Download Optimal Campaigning in Presidential Elections: The Probability of

Optimal Campaigning in Presidential Elections:
The Probability of Being Florida
David Strömberg∗
IIES, Stockholm University
December 8, 2002
Abstract
This paper delivers a precise recommendation for how presidential candidates
should allocate their resources to maximize the probability of gaining a majority
in the Electoral College. A two-candidate, probabilistic-voting model reveals that
more resources should be devoted to states which are likely to be decisive in the
electoral college and, at the same time, have very close state elections. The optimal strategies are empirically estimated using state-level opinion-polls available
in September of the election year. The model’s recommended campaign strategies
closely resemble those used in actual campaigns. The paper also analyses the policy
effects of electoral reform. It finds that resources will be more equally distributed
under Direct Vote than under the Electoral College system, and that both systems
discipline presidents from extracting political rents to an equal extent.
∗
[email protected], IIES, Stockholm University, S-106 91 Stockholm.
I thank Steven Brams, Steve Coate, Antonio Merlo, Torsten Persson, Gerard Roland, Tom Romer, Howard
Rosenthal, Jim Snyder, Jörgen Weibull, and seminar participants at UC Berkeley, Columbia University,
Cornell University, the CEPR/IMOP Conference in Hydra, Georgetown University, the Harvard / MIT
Seminar on Positive Political Economy, IIES, New York University, Stanford University, University of
Pennsylvania, Princeton University, and the Wallis Conference in Rochester. Previous versions have
been circulated under the titles: ”The Lindbeck-Weibull model in the Federal US Structure”, and, ”The
Electoral College and Presidential Resource Allocation”s.
JEL-classification, D72, C50, C72, H50, M37.
Keywords: elections, political campaigns, public expenditures
1. Introduction
The President of the United States is arguably the worlds most powerful political leader,
and the incentives created by his electoral procedure are important. In consequence,
it is not surprising that the Electoral College system1 has been under constant debate.
According to federal historians, there have been more proposals for constitutional amendments to alter or abolish the Electoral College than on any other subject. The issue was
further put in focus by the 2000 election featuring a razor thin victory and a president
elect who lost the popular vote. Still, the effects of the Electoral College system are not
well understood and academic research on the topic is underdeveloped.
This paper attempts to fill this void by providing the first empirically reasonable, fullfledged game-theoretic model of political competition under the Electoral College system.
Moreover, it uses the model to provide precise answers to a range of questions: what would
resource allocations look like if candidates try to maximize the probability of winning,
are they doing this, who would gain and who would lose if the president was instead
elected through a direct national vote, which system creates a more equal distribution of
resources, which system will better protect voters from political rent-seeking?
A major difference between this paper and earlier work is the complete integration of
theory and empirics, tying together theoretical insights with empirical results on actual
campaigns. This is done by constructing a probabilistic-voting model that can both be
explicitly solved, and directly estimated. The explicit solution draws on use of a Central
Limit Theorem approximation, while the possibility of direct estimation draws on a careful
modelling of political preferences and election uncertainty. The uncertainty, which makes
the voting-model probabilistic, is estimated by using the errors of a vote forecast.
The most closely related theoretical work is Snyder’s (1989) model of two-party competition for legislative seats. Although the equilibrium of this model is not explicitly
solved, some important qualitative features are characterized. Snyder (1989) finds that
equilibrium campaign allocations are higher in districts with close to 50-50 vote shares.
Further, if the goal of the parties is to maximize the probability of winning a majority
of seats, then allocations are also higher in districts which are more likely to be pivotal.
Finally, more resources will be spent in safe districts of the advantaged party than in the
safe districts of the other party. Snyder’s paper studies one-member districts and therefore
does not allow for variation in size necessary for the analysis of the Electoral College. Size
effects are instead included in Brams and Davis’ (1974) study of presidential campaigning.
On the other hand, they abstract from differences in vote-shares by assuming that votes
in each state are cast with equal probability for each candidate. Brams and Davis (1974)
find that presidential candidates should allocate resources disproportionately in favor of
large states. Their result is disputed by Colantoni, Levesque and Ordeshook (1975) who
instead argue that a proportional rule, modified to take into account the closeness of the
state election, predicts actual campaign allocations better. Inspired by these results, Nagler and Leighley (1992) empirically investigate state-by-state campaign expenditures on
1
In this system, a direct vote election is held in each state and the winner of the vote is supposed to
get all of that state’s electoral votes. Then all the electoral votes are counted, and the candidate who
receives most votes wins the election. (The fact that Maine and Nebraska organize their presidential
elections by congressional district is disregarded in this paper.)
2
non-network advertising in 1972 and find these to be higher in states with closer elections
and more electoral votes.
A separate theoretical literature has analyzed the policy effects of plurality versus
proportional representation election systems. For example, Persson and Tabellini (1999),
and Lizzeri and Persico (2001), find that under plurality rule governments tend to overprovide redistributive spending, relative to public goods, because its benefits can be more
easily targeted.
The model finally reveals a link between all of the above literature and the literature
concerning ”voting power”, that is, the probability that a vote is decisive in an election.
The statistical properties of voting power have been analyzed extensively; see Banzaf
(1968), Chamberlain and Rothschild (1988), Gelman and Katz (2001), Gelman, King and
Boscardin (1998)), and Merrill (1978). The "voting power" analyzed in this paper is
slightly different since it is conditional on the candidates’ equilibrium strategies.
The estimable probabilistic-voting model developed in this paper is quite general. It
can be used to analyze a range of resource allocation problems in a variety of electoral
settings. As shown in this paper, it can be used to analyze the allocation of campaign
resources as well as redistributive spending and political rents under the Electoral and
Direct Vote systems. With a minor adjustment, it may also be used to analyze the
allocational strategies of parties trying to win a majority of one-member districts, in,
say, US Congressional elections. In Strömberg (2002b), the model is adapted to analyze
endogenous voter turnout.
Section 2 develops the theoretical model. The probability distribution for election
outcomes suggested by the model is empirically estimated in Section 3. These estimates
are used to confront the model’s predictions with actual campaign efforts in Section 4.
Section 5 interprets the equilibrium. Section 6 addresses the effects of a change to a Direct
Vote system. Finally, Section 7 discusses the results and concludes.
2. Model
Two presidential candidates, indexed by superscript R and D, try to maximize their expected probability of winning the election by selecting the number of days, ds , to campaign
in each state s, subject to the constraint
S
X
s=1
dJs ≤ I,
J = R, D. In each state s, there is an election. The candidate who receives a majority of
the votes in a state gets all the es electoral votes of that state. After elections have been
held in all states, the electoral votes are counted, and the candidate who gets more than
half those votes wins the election.
There is a continuum of voters, indexed by subscript i, a mass vs of which live in state
s. Campaigning in a state increases the popularity of the campaigning candidate
¡ J ¢ among
voters in that state, as captured by the increasing and concave function u ds .2 The
2
This paper does not address the question of why campaigning matters. This is an interesting question
in its own right, with many similarities to the question of why advertisements affect consumer choice.
3
voters also care about some fixed characteristics of the candidates, captured by parameters
Ri , η s , and η. The parameter Ri represents an individual-specific ideological preference in
favor of candidate R, and η s and η represent the general popularity of candidate R in
state s and nation, respectively. The voters may vote for candidate R or candidate D,
and voter i in state s will vote for D if
¡ ¢
¡ R¢
∆us = u dD
(2.1)
s − u ds ≥ Ri + η s + η.
At the time when the campaign strategies are chosen, there is uncertainty about the
popularity of the candidates on election day. This uncertainty is captured by the random
variables η s and η. The candidates know that the S state level popularity parameters,
η s , and the national popularity parameter, η, are independently drawn from cumulative
distribution functions Gs = N(0, σ 2s ), and H = N (0, σ 2 ) respectively, but they do not
know the realized values.
of voters’ ideological preferences, Ri , within each state is Fs =
¡The 2distribution
¢
N µs , σ f s , a normal distribution with mean µs and variance σ 2f s . The means of the
states’ ideological distributions may shift over time, but the variance is assumed to remain
constant. The share of votes that candidate D receives in state s is
Fs (∆us − η s − η).
This candidate wins the state if
1
Fs (∆us − η s − η) ≥ ,
2
or, equivalently, if
η s ≤ ∆us − µs − η.
The probability of this event, conditional on the aggregate popularity η, and the campaign
R
visits, dD
s , and ds , is
Gs (∆us − µs − η) .
(2.2)
Let es be the number of votes of state s in the Electoral College. Define stochastic
variables, Ds , indicating whether D wins state s
Ds = 1, with probability Gs (·) ,
Ds = 0, with probability 1 − Gs (·) .
The probability that D wins the election is then
#
"
X
X
1
P D (∆us , η) = Pr
Ds es >
es .
2
s
s
(2.3)
However, it is difficult to find strategies which maximizes the expectation of the above
probability of winning. The reason is that it is a sum of the probabilities of all possible
combinations of state election outcomes which would result in D winning. The number
of such combinations is of the order of 251 , for each of the infinitely many realizations of
η.
4
A way to cut this Gordian knot, and to get a simple analytical solution to this problem, is to assume that the candidates are considering their approximate probabilities of
winning. Since the η s are independent, so are the Ds . Therefore by the Central Limit
Theorem of Liapounov,
P
s Ds es − µ
σE
where
X
µ = µ (∆us , η) =
es Gs (∆us − µs − η) ,
(2.4)
s
and
σ 2E = σ 2E (∆us , η) =
X
s
e2s Gs (·) (1 − Gs (·)) ,
(2.5)
is asymptotically distributed as a standard normal. The mean, µ, is the expected number
of electoral votes. That is, the sum of the electoral votes of each state, multiplied by the
probability of winning that state. The variance, σ 2E , is the sum of the variances of the
state outcomes, which is the e2s multiplied by the usual expression for the variance of a
Bernoulli variable. Using the asymptotic distribution, the approximate probability of D
winning the election, conditional on η, is
¶
µ1 P
e
−
µ
s
s
D
2
.
(2.6)
Pe (∆us , η) = 1 − Φ
σE
The approximate probability of winning the election is
Z
D
P (∆us ) = PeD (∆us , η) h (η) dη.3
(2.7)
Let the set of allowable campaign visits be
(
)
S
X
X = d ∈ <S+ :
ds ≤ I .
s=1
¡
¢
The Nash Equilibrium strategies dD∗ , dR∗ in the competition between the two candidates
are characterized by
¡
¢
¡
¢
¡
¢
P D dD∗ , dR ≥ P D dD∗ , dR∗ ≥ P D dD , dR∗
for all dD , dR ∈ X. This game has a unique interior pure-strategy equilibrium characterized
by the proposition below.
¡
¢
Proposition 1. The unique pair of strategies for the candidates dD , dR that constitute
an interior NE in the game of maximizing the expected probability of winning the election
must satisfy dD = dR = d∗ , and, for all s and for some λ > 0,
Qs u0 (d∗s ) = λ,
(2.8)
3
The error made from using the approximate probability of winning is discussed in Section 5. For
further discussion, see Appendix 6.8 of Stromberg (2002).
5
where
∂P D
.
∂∆us
Qs =
(2.9)
Proof: See Appendix 8.1.
Proposition 1 says that presidential candidates, trying to maximize their probability
of winning the election, should spend more time in states with high values of Qs . This
follows since u0 (d∗s ) is decreasing in d∗s .
Note that Qs can be decomposed into two additively separable parts:
(2.10)
Qs = Qsµ + Qsσ
Z
1
= es
ϕ (x (η)) gs (−µs − η) h (η) dη
σE
¶
µ
Z
e2s
1
− Gs (−µs − η) gs (−µs − η) h (η) dη,
+ 2
ϕ (x (η)) x (η)
σE
2
where
1
2
P
s es
−µ
.
σE
One arises because the candidates have an incentive to influence the expected number
of electoral votes won by D, that is the mean of the normal distribution. The other
arises because the candidates have an incentive to influence the variance in the number
of electoral votes. To evaluate Qs , I now use the structure of the model to estimate the
probability distribution for election outcomes.
x (η) =
3. Estimation
This estimation provides the link between the theoretical probabilistic-voting model above
and the empirical applications discussed below. In equilibrium, both candidates choose
the same allocation, so that ∆us = 0 in all states. The Democratic vote-share in state s
at time t equals
¶
µ
−µst − η st − η t
yst = Fst (−η st − η t ) = Φ
,
(3.1)
σf s
where Φ (·) is the standard normal distribution, or equivalently,
Φ−1 (yst ) = γ st = −
1
(µ + η st + η t ) .
σ f s st
(3.2)
For now, assume that all states have the same variance of preferences, σ2f s = 1, and
the same variance in state-specific shocks, σ 2s .4 The former assumption implies that the
marginal voter density, conditional on the state election being tied, is the same in all
4
These assumptions will be removed in Section 6. However, the estimates become imprecise if separate
values of µst , σ f s , and σ s are estimated for each state using only 14 observations per state. Therefore,
the more restrictive specification will be used for most of the paper.
6
states. Further assume that the mean of the preference distribution, µst , depends on a
set of observable variables Xst . Then we get the following estimable equation,
γ st = − (Xst β + η st + η t ) .
(3.3)
The parameters β, σ s and σ are estimated using a standard maximum-likelihood estimation of the above random-effects model.5
The variables in Xst are basically those used in Campbell (1992). The nation-wide
variables are: the Democratic vote share of the two-party vote share in trial-heat polls
from mid September (all vote-share variables x are transformed by Φ−1 (x)); second quarter economic growth; incumbency; and incumbent president running for re-election. The
state-wide variables for 1948-1984 are: lagged and twice lagged difference from the national mean of the Democratic two-party vote share; the first quarter state economic
growth; the average ADA-scores of each state’s Congress members the year before the
election; the Democratic vote-share of the two-party vote in the midterm state legislative
election; the home state of the president; the home state of the vice president; and dummy
variables described in Campbell (1992). After 1984, state-level opinion-polls were available. For this period, the state-wide variables are: lagged difference from the national
mean of the Democratic vote share of the two-party vote share; the average ADA-scores
of each state’s Congress members the year before the election; and the difference between the state and national polls. Other state-wide variables were insignificant when
state polls were included. The coefficients β and the variance of the state-level popularity shocks, σ2s , are allowed to differ when opinion polls were available and when they
were not. Estimates of equation (3.3) yields forecasts by mid September of the election
year. The data-set contains state elections for the 50 states 1948-2000, except Hawaii and
Alaska which began voting in the 1960 election. During this period there were a total
of 694 state-level presidential election results. Four elections in Alaska and Hawaii were
excluded because there were no lagged vote returns. Nine elections are omitted because
of idiosyncrasies in Presidential voting in Alabama in 1948, and 1964, and in Mississippi
in 1960; see Campbell (1992). This leaves a total of 681 observations.
The estimation results are shown in Table 1. The estimated standard deviation of
the state level shocks after 1984 , σ
bs , equals 0.077, or about 3% in vote shares. This
is more than twice as large as that of the national shocks, σ
b = 0.033. The estimated
state-preference means are
b
µ
bst = Xst β.
The average absolute error in state-election vote-forecasts, Φ (b
µst ), is 3.0 percent and the
wrong winner is predicted in 14 percent of the state elections. This precision is comparable
to the best state-level election-forecast models (Campbell, 1992; Gelman and King, 1993;
5
The model has been extended to include regional swings, see Appendix 6.4 of Strömberg (2002). In
this specification, the democratic vote-share in state s equals
yst = Fst (η st + ηrt + ηt ) ,
where ηrt denotes independent popularity shocks in the Northeast, Midewest, West, and South. However,
taking into account the information of September state-level opinon polls, there are no significant regional
swings. Therefore, the simpler specification without regional swings is used below.
7
Holbrook and DeSart, 1999; Rosenstone, 1983). Given µ
bst , σ
bs and σ
b, Qs can be calculated
using equation 2.10.
4. Relation between Qs and actual campaigns
This section compare the equilibrium campaign strategies, based on the above estimates,
to actual campaign strategies. The first sub-section will investigate presidential candidate
visits to states in last three months of the 2000 election, and also more loosely discuss
visits during the 1988-1996 elections. The second sub-section will study the allocation of
campaign advertisements across media markets in last three months of the 2000 campaign.
4.1. Campaign visits
If one assumes that u (ds ) is of log form, then the optimal allocation, based on equation
(2.8) is,
Q
d∗
Ps ∗ = P s ,
(4.1)
ds
Qs
and the number of days spent in each state should be proportional to Qs .
The Bush and Gore campaigns were very similar to the model-predicted equilibrium
campaign based on September opinion polls. The actual number of year 2000 campaign
visits, after the party conventions, and Qs , are shown in Figure 4.1.6 Campaign visits by
vice presidential candidates are coded as 0.5 visits. The model and the candidates’ actual
campaigns agree on 8 of the 10 states which should receive most attention. Notable
differences between theory and practice are found in Iowa, Illinois and Maine, which
received more campaign visits than predicted, and Colorado, which received less. Perhaps
extra attention was devoted to Maine since its (and Nebraska’s) electoral votes are split
according to district vote outcomes. Other differences could be because the campaigns
had access to information of later date than mid September, and because aspects not
dealt with in this paper matter for the allocation. The raw correlation between campaign
visits and Qs is 0.91. For Republican visits the correlation is 0.90 and for Democratic
visits, 0.88. A tougher comparison is that of campaign visits per electoral vote, ds /es ,
with Qs /es . The correlation between ds /es and Qs /es was 0.81 in 2000.
Finally, I look at the 1996, 1992, and 1988 campaigns. For these campaigns, only
presidential visits are available. The correlation between visits and Qs during those years
are 0.85, 0.64, and 0.76, respectively. But this is mainly a result of presidential candidates
spending more time in large states. For the 1996, 1992, and 1998 elections, the correlation
between ds /es and Qs /es was 0.12, 0.58, and 0.25 respectively. An explanation for the
poor fit in 1996 and 1988 may be that these elections were, ex ante, very uneven. The
expected Democratic vote shares in September of 1996, 1992, and 1988 were 56, 50, and
46 percent. In uneven races, perhaps the candidates have other concerns, such as affecting
the congressional election outcome, rather than maximizing the probability of winning the
presidential election.
6
I am grateful to Daron Shaw for providing me with the campaign data.
8
0
2
4
6
percent
8
10
Florida
Michigan
Pennsylvania
California
Ohio
Missouri
Tennessee
Wisconsin
Washington
Louisiana
Oregon
Illinois
Iowa
Kentucky
Arkansas
Colorado
North Carolina
New Mexico
Georgia
New Hampshire
Arizona
Nevada
Minnesota
New Jersey
Delaware
Connecticut
Mississippi
Virginia
Qs
∑ Qs
West Virginia
Indiana
Maryland
Actual campaign visits
Wyoming
Maine
Montana
North Dakota
South Dakota
Alabama
Alaska
South Carolina
Vermont
New York
Hawaii
(Utah, Texas, Rhode Island, Oklahoma, Texas,
Nebraska, Massachusetts, Kansas, Idaho) ≈ 0 for both series.
Figure 4.1: Actual and equilibrium campaign visits 2000
9
12
4.2. Campaign advertisements
Appendix 8.3 models the decision of presidential candidates to allocate advertisements
across Designated Market Areas (DMAs).7 In that model, two presidential candidates
have a fixed advertising budget I a to spend on am ads in each media market m subject to
M
X
m=1
pm aJm ≤ I a ,
J = R, D, where pm is the price of an advertisement. Media market m contains a mass
vms of voters in state s. Voters are now also affected by campaign advertisements as
captured by the increasing and concave function w(am ). A voter i in media market m in
state s will vote for D if
¡ D¢
¡ R¢
¡ R¢
¡ ¢
u dD
s + w am − u ds − w am ≥ Ri + η s + η.
The equilibrium number of visits is still characterized by equation (2.8). In equilibrium,
both candidates choose the same advertising strategy. Advertising in media market m is
increasing in Qpmm , where
S
X
nms
Qm =
Qs
.
ns
s=1
Qm is the sum of the Qs of the states in the media market weighted by the share of the
population of state s that lives in media market m.
The advertisement data is from the 2000 election and was provided by the Brennan
Center.8 It contains the number and cost of all advertisements relating to the presidential
election, aired in the 75 major media markets between September 1 and Election Day. The
data is disaggregated by whether it supported the Republican, Democrat, or independent
candidate, and by whether it was paid for by the candidate, the party or an independent
group. The cost estimates, pm , are average prices per unit charged in each particular media
market. The estimates are made by the Campaign Media Analysis Group. Advertisements
were aired in 71 markets, where the data set records a total of 174 851 advertisements,
for a total cost of $118 million, making an average price of $680. The Democrats spent
$51 million, while Republicans spent $67 million. To measure total campaign efforts, I
sum together the advertisements by the candidates, the parties and independent groups
supporting the Democratic or Republican candidate.
Figure 4.2 illustrates the predicted and actual advertising. The model and the data
agree on the two media markets where most ads should be aired (Albuquerque - Santa
Fe, and Portland, Oregon). These two markets have the highest effect on the win probability per advertising dollar. In third place the model puts, Orlando - Daytona Beach
- Melbourne, while the data has Detroit (number four in the model). The correlation
7
A DMA is defined by Nielsen Media Research as all counties whose largest viewing share is given to
stations of that same market area. Non-overlapping DMAs cover the entire continental United States,
Hawaii and parts of Alaska.
8
The Brennan Center began compiling this type of data for the 1998 elections. According to them,
no such data exists elsewhere for any other election. This is a new and unique database.
10
Portland, Oregon
7620
Total advertisements
Albuquerque
- Santa Fe
Lexington
0
Denver
0
7620
Qm/pm
Figure 4.2: Total number of advertisements Sept. 1 to election day and Qm /pm , for the
75 largest media markets
between actual campaign advertisement and equilibrium advertisement is 0.75. That few
advertisements were aired in Denver is consistent with the few candidate visits to Colorado (recall Figure 4.1). The few advertisements in Lexington are more surprising, since
candidate visits to Kentucky were close to the equilibrium number.
To see why Albuquerque - Santa Fe gives a large effect per advertising dollar, note
that we can decompose Qpmm into four terms:
Qm X nms es Qs
1
=
.
pm
nm ns es pm /nm
s |{z}
|{z}|{z}| {z }
(o)
(i)
(ii)
(4.2)
(iii)
=
(o) 95 percent of the population of Albuquerque - Santa Fe live in New Mexico ( nnms
m
0.95), and 5 percent in Colorado. (i) Since New Mexico is a small state with only 1.8
million inhabitants, it had a high number of electoral votes per capita. (ii) Since New
Mexico had a forecasted Democratic vote-share of 51.8%, it had a very high value of Qs
per electoral vote. This relationship will be discussed in Section 5, see Figure 5.3 for a
preview. (iii) At the same time, the average cost of an ad per million inhabitants in the
media market is only $209, compared to the average media-market cost, which is $270.
In comparison, the Detroit media market lies entirely in Michigan which had the highest
value of Qs per electoral vote. However, being a fairly large state, Michigan only has 1.8
electoral votes per million inhabitants. Further, the average cost of an ad in Detroit is
higher than in Albuquerque - Santa Fe. Therefore, the marginal impact on the probability
of winning per dollar is lower than in Albuquerque - Santa Fe.
Finally, one can note that since the correlation between price and market size is close
to one (0.92), there is no significant relationship between market size and the number of
ads.Via the price, the size is instead captured in the costs. Assuming log utility, equilibrium expenditures, pm a∗m , are proportional to Qm . Empirically, the simple correlation
between advertisement costs, pm am , and Qm is 0.88.
11
5. Interpretation
This section discusses what Qs measures and why it varies across states. A qualified guess
is that Qs is approximately the joint ”likelihood” that a state is actually decisive in the
Electoral College and, at the same time, has a very close election. I will call states who
are ex post decisive in the Electoral College and have tied state elections decisive swing
states. In the 2000 election, Florida was a decisive swing state. In contrast, neither New
Mexico nor Wyoming were decisive swing states. While New Mexico was a swing state
with a very close state-election outcome, it was not decisive in the Electoral College since
Bush would have won with or without the votes of New Mexico. While Wyoming was
decisive in the Electoral College, since Gore would have won the election, had he won
Wyoming, it was not a swing state.
The above guess is based on the fact that the probability of being a decisive swing
state replaces Qs in the equilibrium condition of the model without the Central Limit
approximation. Further, Appendix 8.2 shows heuristically, why the analytical expression
for Qs approximates the probability of being a decisive swing state. In this Appendix,
Qs is also shown to approximately equal the "voting power" of state s, multiplied by the
marginal voter density conditional on the state election being tied.
To investigate whether Qs is indeed an almost exact approximation of the probability
of being a decisive swing state, one million electoral vote outcomes were simulated for
each election 1988-2000 by using the estimated state-preference means, and drawing state
and national popularity-shocks from their estimated distributions.9 Then, the share of
elections where a state was decisive in the Electoral College and at the same time had a
state election outcome between 49 and 51 percent was recorded. This provides an estimate
which should be roughly equal to Qs , up to a scaling factor, see Appendix 8.2.
Figure 5.1 contains the simulated shares on the y-axis and values computed from the
scaled, analytic expression of Qs , on the x-axis. Large states are trivially more likely to
be decisive. To check that the correlation between Qs and the simulated values is not just
a matter of size, the graph on the right contains the same series divided by the state’s
number of electoral votes. The simple correlation in the diagram to the right is 0.997. So
the two variables are interchangeable, for practical purposes. The 0.003 difference could
result on the Qs -side from using the approximate probability of winning the election,
and on the simulation-side from using a finite number of simulations and recording state
election results between 49 and 51 percent, whereas theoretically it should be exactly 50
percent.
To illustrate how Qs varies across states, I will use the year 2000 election, see Figure
5.2. Based on polls available in mid September, 2000, Florida, Michigan, Pennsylvania,
California, and Ohio were the states most likely to be decisive in the Electoral College and
at the same time have a state election margin of less than 2 percent. This happened in 2.2
percent of the simulated elections in Ohio and 3.4 percent of the simulations in Florida. In
comparison, the scaled, analytic expression for Qs equals 3.5 percent for Florida. (Using
Qs , the probability that Florida would be decisive in the Electoral College and have a
9
Replace
bst in equation (3.1), and draw ηst and ηt from their estimated distri³ µst by
´ the estimated
³
´ µ
2
2
butions N 0, σ
bs and N 0, σ
b , respectively to generate election outcomes yst .
12
.002
0
Pivotal and close per electoral vote
Simulated share pivotal and close elections
.047
0
Qs
.055
0
0
Qs per electoral vote
.002
Figure 5.1: Qs and simulated probability of being a decisive swing state
state margin of victory of 1000 votes may also be calculated. This probability is 1.5 in 10
000. The probability that this would happen in any state is .44 percent.)
The analytic expression for Qs explains exactly why some states are more likely to
be decisive swing states. First, Qs is roughly proportional to the number of electoral
votes.10 While Qsµ and, Qsσ are proportional to electoral votes and electoral votes squared,
respectively, see equation (2.10), Qsσ is generally considerably smaller than Qsµ . This
implies that candidates should, on average, spend more time in large states. However, for
states of equal size there is considerable variation.
To explain differences relative to size, we next study Qs /es . In Figure 5.3, the circular
dots show the share of the simulated elections where a state was decisive in the Electoral
College and at the same time had a state-election outcome between 49 and 51 percent, per
electoral vote. The solid line shows Qsµ /es . Its normal form arises because the candidates
try to affect the expected number of electoral votes, see equation (2.10). This part of Qs /es
accounts for most of the variation in the simulated values. It explains why states like New
York and Texas in a million simulated elections are never decisive in the Electoral College
and at the same time have close state elections. But in Florida, Michigan, Pennsylvania,
and Ohio this happens quite frequently. The solid line is in fact a normal distribution,
multiplied by a constant. It is characterized by three features: its amplitude, its mean,
and its variance.
The amplitude of Qsµ /es is trivially higher when the national election is expected to
be close. This affects all states in a single election in the same way. It explains why the
average Qsµ varies between elections. 11
10
This can be contrasted to the finding that ”voting power”, that is, the probability that a vote is
pivotal in the election is more than proportional to size (Banzaf 1967, Brams and Davis 1974). ”Voting
power” is roughly proportional to Qs , and thus roughly proportional to size. The difference results
from all voters being equally likely to vote for one candidate or the other in their models, while voters
preferences for the candidates in my model are heterogenous and subject to aggregate popularity shocks,
see Chamberlain and Rothschild (1981) for a theoretical discussion and Gelman and Katz (2001) for
empirical results.
11
Formally, define e
ηt P
to be the national popularity-swing which would give equal expected Electoral
Vote shares, µ (e
η) = 12 s es . Then Qsµ is larger when e
ηt is close to zero; See equation (8.6) in the
Appendix.
13
0
0,5
1
1,5
percent
2
2,5
3
3,5
Florida
Michigan
Pennsylvania
California
Ohio
Missouri
Tennessee
Wisconsin
Washington
Louisiana
Oregon
Illinois
Iowa
Kentucky
Arkansas
Colorado
North Carolina
New Mexico
Georgia
New Hampshire
Arizona
Nevada
Minnesota
New Jersey
Delaware
Connecticut
Mississippi
Virginia
West Virginia
Indiana
Maryland
Wyoming
Maine
Montana
North Dakota
South Dakota
Alabama
Alaska
South Carolina
Vermont
New York
(Utah, Texas, Rhode Island, Oklahoma, Texas,
Nebraska, Massachusetts, Kansas, Idaho) = 0
Hawaii
Figure 5.2: Joint probability of being pivotal and having a state margin of victory less
than two percent, based on September 2000 opinion polls.
14
Michigan (18)
Simulated values,
pivotal and close /es
.0015
Qsµ /es
Ohio
.001
.0005
Pennsylvania
California
Wyoming
Texas
New York
0
30
40
50
60
70
µ ∗s
Forecasted democratic vote share
Figure 5.3: Probability of being a decisive swing state per electoral vote.
The mean of Qsµ /es in Figure 5.3 is located slightly above 50%. This position is the
result of a trade-off between average, and timely, influence on state election outcomes.
To get the intuition, suppose that the Democrats are ahead by 60-40 in the national
forecasts. In states with 50-50 forecasts, a candidate’s visit is more likely to influence
the state election outcome. However, states where the Democrats are ahead 60-40 are
likely to have close state elections exactly when the national election is close. Although
the candidates are less likely to influence the state outcome, they are more likely to do
so when it matters. The mean of Qsµ /es will always lie between 50-50 and the national
forecast (60-40).
Consider the example of the 1996 election. In September, Clinton was ahead by 60-40
in the national forecasts, as well as in Pennsylvania, whereas the forecasted outcome in
Texas was 50-50. A visit to Texas was therefore more likely to affect the state outcome
than a visit to Pennsylvania. However, if Texas was a 50-50 state on election day, then
Clinton was probably winning by a landslide, and the electoral votes of Texas were probably not decisive in the electoral college. On the other hand, in the unlikely event that
Pennsylvania was a 50-50 state on election day, the national election is likely to be close,
and the electoral votes of Pennsylvania were likely to be decisive in the electoral college.
Visits only matter if the state is a 50-50 state on election day, and the candidates must
condition their visits on this circumstance.
The model shows how to strike a balance between high average and timely influence.
The less correlated the state election outcomes, the more time should be spent in 50-50
states like Texas. The reason is that without national swings, the state outcomes are not
correlated, and a state being a swing state on election day carries no information about
the outcomes in the other states. In my estimates maximum attention should typically
be given to states in the middle, 55-45 in this example. In September of 2000, Gore was
ahead by 1.3 percentage points. The maximum Qsµ /es was obtained for states where the
expected outcome was a Democratic vote share of 50.8 percent, as illustrated in Figure
15
5.3.12
The variance in Qsµ /es depends on the uncertainty about the election outcome. Better
state-level forecasts lead to a more unequal allocation of campaign resources as the variance of the normal-shaped distribution of Figure 5.3 decreases. States with forecasted vote
shares close to the center of that distribution would gain while states far from the center
would lose. Better national-level forecasts has a similar effect.14 Note that Wyoming and
two other states to the left of the center are noticeably above the normal-shaped curve.
The reason is that I could not find state-level opinion-poll data for these states, and the
forecasts for these states are more uncertain. These states actually lie on a normal-shaped
curve with a higher variance than that drawn in Figure 5.3. These observations illustrate
one effect of improved forecasting on the allocation of resources.
In Figure 5.3, note also that around its peak, the normal-shaped curve is far from the
simulated probabilities of being a decisive swing state per electoral vote. States to the
right of µ∗s , like Michigan and Pennsylvania, generally lie above the curve, while states on
the left, like Ohio, generally lie below. The difference between the simulated values and
Qsµ /es arises because the candidates also have incentives to influence the variance of the
electoral vote distribution, even if this means decreasing the expected number of electoral
votes, see Qsσ in equation (2.10).
To get the intuition of why such behavior is rational, consider the following example
from the world of ice-hockey. One team is trailing by one goal and there is only one
minute left of the game. To increase the probability of scoring an equalizer, the trailing
team pulls out the goalie and puts in an extra offensive player. Most frequently, the result
is that the leading team scores. But the trailing team does not care about this, since they
are losing the game anyway. They only care about increasing the probability that they
score an equalizing goal, which is higher with an extra offensive player. Therefore, it is
12
These points are evident from the analytical form of the mean of equation (5.1), derived in Appendix
8.2. The mean equals
σ2
µ∗st = −
ηt ,
(5.1)
2e
σ 2 + (σ E /a)
where
¡
¢
σ 2E = σ 2E dD = dR , η t = e
ηt ,
X
at =
es gs (−µst − e
η t ) .13
s
The mean always lies between a pro-Republican state bias of µst = 0, which corresponds to a 50%
forecasted Democratic vote share, and µst = −e
η t , which approximately corresponds to the forecasted
national Democratic vote share. (If the Democrats are ahead by 60-40 nationally, then a pro-Republican
swing e
η t , corresponding to about 10%, is needed to draw the election. Therefore µst = −e
ηt corresponds
to 10% pro-democrat bias in a state, that is, a vote share of 60-40.) The smaller the variance of the
national popularity-swings, σ 2 , the closer is the mean to 50-50. In the extreme case where this variance
equals zero, then µ∗s = 0. In the extreme case that σ approaches infinity, µ∗s approaches −e
η.
14
The variance of the normal-form distribution,
σ
e2 = σ 2s + ³
1
1
σ2
+
1
(σ E /a)2
´,
depends on the variance in the state, and national, level popularity shocks, see Appendix 8.2.
16
Variance effect, Qsσ/vs
.0002
Michigan
Pennsylvania
.0001
0
Ohio
-.0001
30
40
50
60
70
Forecasted democratic vote shares
Figure 5.4: Incentive to influence variance
better to increase the variance in goals, even though this decreases net expected goals. In
contrast, if they were allowed, the leading team would like to pull out an offensive player
and put in an extra goalie.
Similarly, a presidential candidates who is behind should try to increase variance in
the election outcome. He can do that by spending more time in large states where he is
behind (putting in an extra offensive player), and less time in states where he is ahead
(pulling the goalie). A candidate who is ahead should instead try to decrease variance
in electoral votes, thus securing his lead, by spending more time in large states where he
is ahead (putting in an extra goalie), and less time in states where he is behind (pulling
out an offensive player). Both candidates thus spend more time in large states where the
expected winner is leading.
To formally see why a trailing candidate increases the variance by spending more time
in states with many electoral votes where he is behind, consider equation (2.5) showing
the variance, conditional on a national shock. The variance in the number of electoral
votes from a state is proportional to these votes squared. Therefore, the effect on the
total variance, per electoral vote, is larger in large states. Further, the variance in a state
outcome is higher the closer the expected result is to a tie. By visiting a state where the
leading candidate is ahead, the trailing candidate moves the expected result closer to a
tie, and increases the variance in election outcome. Similarly, decreasing the number of
visits to a state where the lagging candidate is leading increases the variance
Figure 5.4 illustrates this effect in the year 2000 election. It plots the values of the
analytical expression for Qsσ /es . The lagging candidate (Bush) should put in extra offensive visits in states like Michigan and Pennsylvania, at the cost of weakening the defense
of states like Ohio. The leading candidate (Gore) should increase his defense of states like
Michigan and Pennsylvania, at the cost of offensive visits to Ohio. This resounds with the
result by Snyder (1989) that parties will spend more in safe districts of the advantaged
party than in safe districts of disadvantaged party.
Another way to use the model is to calculate the best-response to the other candidate’s
actual strategy, even though it differs from the equilibrium. According to the model,
17
either candidate could have increased their probability of winning by about 2 percent
compared to their actual strategies.15 The most important feature of the best responses
is that Bush should spend more than the equilibrium time in California, which the Gore
campaign left unguarded with very few visits. The expected democratic vote-share in
California was 56 percent. Fewer visits by Gore moves the expected democratic voteshare closer to 50.8 percent, evaluated at Bush’s equilibrium strategy. This increases the
probability of California being a decisive swing state, which increases Bush’s incentives
to visit California. In fact, Bush spent more than the equilibrium amount of time in
California. This, on the other hand, moves the expected democratic vote-share closer
to 50.8 percent, evaluated at Gore’s equilibrium strategy, increasing Gore’s incentives to
visit California. However, Gore visited California less than the equilibrium number of
times. Roughly speaking, more visits is a best response to more offensive visits (Bush in
California) or fewer defensive visits (Gore in California) by the opponent. Fewer visits is
a best response to fewer offensive visits or more defensive visits by the opponent.
6. Electoral reform
This section will explore the effects of a hotly debated institutional reform, namely, the
change to a direct vote for president. According to federal historians, over 700 proposals
have been introduced in Congress in the last 200 years to reform or eliminate the system.
Indeed, there have been more proposals for constitutional amendments to alter or abolish the Electoral College than on any other subject. The debate intensified as the 2000
presidential election awarded George W. Bush the White House by a razor-thin victory,
despite his losing the popular vote by a 337,000-vote margin. In consequence, a Washington Post/ABC News poll performed shortly after the election suggested that about 6
in 10 Americans would prefer to abandon the Electoral College and switch to a direct
popular vote.
The effects of reform on campaigning and economic policy have also been debated.
Small states have voiced fear that, without the Electoral College, candidates might change
their campaign patterns and shun them altogether. Others have argued that the Electoral
College system creates a bias favoring small states. Finally, Lizzeri and Persico (2001),
and Persson and Tabellini (1999), have argued that under the Electoral College, the
distribution of targeted programs will be more concentrated, while political rents, and
public goods provision will be lower than under Direct Vote. To discuss these issues,
the model will first be modified to discuss economic policy formation under the Electoral
College, and then further modified to discuss policy formation under Direct Vote.
Economic policy Section 4 shows that politicians understand the incentives created
by the Electoral College system. This section explores the consequences if policy is also
influenced by election concerns. To analyze economic policy, one could just re-interpret
the model of Section 4 as describing the incentives to make policy promises to states
15
This result depends on the functional form assumed for u (ds ) . I can not use log form since this is
not defined for zero visits. Instead I use the exponential function u (ds ) = 0.018 ∗ d0.34
, where the two
s
constants were estimated in Strömberg (2002).
18
during the campaign. If a state is important for the election outcome, not only should
the candidates visit that state, they should also make favorable policy promises to that
state. However, the route taken in this section is to analyze the incentives of incumbent
presidents to set policy for re-election concerns.
Suppose each voter in state s receives utility from policy
us = υ (zs ) −
r
+θ
n
where zs is redistributive spending per capita, nr is political rents per capita, and θ is the
incumbent’s competence which is drawn from a known normal distribution with mean
zero. The function υ is increasing and concave. Political rents describe a conflict of
interest between the president and the voters. It could entail party financing, extra
salaries, low effort and waste or outright corruption. The voter observes his total utility
from policy, but not its separate components. So, the voter does not know whether utility
from a government program is high because per capita spending is high, because the
political rents are low, or competence is high. The incumbent president’s preferences are
described by P + (r) where P is the approximate probability of re-election, defined
below, and is an increasing and concave function.
The timing is the following. First the incumbent sets policy. Then his competence and
popularity shocks are realized. Next, the incumbent president runs against an opponent
with unknown competence. After the election, the winner selects a fixed policy and the
utility of the voters is determined by competence, ideology, and popularity.
In this setting, voter i in state s will re-elect the (without loss of generality) democratic
incumbent if
E [θ] ≥ Ri + η s + η 0 .
The voters observe us and form expectations, E [θ] = us − (υ (zs∗ ) − r∗ ) , where zs∗ and r∗
denote equilibrium policy. Therefore, the above equation becomes
∆us = υ (zs ) − r − (υ (zs∗ ) − r∗ ) ≥ Ri + η s + η.
where η = η 0 − θ. The above equation has exactly the same structure as equation (2.1).
The same distributional assumptions are now made regarding the exogenous parameters
Ri , η s , and η, and the approximate probability of winning, P , is again defined by equations
(2.4), (2.5), (2.6), and (2.7).
The incumbent chooses economic policy to
max P +
(r)
subject to a fixed budget constraint
X
ns zs = I.
s
Proposition 2. The incumbent strategy (z ∗ , r∗ ) that maximizes P +
for all s and for some λ > 0,
Qs 0 ∗
υ (zs ) = λ,
ns
19
(r) must satisfy,
(6.1)
0
(r∗ ) =
where
Qs =
1X
Qs .
n s
(6.2)
∂P
∂∆us
is defined by equations (2.9), (2.4) and (2.5).
In equilibrium, the voters policy expectations are correct and ∆us = 0.
Incumbent presidents will provide higher per capita
Predistributive spending to states
with high Qs . Political rents are lower the larger is s Qs , which measures the fall in
re-election probability due to a small decrease in the utility of all voters.
The estimated Qs will in general be different from those relevant for campaigning
decisions. The reason is that decisions regarding economic policy must be taken earlier
than September the election year. The incumbent president must make his policy decisions
before the September opinion poll results are known. Therefore, Qs used for evaluating
the effects on policy will be estimated without opinion poll data.16
Direct Vote Next, suppose the president is elected by a direct national vote. The
number of Democratic votes in state s is then equal to
vs Fs (∆us − η − η s ).
The Democratic candidate wins the election if he receives more than half of the popular
votes:
X
1X
vs Fs (∆us − η − η s ) ≥
vs .
2
s
s
The number of votes won by candidate D is asymptotically normally distributed with
mean and variance


X
∆us − µs − η 
vs Φ  q
µv =
,
(6.3)
σ 2s + σ 2f s
s
σ 2v = σ 2v (∆us , η) .
See Appendix 8.4 for the explicit expression for σ 2v . The probability of a Democratic
victory is
¶
Z µ1 P
s vs − µv
D
2
P =1− Φ
dη.
σv
The incumbent allocates spending and sets political rents to maximize P D +
following proposition characterizes the equilibrium allocation.
16
(r). The
This version of the model has been used in Strömberg (2002a), where it is found that federal civilian
employment is higher when Qs is higher, even using time and state fixed effects and controlling for other
determinants of employment.
20
ME
Estimated marginal voter density
.12
MA
.1 OK
LA
FLNE
SC
GANY
AL
WV KY
.08
SD
NC
PA MD
AZ
MS
RI
NJ
NH
CT
MI
OH
KS
AR
UTTXMN
VA
IN
IL MO
WI
TN
CO DE
WY IA
VT
WA
ND
NV
ORNM
MT
ID
CA
.06
20
30
40
50
60
Share independents 1976-88 (Erikson, Wright and McIver)
Figure 6.1: Marginal voter density and share independents
Proposition 3. The incumbent strategy (z, r) that maximizes P D +
for all s and for some λ > 0,
QDV
s
υ 0 (zs ) = λ,
ns
X
0
(r) =
QDV
s .
(r) must satisfy,
(6.4)
(6.5)
s
measures the likelihood of a draw in the national election, multiplied
The variable QDV
s
by the expected marginal voter density, conditional on the national election being tied,
multiplied by the number of voters in the state; see Appendix 8.4. Since the likelihood
of a draw in the national election is the same in all states, QDV
varies across states only
s
because of differences in the share of marginal voters and the number of voters.
The allocation under Direct Vote depends crucially on the share of marginal voters,
which in turn depends on the estimated variance in the preference distribution, σ2f s .
Therefore, the restriction σ f s = 1 is removed in the maximum likelihood estimation of
equation (3.2), as well as the assumption that σs is the same for all states. However, the
empirical identification of σ f s and σ s is not trivial. If the election outcome in a certain
state varies a lot over time, is this because the state has many marginal voters or is it
because the state has been hit by unusually large shocks shifting voter preferences? The
model solves this problem by identifying σ f s by the response in vote shares to changes that
are common to all states, and observable changes in economic growth at national and state
level, incumbency variables, home state of the president and vice president. States where
the vote share outcome covary strongly with economic growth, etc., are thus estimated to
have many marginal voters. Maine is estimated to have the largest share of marginal voters
while California has the smallest. The estimated share of marginal voters is positively
correlated with the share of independent voters as measured by Erikson, Wright and
McIver (1993), as illustrated in Figure 6.1. The variance in the state popularity-shocks,
σ 2s , is on average larger in southern states and smaller states.
Spending We can now discuss the allocation of spending under the Electoral College
(EC) and Direct Vote (DV ) systems. To structure the discussion, assume log utility so
21
that equilibrium per capita spending under DV equals
zsDV =
QDV
s
ns
1
n
P
QDV
s
I
,
n
whereas the analogous expression for spending under EC is given by just substituting
Qs for QDV
s . The effect of reform is shown in Figure 6.2. The series are 1948-2000
averages, scaled so that 1 denotes equal per capita spending. States above 1 on the y-axis
receive higher than average per capita spending under under the Electoral College system,
whereas states to the right of 1 on the x-axis have higher than average per capita spending
under under the Direct Vote system. Thus, states below the dashed 45 degree line would
gain from reform, those above would lose. Some states, like Maine and New Hampshire,
are well off under both systems while Mississippi is disadvantaged under both. Other
states, like Nevada are among the winners in the present system but among the losers
under Direct Vote. The opposite is true for Rhode Island and Kansas.
The reasons why certain states would gain or lose can be separated into variation in
(i) electoral size per capita and (ii) influence relative to electoral size.
EC :
Qs
es Qs
=
,
ns
ns es
|{z}|{z}
(i)
DV :
(ii)
vs QDV
QDV
s
s
=
.
ns
ns vs
|{z}| {z }
(i)
(ii)
Figure 6.3 plots the 1948-2000 averages of, (i), electoral votes per capita million and
voter turnout to the left, and, (ii), the probability of being a pivotal swing state per
electoral vote and the share of marginal voters to the right. The latter have been scaled
so that 1 denotes equal per capita influence relative to size. Nevada and Delaware would
lose from reform primarily because of their heavy endowment of electoral votes relative
to popular votes, while Ohio would lose primarily because it is likely to be a pivotal
swing state, but does not have many marginal voters. On the winning side, Rhode Island
and Massachusetts would gain because of their many marginal voters, while Kansas and
Nebraska would gain because of their low probabilities of being decisive swing states in
the present system.
Small states have not, on average, been advantaged by the Electoral College system.
Although small states are over-represented in terms of electoral votes, they have more
often had lop-sided elections and have larger state-level uncertainty, σ 2s . On net, Qs /ns is
not correlated with size.17 It is also the case that QDV
s /ns is uncorrelated with state size.
So small states, as a group, neither gain nor lose from electoral reform.
Which political system creates a more unequal distribution of resources? The extreme
variation in the probability of being a pivotal swing state per electoral vote creates very
strong incentives for unequal distribution under the EC. The average probability that
17
I thank Andrew Gelman for pointing this out to me.
22
DE
NV
2.5
Electoral College
OHVT
MT
2
NM
NH
ME
MI
WY WA
IL
CT
MO
WI
NJ
SD
MD
HI
IA NY
KY WV
AK
CO
OR
ND
MN
VA
AL
OK
ID
TN
UT
CA
TX
FL
NC
RI
MA
LA
AR
AZ
IN
SC
GA
MS
NE
KS
PA
1.5
1
.5
0
0
.5
1
1.5
2
2.5
Direct Vote
Figure 6.2: Redistributive spending per capita, relative to national average, under the
Electoral College and Direct Vote
(ii)
WY
8
2.5
NV
AK
Qs/es : Pr(decisive swing state)/es
Electoral votes per capita million
(i)
VT
6
DE
ND
MT
NH SD
ID
DC
HI
4
AR
SC
2
.3
MS
GA
TN NC
VA
ALTXLA FL
AZ
NM
KY
MD
RI
ME
UT
NEWV
KS
IA
OK
CO
OR
CT MN
WI
WA
MO MA
PA OH
NJ
MI INIL
NY
CA
.5
Voter turnout
OH
2
PA
1.5
NM
MT
CA OR
NV
1
.5
MS
0
.5
.7
ND
ID AK
IL
WA
MO
WI
MI
CT
DE
NY
MD
KY
NH
IA
VA FL
AL
CO TX
WV
VT
HI
TN
LA
NC MN
SD
OK
AR SCGA
IN
WY
AZ
UT
NJ
ME
MA
RI
KS NE
1
QsDV/vs : Marginal voter density
1.5
Figure 6.3: Variables affecting distribution under Electoral College and Direct Vote
23
Ohio is a pivotal swing state is more than twenty times that of Kansas. By comparison,
Maine has only twice as high marginal voter density as California. To make things worse,
the variation in electoral votes per capita is also higher than the variation in voter turnout.
While Wyoming has four times as many electoral votes per capita as California, Minnesota
has less than twice times the voter turnout of Hawaii. Given this, it is not surprising that
the equilibrium allocation of redistributive expenditures is much less equal under EC.
The Lorenz-curve of spending under EC is strictly below that of spending under the
DV . This finding is consistent with Persson, and Tabellini (2000) and Lizzeri and Persico
(2001) who conclude that spending will be more narrowly targeted under majoritarian
elections. Persson and Tabellini (2000) analyze electoral competition in an election with
three electoral districts. Their result is driven by the assumption that the district with
the highest marginal voter density is always decisive in the electoral college (majoritarian
election), while under proportional elections candidates internalize marginal voter density
across all states, which is more equally distributed. Lizzeri and Persico (2001) use a
very different framework, assuming no uncertainty about the election outcome, given
candidate strategies. In consequence, candidates use completely mixed strategies and all
states receive equal expected treatment.
Political rents First a theoretical point: there is no a priori reason to believe that
political rents would depend on the size or number of states. Rents are decreasing in
X ∂P D
X
X es Qs
=
Qs =
vs
,
∂∆us
vs es
s
s
s
which measures how the probability of re-election falls when the utility of all voters is
decreased marginally. If Qs was more than proportionally increasing in the number of
electoral votes, as suggested by Brams and Davis (1974), larger states would contribute
more than proportionally to keeping rents low and the best electoral system would be to
have just one state of maximum size (Direct Vote). However, Qs is roughly proportional to
the number of electoral votes, see equation (2.10) and Figure (5.3), so all states contribute
proportionally and there is no a priori reason to believe that size and number of states
matter.
Empirically, the ability to discipline politicians and
Pkeep political
P rents low is about
the same for both electoral systems. Figure 6.4 plots s Qs and s QDV
s , multiplied by
an increase in political rents ∆r causing an average 5 percent fall in vote support.18 This
fall is trivially higher in elections which are ex ante close. Therefore, equilibrium political
rents are lower in 2000 than in 1984, under both electoral systems. Further, the average
fall is about the same for the two systems and politicians would be disciplined to an equal
extent.
This depends on two factors, how the probability of re-election depends on a uniform
fall in voter support across all states, and how the fall is likely to be distributed across
states. First, the fall in the probability of being elected from a uniform loss of 5 percent
∆r = 0.05/average ϕ(·)
σ f s . The figure describes the marginal incentives for political rents. As a measure
of the probability of losing the election, there is an approximation error since Qs and QDV
measure
s
marginal changes.
18
24
Electoral College
.6
1988
1976
1960
2000
1952
.5
.4
1968
.3
1956
1992
1948
1996
.2
1964
.1
1980
1984
1972
1940
1944
.1
.2
.3
.4
.5
.6
Direct Vote
Figure 6.4: Probability that an incumbent would loose the election because of a corruption
scandal (5 percent fall in support) under EC and DV
support in all states is around 33 percent both under EC and DV .19 This similarity
is far from obvious since the frequency of quite different events are measured: that the
incumbent loses by less than 5 percent in any state that is decisive in the electoral college;
and that the incumbent loses the national election by less than 5 percent. Appendix 8.4
shows that this response is proportional to average "voting power", that is, the probability
that one randomly drawn voter decides the election. The result is therefore consistent
with the finding of Gelman and Katz (2001) that average voting power is about the same
under EC and DV .
However, the fall in support will not be uniform, it will be higher in states with many
marginal voters. If the fall is larger in states (like Florida, Michigan, and Pennsylvania
in 2000) who are likely to be decisive swing states, then the outcome under EC will be
more sensitive to the corruption scandal than under DV . This turns out not to be true
empirically, as these two variables are uncorrelated. A more direct measure of the share
of marginal voters, the share of independent voters in Erikson, Wright and McIver (1993),
is also not related to the likelihood of being a decisive swing state.
In contrast, Persson and Tabellini (1999) conclude that there will be more corruption
under Direct Vote than if the electorate is divided into three districts, a separate vote is
held in each district, and the candidate with the most districts wins the election. They
reach their conclusion by assuming that there are more marginal voters in the one district
that is always decisive. This assumption, translated to our framework, is not valid in the
case of the US as there are not more marginal voters in states more likely to be decisive
swing states.
Election statistics Finally, some argue that razor-thin victories, and presidents without a majority of the popular vote are unattractive features of an electoral system. The
model can be used to estimate the probability of these events. Given that the elections
1948-2000 are representative of future elections, this can be done by simulating elections
19
This is estimated by simulations.
25
and recording event frequencies. The likelihood of a winning margin less than 1000 votes is
about 50 times higher under the Electoral College system (0.4 percent compared to 0.008
percent under Direct Vote). The reason is that a margin of victory of, say, one percent,
is equally likely in the two systems, but, since there are 50 states, one percent contains
on average 50 times more voters under Direct Vote. This is consistent with equal average
voting power under both systems. Every time a single voter is pivotal under DV , then all
voters in the nation are pivotal. Every time a single voter is decisive under EC, then all
voters in that state are decisive. The latter event occurs 50 times more often but involves
1/50 as many voters, so average voting power is the same. The probability of electing a
president without a majority of the popular vote is about 4 percent. This implies that we
should expect this outcome about once in every hundred years. Historically, this event
has happened around three (perhaps four) times in the last 200 years: 1824 (perhaps),
1876, 1888, and 2000. Arguably, however, the outcomes in 1824 and 1876 had to do with
peculiarities in the aggregation of votes.20
The above results support a reform of the Electoral College system in favor of Direct
Vote. Such a reform would decrease political incentives to allocate resources unequally
across states for electoral reasons. It would also decrease the probability of razor thin
winning margins and, of course, the probability of electing presidents without support
of a majority of the popular votes. There would be no cost in terms of less discipline
on politicians. Also, small states need not fear reform. As a group, they are neither
advantaged or disadvantaged by the present system, nor will they be under Direct Vote.
7. Conclusion and discussion
This paper develops a game-theoretic model of how presidential candidates should allocate
resources across states in order to get a majority of the electoral votes. After applying it
empirically to the US, a large set of questions have been answered.
What would resource allocations look like if candidates try to maximize the probability
of winning a majority in the electoral college? Some theoretical results deserves mentioning. First, more resources should be devoted to states who are likely to be decisive swing
states, that is, states who are decisive in the electoral college and, at the same time, have
tied state elections. Second, the probability of being a decisive swing state equals the
"voting power" in the state, multiplied by the marginal voter density conditional on the
state election being tied. Third, the probability of being a decisive swing state is roughly
proportional to the number of electoral votes. Fourth, this probability per electoral vote is
highest for states who have a forecasted state election outcome which lies between a draw
and the forecasted national election outcome. Fifth, more precise state-election forecasts
make the optimal allocation of resources more concentrated. Sixth, the presidential candidates who is lagging behind should try to increase variance in electoral votes. This is
done by spending more time in large states where this candidate is behind, and less time
in large states where this candidate is ahead.
20
In 1824, six of the twenty four States at the time still chose their Electors in the State legislature, so
the popular vote outcome is not known. In the chaotic election of 1876, each State delivered to Congress
two slates of electors and a special commission accepted slates favoring of the popular vote loser. See "A
Brief History of the Electoral College" in http://www.fec.gov/elections.html.
26
The second, fourth and fifth points are new, to the best of my knowledge. The
first point shows more precisely what people have conjectured or partly demonstrated
earlier, see, for example, Snyder (1989). The third point has not previously been shown
theoretically, although (because of the second point) it is closely related to the empirical
finding of Gelman and Katz (2001) that "voting power" is proportional to size under
reasonable assumptions. The sixth point makes more precise the statement of Snyder
(1989) that more resources will be spent in safe districts of the advantaged party than in
the safe districts of the other party.
More importantly, this paper quantifies these effects and shows that presidential candidates’ actual strategies very closely resemble the optimal strategies. The model is applied
to presidential campaign visits across states during the 1988-2000 presidential elections,
and to presidential campaign advertisements across media markets in the 2000 election.
The actual allocation of these resources closely resembles the optimal allocation in the
model. In the 2000 election, the correlation between optimal and actual visits by state
is 0.91, and the correlation between optimal and actual advertisement expenditures by
advertising market is 0.88.
Who would gain and who would lose if the president was instead elected through
a direct national vote? The gainers and losers from reform are identified as displayed
in Figure 6.2. States like Kansas and Nebraska, would gain, while Nevada and Delaware
would lose. Small states are, as a group, not favored by the Electoral College system. The
distribution of redistributive spending is much more unequal under the Electoral College
than under Direct Vote. This is basically because the probability of being a decisive swing
state is extremely unevenly distributed. The same conclusion was reached by Lizzeri and
Persico (2001) and Persson, and Tabellini (2000), but for different reasons. Third, unlike
Persson and Tabellini (1999), this paper finds that both systems are about equally fit to
discipline politicians from corruption or other activities which benefit them but hurt the
voters.
The main difference between this paper and the earlier work is the complete integration
of theory and empirics, tying together theoretical insights with empirical results. Since
the modelling framework is quite general, this integration can be carried over to the study
of other resource allocation problems in other electoral settings.
27
References
[1] Anderson, Gary M. and Robert D. Tollison, 1991, “Congressional Influence and Patterns of New Deal Spending, 1933-1939”, Journal of Law and Economics, 34, 161-175.
[2] Banzhaf, John R., 1968, ”One Man, 3.312 Votes: A Mathematical Analysis of the
Electoral College”, Villanova Law Review 13, 304-332.
[3] Brams, Steven J. and Morton D. Davis, 1974, ”The 3/2 Rule in Presidential Campaigning.” American Political Science Review, 68(1), 113-34.
[4] Campbell, James E., 1992, Forecasting the Presidential Vote in the States. American
Journal of Political Science, 36, 386-407.
[5] Chamberlain, Gary and Michael Rothchild, 1981, ”A Note on the Probability of
Casting a Decisive Vote”, Journal of Economic Theory 25, 152-162.
[6] Cohen, Jeffrey E., 1998, ”State-Level Public Opinion Polls as Predictors of Presidential Election Results”, American Politics Quarterly, Vol. 26 Issue 2, 139.
[7] Colantoni, Claude S., Terrence J. Levesque, and Peter C. Ordeshook, 1975, ”Campaign Resource Allocation under the Electoral College (with discussion)”, American
Political Science Review 69, 141-161.
[8] Erikson, Robert S., Gerald C. Wright and John P. McIver, 1993, "Statehouse Democracy", Cambridge University Press.
[9] Gelman, Andrew, and Jonathan N. Katz, 2001, ”How Much Does a Vote Count?
Voting Power, Coalitions, and the Electoral College”, Social Science Working Paper
1121, California Institute of Technology.
[10] Gelman, Andrew, and Gary King, 1993, ”Why are American Presidential Election
Campaign Polls so Variable when Votes are so Predictable?”, British Journal of
Political Science, 23, 409-451.
[11] Gelman, Andrew, Gary King, and W. John Boscardin, 1998, ”Estimating the Probability of Events that have Never Occurred: when is your vote decisive?”, Journal of
the American Statistical Association, 93(441), 1-9.
[12] Hobrook, Thomas M. and Jay A. DeSart, 1999, ”Using State Polls to Forecast Presidential Election Outcomes in the American States”, International Journal of Forecasting, 15, 137-142.
[13] Lindbeck, Assar and Jörgen W. Weibull, 1987, ”Balanced-Budget Redistribution as
the Outcome of Political Competition”, Public Choice, 52, 273-97.
[14] Lizzeri, Allesandro and Nicola Persico, 2001, "The Provision of Public Goods under
Alternative Electoral Incentives", American Economic Review, 91(1), 225-239.
28
[15] Merrill, Samuel III, 1978, ”Citizen Voting Power Under the Electoral College: A
Stochastic Voting Model Based on State Voting Patterns”, SIAM Journal on Applied
Mathematics, 34(2), 376-390.
[16] Nagler, Jonathan, and Jan Leighley 1992, ”Presidential Campaign Expenditures:
Evidence on Allocations and Effects”, Public Choice 72„ 319-333.
[17] Persson, Torsten and Guido Tabellini, 1999, ”The Size and Scope of Government:
Comparative Politics with Rational Politicians,” European Economic Review 43,
699-735.
[18] Persson, Torsten, and Guido Tabellini, 2000, Political Economics: Explaining Economic Policy, MIT Press.
[19] Rosenstone, Steven J., 1983, ”Forecasting Presidential Elections”, New Haven: Yale
University Press.
[20] Shachar, Ron and Barry Nalebuff, 1999, ”Follow the Leader: Theory and Evidence
on Political Participation”, American Economic Review 89(3), 525-547.
[21] Shaw, Daron R., 1999, ”The Effect of TV Ads and Candidate Appearances on
Statewide Presidential Votes, 1988-96, American Political Science Review 93(2), 345361.
[22] Strömberg, David, 2002, ”Optimal Campaigning in Presidential Elections: The Probability of Being Florida”, IIES working paper 706.
[23] Strömberg, David, 2002a, ”The Electoral College and the Distribution of Federal
Employment”, Stockholm university, in preparation.
[24] Strömberg, David, 2002b, ”Follow the Leader 2.0: Theory and Evidence on Political
Participation in Presidential Elections”, Stockholm university, in preparation.
[25] Snyder, James M., 1989, ”Election Goals and the Allocation of Campaign Resources”,
Econometrica 57(3), 637-660.
[26] Wallis, John J., 1996, “What Determines the Allocation of National Government
Grants to the States?”, NBER Historical Paper 90.
[27] Wright, Gavin, 1974, ”The Political Economy of New Deal Spending: An Econometric Analysis”, Review of Economics and Statistics 56, 30-38.
29
8. Appendix
8.1. Proof of Proposition 1
¡
¢
Since P D dD∗ , dR is continuous and differentiable, a necessary condition for an interior
NE is
¡
¢
¡ ¢
∂P D dD∗ , dR∗
= Qs u0 dD∗
(8.1)
= λD ,
s
D
∂ds
¡
¢
¡ R∗ ¢
∂P D dD∗ , dR∗
0
=
Q
u
(8.2)
ds = λR .
s
R
∂ds
Therefore,
¡ ¢
u0 dD∗
λD
s
= R,
(8.3)
u0 (dR∗
λ
s )
R∗
for all s. Suppose that dD∗ 6= dR∗ . This means that dD∗
s < ds for some s, implying that
λD > λR by equation (8.3). Because of the budget constraint, it must be the case that
D
R∗
0
dD∗
< λR , a contradiction. Therefore, λD = λR ,
s0 > ds0 for some s , which implies λ
D∗
R∗
which implies ds = ds for all s.
Uniqueness: Suppose there are two equilibria with equilibrium strategies d and d0
corresponding to λ > λ0 . The condition on the Lagrange multipliers implies ds > d0s for all
s, which violates the budget constraint. Therefore, the only possibility is λ = λ0 , which
implies ds = d0s for all s.
8.2. Derivation and interpretation of Qs
To see, heuristically, why the analytical expression for Qs approximates the joint probability of a state being decisive in the electoral college at the same time as having a tied
election, note that state s can only be decisive if the margin of victory in the electoral
college is less than its electoral votes, es . To a second order approximation, this happens
with probability
µ1 P
¶
µ1 P
¶
s0 es0 + es − µ
s0 es0 − es − µ
2
2
Φ
−Φ
σE
σE
¶¶
µ
µ
2
es
1
es
= 2pECD (η) ,
+
ϕ (x (η)) x (η) Gs (·) −
≈ 2 ϕ (x (η))
σ E σ 2E
2
where the last equality defines pECD (η). However, s is only decisive when D wins nationally and in state s, or R wins nationally and in state s. This happens about half of
the times the margin of victory is less than es , so the probability that a state is Electoral
College Decisive is approximately half the above expression. Second, given a national
shock, η, the state outcome is a tie if Fs (−η s − η) = 12 which happens with likelihood
gs (−µs − η) . Conditional on the national shock, the joint likelihood of being decisive
and having a tied election is thus roughly pECD (η) gs (−µs − η), and the unconditional
probability approximately Qs .21
21
A stricter, and longer, version of this argument can be made. Please contact the author for details.
30
To compare Qs with simulated frequencies of being EC decisive and having a state
election margin of less than two percent, Qs must be scaled. The probability of a winning
margin of less than x percent equals
1
1
x
x
−
≤ Fs (−η s − η) ≤ +
.
2 200
2 200
The probability that the state-level shock falls in this region is approximately
gs (−µs − η)
σf s x
.
ϕ (0) 100
The unconditional probability of being pivotal and having a state-election margin of x
percent is therefore approximately
Qs
σf s x
.
ϕ (0) 100
The scaled values of Qs are very similar to the simulated frequencies of the corresponding events. In 3.4 percent of the 1 million simulated elections, Florida was decisive
in the Electoral College and had a state margin of victory of less than 2 percent. The
scaled Qs for Florida was 3.5 percent. To get the probability that the state margin of
σf s
victory is within, say 1000, votes, Qs should be scaled by 1000
. Using this formula,
vs ϕ(0)
the probability of a state being decisive in the Electoral College, and at the same time
having an election result with a state margin of victory less than 1000 was 0.00015 in
Florida in the 2000 election. The probability that this would happen in any state was
.0044. The probability of a victory margin of one vote in Florida is 0.15 per million, and
the probability of this happening in any state is 4.4 per million. The state where one vote
is most likely to be decisive is Delaware, where it is decisive .4 times in a million elections.
It is important to understand how the scaling of Qs relates to its interpretation.
1
Qs =
0.02
µ
¶
ϕ (0)
σf s
.
Qs
0.02
ϕ (0)
σf s
{z
}
|
| {z }
Pr(EC decisive and
marginal voter density,
margin of two percent) conditional on tied state election
(8.4)
or, similarly,
µ
¶
ϕ (0)
σf s 1
.
Qs = vs Qs
ϕ (0) vs
σf s
{z
}
|
| {z }
”voting
marginal voter density,
power"
conditional on tied state election
(8.5)
When all states are assumed to have the same variance of preferences σ f s (until Section
6), Qs is proportional to the probability of being EC decisive and having a state margin
of victory of less than two percent.
31
tfi
pfi
.014273
.000531
-.027613
.101768
nytt
Figure 8.1:
To simplify the analytical form of Qsµ , do a first order Taylor expansion of theP
mean
of the expected number of electoral votes µ (η) around η = e
η for which µ (e
η ) = 12 s es ,
that is the value of the national shock which makes the expected outcome a draw. With
this approximation
µ (η) =
X
s
es Gs (−µs − η) ≈
a=
X
s
1X
es − a (η − e
η) ,
2 s
es gs (−µs − e
η) .
Since the mean of the electoral votes, µ (η) , is much more sensitive to national shocks
than is the variance σ E (η), the latter is assumed fixed
X
e2s Gs (−µs − e
η ) (1 − Gs (−µs − e
η )) .
σ 2E (η) = σ 2E =
s
Then
1
ϕ
σ E (η)
µ1 P
2
−µ
σ E (η)
s es
¶
¶
µ
1
η −e
η
≈
.
ϕ
σE
σ E /a
This approximation is very good. Figure 8.1 shows the true, tfi, and the approximated
functions, pfi. The values are calculated for an interval of four standard deviations centered around e
η in the 2000 presidential election. We now have
¶
Z µ
1
η−e
η
gs (−µs − η) h (η) dη.
Qsµ ≈ es
ϕ
σE
σ E /a
Integrating over η,
Qsµ
where
!
Ã
ω
η 2 + (σ E /a)2 µ2s + σ 2 (e
η + µs )2
1 σ 2s e
≈ es exp −
2π
2 σ 2s σ 2 + (σ E /a)2 σ 2 + (σ E /a)2 σ 2s
2
ω =
µ
1
1
1
2 + 2 + 2
σs σ
(σ E /a)
32
¶−1
.
(8.6)
Qsµ is larger when e
η is close to zero. This effects all states in the same way, but varies
across elections. To clarify the differences between states with different µs , rewrite
!
Ã
ω
1 c + (µs − µ∗s )2
,
Qsµ ≈ es exp −
2π
2
σ
e2
where
µ∗s = −
and
σ2
e
η,
σ 2 + (σ E /a)2
σ
e2 = σ 2s + ³
1
1
σ2
+
1
(σ E /a)2
Qsσ is calculated using numerical integration.
´.
8.3. Campaign advertisements
This Appendix analyses the decision of presidential candidates to allocate advertisements
across media markets. Two presidential candidates: R and D, select the number of ads,
am , in each media market m and the number of visits ds in each state s, subject to
M
X
m=1
pm aJm ≤ I a ,
S
X
s=1
22
dJs ≤ I,
J = R, D. Media market m contains a mass vms of voters in state s. Voters reactions
to campaign advertisements are captured by the increasing and concave function w(am ).
A voter i in media market m in state s will vote for D if
¡ ¢
¡ R¢
¡ D¢
¡ R¢
∆uwms = u dD
s − u ds + w am − w am ≥ Ri + η s + η.
In each media market m in state s, the individual specific preferences for candidates,
Ri , are distributed with cumulative density function Fs . The state and national-level
popularity-swings are drawn from the same distributions as before.
The share D votes in media market m in state s equals
Fs (∆uwms − η s − η).
D wins the state if
X
m
vms Fs (∆uwms − η s − η) ≥
22
vs
.
2
For simplicity, it is assumed that both parties have the same resources for buying advertisements.
The asymmetric equilibrium with unequal resources can be characterized, theoretically and empirically,
given assumptions about w (a), the effectiveness of advertisements in changing votes.
33
Define the total (state and national) swing which causes a draw in the state:
¡
¢ X
¡
¢
vs
η s dD , aD , dR , aR :
vms Fs (∆uwms − η s dD , aD , dR , aR ) = .
2
m
(8.7)
D wins state s if
η s + η ≤ η s (·) .
Conditional on the aggregate shock η, and the strategies, dD , aD , dR , aR , this happens
with probability
¡ ¡
¢
¢
Gs η s dD , aD , dR , aR − η .
The function η s (·) now plays the same role as ∆us − µs in Section 2, see equation (2.2).
The rest of the analysis is in that section, only exchanging η s (·) for ∆us − µs . The best
reply functions of D is characterized by
∂η (·)
(8.8)
Qs s D = λD ,
∂ds
S
X
s=1
Qs
∂η s (·)
= µD pm .
∂aD
m
(8.9)
Similarly, the best reply function of R is characterized by equations (8.8) and (8.9),
replacing superscripts D by R. Because of the fixed budget and time constraints, the
allocations must be symmetric so that λD = λR = λ, and µD = µR = µ. Differentiating
equation (8.7), and evaluating it in equilibrium yields
¡ D¢
∂η s (·)
0
=
u
ds ,
∂dD
s
vms 0 ¡ D ¢
∂η s (·)
=
w am .
∂aD
vs
m
¢
¡
Proposition 4. A pair of strategies for the parties dD , dR , aD , aR that constitute a NE
in the game of maximizing the expected probability of winning the election must satisfy
dD = dR = d∗ , and aD = aR = a∗ , and for all s and for some λ, µ > 0
Qs u0 (d∗s ) = λ,
(8.10)
Qm w0 (a∗m ) = µpm ,
(8.11)
where
Qm =
S
X
Qs
s=1
vms
.
vs
Qm is the sum of the Qs of the states in the media market weighted by the share of the
voting population of state s that lives in media market m.
Last, assuming that votes per capita
vms
= ts
nms
is the same for all media markets within each state,
Qm =
S
X
s=1
34
Qs
nms
.
ns
8.4. Direct presidential vote
First, the approximate probability of winning the election is derived under DV . Conditional on η, the expected vote share of D in state s is
Z
µvs (∆us , η) = Fs (∆us − η − η s )gs (η s ) dη s


∆us − η − µs 
= Φ q
,
2
2
σs + σf s
and the expected national vote of D is
µv (∆us , η) =
X
vs µvs (∆us , η) .
s
Again conditional on η, the variance in D’s votes in state s is,
¶
¶2
Z µ µ
∆us − η − η s − µs
2
2
− µvs (∆us , η) gs (η s ) dη s ,
σ vs = vs
Φ
σf s
and the variance in the national votes of D is
X
σ 2vs .
σ 2v =
s
The approximate probability of D winning the election is
¶
Z µ1 P
s vs − µv
D
2
h (η) dη.
P (∆us ) = 1 − Φ
σv
The equilibrium strategies of Proposition 3 depend crucially on
∂P D
∂P D ∂µv
∂P D ∂σ v
DV
= QDV
=
+
= QDV
s
sµ + Qsσ ,
∂∆us
∂µv ∂∆us
∂σ v ∂∆us
where
QDV
sµ =
v
q s
σ v σ 2s + σ 2f s
QDV
sσ
=
Z
ϕ
Z µ
µ1 P
2
∂
Φ
∂σ v
s vs − µv
σv
µ1 P
2
s
¶


∆us − µs − η 
ϕ q
h (η) dη,
σ 2s + σ 2f s
vs − µv
σv
¶¶
(8.12)
∂σ v
h (η) dη.
∂∆us
DV
DV
DV
Empirically, QDV
sµ À Qsσ , the size of Qsσ is negligible compared to Qsµ .
DV
The interpretation of Qsµ is
QDV
sµ = vs (pdf of tied election) E [fs | election tied] .
35
(8.13)
This follows since, conditional on η, the expected marginal voter density in state s is


Z ∞
1
∆us − µs − η 
fs (∆us − η − η s )gs (η s ) dη s = q
ϕ q
.
−∞
σ 2s + σ 2f s
σ 2s + σ 2f s
The probability density function of a tied national election, conditional on η, is
¶
µ1 P
1
s vs − µv
2
,
ϕ
σv
σv
and the unconditional
pdf of tied election =
Z
1
ϕ
σv
µ1 P
2
s
vs − µv
σv
¶
h (η) dη.
(8.14)
Therefore, the marginal voter density, conditional on a tied election, is
(8.15)
E [fs | election tied]
³1P
´




v
−µ
s
Z
ϕ 2 sσv v
∆us − µs − η 

q 1
ϕ q
=
h (η) dη.
R ³ 12 P s vs −µv ´
2 + σ2
2 + σ2
σ
σ
ϕ
h
(η)
dη
s
s
fs
fs
σv
Inserting equations (8.14) and (8.15) into equation (8.13) yields equation (8.12).
To see the relationship to "voting power" note that the probability of a national
election margin of x votes or less equals
¶
µ1 P
¶
Z µ1 P
x
x
s vs + 2 − µv
s vs − 2 − µv
DV
2
2
px
−Φ
h (η) dη
(8.16)
=
Φ
σv
σv
¶
µ1 P
Z
1
s vs − µv
2
≈ x
h (η) dη.
ϕ
σv
σv
DV
Voting power is by definition pDV
≈ pdf of tied
1 . By equations (8.14) and (8.16), p1
election, therefore equation (8.13) may be written
QDV
sµ ≈ vs ("voting power") E [fs | election tied] .
(8.17)
This is the equivalent to equation (8.5) under the Electoral College.
Rents The response in re-election probability to a uniform fall in vote share support
across states is proportional to average "voting power". To see this, first define ∆uFs as
the change in voter utility in state s necessary to induce a one percent fall in support:
∂F (∆us ) F
∆us = 1,
∂∆us
or equivalently,
∆uFs =
36
1
.
fs
Next, define the response in the probability of winning representing an equal fall in vote
shares as
∂P D
Qs
∂P D
=
∆uFs =
F
∂∆us
∂∆us
fs
Under EC, voting power equals
Qs
,
vs fs (0)
≈
pEC
1
see equation (8.5). Average voting power therefore equals the per capita response in
re-election probability to a uniform one percent fall in vote share support
X vs
s
v
pEC
≈
1
1 X ∂P
.
v s ∂∆uFs
Similarly, under DV
QDV
1 X ∂P
1X
s
≈ pDV
=
1 ,
v s ∂∆uFs
v s E [fs | national election tied]
DV
since QDV
sµ À Qsσ , and using equation (8.17).
It is reasonable that 1000 votes decide the election about 50 times more often under
the Electoral College. To see this, note that the probability of 1000 votes in any state
deciding the election under EC and DV is approximately
X 1000 µ ϕ (0) ¶
EC
,
p1000 ≈
Qs
/
v
σ
s
f
s
s
and
pDV
1000 ≈
X
s
QDV
s
1000 X vs
/
E [fs | national election tied] ,
v
v
s
where the latter equation was derived by inserting pDV
1000 /1000 instead of (pdf tied election)
into equation (8.13), summing over s, and solving for pDV
1000 . Assume that the marginal
1
and
voter density is the same fs = 1 and that all states are of equal size, then vvs = 50
pDV
1000 =
1 EC
p .
50 1000
Since Qs /vs is not strongly correlated with vs , the actual relation is not far from this.
8.5. Data definitions and sources
• Dmvote: state Democratic percentage of the two-party presidential vote. Source:
1940-1944, ICPSR Study 0019; 1948-1988, Campbell; 1992, 1996, Statistical Abstract of the United States 2000; 2000, Federal Election Commission, 2000 OFFICIAL PRESIDENTIAL GENERAL ELECTION RESULTS.
• Electoral votes won (by state). Source: National Archives and Records Administration.
37
• National trial-heat poll results. Source: 1948-1996, Campbell (2000); 2000, Gallup.
• Second quarter national economic growth, multiplied by 1 if Democratic incumbent
president and -1 if Republican incumbent president. Source: August or September
election year issue of the Survey of Current Business, U.S. Department of Commerce,
Bureau of Economic Analysis.
• Growth in personal state’s total personal income between the prior year’s fourth
quarter and the first quarter of the election year, standardized across states in
each year, multiplied by 1 if Democratic incumbent president and -1 if Republican
incumbent president. Source: Survey of Current Business, U.S. Department of
Commerce, Bureau of Economic Analysis.
• Incumbent: 1 if incumbent president Democrat, -1 if incumbent president Republican.
• Presinc: 1 if incumbent Democratic president seeking re-election, -1 if incumbent
Republican president seeking re-election.
• President’s home state: 1 if Democratic president home state, -1 if Republican (0.5
and -0.5 for large states (New York, Illinois, California). Source: Campbell 19481988; 1992-2000: Dave Leip’s Atlas of U.S. Presidential Elections.
• Vice president’s home state: 1 if Democratic president home state, -1 if Republican
(0.5 and -0.5 for large states (New York, Illinois, California). Source: Campbell
1948-1988; 1992-2000: Dave Leip’s Atlas of U.S. Presidential Elections.
• Average ADA-scores: Average ADA-scores of state’s members in Congress year
before election. Source: Tim Groseclose
(http://faculty-gsb.stanford.edu/groseclose/homepage.htm).
• Legis: Partisan division of the lower chamber of the state legislature after the previous midterm election. Index is Democratic share of state legislative seats above the
50% mark. Two states, Nebraska and Minnesota, held nonpartisan state legislative
elections for all (Nebraska) or part (Minnesota of the period under study. In the
case of Nebraska, the state legislative division was estimated based on the ranking
of states of Wright, Erikson, and McIver’s state partisan rankings based on public
opinion data. Using this index, Nebraska was assigned the mean partisan division of
the state most similar to it on the public opinion index, the nearly equally Republican state of North Dakota. The partisan division of the Minnesota legislature in its
nonpartisan years (before 1972) is coded as the mean of its partisan division once it
reformed to partisan elections (62% Democratic). Washington D.C. was as having
the same partisan division as Maryland. Source: 1948-1988, Campbell; 1992-2000,
Statistical Abstract of the United States.
• State-level opinion polls. Democratic share of two party vote. Source: Pre-election
issues of the Hotline (www.nationaljournal.com).
38
• Regional dummy variables, see Campbell.
• Participation rate: 1948-1988, Shachar and Nalebuff (1999), voting age population
1992-2000, and votes cast 2000, Federal Election Commission, votes cast 1992, 1996,
Statistical Abstract of the United States.
39