could be due to a
large fraction of Independents (πI ), large swings in the preferences of the median Independent (ε),
or a tight distribution of ideology among Independents (τ ), in which case relatively small swings in
the preferences of the median Independent translate into relatively large swings in voting outcomes.
While we cannot separately identify these preference parameters, it is possible to identify their
ratio (α = ε/τ ), with data on the statewide fraction of Independents (πI ) and the statewide
standard deviation of the voting distribution (σ) as follows:
α=
√
2 3σ
ε
.
=
τ
πI
(23)
Further, using the theoretical restriction on the sum of these preference parameters + τ ≤ 1/2,
we can place an upper bound on the dispersion in the preferences of Independents:
τ≤
1
,
2(1 + α)
(24)
and, we can in turn place an upper bound on the optimal degree of responsiveness [ro = 2τ <
1/(1 + α)]. In the baseline analysis of the empirical application to follow, we assume that the
11
optimal degree of responsiveness equals this upper bound. In addition, as a robustness check, we
allow the optimal responsiveness to fall in a range below this upper bound.
3.3
Verifying the conditions for implementation
The necessary and sufficient condition for implementability presented in Proposition 2 can not
be verified directly without information on the underlying preference parameters (ε, τ ). As just
noted however, with outside information on the fraction of Independents, we can identify the
ratio α = ε/τ . We can use information on this ratio to place an upper bound on the coefficient
associated with the implementability of the optimal seat-vote curve. In particular, we can show
that
(
ε
ε
+ ε − (τ + ε) ln(1 + )) ≤ α/2,
2τ
τ
(25)
and this implies that a sufficient condition for implementability is:
πI ≤
2 min(πD , πR )
.
α
(26)
Clearly as the ratio α approaches unity, this sufficient condition converges to πI ≤ 2 min(πD , πR ),
which is just that given in Section 2.3. For any given α < 1, however, this ratio provides additional
information to the researcher attempting to verify the conditions for implementation. Substituting
in the expression for α from equation (23) and using the fact that µ = πD + πI /2, this sufficient
condition can be re-written as
πI ≤ 2 min(µ −
√
√
3σ, 1 − µ − 3σ).
(27)
Thus, the fraction of Independents must be below a critical value, the calculation of which only
requires information on the statewide mean and standard deviation of the Democratic vote share.
3.4
Estimating the welfare gains from socially optimal districting
Given that we only observe voting outcomes, which do not reveal the intensity of voter preferences
for one party over another, the parameters of the surplus expression (β, γ) in equation (12) are
clearly not identified in the empirical analysis. We can, however, use the theoretical restriction on
the ratio of these parameters in order to calculate the range of proportionate welfare gains.
To this end, first note that the percentage increase in aggregate welfare from socially optimal
districting can be written as follows:
12
∆G =
E[W (S o (V ))] − E[W (S(V ))]
E[W (S(V ))]
(28)
Then, using the expressions from equations (12) and (13) and dividing through by β, we have
that:
γ
o
2
γ
β E[(S (V ) − S(V )) ]
∆G( ) =
β
1 − βγ {c + E[S(V )2 ] − 2E[S(V )S o (V )]}
(29)
Recall from the theoretical section that the ratio γ/β is the fraction of the surplus a partisan
obtains from having a perfectly congruent legislature that is dissipated by having a legislature
composed entirely of the opposition party. When parties are not that polarized in terms of their
underlying ideologies or when the legislature is responsible for choosing only policies on which
there is little disagreement across ideologies (for example, spending on public safety and highway
maintenance) this ratio may be close to zero. When parties are polarized and are choosing policies
on which there is strong ideological disagreement (such as the level of transfer payments for the
poor or the regulation of abortion), this ratio may be close to one. In the former case, districting
is not very important, while in the latter case it is crucial to citizen welfare. Using the restriction
that γ/β ∈ [0, 1], we can thus bound these proportionate welfare gains as follows:
γ
0 ≤ ∆G( ) ≤ ∆G(1).
β
(30)
Because the upper bound will only be relevant for legislatures in states in which parties are
polarized and which choose policies on which there is strong disagreement, we will provide welfare
calculations for different values of this key ratio (γ/β) in the empirical application to follow.
We next turn to the measurement of this upper bound. Inserting equation (21) into equation
(29), we have that
( βγ ){(ξ o )2 + 2ξ o ro µ + (ro )2 (µ2 + σ 2 ) − 2E[(ξ o + ro V )S(V )] + E[S(V )2 ]}
γ
∆G( ) =
β
1 − ( βγ ){c + E[S(V )2 ] − 2ξ o E[S(V )] − 2ro E[V S(V )]}
(31)
where ro = 2τ represents optimal responsiveness and ξ o = µ(1 − 2τ ) represents the vertical
intercept of the optimal seat-vote curve.
Notice that we can express the constant as c = µ +
√
2 3τ σ 2
[ε /3
ε
+ τ 2 /3 − 1/4], and hence,
given a particular value of the ratio γ/β and the parameter τ , this expression can be evaluated by
computing three moments of the equilibrium seat-vote curve: E[S(V )], E[S(V )2 ], and E[V S(V )].
13
We next show that these three expectations can be expressed as functions of the district-specific
vote thresholds. To this end, let the associated range of average votes be [V , V ] and the districtspecific vote thresholds be (Vi∗ ), and let i be the smallest district such that Vi∗ ≥ V and i be the
largest district such that Vi∗ ≤ V .4 Then, using straightforward area calculations, we can convert
the three key expectations into functions of the district-specific vote threshholds as follows:
i+1/n
E[S(V )] =
X
i=i
(i − 1/n)
∆Vi∗
,
V −V
i+1/n
E[S(V )2 ] =
X
(32)
∆Vi∗
,
V −V
(33)
X 1
∆Vi∗2
(i − 1/n)
2
V −V
i=i
(34)
i=i
(i − 1/n)2
and
i+1/n
E[V S(V )] =
∗
∗
= V − Vi∗ , and ∆Vi∗ = Vi+1/n
− Vi∗ for i = i + 1/n, ...., i.
where ∆Vi∗ = Vi∗ − V , ∆Vi+1/n
4
Application to U.S. state legislatures
To illustrate the approach, we study redistricting plans for elections to U.S. state legislatures. We
prefer this setting over the U.S. House because, in federal redistricting, state officials control the
redistricting process and each redistricting plan thus only partially contributes to the resulting
allocation of national seats across parties in Congress. Redistricting plans for state legislatures are
also controlled by state officials, and redistricting plans thus perfectly correspond to changes in
seats in state legislatures.
In the empirical application, we examine state districting plans put in place during the early
1990s. For consistency with the theoretical framework, we focus on states with single-member
districts; many states elect multiple members from each district to state legislatures. In addition,
given the bicameral nature of state legislatures, we follow the existing empirical literature on
redistricting and focus on the elections to the lower house. As shown in Table 1, we have complete
data for 28 states that elect representatives from single-member districts, and most of these states
adopted redistricting plans in 1992 and then again in 2002; for these states, we thus examine state
4
As pointed out in Coate and Knight (2005), any seat-vote curve S(V ) may be equivalently described by a triple
{i, i, (Vi∗ )ii=i } - the so-called inverse seat-vote curve. The interpretation is that districts i = 1/n, .., i − 1/n are safe
Democrat; i = i + 1/n, .., 1 are safe Republican and Vi∗ is the vote threshold for competitive district i ∈ {i, .., i}.
14
legislative elections held in 1992, 1994, 1996, 1998, and 2000. States deviating from this pattern
of elections include Virginia, which has elections in odd years and adopted redistricting plans in
1991 and 2001, and Colorado, whose district lines were redrawn in 1998 following litigation over
the representation of minority groups in the state legislature.
4.1
Data sources
Recall from the empirical methodology section that our approach requires data on district-specific
voting returns and the statewide fraction of voters identifying as Independents. We use data from
Ansolabehere and Snyder (2002) on state legislative election returns, focusing only on returns
for the two major party candidates and computing support for the Democratic candidate as the
fraction of the two-party vote. We also use estimates of the fraction of Independent voters in each
state. These data are derived from annual New York Times surveys in which voters are asked to
self-identify as a Republican, Democrat, or Independent.5 In order to compute the time-invariant
fraction of Independents for each state, we take averages across the years listed in Table 1.
4.2
Estimating the moments of the voting distributions
Consider some legislative district i in a particular state. Suppose that the state has legislative
elections in 1992, 1994, 1996, 1998, and 2000. Assume that all these elections are contested in
district i and let the vote share received by the Democratic candidate in year t be Vit . Then, we
form our estimates of µi and σi by just taking the mean and standard deviation of these vote
5
These data were downloaded from the website http://php.indiana.edu/˜wright1/cbs7603 pct.zip.
15
shares; that is,6
P
µ
bi = t V5it
.
q
P (Vit −b
µi )2
σ
bi =
t
5
(35)
A problem that arises with this procedure is that many elections are in fact uncontested. In
our data, roughly 30 percent of state legislative elections are uncontested, and the support for the
Democratic candidate is thus censored for these elections. One potential solution to this problem
would be to simply assume that uncontested Democratic victories correspond to safe Democratic
seats and that uncontested Republican victories correspond to safe Republican seats. After all,
it is only the vote thresholds for competitive districts that matter for the seat-vote curve. The
primary drawback of this approach is that both the equilibrium and optimal seat-vote curves
depend upon the statewide moments (µ, σ), which require explicit measures of the district-specific
moments (µi , σi ) for both contested and uncontested districts.
To get around this problem, for districts with at least two contested races, we simply estimate
these two moments using voting data from these contested elections; that is, we use the formulas
in equation (35), but with a smaller number of elections. For the 25 percent of districts that have
either zero or only one contested race, we predict the district-specific moments by first regressing
the mean and standard deviation of vote shares in contested districts on voter characteristics taken
from Census data. These data, which are published in Barone et al (1998) are available for state
legislative districts and include the fraction of residents living in urban areas, the fraction living
in suburban areas, household income, percent of residents with a college degree, percent over age
65, percent African American, and percent Hispanic.
To implement this prediction approach, we assume that the mean Democratic vote share in
6 An obvious limitation of this approach is that because of the limited time dimension of the panel, our estimates
of the district-specific moments will be noisy. As an alternative approach, which we leave for future work, one
could estimate statistical models in which the moments of the voting distribution are parameterized in terms of
observed and unobserved district characteristics; the number of parameters to be estimated is smaller and hence
estimates of district-specific moments should entail less noise in this case. For example, using the definitions
of
√
the district-specific moments, we can re-write the district-specific vote as Vi = µi + σi w,where w = 3 (1/2 − m)
and is distributed uniform with mean 0 and variance 1. Adding a time dimension (t) to the model, we have that
Vit = µi + σi wt . Then, we parameterize the mean (µi = Xi0 β + ηi ) and variance (σi2 = exp(Xi0 γ + υi )) of the
voting distribution, where (ηi ,υi ) are district-specific random effects, which are normally distributed with variances
equal to ση2 and σν2 , respectively. Substituting in these parameterizations, we have that Vit = Xi0 β + ηi + uit ,
p
exp(Xi0 γ + υi )wt . Using standard panel data estimation techniques, the researcher could then
where uit =
estimate the parameters of the model (β, γ,ση2 and σν2 ). In this framework, the district-specific moments, and
hence the equilibrium seat-vote curve, are random from the econometrician’s point of view. Simulating the model
by drawing random effects from the normal distribution, the researcher could then make statements about features
of the distribution of the seat-vote curve and the associated welfare gains from socially optimal districting.
16
district i takes the following form, which restricts the value to be between zero and one:
µi =
exp(Xi0 β)
1 + exp(Xi0 β)
(36)
where Xi are voter characteristics and β represents parameters to be estimated. For estimation
purposes, this equation can be written in log-odds form:
ln(µi /1 − µi ) = Xi0 β
(37)
Similarly, we parameterize the standard deviation of the voting distribution as follows:
ln(σi ) = Xi0 γ
(38)
In order to increase the power associated with this prediction approach, we run this regression
across all contested districts in the United States, implicitly assuming that the mapping from
observed characteristics into partisan affiliation is homogeneous across states. While this assumption is somewhat restrictive, we do include a set of state dummy variables, thereby allowing two
districts with identical observable characteristics but in different states to have different fractions
of voter types.
Table 2 provides the results from these two prediction equations. As shown in column 1,
the partisan advantage for the Democratic party (µi ) in contested districts is increasing in the
percent urban and suburban (both of these are relative to the omitted category - percent rural),
percent with a college degree, percent over age 65, percent African American, and percent Hispanic
but is decreasing household income. Interestingly, the coefficients from the regression of these
covariates on the standard deviation of the voting distribution, which is proportional to fraction
of Independent votes (µi ), are mostly reversed. As shown in column 2, the likelihood of being an
Independent voter is decreasing in all of the observable measures.
These coefficients and state-specific constant terms, which are not reported in Table 2, are
then used in order to predict these two moments in uncontested districts. With our measured
district-specific moments in contested districts and predicted moments in uncontested districts,
we can now estimate the statewide moments in a straightforward manner:
n
µ
b =
1X
µ
bi
n i=1
17
(39)
n
σ
b
=
1X
σ
bi
n i=1
With estimates of the district-specific and statewide voting moments in hand, we proceed to
estimation of the equilibrium seat-vote curve.
4.3
Equilibrium seat-vote curves
Figure 1 plots the equilibrium seat-vote curve as estimated for each state in the sample. As shown,
the range of possible statewide support for the Democratic Party seems reasonable. For example,
in New York, typically considered a heavily Democratic state, the party receives support on the
range [0.57,0.72], while in Utah, typically considered a heavily Republican state, the Democrats
receive support on the range [0.34,0.51]. Notice also that the equilibrium seat-vote curve is close
to linear in some states, such as Mississippi, New York, and Ohio, while it has important nonlinearities in other states, such as Delaware, Florida, and Montana. These differences in the degree
of non-linearity across states highlight the importance of allowing for a general functional form in
estimating equilibrium seat-vote curves.
Column 1 of Table 3 reports a measure of the responsiveness of each of these equilibrium seatvote curves. This measure is obtained by computing the slope of the linear seat-vote curve that
best approximates the equilibrium seat-vote curve.7
Notice that in every state, responsiveness
exceeds 1, and, in the vast majority of states, it exceeds 2.
Column 3 of Table 3 reports the partisan bias associated with the equilibrium seat-vote curves.
This is measured by S(1/2) − 1/2 and is the difference between the fraction of seats that the
Democrats hold when they get half the statewide vote and 1/2. Notice that in the four states
for which V = 1/2 does not lie in the range of possible Democratic vote shares, this measure
cannot be computed. The interesting thing to note is that in all but three states, the equilibrium
seat-vote curve is biased towards the Republicans. The average bias of 4.27 percent implies that,
when voters are equally split, Republicans would secure 54.27 of the seats on average, relative to
45.73 percent for Democrats, and would thus hold a significant advantage of 8.54 percent in the
legislature.
7 To be more precise, define the linearized equilibrium seat-vote curve to be S l (V ) = 1/2 + b + r(V − 1/2), where
b and r are chosen in order to minimize the expected square distance between the equilibrium and the linearized
equilibrium seat-vote curves. That is: r =cov[V, S(V )]/var(V ) and b = E[S(V )] − 1/2 − r[E(V ) − 1/2]. Then our
measure of responsiveness is r.
18
4.4
Optimal seat-vote curves
For comparison purposes, Figure 1 also includes the optimal seat-vote curves. These are plotted
under the assumption that responsiveness, 2τ , is at its maximal level 1/(1 + α) in each state
(see equation (24)). It is apparent from the Figures, that the equilibrium seat-vote curve is more
responsiveness than the optimal seat-vote curve in all cases, suggesting that, if anything, U.S.
state legislative elections are too competitive and that there are too few safe seats from a social
welfare perspective! Column 2 of Table 3 reports the responsiveness of the optimal seat-vote
curve. As noted in the theoretical section, optimal responsiveness is always below one, while the
responsiveness associated with the equilibrium seat-vote curves substantially exceeds one in all
cases.
Column 4 of Table 3 reports the partisan bias associated with the optimal seat-vote curve,
which is defined as S o (1/2) − 1/2. It is notable that in all but two states the bias associated with
the equilibrium seat-vote curve is less than that associated with the optimal seat-vote curve. Thus,
the partisan bias towards the Republicans exhibited in the equilibrium seat-vote curves, cannot
be justified as “optimal partisan bias”. Indeed, in twelve states, the equilibrium seat-vote curve
is biased towards the Republicans, when it should optimally be biased towards the Democrats!
A reasonable objection to this comparison of partisan bias under the optimal and equilibrium
seat-vote curves is that it just tells us about the properties of the curves at the vote share V =
1/2. For a more global comparison, we computed the expected Democratic seat share (E(S(V )))
under the equilibrium and optimal seat-vote curves (E(S(V )) and E(S o (V ))). Column 6 of Table
3 reports the difference in this expected seat-share under the equilibrium and optimal curves.
When this difference is positive, the expected Democratic seat share is higher than optimal. The
interesting thing to note is that, in this expected seat sense, there appears to be no obvious bias in
one or the other direction. In almost half the states, the expected Democratic seat share is higher
than optimal and the average difference is very close to zero. This makes us hesitant to draw any
strong conclusions concerning the general direction of bias in U.S. state legislative elections.
4.5
Verifying the conditions for implementation
As shown in Table 4, the conditions for implementation, the restriction that the fraction of Independents must be below a maximal level is satisfied in every state. The state closest to not
satisfying the requirements is Rhode Island, which is reported to have 51% Independents, just
19
slightly below the maximal level of 55%. The next closest state is Connecticut, which is reported
to have 42% Independents, relative to the maximal level of 65%. In summary, these results
demonstrate that the conditions for implementability of the optimal seat-vote curve are indeed
permissive, being satisfied in all of 28 states included in our analysis and by a large margin in all
cases except for the state of Rhode Island.8
4.6
Welfare gains
Given that the optimal seat-vote curve is implementable in all states, we next turn to measuring
the welfare gains associated with socially optimal districting. To begin with, we compute the
percentage welfare gains under the assumptions that the ratio γ/β is at its maximal level (i.e., 1)
and that optimal responsiveness is at its maximal level in each state (i.e., 1/(1 + α)). As shown
in Table 5, the percentage welfare gains to socially optimal districting vary from just 0.31% in
Pennsylvania to 6.5% in Rhode Island. A visual comparison of the equilibrium and optimal seatvote curves in Figure 1 supports these welfare calculations as the optimal and equilibrium seat-vote
curves are similar in states with low potential welfare gains but quite different in states with large
potential welfare gains.9
On average, states could increase welfare by roughly 2%, suggesting
that the gains from socially optimal districting are typically small relative to the surplus that is
generated by state legislatures. Of course, if this surplus is itself very large, then these gains could
be quite large in monetary terms. But without further assumptions on the underlying welfare
parameters β and γ, these percentage gains in welfare cannot be converted into monetary terms.
For a sense of how these results depend on the specific assumptions about the parameters
γ/β and 2τ , Table 6 reports the average welfare gains associated with socially optimal districting
8 Note that in evaluating the conditions for implementation, we have not used external information on the
statewide fraction of Democrats and Republican. It is comforting to note, however, that the implied fraction of
partisans from voting behavior are highly correlated with the fraction of voters self-reporting as partisans. Using
the fact that the implied statewide fraction of Democrats and Republicans are given by πD = µ − πI /2 and
πR = 1 − µ − πI /2, we calculate correlation between the implied and reported fraction of voters of roughly 0.8 for
both Republicans and Democrats.
9 While a welfare comparison of the optimal seat-vote curve, which is continuous, and the measured seat-vote
curve, which is a discrete step-function, is somewhat artificial, we do so in order to apply the results from the
companion theoretical paper, which assumed a continuous seat-vote curve, in order to verify that the optimal seatvote curve is implementable. To provide a sense of the error associated with this approximation, we have derived a
discrete optimal seat-vote curve, which is a step-function approximation of the continuous optimal seat-vote curve.
Welfare associated with this discrete optimal seat-vote curve is very similar to welfare under the continuous optimal
seat-vote curve, and the approximation error associated with the use of a continuous optimal seat-vote curve is
thus small in practice. The small size of this error should not be surprising given that the discrete and continuous
optimal seat-vote curve converge as the number of districts grows large. As shown in table 1, states have a large
number of districts, averaging over 100 districts and ranging from 41 in Delaware to 203 in Pennsylvania.
20
arising under different parameter values. We allow the ratio γ/β to vary from 0.25 to 1 and allow
2τ to vary from 0.25/(1 + α) to 1/(1 + α) in each state. The notation η refers to the numerator
in the ratio η/(1 + α), so that the case in which 2τ is set equal to 0.25/(1 + α) in each state
corresponds to η = 0.25; the case in which 2τ is set equal to 0.5/(1 + α) in each state corresponds
to η = 0.5; etc. As shown, holding the ratio γ/β constant, the welfare gains to socially optimal
districting, averaged across all states, are uniformly increasing as the parameter η is reduced.
This pattern reflects the fact that reductions in the parameter η are associated with reductions
in optimal responsiveness, which, as shown in Table 3, was already below the responsiveness
associated with the equilibrium seat-vote curve in the baseline analysis. Holding the parameter η
constant and reducing the ratio γ/β, we have that welfare is uniformly decreasing. As explained
above, districting matters less for welfare as this ratio decreases, reflecting the fact that the conflict
between parties over the available policy choices is less severe.
Perhaps the most noteworthy lesson to be drawn from Tables 5 and 6 is that the welfare
gains from socially optimal districting are relatively small as a proportion of the total surplus
generated by state legislatures. In principle, there may be two reasons for this. The first is that
the districting plans that states actually implement are relatively close to optimal plans. The
second is that, because of the diverse ideological make up of the U.S. states, the aggregate gains
that can be obtained by tinkering with districting plans are relatively small. To get a feel for
which of these views is correct, we have computed the proportionate welfare gains that would
arise from implementing the optimal seat-vote curve over the welfare minimizing seat-vote curve.
We take the latter to be the seat-vote curve that makes the seat share of the party with the
smallest partisan base equal to 1.10
The idea is that if these gains are small, then the second
view is correct. Table 7 reports the results under the same parametric assumptions that underlie
Table 5. They strongly suggest that it is the first view that is correct: the states seem to be doing
districting in a way that is generating seat-vote curves that are relatively close to optimal.
A further interesting benchmark of comparison is the seat-vote curve that would be generated
by a proportional representation electoral system. Under such a system, seats are allocated in
strict proportion to statewide votes and hence the implied PR seat-vote curve is S(V ) = V .
As shown in Column 1 of Table 8, all states would experience a welfare gain by implementing
10 An important objection to this, is that such a seat-vote curve may not be implementable. We agree with this
objection, but time constraints have prevented the computation of the welfare minimizing implementable seat-vote
curve.
21
the PR seat-vote curve, relative to the welfare level experienced under the equilibrium seat-vote
curve. This result in part reflects the fact that the PR seat-vote curve has responsiveness of 1,
while responsiveness associated with the equilibrium seat-vote exceeds 1 in all cases. As noted
previously, optimal responsiveness is less than 1. Moreover, the PR seat-vote curve is linear, a
feature shared by the optimal seat-vote curve. Perhaps more surprisingly, however, the additional
gains from implementing the optimal seat-vote curve, relative to PR, are very small. As shown
in Column 2 of Table 8, the welfare gains to optimal districting from PR vary between 0.01 and
0.11 percent.
5
Relation to the literature
As noted in the introduction, there is a long history of estimating seat-vote curves, the key
construct in our empirical analysis, in the existing literature. Early studies used the uniform
partisan swing model, first developed by Butler (1951); this method uses district-specific voting
returns and then uncovers a seat-vote curve by simulating how the distribution of seats changes
as national support for the Democrats increases and decreases, implicitly assuming that districtspecific votes move uniformly. For example, if national support for Democratic candidates was
recorded at 55 percent, consider an increase in national support to 60 percent. The approach
assumes that support for Democratic candidates in each district would increase by 5 percentage
points so that seats won by Republicans with between 45 and 50 percent of the recorded vote
would be predicted to flip to the Democrats. This approach was later critiqued for both ignoring
district-specific factors and for not restricting district vote shares to be between zero and one
(King, 1989).
Other early studies, such as Tufte (1973), posit a linear seat-vote curve and estimate the
intercept and slope of this curve by regressing the number of seats for Democrats in a state on
their share of the vote across all districts. One critique of this approach is that the regression line
is not constrained to pass through the points (0,0) and (1,1). To address this concern, King and
Browning (1987) develop a non-linear bilogit model:
S(V ) =
1
.
V
1 + exp(−b − r ln( 1−V
))
(40)
This curve is characterized by the bias parameter (b) and responsiveness parameter (r) and, in
22
contrast to the linear seat-vote curve, is constrained to pass through the points (0,0) and (1,1).
Gelman and King (1990, 1994) advance the literature by developing an approach that does
not rely on particular functional form assumptions. They instead begin with an assumption over
the statistical distribution of the vote-generating process and then estimate the parameters of this
distribution. Using these parameter estimates, they simulate how seats change given hypothetical
changes in support for parties. In their initial paper, they posit a normal distribution for underlying
support for the Democratic party (ui ∼ N (αi , σ 2 ) in district i, where this underlying support
is related to votes (Vi ) according to Vi = exp(ui )/(1 + exp(ui )). Importantly, the variance of
underlying support is assumed to be constant across districts. The authors then make assumptions
over the distribution over the district effect αi and estimate the parameters of that distribution.
In a follow-up paper, Gelman and King (1994) incorporate observable candidate characteristics
(Xi ), such as incumbency status and party control of the district, and estimate the parameters of
the following regression equation:
Vi = Xi β + ui .
(41)
where ui ∼ N (0, σ 2 ). Again, the variance in support for the Democratic party is assumed to be
constant across districts.
Given our goal of evaluating districting plans from a welfare perspective, our approach begins
with underlying citizen preferences over policies, and hence candidates, rather than a functional
form for the seat-vote curve or an assumed vote-generating process. These preferences and associated voting behavior in turn lead to a vote-generating process and ultimately a seat-vote curve.
Similarly to Butler, our approach does not incorporate a district-specific preference shock. It is
important to note, however, that our approach assumes that only preferences, and not necessarily
votes, swing in a uniform manner. As shown in equation (1), these preference shocks translate
into vote shares for the Democrat candidate heterogeneously and in proportion to the fraction of
Independent voters. Moreover, our approach, unlike Butler’s, restricts district-specific vote shares
to be between 0 and 1, as can also be seen in equation (1). Relative to the work of Gelman and
King (1990, 1994), there are two key differences in our approach. First, while our approach allows
for the variance of the vote-generating process to vary across districts, their approach assumes
that this variance is constant across districts; in the context of our model, this assumption would
require that all districts have an equal fraction of Independent voters. Second, in explaining differences in voting patterns across districts, Gelman and King (1994) rely on observable candidate
23
characteristics; this approach is inappropriate for our framework, in which candidates within a
party are homogenous across districts and thus differences in voting outcomes across districts can
be generated only by differences in voter preferences. Developing an approach that incorporates
both candidate and voter characteristics would be desirable, and while beyond the scope of the
current analysis, we hope to incorporate such heterogeneity in candidate characteristics, such as
incumbency and race, in future extensions of this methodology.
The only other work that we are aware of that explores the micro-foundations of seat-vote
curves and their significance for welfare is the independent work of Besley and Preston (2005).
They develop an alternative micro-founded model that generates an equilibrium relationship between seats and votes. They use their model to solve for what the distribution of voter types
must be across districts if the equilibrium seat-vote curve is to be of the bilogit form. Their main
theoretical point is to show that this distribution, and hence the shape of the seat-vote curve, is
a key determinant of parties’ electoral incentives to put in effort on the part of their constituents.
They provide empirical evidence in favor of their theory by showing that local government performance in the U.K. is related to the parameters of the local seat-vote curve in the way the theory
suggests. Their work therefore suggests a novel theoretical mechanism why the form of the seatvote curve (and hence districting) matters for citizens’ welfare and provides evidence for this. By
contrast, our model reflects the conventional view that districting matters because it determines
which party gets the most seats and hence the ideological composition of the legislature. Clearly,
it would be desirable to develop an analysis in which both forces are present.
While our methods differ somewhat from those in the existing literature, it is important to note
that our substantive findings are in many respects quite similar. Our finding that responsiveness
associated with the equilibrium seat-vote curve averages 2.36 is similar to that found in the existing
literature. Although using data from a slightly earlier period, Gelman and King (1994) examine
legislative elections during the 1970s and 1980s for a similar set of U.S. states and report an average
responsiveness of just below two. Both of these estimates appear to be at the low end of the range
of estimated responsiveness parameters in the literature, which has found parameters between two
and four, with three being a focal point, a finding that has become known as the “cube law” (King
and Browning, 1987). Regarding partisan bias, our results are again qualitatively similar to those
in the existing literature: Gelman and King (1994) report bias parameters that average roughly
3 percent in absolute value across all states and years, while we find that bias associated with
24
the equilibrium seat-vote curve averages roughly 5 percent across states. However, while Gelman
and King find that on average the direction of partisan bias favors the Democrats, we find that
it favors the Republicans. This discrepancy may be explained by the fact that we are using data
from the 1990s districting period. In summary, the general similarity in our findings to those in
the existing literature suggest that the social welfare implications of our results may be robust to
alternative datasets and approaches to estimating equilibrium seat-vote curves and are not being
driven by anomalous estimated bias and responsiveness parameters.
6
Conclusion
This paper has explored the development of a methodology for comparing actual and optimal
districting plans and evaluating the welfare gains from socially optimal districting. We start by
showing that the equilibrium seat-vote curve, which is generated by actual districting plans, can
be expressed as a function of the district-specific and statewide moments, namely the mean and
variance, of the voting distribution, suggesting an empirical approach that only requires estimation
of these first two moments. We then develop empirical expressions for the optimal seat-vote curve,
which can be uncovered using the statewide moments of the voting distribution along with data on
the fraction of Independent voters across all districts. In addition, the conditions for implementing
the optimal seat-vote curve can be expressed as an upper bound on the fraction of Independent
voters and can thus be verified using similar data on these statewide moments and the fraction
of Independent voters. Finally, we develop expressions for the welfare gains from socially optimal
districting.
To illustrate the approach, this methodology is then applied to voting returns from U.S. state
legislatures during the 1990s. We find that responsiveness tends to be too large, relative to optimal
responsiveness, suggesting that, if anything, U.S. state legislative elections are too competitive and
that there are too few safe seats from a social welfare perspective. Encouragingly, the conditions
for implementing the optimal seat-vote curve are permissive and are satisfied in all states and by
a large margin in most states. While there is significant variation across states, the welfare gains
to socially optimal districting, relative to the overall surplus generated by state legislatures, are
on average quite small. This appears to be because state districting plans are reasonably close
to optimal rather than because there are little aggregate gains to be had from varying districting
plans. We also find that implementing proportional representation would lead to welfare gains
25
in all states and would produce welfare levels quite close to those achieved with socially optimal
districting.
While we feel that this approach provides a significant advance forward, we recognize its
limitations and hope to make advances both in terms of statistical techniques as well as in the
incorporation of several important real-world features of U.S. legislatures in future extensions
of the methodology. Regarding alternative statistical techniques, one step forward would be
the incorporation of statistical models for estimating the district-specific moments of the voting
distribution. This approach would be similar to that used Gelman and King (1990,1994) and
would require simulation of both the positive features, such as the equilibrium seat-vote curve,
and normative features, such as the welfare gains from socially optimal districting, of the analysis.
Second, in order to better fit the voting data, it would be useful to incorporate a district-specific
shock in the preferences of Independent voters into the empirical approach. With a district-specific
shock, district-specific support for parties is no longer a deterministic function of statewide support
for parties, as expressed in equation (4), and the positive and normative analyses will thus need
to modified accordingly. Regarding institutional features of legislatures, it would be interesting
to incorporate incumbency into the framework; this extension is challenging as it would obviously
require a dynamic political economy model. A second extension would involve race, a key focus
of current policy debates over representation and the redistricting process. One could consider
a model in which voters and candidates are characterized by both their ideology and their race.
Achieving the optimal racial representation and the optimal ideological representation in the
legislature may not be feasible and thus there may be a trade-off between race and ideology.
Finally, this model could be used to analyze partisan gerrymandering, under which districting
officials aim to further partisan, rather than social, objectives. This positive approach may provide
further insights into the differences between optimal and equilibrium seat-vote curves documented
in this paper as well as the sources of the measured welfare gains from socially optimal districting.
26
References
Ansolabehere, Stephen and James Snyder, [2002], “The Incumbency Advantage in U.S.
Elections: An Analysis of State and Federal Offices: 1942-200,” Election Law Journal, 1,
315-338.
Barone, Michael, William Lilley III, and Laurance DeFranco, [1998], State Legislative Elections: Voting Patterns and Demographics, Washington D.C: Congressional Quarterly.
Besley, Timothy and Ian Preston, [2005], “Electoral Bias and Policy Choices: Theory and
Evidence,” mimeo, London School of Economics.
Coate, Stephen and Brian Knight, [2005], “Socially Optimal Districting,” NBER working
paper 11462.
Gelman, Andrew and Gary King, [1990], “Estimating the Electoral Consequences of Legislative Redistricting,” Journal of the American Statistical Association, 85, 274-282.
Gelman, Andrew and Gary King, [1994], “Enhancing Democracy through Legislative Redistricting,” American Political Science Review, 88(3), 541-559.
King, Gary, [1989], “Representation through Legislative Redistricting: A Stochastic Model,”
American Journal of Political Science, 33(4), 787-824.
King, Gary and Robert Browning, [1987], “Democratic Representation and Partisan Bias in
Congressional Elections,” American Political Science Review, 81(4), 1251-1273.
Tufte, Edward, [1973], “The Relationship Between Seats and Votes in Two-Party Systems,”
American Political Science Review, 67(2), 540-554.
27
7
Appendix
Proof of Proposition 1: Note first that ∂ 2 W (S; m)/∂S 2 < 0, so that W (·; m) is strictly concave.
Thus, the first order condition is sufficient for S o to be optimal. Differentiating equation (7) yields
∂W (S o ; m(V ))/∂S = 2γ{πD + πI (1 − m) − S}.
Clearly, ∂W (S; m)/∂S = 0 if and only if
S = πD + πI (1 − m).
Substituting in the expression for m(V ) from (3), we obtain
S o (V ) = 1/2 + (πD − πR )(1/2 − τ ) + 2τ (V − 1/2)
as required. QED
Proof of Lemma: From (7), aggregate welfare when the median Independent has ideology m
and the Democrats have seat share S is given by:
W (S; m) = β − γ[πD (1 − S)2 + πR S 2 + πI
Using the fact that
R m+τ
m−τ
Z
m+τ
m−τ
(1 − S − x)2
dx
].
2τ
2
2
(1 − S − x)2 dx
2τ = (1 − S − m) + τ /3, we can re-write welfare as follows:
W (S; m) = β − γ{πD (1 − S)2 + πR S 2 + πI [(1 − S − m)2 + τ 2 /3]}
Reorganizing terms, we have that:
W (S; m) = β − γ{S 2 − 2S[πD + πI (1 − m)] + πD + πI [(1 − m)2 + τ 2 /3]}
Using the fact that πD + πI (1 − m) = S o , we have:
W (S; m) = β − γ{S 2 − 2SS o + πD + πI [(1 − m)2 + τ 2 /3]}
Finally, taking expected values with respect to V and using the fact that E[(1 − m)2 ] = E(m2 ) =
1/4 + ε2 /3, we have that:
E[W (S; m)] = β − γ{c + E(S 2 ) − 2E(SS o )}
where the constant is given by c = πD + πI [1/4 + ε2 /3 + τ 2 /3]. QED
28
.8
DE
.6
.7
.45
.55 .6 .65
.4
.5
.55
.2
.4
.5
.6
.7
.6
.8
.5
.6
KY
.6
.4
.6
.4 5
.5
.55
.6
.4
.4 5 .5 .55 .6
.2
.4
.2
.2
.2
.4
.4
.4
.4
KS
.6
.6
.45
IL
.8
IA
.8
FL
.5
1
.65
.8
.6
.8
.5 5
.2
.4
.5
.3
.4
.4
.5
.4
.6
.5
.7
.6
.6
.6
.7
.9
.8
CT
.8
CO
.7
CA
.8
AL
.45 .5 .55 .6 .65
.35 .4 .45 .5 .55
Figure 1a: Seat-Vote Curves
S( V)
Graphs by state
Optimal S(V)
.45
.5
.55
.6
MT
.6
.8
.6
.7
.2
.6
.3
.2
.4
.4
.4
.4
.6
.5
.8
.6
MS
.8
MO
.9
MI
.8
ME
.5
.6
.7
.45
.5 5
.6
.45 .5 .5 5 .6 .65
NV
1
.6
.65
.7
.35 .4 .45 .5 .55
OH
OK
.7
.6
.45
.5
.55
.6
.4
.6
.55
.6
.65
.7
.2
.3
.6
.2
.4
.5
.4
.4
.4
.6
.6
.5
.7
.8
.8
.8
.55
NY
.8
NM
.5
.6
.4
.4
.5
Figure 1b: Seat-Vote Curves
S( V)
Graphs by state
Optimal S(V)
.6
.45 .5 .55 .6 .65
1
.6
.4
.45
.5
.5
.4
.4
.6
Figure 1c: Seat-Vote Curves
S( V)
Graphs by state
.55
.6
.3 .4 .5 .6 .7
.8
.6
.4
.2
.35
.5
WI
.3
.1
.2
.4 .5 .6
.45 .5 .55 .6 .65
.4 5
VA
.4
.7 .8
.55 .6 .65 .7 .75
UT
.5
TN
.5
.3
.4
.3
.4
.6
.6 .7 .8 .9
.5
.6
.4
.2
.4 .45 .5 .55 .6
SC
.7
RI
.7
PA
.6
.8
OR
Optimal S(V)
.4 .45 .5 .55 .6
Table 1: States and Years Included in Analysis
state
AL
CA
CO
CT
DE
FL
IA
IL
KS
KY
ME
MI
MO
MS
MT
NM
NV
NY
OH
OK
OR
PA
RI
SC
TN
UT
VA
WI
first redistricting
1994
1992
1992
1992
1992
1994
1992
1992
1992
1996
1994
1992
1992
1995
1994
1992
1992
1992
1992
1992
1992
1992
1992
1992
1994
1992
1991
1992
subsequent redistricting
2002
2002
1998
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
2002
1998
2002
2002
2001
2002
number districts
105
80
65
151
41
120
100
118
125
100
151
110
163
122
100
70
42
150
99
101
60
203
100
124
99
75
100
99
Table 2: Predicting Moments of Voting Distribution for Uncontested Districts
(all regressions include state-specific constant terms, standard errors in parentheses)
Dependent Variable
percent urban
percent suburban
household income (thousands)
percent with college degree
percent over age 65
percent African American
percent Hispanic
R-squared
Number of observations
ln(µ/1−µ)
0.2744
(0.0357)
0.1533
(0.0376)
-0.0152
(0.0016)
0.7805
(0.1624)
1.4364
(0.1960)
2.7355
(0.0876)
1.6590
(0.1441)
0.5689
2287
ln(σ)
-0.1313
(0.0539)
-0.1107
0.0569
-0.0020
(0.0024)
-0.5399
(0.2457)
-0.2967
(0.2965)
-1.2195
(0.1325)
-0.3654
(0.2180)
0.0971
2287
Table 3: Properties of Equilibrium and Optimal Seat-Vote Curves
state
AL
CA
CO
CT
DE
FL
IA
IL
KS
KY
ME
MI
MO
MS
MT
NM
NV
NY
OH
OK
OR
PA
RI
SC
TN
UT
VA
WI
Average
r
3.3018
1.5723
2.3034
2.0138
2.0933
2.9840
3.2076
1.6746
2.2100
4.9837
2.9785
1.4308
2.4998
2.5952
2.9718
2.1481
2.4760
1.2115
1.4504
3.4141
2.4147
1.3501
1.5070
2.3961
2.7522
1.8733
2.1171
2.0200
2.3554
optimal r
0.6947
0.6055
0.6823
0.6190
0.6007
0.6227
0.6867
0.6352
0.5938
0.5906
0.6557
0.6760
0.6570
0.7206
0.6890
0.6223
0.6027
0.6708
0.6199
0.5066
0.6109
0.5489
0.7329
0.6790
0.6903
0.6602
0.6397
0.6739
0.6424
b
n/a
-1.25%
-3.85%
-2.32%
-10.98%
-10.83%
-4.00%
-5.08%
-7.60%
-8.00%
0.33%
-8.18%
-7.06%
n/a
-6.00%
4.29%
11.90%
n/a
-7.58%
-5.45%
-1.67%
-2.71%
n/a
-9.68%
-2.53%
-4.67%
-2.00%
-7.58%
-4.27%
optimal b difference in expected seats
n/a
10.37%
1.24%
2.37%
-1.45%
-5.26%
1.68%
3.01%
-0.54%
-8.32%
0.66%
-3.63%
-0.24%
-3.51%
1.94%
-1.98%
-1.72%
-6.90%
1.04%
5.57%
0.55%
2.08%
0.11%
-4.06%
1.03%
1.52%
n/a
13.62%
-1.79%
-15.49%
0.74%
6.24%
0.77%
8.39%
n/a
6.95%
0.10%
-5.94%
1.67%
6.47%
-0.03%
-4.40%
0.77%
-1.18%
n/a
22.55%
-0.01%
-3.55%
1.32%
3.51%
-3.05%
-12.58%
0.44%
-1.24%
-0.23%
-4.32%
0.21%
0.37%
Table 4: Conditions for Implementability
state
AL
CA
CO
CT
DE
FL
IA
IL
KS
KY
ME
MI
MO
MS
MT
NM
NV
NY
OH
OK
OR
PA
RI
SC
TN
UT
VA
WI
% independents
29.44%
26.01%
39.12%
41.89%
36.20%
27.59%
40.46%
33.64%
31.18%
22.55%
45.71%
35.16%
37.15%
24.94%
36.36%
29.43%
29.78%
30.92%
32.41%
18.79%
30.99%
24.59%
51.19%
33.02%
33.94%
33.91%
34.42%
36.43%
maximum % independents
65.13%
71.89%
76.70%
64.83%
74.87%
79.51%
81.01%
69.07%
70.90%
78.80%
72.72%
75.95%
72.87%
64.07%
73.05%
77.12%
74.46%
56.84%
78.63%
74.59%
80.07%
76.03%
54.72%
81.60%
75.27%
67.75%
77.76%
80.55%
Table 5: Welfare Gains to Optimal Districting
state
AL
CA
CO
CT
DE
FL
IA
IL
KS
KY
ME
MI
MO
MS
MT
NM
NV
NY
OH
OK
OR
PA
RI
SC
TN
UT
VA
WI
Average
% welfare gains
2.59%
0.37%
1.28%
1.47%
2.40%
2.02%
2.40%
0.54%
2.03%
5.82%
3.33%
0.40%
1.37%
2.69%
4.76%
1.28%
2.48%
0.66%
0.73%
3.70%
1.60%
0.31%
6.50%
0.97%
1.21%
2.47%
0.95%
0.83%
2.04%
Table 6: Welfare Gains Under Alternative Parameter Values
η=1
η=0.75
η=0.50
η=0.25
γ/β=1
2.04%
2.22%
2.43%
2.66%
γ/β=0.75
1.44%
1.57%
1.72%
1.88%
γ/β=0.50
0.91%
0.99%
1.09%
1.19%
γ/β=0.25
0.43%
0.47%
0.52%
0.56%
Table 7: Welfare Gains From Worst Seat-Vote Curve
state
AL
CA
CO
CT
DE
FL
IA
IL
KS
KY
ME
MI
MO
MS
MT
NM
NV
NY
OH
OK
OR
PA
RI
SC
TN
UT
VA
WI
Average
% welfare gains (relative to worst S(V))
82.79%
62.22%
50.08%
55.86%
45.15%
50.22%
43.85%
61.18%
55.50%
53.84%
46.58%
53.85%
54.27%
95.31%
58.84%
51.68%
52.93%
97.02%
46.27%
57.09%
44.79%
50.76%
85.82%
47.92%
57.70%
66.74%
47.77%
46.09%
57.93%
Table 8: Welfare Gains to PR
state
AL
CA
CO
CT
DE
FL
IA
IL
KS
KY
ME
MI
MO
MS
MT
NM
NV
NY
OH
OK
OR
PA
RI
SC
TN
UT
VA
WI
Average
% welfare gains from implementing PR
2.57%
0.32%
1.24%
1.37%
2.28%
1.98%
2.37%
0.49%
1.95%
5.78%
3.25%
0.37%
1.33%
2.68%
4.73%
1.23%
2.40%
0.63%
0.66%
3.61%
1.54%
0.22%
6.48%
0.94%
1.18%
2.43%
0.90%
0.80%
1.99%
% welfare gains to optimal districting from PR
0.02%
0.05%
0.04%
0.10%
0.11%
0.04%
0.04%
0.05%
0.08%
0.04%
0.07%
0.03%
0.05%
0.01%
0.03%
0.05%
0.08%
0.03%
0.06%
0.09%
0.06%
0.09%
0.03%
0.03%
0.03%
0.04%
0.05%
0.04%
0.05%