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 candidatesshould allocate their resources to maximize the probability of gaining a majorityin the Electoral College. A two-candidate, probabilistic-voting model reveals thatmore resources should be devoted to states which are likely to be decisive in theelectoral college and, at the same time, have very close state elections. The op-timal strategies are empirically estimated using state-level opinion-polls availablein September of the election year. The model’s recommended campaign strategiesclosely resemble those used in actual campaigns. The paper also analyses the policyeffects of electoral reform. It finds that resources will be more equally distributedunder Direct Vote than under the Electoral College system, and that both systemsdiscipline presidents from extracting political rents to an equal extent.

∗david.stromberg@iies.su.se, IIES, Stockholm University, S-106 91 Stockholm.I thank Steven Brams, Steve Coate, Antonio Merlo, Torsten Persson, Gerard Roland, Tom Romer, HowardRosenthal, Jim Snyder, Jörgen Weibull, and seminar participants at UC Berkeley, Columbia University,Cornell University, the CEPR/IMOP Conference in Hydra, Georgetown University, the Harvard / MITSeminar on Positive Political Economy, IIES, New York University, Stanford University, University ofPennsylvania, Princeton University, and the Wallis Conference in Rochester. Previous versions havebeen circulated under the titles: ”The Lindbeck-Weibull model in the Federal US Structure”, and, ”TheElectoral 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 amend-ments to alter or abolish the Electoral College than on any other subject. The issue wasfurther put in focus by the 2000 election featuring a razor thin victory and a presidentelect who lost the popular vote. Still, the effects of the Electoral College system are notwell understood and academic research on the topic is underdeveloped.This paper attempts to fill this void by providing the first empirically reasonable, full-

fledged 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 wouldresource 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 insteadelected through a direct national vote, which system creates a more equal distribution ofresources, 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 actualcampaigns. This is done by constructing a probabilistic-voting model that can both beexplicitly solved, and directly estimated. The explicit solution draws on use of a CentralLimit Theorem approximation, while the possibility of direct estimation draws on a carefulmodelling of political preferences and election uncertainty. The uncertainty, which makesthe 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 com-

petition for legislative seats. Although the equilibrium of this model is not explicitlysolved, some important qualitative features are characterized. Snyder (1989) finds thatequilibrium 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 majorityof 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 thesafe districts of the other party. Snyder’s paper studies one-member districts and thereforedoes not allow for variation in size necessary for the analysis of the Electoral College. Sizeeffects 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 votesin each state are cast with equal probability for each candidate. Brams and Davis (1974)find that presidential candidates should allocate resources disproportionately in favor oflarge states. Their result is disputed by Colantoni, Levesque and Ordeshook (1975) whoinstead argue that a proportional rule, modified to take into account the closeness of thestate election, predicts actual campaign allocations better. Inspired by these results, Na-gler and Leighley (1992) empirically investigate state-by-state campaign expenditures on

1In this system, a direct vote election is held in each state and the winner of the vote is supposed toget all of that state’s electoral votes. Then all the electoral votes are counted, and the candidate whoreceives most votes wins the election. (The fact that Maine and Nebraska organize their presidentialelections by congressional district is disregarded in this paper.)

2

non-network advertising in 1972 and find these to be higher in states with closer electionsand 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 over-provide redistributive spending, relative to public goods, because its benefits can be moreeasily 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 andBoscardin (1998)), and Merrill (1978). The "voting power" analyzed in this paper isslightly 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 electoralsettings. As shown in this paper, it can be used to analyze the allocation of campaignresources as well as redistributive spending and political rents under the Electoral andDirect Vote systems. With a minor adjustment, it may also be used to analyze theallocational 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 analyzeendogenous 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 estimatesare 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 DirectVote system. Finally, Section 7 discusses the results and concludes.

2. Model

Two presidential candidates, indexed by superscript R and D, try to maximize their ex-pected probability of winning the election by selecting the number of days, ds, to campaignin each state s, subject to the constraint

SXs=1

dJs ≤ I,

J = R,D. In each state s, there is an election. The candidate who receives a majority ofthe votes in a state gets all the es electoral votes of that state. After elections have beenheld in all states, the electoral votes are counted, and the candidate who gets more thanhalf 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 amongvoters in that state, as captured by the increasing and concave function u

¡dJs¢.2 The

2This paper does not address the question of why campaigning matters. This is an interesting questionin 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 parametersRi, ηs, and η. The parameter Ri represents an individual-specific ideological preference infavor of candidate R, and ηs and η represent the general popularity of candidate R instate 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

∆us = u¡dDs¢− u

¡dRs¢ ≥ Ri + ηs + η. (2.1)

At the time when the campaign strategies are chosen, there is uncertainty about thepopularity of the candidates on election day. This uncertainty is captured by the randomvariables ηs and η. The candidates know that the S state level popularity parameters,ηs, and the national popularity parameter, η, are independently drawn from cumulativedistribution functions Gs = N(0, σ2s), and H = N (0, σ2) respectively, but they do notknow the realized values.The distribution of voters’ ideological preferences, Ri, within each state is Fs =

N¡µs, σ

2fs

¢, a normal distribution with mean µs and variance σ2fs. The means of the

states’ ideological distributions may shift over time, but the variance is assumed to remainconstant. The share of votes that candidate D receives in state s is

Fs(∆us − ηs − η).

This candidate wins the state if

Fs(∆us − ηs − η) ≥ 12,

or, equivalently, ifηs ≤ ∆us − µs − η.

The probability of this event, conditional on the aggregate popularity η, and the campaignvisits, dDs , and dRs , is

Gs (∆us − µs − η) . (2.2)

Let es be the number of votes of state s in the Electoral College. Define stochasticvariables, 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

PD (∆us, η) = Pr

"Xs

Dses >1

2

Xs

es

#. (2.3)

However, it is difficult to find strategies which maximizes the expectation of the aboveprobability of winning. The reason is that it is a sum of the probabilities of all possiblecombinations of state election outcomes which would result in D winning. The numberof such combinations is of the order of 251, for each of the infinitely many realizations ofη.

4

A way to cut this Gordian knot, and to get a simple analytical solution to this prob-lem, is to assume that the candidates are considering their approximate probabilities ofwinning. Since the ηs are independent, so are the Ds. Therefore by the Central LimitTheorem of Liapounov, P

sDses − µ

σE

whereµ = µ (∆us, η) =

Xs

esGs (∆us − µs − η) , (2.4)

andσ2E = σ2E (∆us, η) =

Xs

e2sGs (·) (1−Gs (·)) , (2.5)

is asymptotically distributed as a standard normal. The mean, µ, is the expected numberof electoral votes. That is, the sum of the electoral votes of each state, multiplied by theprobability of winning that state. The variance, σ2E, is the sum of the variances of thestate outcomes, which is the e2s multiplied by the usual expression for the variance of aBernoulli variable. Using the asymptotic distribution, the approximate probability of Dwinning the election, conditional on η, is

ePD (∆us, η) = 1− Φ

µ 12

Ps es − µ

σE

¶. (2.6)

The approximate probability of winning the election is

PD (∆us) =

Z ePD (∆us, η)h (η) dη.3 (2.7)

Let the set of allowable campaign visits be

X =

(d ∈ <S

+ :SXs=1

ds ≤ I

).

The Nash Equilibrium strategies¡dD∗, dR∗

¢in the competition between the two candidates

are characterized by

PD¡dD∗, dR

¢ ≥ PD¡dD∗, dR∗

¢ ≥ PD¡dD, dR∗

¢for all dD, dR ∈ X. This game has a unique interior pure-strategy equilibrium characterizedby 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 electionmust satisfy dD = dR = d∗, and, for all s and for some λ > 0,

Qsu0 (d∗s) = λ, (2.8)

3The error made from using the approximate probability of winning is discussed in Section 5. Forfurther discussion, see Appendix 6.8 of Stromberg (2002).

5

where

Qs =∂PD

∂∆us. (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. Thisfollows since u0 (d∗s) is decreasing in d∗s.Note that Qs can be decomposed into two additively separable parts:

Qs = Qsµ +Qsσ (2.10)

= es

Z1

σEϕ (x (η)) gs (−µs − η)h (η) dη

+e2sσ2E

Zϕ (x (η))x (η)

µ1

2−Gs (−µs − η)

¶gs (−µs − η)h (η) dη,

where

x (η) =12

Ps es − µ

σE.

One arises because the candidates have an incentive to influence the expected numberof electoral votes won by D, that is the mean of the normal distribution. The otherarises because the candidates have an incentive to influence the variance in the numberof electoral votes. To evaluate Qs, I now use the structure of the model to estimate theprobability distribution for election outcomes.

3. Estimation

This estimation provides the link between the theoretical probabilistic-voting model aboveand the empirical applications discussed below. In equilibrium, both candidates choosethe same allocation, so that ∆us = 0 in all states. The Democratic vote-share in state sat time t equals

yst = Fst (−ηst − ηt) = Φ

µ−µst − ηst − ηtσfs

¶, (3.1)

where Φ (·) is the standard normal distribution, or equivalently,

Φ−1(yst) = γst = −1

σfs(µst + ηst + ηt) . (3.2)

For now, assume that all states have the same variance of preferences, σ2fs = 1, andthe same variance in state-specific shocks, σ2s.

4 The former assumption implies that themarginal voter density, conditional on the state election being tied, is the same in all

4These assumptions will be removed in Section 6. However, the estimates become imprecise if separatevalues of µst, σfs, and σs 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 aset of observable variables Xst. Then we get the following estimable equation,

γst = − (Xstβ + ηst + ηt) . (3.3)

The parameters β, σs and σ are estimated using a standard maximum-likelihood estima-tion of the above random-effects model.5

The variables in Xst are basically those used in Campbell (1992). The nation-widevariables are: the Democratic vote share of the two-party vote share in trial-heat pollsfrom mid September (all vote-share variables x are transformed by Φ−1(x)); second quar-ter economic growth; incumbency; and incumbent president running for re-election. Thestate-wide variables for 1948-1984 are: lagged and twice lagged difference from the na-tional mean of the Democratic two-party vote share; the first quarter state economicgrowth; the average ADA-scores of each state’s Congress members the year before theelection; the Democratic vote-share of the two-party vote in the midterm state legislativeelection; the home state of the president; the home state of the vice president; and dummyvariables described in Campbell (1992). After 1984, state-level opinion-polls were avail-able. For this period, the state-wide variables are: lagged difference from the nationalmean of the Democratic vote share of the two-party vote share; the average ADA-scoresof each state’s Congress members the year before the election; and the difference be-tween the state and national polls. Other state-wide variables were insignificant whenstate polls were included. The coefficients β and the variance of the state-level popu-larity shocks, σ2s, are allowed to differ when opinion polls were available and when theywere not. Estimates of equation (3.3) yields forecasts by mid September of the electionyear. The data-set contains state elections for the 50 states 1948-2000, except Hawaii andAlaska which began voting in the 1960 election. During this period there were a totalof 694 state-level presidential election results. Four elections in Alaska and Hawaii wereexcluded because there were no lagged vote returns. Nine elections are omitted becauseof idiosyncrasies in Presidential voting in Alabama in 1948, and 1964, and in Mississippiin 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 , bσs, equals 0.077, or about 3% in vote shares. Thisis more than twice as large as that of the national shocks, bσ = 0.033. The estimatedstate-preference means are bµst = Xst

bβ.The average absolute error in state-election vote-forecasts, Φ (bµst), is 3.0 percent and thewrong winner is predicted in 14 percent of the state elections. This precision is comparableto the best state-level election-forecast models (Campbell, 1992; Gelman and King, 1993;

5The model has been extended to include regional swings, see Appendix 6.4 of Strömberg (2002). Inthis specification, the democratic vote-share in state s equals

yst = Fst (ηst + 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 regionalswings. Therefore, the simpler specification without regional swings is used below.

7

Holbrook and DeSart, 1999; Rosenstone, 1983). Given bµst, bσs and bσ, Qs can be calculatedusing 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 candidatevisits to states in last three months of the 2000 election, and also more loosely discussvisits during the 1988-1996 elections. The second sub-section will study the allocation ofcampaign 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,

d∗sPd∗s=

QsPQs

, (4.1)

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 campaignvisits, after the party conventions, and Qs, are shown in Figure 4.1.6 Campaign visits byvice presidential candidates are coded as 0.5 visits. The model and the candidates’ actualcampaigns agree on 8 of the 10 states which should receive most attention. Notabledifferences between theory and practice are found in Iowa, Illinois and Maine, whichreceived more campaign visits than predicted, and Colorado, which received less. Perhapsextra attention was devoted to Maine since its (and Nebraska’s) electoral votes are splitaccording to district vote outcomes. Other differences could be because the campaignshad access to information of later date than mid September, and because aspects notdealt with in this paper matter for the allocation. The raw correlation between campaignvisits and Qs is 0.91. For Republican visits the correlation is 0.90 and for Democraticvisits, 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 yearsare 0.85, 0.64, and 0.76, respectively. But this is mainly a result of presidential candidatesspending more time in large states. For the 1996, 1992, and 1998 elections, the correlationbetween ds/es and Qs/es was 0.12, 0.58, and 0.25 respectively. An explanation for thepoor fit in 1996 and 1988 may be that these elections were, ex ante, very uneven. Theexpected Democratic vote shares in September of 1996, 1992, and 1988 were 56, 50, and46 percent. In uneven races, perhaps the candidates have other concerns, such as affectingthe congressional election outcome, rather than maximizing the probability of winning thepresidential election.

6I am grateful to Daron Shaw for providing me with the campaign data.

8

0 2 4 6 8 10 12

Hawaii

New York

Vermont

South Carolina

Alaska

Alabama

South Dakota

North Dakota

Montana

Maine

Wyoming

Maryland

Indiana

West Virginia

Virginia

Mississippi

Connecticut

Delaware

New Jersey

Minnesota

Nevada

Arizona

New Hampshire

Georgia

New Mexico

North Carolina

Colorado

Arkansas

Kentucky

Iowa

Illinois

Oregon

Louisiana

Washington

Wisconsin

Tennessee

Missouri

Ohio

California

Pennsylvania

Michigan

Florida

percent

∑ s

s

QQ

(Utah, Texas, Rhode Island, Oklahoma, Texas, Nebraska, Massachusetts, Kansas, Idaho) ≈ 0 for both series.

Actual campaign visits

Figure 4.1: Actual and equilibrium campaign visits 2000

9

4.2. Campaign advertisements

Appendix 8.3 models the decision of presidential candidates to allocate advertisementsacross Designated Market Areas (DMAs).7 In that model, two presidential candidateshave a fixed advertising budget Ia to spend on am ads in each media market m subject to

MXm=1

pmaJm ≤ Ia,

J = R,D, where pm is the price of an advertisement. Media market m contains a massvms of voters in state s. Voters are now also affected by campaign advertisements ascaptured by the increasing and concave function w(am). A voter i in media market m instate s will vote for D if

u¡dDs¢+ w

¡aDm¢− u

¡dRs¢− w

¡aRm¢ ≥ Ri + ηs + 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 isincreasing in Qm

pm, where

Qm =SXs=1

Qsnms

ns.

Qm is the sum of the Qs of the states in the media market weighted by the share of thepopulation 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 presidentialelection, aired in the 75 major media markets between September 1 and Election Day. Thedata is disaggregated by whether it supported the Republican, Democrat, or independentcandidate, and by whether it was paid for by the candidate, the party or an independentgroup. The cost estimates, pm, are average prices per unit charged in each particular mediamarket. The estimates are made by the CampaignMedia Analysis Group. Advertisementswere 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, Isum together the advertisements by the candidates, the parties and independent groupssupporting 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 - SantaFe, and Portland, Oregon). These two markets have the highest effect on the win prob-ability 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

7A DMA is defined by Nielsen Media Research as all counties whose largest viewing share is given tostations of that same market area. Non-overlapping DMAs cover the entire continental United States,Hawaii and parts of Alaska.

8The 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

0 76200

7620

Tota

l adv

ertis

emen

ts

Qm/pm

Albuquerque - Santa Fe

Portland, Oregon

DenverLexington

Figure 4.2: Total number of advertisements Sept. 1 to election day and Qm/pm, for the75 largest media markets

between actual campaign advertisement and equilibrium advertisement is 0.75. That fewadvertisements were aired in Denver is consistent with the few candidate visits to Col-orado (recall Figure 4.1). The few advertisements in Lexington are more surprising, sincecandidate 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 Qm

pminto four terms:

Qm

pm=Xs

nms

nm|{z}(o)

esns|{z}(i)

Qs

es|{z}(ii)

1

pm/nm| {z }(iii)

. (4.2)

(o) 95 percent of the population of Albuquerque - Santa Fe live in New Mexico (nms

nm=

0.95), and 5 percent in Colorado. (i) Since New Mexico is a small state with only 1.8million inhabitants, it had a high number of electoral votes per capita. (ii) Since NewMexico 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 apreview. (iii) At the same time, the average cost of an ad per million inhabitants in themedia 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 highestvalue of Qs per electoral vote. However, being a fairly large state, Michigan only has 1.8electoral votes per million inhabitants. Further, the average cost of an ad in Detroit ishigher than in Albuquerque - Santa Fe. Therefore, the marginal impact on the probabilityof 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 ofads.Via the price, the size is instead captured in the costs. Assuming log utility, equi-librium expenditures, pma∗m, are proportional to Qm. Empirically, the simple correlationbetween advertisement costs, pmam, and Qm is 0.88.

11

5. Interpretation

This section discusses what Qs measures and why it varies across states. A qualified guessis that Qs is approximately the joint ”likelihood” that a state is actually decisive in theElectoral College and, at the same time, has a very close election. I will call states whoare ex post decisive in the Electoral College and have tied state elections decisive swingstates. In the 2000 election, Florida was a decisive swing state. In contrast, neither NewMexico nor Wyoming were decisive swing states. While New Mexico was a swing statewith a very close state-election outcome, it was not decisive in the Electoral College sinceBush would have won with or without the votes of New Mexico. While Wyoming wasdecisive in the Electoral College, since Gore would have won the election, had he wonWyoming, 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 Limitapproximation. Further, Appendix 8.2 shows heuristically, why the analytical expressionfor 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 themarginal 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 foreach election 1988-2000 by using the estimated state-preference means, and drawing stateand national popularity-shocks from their estimated distributions.9 Then, the share ofelections where a state was decisive in the Electoral College and at the same time had astate election outcome between 49 and 51 percent was recorded. This provides an estimatewhich 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 tobe decisive. To check that the correlation between Qs and the simulated values is not justa matter of size, the graph on the right contains the same series divided by the state’snumber of electoral votes. The simple correlation in the diagram to the right is 0.997. Sothe two variables are interchangeable, for practical purposes. The 0.003 difference couldresult 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 stateelection results between 49 and 51 percent, whereas theoretically it should be exactly 50percent.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 andat the same time have a state election margin of less than 2 percent. This happened in 2.2percent of the simulated elections in Ohio and 3.4 percent of the simulations in Florida. Incomparison, the scaled, analytic expression for Qs equals 3.5 percent for Florida. (UsingQs, the probability that Florida would be decisive in the Electoral College and have a

9Replace µst by the estimated bµst in equation (3.1), and draw ηst and ηt from their estimated distri-

butions N³0, bσ2s´ and N

³0, bσ2´, respectively to generate election outcomes yst.

12

Pivo

tal a

nd c

lose

per

ele

ctor

al v

ote

Qs per electoral vote0 .0020

.002

Sim

ulat

ed s

hare

piv

otal

and

clo

se e

lect

ions

Qs0 .0550

.047

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 10000. 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 tobe decisive swing states. First, Qs is roughly proportional to the number of electoralvotes.10 WhileQsµ and,Qsσ are proportional to electoral votes and electoral votes squared,respectively, see equation (2.10), Qsσ is generally considerably smaller than Qsµ. Thisimplies that candidates should, on average, spend more time in large states. However, forstates 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 ElectoralCollege and at the same time had a state-election outcome between 49 and 51 percent, perelectoral vote. The solid line shows Qsµ/es. Its normal form arises because the candidatestry to affect the expected number of electoral votes, see equation (2.10). This part ofQs/esaccounts for most of the variation in the simulated values. It explains why states like NewYork and Texas in a million simulated elections are never decisive in the Electoral Collegeand 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 theaverage Qsµ varies between elections. 11

10This can be contrasted to the finding that ”voting power”, that is, the probability that a vote ispivotal in the election is more than proportional to size (Banzaf 1967, Brams and Davis 1974). ”Votingpower” is roughly proportional to Qs, and thus roughly proportional to size. The difference resultsfrom all voters being equally likely to vote for one candidate or the other in their models, while voterspreferences 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) forempirical results.11Formally, define eηt to be the national popularity-swing which would give equal expected Electoral

Vote shares, µ (eη) = 12

Ps es. Then Qsµ is larger when eηt is close to zero; See equation (8.6) in the

Appendix.

13

0 0,5 1 1,5 2 2,5 3 3,5

Hawaii

New York

Vermont

South Carolina

Alaska

Alabama

South Dakota

North Dakota

Montana

Maine

Wyoming

Maryland

Indiana

West Virginia

Virginia

Mississippi

Connecticut

Delaware

New Jersey

Minnesota

Nevada

Arizona

New Hampshire

Georgia

New Mexico

North Carolina

Colorado

Arkansas

Kentucky

Iowa

Illinois

Oregon

Louisiana

Washington

Wisconsin

Tennessee

Missouri

Ohio

California

Pennsylvania

Michigan

Florida

percent

(Utah, Texas, Rhode Island, Oklahoma, Texas, Nebraska, Massachusetts, Kansas, Idaho) = 0

Figure 5.2: Joint probability of being pivotal and having a state margin of victory lessthan two percent, based on September 2000 opinion polls.

14

Simulated values, pivotal and close /es

Forecasted democratic vote share

30 40 50 60 700

.0005

.001

.0015

Michigan (18)

PennsylvaniaOhio

California

New YorkTexasWyoming

µ∗s

Qsµ /es

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 theresult 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 nationalforecasts. In states with 50-50 forecasts, a candidate’s visit is more likely to influencethe state election outcome. However, states where the Democrats are ahead 60-40 arelikely to have close state elections exactly when the national election is close. Althoughthe candidates are less likely to influence the state outcome, they are more likely to doso when it matters. The mean of Qsµ/es will always lie between 50-50 and the nationalforecast (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 inTexas was 50-50. A visit to Texas was therefore more likely to affect the state outcomethan a visit to Pennsylvania. However, if Texas was a 50-50 state on election day, thenClinton was probably winning by a landslide, and the electoral votes of Texas were prob-ably not decisive in the electoral college. On the other hand, in the unlikely event thatPennsylvania 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 mustcondition 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 50-50states like Texas. The reason is that without national swings, the state outcomes are notcorrelated, and a state being a swing state on election day carries no information aboutthe outcomes in the other states. In my estimates maximum attention should typicallybe given to states in the middle, 55-45 in this example. In September of 2000, Gore wasahead by 1.3 percentage points. The maximum Qsµ/es was obtained for states where theexpected outcome was a Democratic vote share of 50.8 percent, as illustrated in Figure

15

5.3.12

The variance inQsµ/es depends on the uncertainty about the election outcome. Betterstate-level forecasts lead to a more unequal allocation of campaign resources as the vari-ance of the normal-shaped distribution of Figure 5.3 decreases. States with forecasted voteshares close to the center of that distribution would gain while states far from the centerwould lose. Better national-level forecasts has a similar effect.14 Note that Wyoming andtwo 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 theforecasts for these states are more uncertain. These states actually lie on a normal-shapedcurve with a higher variance than that drawn in Figure 5.3. These observations illustrateone 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 theright of µ∗s, like Michigan and Pennsylvania, generally lie above the curve, while states onthe left, like Ohio, generally lie below. The difference between the simulated values andQsµ/es arises because the candidates also have incentives to influence the variance of theelectoral vote distribution, even if this means decreasing the expected number of electoralvotes, 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 oneminute left of the game. To increase the probability of scoring an equalizer, the trailingteam pulls out the goalie and puts in an extra offensive player. Most frequently, the resultis that the leading team scores. But the trailing team does not care about this, since theyare losing the game anyway. They only care about increasing the probability that theyscore an equalizing goal, which is higher with an extra offensive player. Therefore, it is

12These points are evident from the analytical form of the mean of equation (5.1), derived in Appendix8.2. The mean equals

µ∗st = −σ2

σ2 + (σE/a)2eηt, (5.1)

whereσ2E = σ2E

¡dD = dR, ηt = eηt¢ ,

at =Xs

esgs (−µst − eηt) .13The mean always lies between a pro-Republican state bias of µst = 0, which corresponds to a 50%forecasted Democratic vote share, and µst = −eηt, which approximately corresponds to the forecastednational Democratic vote share. (If the Democrats are ahead by 60-40 nationally, then a pro-Republicanswing eηt, corresponding to about 10%, is needed to draw the election. Therefore µst = −eηt correspondsto 10% pro-democrat bias in a state, that is, a vote share of 60-40.) The smaller the variance of thenational popularity-swings, σ2, the closer is the mean to 50-50. In the extreme case where this varianceequals zero, then µ∗s = 0. In the extreme case that σ approaches infinity, µ

∗s approaches −eη.

14The variance of the normal-form distribution,

eσ2 = σ2s +1³

1σ2 +

1(σE/a)

2

´ ,depends on the variance in the state, and national, level popularity shocks, see Appendix 8.2.

16

30 40 50 60 70-.0001

0

.0001

.0002

Michigan Pennsylvania

Ohio

Varia

nce

effe

ct, Q

sσ/v

s

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. Incontrast, if they were allowed, the leading team would like to pull out an offensive playerand 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 isbehind (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 variancein electoral votes, thus securing his lead, by spending more time in large states where heis ahead (putting in an extra goalie), and less time in states where he is behind (pullingout an offensive player). Both candidates thus spend more time in large states where theexpected 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) showingthe variance, conditional on a national shock. The variance in the number of electoralvotes from a state is proportional to these votes squared. Therefore, the effect on thetotal variance, per electoral vote, is larger in large states. Further, the variance in a stateoutcome is higher the closer the expected result is to a tie. By visiting a state where theleading candidate is ahead, the trailing candidate moves the expected result closer to atie, and increases the variance in election outcome. Similarly, decreasing the number ofvisits to a state where the lagging candidate is leading increases the varianceFigure 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 offen-sive visits in states like Michigan and Pennsylvania, at the cost of weakening the defenseof states like Ohio. The leading candidate (Gore) should increase his defense of states likeMichigan and Pennsylvania, at the cost of offensive visits to Ohio. This resounds with theresult by Snyder (1989) that parties will spend more in safe districts of the advantagedparty 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 percentcompared to their actual strategies.15 The most important feature of the best responsesis that Bush should spend more than the equilibrium time in California, which the Gorecampaign left unguarded with very few visits. The expected democratic vote-share inCalifornia was 56 percent. Fewer visits by Gore moves the expected democratic vote-share closer to 50.8 percent, evaluated at Bush’s equilibrium strategy. This increases theprobability of California being a decisive swing state, which increases Bush’s incentivesto visit California. In fact, Bush spent more than the equilibrium amount of time inCalifornia. This, on the other hand, moves the expected democratic vote-share closerto 50.8 percent, evaluated at Gore’s equilibrium strategy, increasing Gore’s incentives tovisit California. However, Gore visited California less than the equilibrium number oftimes. Roughly speaking, more visits is a best response to more offensive visits (Bush inCalifornia) or fewer defensive visits (Gore in California) by the opponent. Fewer visits isa 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, thechange to a direct vote for president. According to federal historians, over 700 proposalshave 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 abol-ish the Electoral College than on any other subject. The debate intensified as the 2000presidential 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 Wash-ington Post/ABC News poll performed shortly after the election suggested that about 6in 10 Americans would prefer to abandon the Electoral College and switch to a directpopular 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 changetheir campaign patterns and shun them altogether. Others have argued that the ElectoralCollege system creates a bias favoring small states. Finally, Lizzeri and Persico (2001),and Persson and Tabellini (1999), have argued that under the Electoral College, thedistribution of targeted programs will be more concentrated, while political rents, andpublic 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 ElectoralCollege, and then further modified to discuss policy formation under Direct Vote.

Economic policy Section 4 shows that politicians understand the incentives createdby the Electoral College system. This section explores the consequences if policy is alsoinfluenced by election concerns. To analyze economic policy, one could just re-interpretthe model of Section 4 as describing the incentives to make policy promises to states

15This result depends on the functional form assumed for u (ds) . I can not use log form since this isnot defined for zero visits. Instead I use the exponential function u (ds) = 0.018 ∗ d0.34s , where the twoconstants were estimated in Strömberg (2002).

18

during the campaign. If a state is important for the election outcome, not only shouldthe candidates visit that state, they should also make favorable policy promises to thatstate. However, the route taken in this section is to analyze the incentives of incumbentpresidents to set policy for re-election concerns.Suppose each voter in state s receives utility from policy

us = υ (zs)− r

n+ θ

where zs is redistributive spending per capita, rnis political rents per capita, and θ is the

incumbent’s competence which is drawn from a known normal distribution with meanzero. The function υ is increasing and concave. Political rents describe a conflict ofinterest between the president and the voters. It could entail party financing, extrasalaries, low effort and waste or outright corruption. The voter observes his total utilityfrom policy, but not its separate components. So, the voter does not know whether utilityfrom a government program is high because per capita spending is high, because thepolitical rents are low, or competence is high. The incumbent president’s preferences aredescribed by P + (r) where P is the approximate probability of re-election, definedbelow, 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 opponentwith unknown competence. After the election, the winner selects a fixed policy and theutility 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 ifE [θ] ≥ Ri + ηs + η0.

The voters observe us and form expectations, E [θ] = us − (υ (z∗s)− r∗) , where z∗s and r∗

denote equilibrium policy. Therefore, the above equation becomes

∆us = υ (zs)− r − (υ (z∗s)− r∗) ≥ Ri + ηs + η.

where η = η0 − θ. The above equation has exactly the same structure as equation (2.1).The same distributional assumptions are now made regarding the exogenous parametersRi, ηs, 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

maxP + (r)

subject to a fixed budget constraint Xs

nszs = I.

Proposition 2. The incumbent strategy (z∗, r∗) that maximizes P + (r) must satisfy,for all s and for some λ > 0,

Qs

nsυ0 (z∗s) = λ, (6.1)

19

0 (r∗) =1

n

Xs

Qs. (6.2)

where

Qs =∂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 redistributive spending to states

with high Qs. Political rents are lower the larger isP

sQs, which measures the fall inre-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 earlierthan September the election year. The incumbent president must make his policy decisionsbefore the September opinion poll results are known. Therefore, Qs used for evaluatingthe effects on policy will be estimated without opinion poll data.16

Direct Vote Next, suppose the president is elected by a direct national vote. Thenumber of Democratic votes in state s is then equal to

vsFs(∆us − η − ηs).

The Democratic candidate wins the election if he receives more than half of the popularvotes: X

s

vsFs(∆us − η − ηs) ≥1

2

Xs

vs.

The number of votes won by candidate D is asymptotically normally distributed withmean and variance

µv =Xs

vsΦ

∆us − µs − ηqσ2s + σ2fs

, (6.3)

σ2v = σ2v (∆us, η) .

See Appendix 8.4 for the explicit expression for σ2v. The probability of a Democraticvictory is

PD = 1−Z

Φ

µ 12

Ps vs − µvσv

¶dη.

The incumbent allocates spending and sets political rents to maximize PD + (r). Thefollowing proposition characterizes the equilibrium allocation.

16This version of the model has been used in Strömberg (2002a), where it is found that federal civilianemployment is higher when Qs is higher, even using time and state fixed effects and controlling for otherdeterminants of employment.

20

20 30 40 50 60.06

.08

.1

.12

CT

ME

MA

NH

RI

VT

DE

NJ

NY

PA

ILIN

MIOH

WIIA

KS MN

MO

NE

ND

SD

VA

AL

AR

FL

GA

LA

MS

NC

SC

TXKY

MD

OK

TN

WV

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UT

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Estim

ated

mar

gina

l vot

er d

ensi

ty

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 PD + (r) must satisfy,for all s and for some λ > 0,

QDVs

nsυ0 (zs) = λ, (6.4)

0 (r) =Xs

QDVs . (6.5)

The variableQDVs measures the likelihood of a draw in the national election, multiplied

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 likelihoodof a draw in the national election is the same in all states, QDV

s varies across states onlybecause 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, σ2fs.Therefore, the restriction σfs = 1 is removed in the maximum likelihood estimation ofequation (3.2), as well as the assumption that σs is the same for all states. However, theempirical identification of σfs and σs is not trivial. If the election outcome in a certainstate varies a lot over time, is this because the state has many marginal voters or is itbecause the state has been hit by unusually large shocks shifting voter preferences? Themodel solves this problem by identifying σfs by the response in vote shares to changes thatare common to all states, and observable changes in economic growth at national and statelevel, incumbency variables, home state of the president and vice president. States wherethe vote share outcome covary strongly with economic growth, etc., are thus estimated tohave many marginal voters. Maine is estimated to have the largest share of marginal voterswhile California has the smallest. The estimated share of marginal voters is positivelycorrelated with the share of independent voters as measured by Erikson, Wright andMcIver (1993), as illustrated in Figure 6.1. The variance in the state popularity-shocks,σ2s, 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

zDVs =

QDVs

ns1n

PQDV

s

I

n,

whereas the analogous expression for spending under EC is given by just substitutingQs for QDV

s . The effect of reform is shown in Figure 6.2. The series are 1948-2000averages, scaled so that 1 denotes equal per capita spending. States above 1 on the y-axisreceive 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 spendingunder under the Direct Vote system. Thus, states below the dashed 45 degree line wouldgain 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. Otherstates, like Nevada are among the winners in the present system but among the losersunder 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

ns=

esns|{z}(i)

Qs

es|{z}(ii)

,

DV :QDVs

ns=

vsns|{z}(i)

QDVs

vs| {z }(ii)

.

Figure 6.3 plots the 1948-2000 averages of, (i), electoral votes per capita million andvoter turnout to the left, and, (ii), the probability of being a pivotal swing state perelectoral vote and the share of marginal voters to the right. The latter have been scaledso that 1 denotes equal per capita influence relative to size. Nevada and Delaware wouldlose from reform primarily because of their heavy endowment of electoral votes relativeto popular votes, while Ohio would lose primarily because it is likely to be a pivotalswing state, but does not have many marginal voters. On the winning side, Rhode Islandand Massachusetts would gain because of their many marginal voters, while Kansas andNebraska would gain because of their low probabilities of being decisive swing states inthe 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 moreoften had lop-sided elections and have larger state-level uncertainty, σ2s. On net, Qs/ns isnot 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 verystrong incentives for unequal distribution under the EC. The average probability that

17I thank Andrew Gelman for pointing this out to me.

22

0 .5 1 1.5 2 2.50

.5

1

1.5

2

2.5

CT

ME

MA

NH

RI

VT

DE

NJ

NY

PA

IL

IN

MI

OH

WI

IA

KS

MN

MO

NE

ND

SD

VAAL

ARFL

GA

LA

MS

NCSC

TX

KYMD

OKTN

WV

AZ

CO

ID

MT

NV

NM

UT

WY

CA

OR

WA

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Direct Vote

Elec

tora

l Col

lege

Figure 6.2: Redistributive spending per capita, relative to national average, under theElectoral College and Direct Vote

Voter turnout.3 .5 .7

2

4

6

8

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QsDV/vs : Marginal voter density

Qs/e

s : P

r(dec

isiv

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stat

e)/e

s

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es p

er c

apita

milli

on

(i) (ii)

.5 1 1.50

.5

1

1.5

2

2.5

UTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUT

MNMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNSDSDSDSDSDSDSDSDSDSDSDSDSDSDSDSD

NDNDNDNDNDNDNDNDNDNDNDNDNDNDNDND

CTCTCTCTCTCTCTCTCTCTCTCTCTCTCTCT

IAIAIAIAIAIAIAIAIAIAIAIAIAIAIAIA NHNHNHNHNHNHNHNHNHNHNHNHNHNHNHNH

IDIDIDIDIDIDIDIDIDIDIDIDIDIDIDID

WIWIWIWIWIWIWIWIWIWIWIWIWIWIWIWI

ILILILILILILILILILILILILILILILIL

MTMTMTMTMTMTMTMTMTMTMTMTMTMTMTMT

MAMAMAMAMAMAMAMAMAMAMAMAMAMAMAMAININININININININININININININININ

DEDEDEDEDEDEDEDEDEDEDEDEDEDEDEDE

RIRIRIRIRIRIRIRIRIRIRIRIRIRIRIRI

WVWVWVWVWVWVWVWVWVWVWVWVWVWVWVWV

WYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWY

COCOCOCOCOCOCOCOCOCOCOCOCOCOCOCO

MOMOMOMOMOMOMOMOMOMOMOMOMOMOMOMO

MEMEMEMEMEMEMEMEMEMEMEMEMEMEMEME

OROROROROROROROROROROROROROROROR

KSKSKSKSKSKSKSKSKSKSKSKSKSKSKSKS

WAWAWAWAWAWAWAWAWAWAWAWAWAWAWAWAMIMIMIMIMIMIMIMIMIMIMIMIMIMIMIMI

OHOHOHOHOHOHOHOHOHOHOHOHOHOHOHOH

NENENENENENENENENENENENENENENENE

NJNJNJNJNJNJNJNJNJNJNJNJNJNJNJNJ

VTVTVTVTVTVTVTVTVTVTVTVTVTVTVTVT

PAPAPAPAPAPAPAPAPAPAPAPAPAPAPAPA

NYNYNYNYNYNYNYNYNYNYNYNYNYNYNYNY

CACACACACACACACACACACACACACACACA

OKOKOKOKOKOKOKOKOKOKOKOKOKOKOKOK

NMNMNMNMNMNMNMNMNMNMNMNMNMNMNMNM

NVNVNVNVNVNVNVNVNVNVNVNVNVNVNVNV

KYKYKYKYKYKYKYKYKYKYKYKYKYKYKYKY

AKAKAKAKAKAKAKAKAKAKAKAKAK

MDMDMDMDMDMDMDMDMDMDMDMDMDMDMDMD

AZAZAZAZAZAZAZAZAZAZAZAZAZAZAZAZ

HIHIHIHIHIHIHIHIHIHIHIHIHINCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCFLFLFLFLFLFLFLFLFLFLFLFLFLFLFLFL

TNTNTNTNTNTNTNTNTNTNTNTNTNTNTNTN LALALALALALALALALALALALALALALALAARARARARARARARARARARARARARARARAR

TXTXTXTXTXTXTXTXTXTXTXTXTXTXTXTXALALALALALALALALALALALALALALVAVAVAVAVAVAVAVAVAVAVAVAVAVAVAVA

GAGAGAGAGAGAGAGAGAGAGAGAGAGAGAGAMSMSMSMSMSMSMSMSMSMSMSMSMSMSMS

SCSCSCSCSCSCSCSCSCSCSCSCSCSCSCSC

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, Minnesotahas less than twice times the voter turnout of Hawaii. Given this, it is not surprising thatthe 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 theDV . This finding is consistent with Persson, and Tabellini (2000) and Lizzeri and Persico(2001) who conclude that spending will be more narrowly targeted under majoritarianelections. Persson and Tabellini (2000) analyze electoral competition in an election withthree electoral districts. Their result is driven by the assumption that the district withthe highest marginal voter density is always decisive in the electoral college (majoritarianelection), while under proportional elections candidates internalize marginal voter densityacross all states, which is more equally distributed. Lizzeri and Persico (2001) use avery different framework, assuming no uncertainty about the election outcome, givencandidate strategies. In consequence, candidates use completely mixed strategies and allstates receive equal expected treatment.

Political rents First a theoretical point: there is no a priori reason to believe thatpolitical rents would depend on the size or number of states. Rents are decreasing inX

s

∂PD

∂∆us=Xs

Qs =Xs

vsesvs

Qs

es,

which measures how the probability of re-election falls when the utility of all voters isdecreased marginally. If Qs was more than proportionally increasing in the number ofelectoral votes, as suggested by Brams and Davis (1974), larger states would contributemore than proportionally to keeping rents low and the best electoral system would be tohave just one state of maximum size (Direct Vote). However, Qs is roughly proportional tothe number of electoral votes, see equation (2.10) and Figure (5.3), so all states contributeproportionally and there is no a priori reason to believe that size and number of statesmatter.Empirically, the ability to discipline politicians and keep political rents low is about

the same for both electoral systems. Figure 6.4 plotsP

sQs andP

sQDVs , multiplied by

an increase in political rents ∆r causing an average 5 percent fall in vote support.18 Thisfall is trivially higher in elections which are ex ante close. Therefore, equilibrium politicalrents are lower in 2000 than in 1984, under both electoral systems. Further, the averagefall is about the same for the two systems and politicians would be disciplined to an equalextent.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 acrossstates. First, the fall in the probability of being elected from a uniform loss of 5 percent

18∆r = 0.05/average ϕ(·)σfs

. 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 QDVs measure

marginal changes.

24

Elec

tora

l Col

lege

Direct Vote.1 .2 .3 .4 .5 .6

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.2

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Figure 6.4: Probability that an incumbent would loose the election because of a corruptionscandal (5 percent fall in support) under EC and DV

support in all states is around 33 percent both under EC and DV .19 This similarityis far from obvious since the frequency of quite different events are measured: that theincumbent 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 8.4shows that this response is proportional to average "voting power", that is, the probabilitythat one randomly drawn voter decides the election. The result is therefore consistentwith the finding of Gelman and Katz (2001) that average voting power is about the sameunder 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 Pennsylvaniain 2000) who are likely to be decisive swing states, then the outcome under EC will bemore sensitive to the corruption scandal than under DV . This turns out not to be trueempirically, as these two variables are uncorrelated. A more direct measure of the shareof 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 isheld in each district, and the candidate with the most districts wins the election. Theyreach their conclusion by assuming that there are more marginal voters in the one districtthat is always decisive. This assumption, translated to our framework, is not valid in thecase of the US as there are not more marginal voters in states more likely to be decisiveswing states.

Election statistics Finally, some argue that razor-thin victories, and presidents with-out a majority of the popular vote are unattractive features of an electoral system. Themodel can be used to estimate the probability of these events. Given that the elections1948-2000 are representative of future elections, this can be done by simulating elections

19This is estimated by simulations.

25

and recording event frequencies. The likelihood of a winning margin less than 1000 votes isabout 50 times higher under the Electoral College system (0.4 percent compared to 0.008percent 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 containson average 50 times more voters under Direct Vote. This is consistent with equal averagevoting power under both systems. Every time a single voter is pivotal under DV , then allvoters in the nation are pivotal. Every time a single voter is decisive under EC, then allvoters in that state are decisive. The latter event occurs 50 times more often but involves1/50 as many voters, so average voting power is the same. The probability of electing apresident without a majority of the popular vote is about 4 percent. This implies that weshould expect this outcome about once in every hundred years. Historically, this eventhas 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 withpeculiarities in the aggregation of votes.20

The above results support a reform of the Electoral College system in favor of DirectVote. Such a reform would decrease political incentives to allocate resources unequallyacross states for electoral reasons. It would also decrease the probability of razor thinwinning margins and, of course, the probability of electing presidents without supportof a majority of the popular votes. There would be no cost in terms of less disciplineon politicians. Also, small states need not fear reform. As a group, they are neitheradvantaged 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 allocateresources across states in order to get a majority of the electoral votes. After applying itempirically 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 mention-ing. First, more resources should be devoted to states who are likely to be decisive swingstates, that is, states who are decisive in the electoral college and, at the same time, havetied 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 thestate election being tied. Third, the probability of being a decisive swing state is roughlyproportional to the number of electoral votes. Fourth, this probability per electoral vote ishighest for states who have a forecasted state election outcome which lies between a drawand the forecasted national election outcome. Fifth, more precise state-election forecastsmake the optimal allocation of resources more concentrated. Sixth, the presidential can-didates who is lagging behind should try to increase variance in electoral votes. This isdone by spending more time in large states where this candidate is behind, and less timein large states where this candidate is ahead.

20In 1824, six of the twenty four States at the time still chose their Electors in the State legislature, sothe popular vote outcome is not known. In the chaotic election of 1876, each State delivered to Congresstwo slates of electors and a special commission accepted slates favoring of the popular vote loser. See "ABrief 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. Thefirst point shows more precisely what people have conjectured or partly demonstratedearlier, see, for example, Snyder (1989). The third point has not previously been showntheoretically, although (because of the second point) it is closely related to the empiricalfinding of Gelman and Katz (2001) that "voting power" is proportional to size underreasonable 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 inthe safe districts of the other party.More importantly, this paper quantifies these effects and shows that presidential candi-

dates’ actual strategies very closely resemble the optimal strategies. The model is appliedto 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 themodel. In the 2000 election, the correlation between optimal and actual visits by stateis 0.91, and the correlation between optimal and actual advertisement expenditures byadvertising 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 displayedin Figure 6.2. States like Kansas and Nebraska, would gain, while Nevada and Delawarewould lose. Small states are, as a group, not favored by the Electoral College system. Thedistribution of redistributive spending is much more unequal under the Electoral Collegethan under Direct Vote. This is basically because the probability of being a decisive swingstate is extremely unevenly distributed. The same conclusion was reached by Lizzeri andPersico (2001) and Persson, and Tabellini (2000), but for different reasons. Third, unlikePersson and Tabellini (1999), this paper finds that both systems are about equally fit todiscipline politicians from corruption or other activities which benefit them but hurt thevoters.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. Sincethe modelling framework is quite general, this integration can be carried over to the studyof other resource allocation problems in other electoral settings.

27

References

[1] Anderson, Gary M. and Robert D. Tollison, 1991, “Congressional Influence and Pat-terns 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 theElectoral College”, Villanova Law Review 13, 304-332.

[3] Brams, Steven J. and Morton D. Davis, 1974, ”The 3/2 Rule in Presidential Cam-paigning.” American Political Science Review, 68(1), 113-34.

[4] Campbell, James E., 1992, Forecasting the Presidential Vote in the States. AmericanJournal of Political Science, 36, 386-407.

[5] Chamberlain, Gary and Michael Rothchild, 1981, ”A Note on the Probability ofCasting a Decisive Vote”, Journal of Economic Theory 25, 152-162.

[6] Cohen, Jeffrey E., 1998, ”State-Level Public Opinion Polls as Predictors of Presiden-tial Election Results”, American Politics Quarterly, Vol. 26 Issue 2, 139.

[7] Colantoni, Claude S., Terrence J. Levesque, and Peter C. Ordeshook, 1975, ”Cam-paign Resource Allocation under the Electoral College (with discussion)”, AmericanPolitical Science Review 69, 141-161.

[8] Erikson, Robert S., Gerald C. Wright and John P. McIver, 1993, "Statehouse Democ-racy", 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 Paper1121, California Institute of Technology.

[10] Gelman, Andrew, and Gary King, 1993, ”Why are American Presidential ElectionCampaign Polls so Variable when Votes are so Predictable?”, British Journal ofPolitical Science, 23, 409-451.

[11] Gelman, Andrew, Gary King, and W. John Boscardin, 1998, ”Estimating the Prob-ability of Events that have Never Occurred: when is your vote decisive?”, Journal ofthe American Statistical Association, 93(441), 1-9.

[12] Hobrook, Thomas M. and Jay A. DeSart, 1999, ”Using State Polls to Forecast Pres-idential Election Outcomes in the American States”, International Journal of Fore-casting, 15, 137-142.

[13] Lindbeck, Assar and Jörgen W. Weibull, 1987, ”Balanced-Budget Redistribution asthe Outcome of Political Competition”, Public Choice, 52, 273-97.

[14] Lizzeri, Allesandro and Nicola Persico, 2001, "The Provision of Public Goods underAlternative Electoral Incentives", American Economic Review, 91(1), 225-239.

28

[15] Merrill, Samuel III, 1978, ”Citizen Voting Power Under the Electoral College: AStochastic Voting Model Based on State Voting Patterns”, SIAM Journal on AppliedMathematics, 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 Eco-nomic Policy, MIT Press.

[19] Rosenstone, Steven J., 1983, ”Forecasting Presidential Elections”, New Haven: YaleUniversity Press.

[20] Shachar, Ron and Barry Nalebuff, 1999, ”Follow the Leader: Theory and Evidenceon Political Participation”, American Economic Review 89(3), 525-547.

[21] Shaw, Daron R., 1999, ”The Effect of TV Ads and Candidate Appearances onStatewide Presidential Votes, 1988-96, American Political Science Review 93(2), 345-361.

[22] Strömberg, David, 2002, ”Optimal Campaigning in Presidential Elections: The Prob-ability of Being Florida”, IIES working paper 706.

[23] Strömberg, David, 2002a, ”The Electoral College and the Distribution of FederalEmployment”, Stockholm university, in preparation.

[24] Strömberg, David, 2002b, ”Follow the Leader 2.0: Theory and Evidence on PoliticalParticipation 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 GovernmentGrants to the States?”, NBER Historical Paper 90.

[27] Wright, Gavin, 1974, ”The Political Economy of New Deal Spending: An Economet-ric Analysis”, Review of Economics and Statistics 56, 30-38.

29

8. Appendix

8.1. Proof of Proposition 1

Since PD¡dD∗, dR

¢is continuous and differentiable, a necessary condition for an interior

NE is∂PD

¡dD∗, dR∗

¢∂dDs

= Qsu0 ¡dD∗s ¢ = λD, (8.1)

∂PD¡dD∗, dR∗

¢∂dRs

= Qsu0 ¡dR∗s ¢ = λR. (8.2)

Therefore,u0¡dD∗s¢

u0 (dR∗s )=

λD

λR, (8.3)

for all s. Suppose that dD∗ 6= dR∗. This means that dD∗s < dR∗s for some s, implying thatλD > λR by equation (8.3). Because of the budget constraint, it must be the case thatdD∗s0 > dR∗s0 for some s

0, which implies λD < λR, a contradiction. Therefore, λD = λR,which implies dD∗s = dR∗s 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 alls, which violates the budget constraint. Therefore, the only possibility is λ = λ0, whichimplies ds = d0s for all s.

8.2. Derivation and interpretation of Qs

To see, heuristically, why the analytical expression for Qs approximates the joint proba-bility of a state being decisive in the electoral college at the same time as having a tiedelection, note that state s can only be decisive if the margin of victory in the electoralcollege is less than its electoral votes, es. To a second order approximation, this happenswith probability

Φ

µ 12

Ps0 es0 + es − µ

σE

¶− Φ

µ 12

Ps0 es0 − es − µ

σE

¶≈ 2

µϕ (x (η))

esσE+

e2sσ2E

ϕ (x (η))x (η)

µGs (·)− 1

2

¶¶= 2pECD (η) ,

where the last equality defines pECD (η). However, s is only decisive when D wins na-tionally and in state s, or R wins nationally and in state s. This happens about half ofthe times the margin of victory is less than es, so the probability that a state is ElectoralCollege Decisive is approximately half the above expression. Second, given a nationalshock, η, the state outcome is a tie if Fs(−ηs − η) = 1

2which happens with likelihood

gs (−µs − η) . Conditional on the national shock, the joint likelihood of being decisiveand having a tied election is thus roughly pECD (η) gs (−µs − η), and the unconditionalprobability approximately Qs.21

21A 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 stateelection margin of less than two percent, Qs must be scaled. The probability of a winningmargin of less than x percent equals

1

2− x

200≤ Fs(−ηs − η) ≤ 1

2+

x

200.

The probability that the state-level shock falls in this region is approximately

gs (−µs − η)σfsϕ (0)

x

100.

The unconditional probability of being pivotal and having a state-election margin of xpercent is therefore approximately

Qsσfsϕ (0)

x

100.

The scaled values of Qs are very similar to the simulated frequencies of the corre-sponding events. In 3.4 percent of the 1 million simulated elections, Florida was decisivein the Electoral College and had a state margin of victory of less than 2 percent. Thescaled Qs for Florida was 3.5 percent. To get the probability that the state margin ofvictory is within, say 1000, votes, Qs should be scaled by 1000

vs

σfsϕ(0)

. Using this formula,the probability of a state being decisive in the Electoral College, and at the same timehaving an election result with a state margin of victory less than 1000 was 0.00015 inFlorida 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, andthe probability of this happening in any state is 4.4 per million. The state where one voteis 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.

Qs =1

0.02Qs

µσfsϕ (0)

0.02

¶| {z }

Pr(EC decisive andmargin of two percent)

ϕ (0)

σfs| {z } .marginal voter density,

conditional on tied state election

(8.4)

or, similarly,

Qs = vsQs

µσfsϕ (0)

1

vs

¶| {z }”votingpower"

ϕ (0)

σfs| {z } .marginal voter density,

conditional on tied state election

(8.5)

When all states are assumed to have the same variance of preferences σfs (until Section6), Qs is proportional to the probability of being EC decisive and having a state marginof victory of less than two percent.

31

nytt

tfi pfi

-.027613 .101768

.000531

.014273

Figure 8.1:

To simplify the analytical form of Qsµ, do a first order Taylor expansion of the meanof the expected number of electoral votes µ (η) around η = eη for which µ (eη) = 1

2

Ps es,

that is the value of the national shock which makes the expected outcome a draw. Withthis approximation

µ (η) =Xs

esGs (−µs − η) ≈ 12

Xs

es − a (η − eη) ,a =

Xs

esgs (−µs − eη) .Since the mean of the electoral votes, µ (η) , is much more sensitive to national shocksthan is the variance σE (η), the latter is assumed fixed

σ2E (η) = σ2E =Xs

e2sGs (−µs − eη) (1−Gs (−µs − eη)) .Then

1

σE (η)ϕ

µ 12

Ps es − µ

σE (η)

¶≈ 1

σEϕ

µη − eησE/a

¶.

This approximation is very good. Figure 8.1 shows the true, tfi, and the approximatedfunctions, pfi. The values are calculated for an interval of four standard deviations cen-tered around eη in the 2000 presidential election. We now have

Qsµ ≈ es1

σE

Zϕ

µη − eησE/a

¶gs (−µs − η)h (η) dη.

Integrating over η,

Qsµ ≈ esω

2πexp

Ã−12

σ2seη2 + (σE/a)2 µ2s + σ2 (eη + µs)2

σ2sσ2 + (σE/a)

2 σ2 + (σE/a)2 σ2s

!(8.6)

where

ω2 =

µ1

(σE/a)2 +

1

σ2s+1

σ2

¶−1.

32

Qsµ is larger when eη is close to zero. This effects all states in the same way, but variesacross elections. To clarify the differences between states with different µs, rewrite

Qsµ ≈ esω

2πexp

Ã−12

c+ (µs − µ∗s)2

eσ2!,

where

µ∗s = −σ2

σ2 + (σE/a)2eη,

and eσ2 = σ2s +1³

1σ2+ 1

(σE/a)2

´ .Qsσ is calculated using numerical integration.

8.3. Campaign advertisements

This Appendix analyses the decision of presidential candidates to allocate advertisementsacross 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

MXm=1

pmaJm ≤ Ia,

SXs=1

dJs ≤ I,

J = R,D.22 Media market m contains a mass vms of voters in state s. Voters reactionsto 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

∆uwms = u¡dDs¢− u

¡dRs¢+ w

¡aDm¢− w

¡aRm¢ ≥ Ri + ηs + 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-levelpopularity-swings are drawn from the same distributions as before.The share D votes in media market m in state s equals

Fs(∆uwms − ηs − η).

D wins the state if Xm

vmsFs(∆uwms − ηs − η) ≥ vs2.

22For 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:

ηs¡dD, aD, dR, aR

¢:Xm

vmsFs(∆uwms − ηs¡dD, aD, dR, aR

¢) =

vs2. (8.7)

D wins state s ifηs + η ≤ ηs (·) .

Conditional on the aggregate shock η, and the strategies, dD, aD, dR, aR, this happenswith probability

Gs

¡ηs¡dD, aD, dR, aR

¢− η¢.

The function ηs (·) 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 ηs (·) for ∆us − µs. The bestreply functions of D is characterized by

Qs∂ηs (·)∂dDs

= λD, (8.8)

SXs=1

Qs∂ηs (·)∂aDm

= µDpm. (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, theallocations must be symmetric so that λD = λR = λ, and µD = µR = µ. Differentiatingequation (8.7), and evaluating it in equilibrium yields

∂ηs (·)∂dDs

= u0¡dDs¢,

∂ηs (·)∂aDm

=vms

vsw0¡aDm¢.

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 satisfydD = dR = d∗, and aD = aR = a∗, and for all s and for some λ, µ > 0

Qsu0 (d∗s) = λ, (8.10)

Qmw0 (a∗m) = µpm, (8.11)

where

Qm =SXs=1

Qsvms

vs.

Qm is the sum of the Qs of the states in the media market weighted by the share of thevoting population of state s that lives in media market m.Last, assuming that votes per capita

vms

nms= ts

is the same for all media markets within each state,

Qm =SXs=1

Qsnms

ns.

34

8.4. Direct presidential vote

First, the approximate probability of winning the election is derived under DV . Condi-tional on η, the expected vote share of D in state s is

µvs (∆us, η) =

ZFs(∆us − η − ηs)gs (ηs) dηs

= Φ

∆us − η − µsqσ2s + σ2fs

,

and the expected national vote of D is

µv (∆us, η) =Xs

vsµvs (∆us, η) .

Again conditional on η, the variance in D’s votes in state s is,

σ2vs = v2s

Z µΦ

µ∆us − η − ηs − µs

σfs

¶− µvs (∆us, η)

¶2gs (ηs) dηs,

and the variance in the national votes of D is

σ2v =Xs

σ2vs.

The approximate probability of D winning the election is

PD (∆us) = 1−Z

Φ

µ 12

Ps vs − µvσv

¶h (η) dη.

The equilibrium strategies of Proposition 3 depend crucially on

∂PD

∂∆us= QDV

s =∂PD

∂µv

∂µv∂∆us

+∂PD

∂σv

∂σv∂∆us

= QDVsµ +QDV

sσ ,

where

QDVsµ =

vs

σvqσ2s + σ2fs

Zϕ

µ 12

Ps vs − µvσv

¶ϕ

∆us − µs − ηqσ2s + σ2fs

h (η) dη, (8.12)

QDVsσ =

Z µ∂

∂σvΦ

µ 12

Ps vs − µvσv

¶¶∂σv∂∆us

h (η) dη.

Empirically, QDVsµ À QDV

sσ , the size of QDVsσ is negligible compared to QDV

sµ .The interpretation of QDV

sµ is

QDVsµ = vs (pdf of tied election) E [fs | election tied] . (8.13)

35

This follows since, conditional on η, the expected marginal voter density in state s isZ ∞

−∞fs(∆us − η − ηs)gs (ηs) dηs =

1qσ2s + σ2fs

ϕ

∆us − µs − ηqσ2s + σ2fs

.

The probability density function of a tied national election, conditional on η, is

1

σvϕ

µ 12

Ps vs − µvσv

¶,

and the unconditional

pdf of tied election =Z

1

σvϕ

µ 12

Ps vs − µvσv

¶h (η) dη. (8.14)

Therefore, the marginal voter density, conditional on a tied election, is

E [fs | election tied] (8.15)

=

Z ϕ³

12

Ps vs−µvσv

´Rϕ³

12

Ps vs−µvσv

´h (η) dη

1qσ2s + σ2fs

ϕ

∆us − µs − ηqσ2s + σ2fs

h (η) dη.

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

pDVx =

ZΦ

µ 12

Ps vs +

x2− µv

σv

¶− Φ

µ 12

Ps vs − x

2− µv

σv

¶h (η) dη (8.16)

≈ x

Z1

σvϕ

µ 12

Ps vs − µvσv

¶h (η) dη.

Voting power is by definition pDV1 . By equations (8.14) and (8.16), pDV

1 ≈ pdf of tiedelection, therefore equation (8.13) may be written

QDVsµ ≈ 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 supportacross states is proportional to average "voting power". To see this, first define ∆uFs asthe change in voter utility in state s necessary to induce a one percent fall in support:

∂F (∆us)

∂∆us∆uFs = 1,

or equivalently,

∆uFs =1

fs.

36

Next, define the response in the probability of winning representing an equal fall in voteshares as

∂PD

∂∆uFs=

∂PD

∂∆us∆uFs =

Qs

fs

Under EC, voting power equals

pEC1 ≈ Qs

vsfs (0),

see equation (8.5). Average voting power therefore equals the per capita response inre-election probability to a uniform one percent fall in vote share supportX

s

vsvpEC1 ≈ 1

v

Xs

∂P

∂∆uFs.

Similarly, under DV

1

v

Xs

∂P

∂∆uFs=1

v

Xs

QDVs

E [fs | national election tied] ≈ pDV1 ,

since QDVsµ À QDV

sσ , 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 statedeciding the election under EC and DV is approximately

pEC1000 ≈Xs

Qs1000

vs/

µϕ (0)

σfs

¶,

andpDV1000 ≈

Xs

QDVs

1000

v/Xs

vsvE [fs | national election tied] ,

where the latter equation was derived by inserting pDV1000/1000 instead of (pdf tied election)

into equation (8.13), summing over s, and solving for pDV1000. Assume that the marginal

voter density is the same fs = 1 and that all states are of equal size, then vsv= 1

50and

pDV1000 =

1

50pEC1000.

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 Ab-stract of the United States 2000; 2000, Federal Election Commission, 2000 OFFI-CIAL PRESIDENTIAL GENERAL ELECTION RESULTS.

• Electoral votes won (by state). Source: National Archives and Records Administra-tion.

37

• National trial-heat poll results. Source: 1948-1996, Campbell (2000); 2000, Gallup.• Second quarter national economic growth, multiplied by 1 if Democratic incumbentpresident and -1 if Republican incumbent president. Source: August or Septemberelection 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 fourthquarter and the first quarter of the election year, standardized across states ineach year, multiplied by 1 if Democratic incumbent president and -1 if Republicanincumbent president. Source: Survey of Current Business, U.S. Department ofCommerce, Bureau of Economic Analysis.

• Incumbent: 1 if incumbent president Democrat, -1 if incumbent president Republi-can.

• Presinc: 1 if incumbent Democratic president seeking re-election, -1 if incumbentRepublican president seeking re-election.

• President’s home state: 1 if Democratic president home state, -1 if Republican (0.5and -0.5 for large states (New York, Illinois, California). Source: Campbell 1948-1988; 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: Campbell1948-1988; 1992-2000: Dave Leip’s Atlas of U.S. Presidential Elections.

• Average ADA-scores: Average ADA-scores of state’s members in Congress yearbefore 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 previ-ous midterm election. Index is Democratic share of state legislative seats above the50% mark. Two states, Nebraska and Minnesota, held nonpartisan state legislativeelections for all (Nebraska) or part (Minnesota of the period under study. In thecase of Nebraska, the state legislative division was estimated based on the rankingof states of Wright, Erikson, and McIver’s state partisan rankings based on publicopinion data. Using this index, Nebraska was assigned the mean partisan division ofthe state most similar to it on the public opinion index, the nearly equally Republi-can state of North Dakota. The partisan division of the Minnesota legislature in itsnonpartisan years (before 1972) is coded as the mean of its partisan division once itreformed to partisan elections (62% Democratic). Washington D.C. was as havingthe 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-electionissues of the Hotline (www.nationaljournal.com).

38

• Regional dummy variables, see Campbell.• Participation rate: 1948-1988, Shachar and Nalebuff (1999), voting age population1992-2000, and votes cast 2000, Federal Election Commission, votes cast 1992, 1996,Statistical Abstract of the United States.

39