ANALYZING THE
ELECTORAL COLLEGE
Nicholas R. Miller
Political Science, UMBC
INFORMS Meeting
October 14, 2008
http://userpages.umbc.edu/~nmiller/ELECTCOLLEGE.html
Preface
• Polsby’s Law: What’s bad for the political system
is good for political science, and vice versa.
• George C. Edwards, WHY THE ELECTORAL
COLLEGE IS BAD FOR AMERICA (Yale, 2004)
• Deduction: The Electoral College is good for
Political Science.
Problematic Features of the Electoral College
• The Voting Power Problem. Does the Electoral College
system (as it presently operates) give voters in different
states unequal voting power?
– If so, voters in which states are favored and which
disfavored and by how much?
• The Election Reversal Problem. The candidate who wins
the most popular votes nationwide may fail to be elected.
– The election 2000 provides an example (provided we
take the official popular vote in FL at face value).
• The Electoral College Deadlock Problem, i.e., the House
contingent procedure.
• Here I present some analytic results pertaining to the
first and second problems of the existing Electoral
College as well as variants of the EC.
The Voting Power Problem
• As a first step, we need to distinguish between
– voting weight and
– voting power.
• We also need to distinguish between two distinct issues:
– how electoral votes are apportioned among the states
(which determines voting weight), and
– how electoral votes are cast within states (which, in
conjunction with the apportionment of voting weight,
determines voting power).
The Apportionment of Electoral Votes
• The apportionment of electoral votes is fixed in the
Constitution,
– except that Congress can by law change the size of
the House of Representatives, and Congress can
therefore also change
• the number of electoral votes, and
• the ratio
―Senatorial‖ electoral votes
Total electoral votes
• which reflects the magnitude of the small-state
advantage in apportionment.
Chart 1. The Small-State EV Apportionment
Advantage
The Casting of Electoral Votes
• How electoral votes are cast within states is
determined by state law.
– But, with few exceptions, since about 1836 states
have cast their electoral votes on a winner-take-all
basis.
• By standard voting power calculations,
– the winner-take-all practice produces a large-state
advantage
– that more than balances out the small-state
advantage in electoral vote apportionment.
A Priori Voting Power
• A measure of a priori voting power is a measure that
– takes account of the structure of the voting rules
– but of nothing else (e.g., demographics, historic voting patterns,
ideology, poll results, etc.).
• The standard measure of a priori voting power is the
Absolute Banzhaf (or Penrose) Measure.
– Dan Felsenthal and Moshe Machover, The Measure of Voting
Power: Theory and Practice, Problems and Paradoxes, 1998
• A voter’s absolute Banzhaf voting power is
– the probability that the voter’s vote is decisive (i.e., determines
the outcome the election),
– given that all other voters vote by independently flipping fair
coins (i.e., given a Bernoulli probability space producing a
Bernoulli election).
A Priori Individual Voting Power
• In a simple one person, one vote majority rule election
with n voters,
– the a priori voting power of an individual voter is the probability
that his vote is decisive, i.e.,
• the probability that the vote is otherwise tied (if n is odd), or
• one half the probability the vote is otherwise within one vote
of a tie (if n is even).
• Provided n is larger than about 25, this probability is very
well approximated by √ (2 / πn),
– Which implies that that individual voting power is inversely
proportional to the square root of the number of voters.
Calculating Power Index Values
• There are other mathematical formulas and algorithms
that for calculating or approximating voting power in
weighted voting games, i.e.,
– in which voters cast (unequal) blocs of votes.
• Various website make these algorithms readily available.
• One of the best of these is the website created by
Dennis Leech (University of Warwick and another VPP
Board member): Computer Algorithms for Voting Power
Analysis,
http://www.warwick.ac.uk/~ecaae/#Progam_List
which was used in making most of the calculations that
follow.
A Priori State Voting Power in the Electoral
College (with Winner-Take-All)
• A state’s a priori voting power is
– the probability that the state’s block of electoral votes is decisive
(i.e., determines the outcome the election),
– given that all other states cast their blocs of electoral votes by
independently flipping fair coins.
• For example (using Leech’s website), the a priori voting
power of CA (with 55 EV out of 583) = .475 .
– This means if every other state’s vote is determined
by a flip of a coin,
• 52.5% of the time one or other candidate will have at least
270 electoral votes before CA casts its 55 votes, but
• 47.5% of the time CA’s 55 votes will determine the outcome.
Chart 2. Share of Voting Power by Share of
Electoral Votes
Chart 3. Share of Voting Power by Share of
Population
Individual Voting Power
in the Electoral College System
• The a priori voting power of an individual voter in the
Electoral College system (as it works in practice) is
the probability that the individual voter is
decisive in his state
multiplied by
the probability that the bloc of votes cast by the voter’s
state is decisive in the Electoral College
or equivalently
individual voting power in the state
multiplied by
state voting power in the Electoral College
The Banzhaf Effect
• (1) Individual voting power within each state is (almost
exactly) inversely proportional to the square root to the
number of voters in the state.
• (2) As shown in Chart 2, state voting power in the
Electoral College is approximately proportional to its
voting weight (number of electoral votes).
• (3) As shown in Chart 1, the voting weight of states in
turn is approximately (apart from the small-state
apportionment advantage) proportional to population
(and number voters).
• (4) As shown in Chart 3, putting together (2) and (3),
state voting power is approximately proportional to
population.
• (5) So putting together (1) and (4), individual a priori
voting power is approximately proportional to the square
root of the number of voters in a state.
– However this large-state advantage is counterbalanced in some degree
by the small-state apportionment advantage, as shown in the Chart 4.
Banzhaf Effect in Bernoulli Elections
Individual Voting Power Under the Existing EC
• The following Chart 4 shows how a priori individual voting power
under the existing Electoral College varies by state population.
• It also shows:
– mean individual voting power nationwide, and
– individual voting power under direct popular vote (calculated in
the same manner as individual voting power within a state).
• Note that it is substantially greater than mean individual voting
power under the Electoral College.
– Indeed, it is greater than individual voting power in every state
except California.
– By the criterion of a priori voting power, only voters in California
would be hurt if the existing Electoral College were replaced by a
direct popular vote.
Methodological note: in most of the following charts, individual voting power is
scaled so that the voters in the least favored state have a value of 1.000, so
– numerical values are not comparable from chart to chart, and
– the scaled value of individual voting power under direct popular vote
changes from chart to chart.
The number of voters in each state is assumed to be a constant fraction (.4337)
of state population.
Individual Voting Power By State Population:
Existing Electoral College
The Interpretation of a Priori Voting Power
• Remember that Chart 4 displays individual a priori voting
power in states with different populations,
– which takes account of the Electoral College voting rules but
nothing else.
– A priori, a voter in California has about three times the probability
of casting a decisive vote than one in New Hampshire.
– But if we take account of recent voting patterns, current poll
results, and other information, a voter in New Hampshire may
have a greater empirical (or a posteriori) probability of
decisiveness in the upcoming election, and accordingly get more
attention from the candidates and party organizations, than one
in California.
– But if California and New Hampshire had equal ―battleground‖
status, the California’s a priori advantage would be reflected in
its a posteriori voting power as well.
Winner’s Margin by State Size
Interpretation of A Priori Voting Power (cont.)
• If it is only weakly related to empirical voting power in
any particular election, the question arises of whether a
priori voting power and the Banzhaf effect should be of
concern to political science and practice.
• Constitution-makers arguably should — and to some
extent must — design political institutions from behind a
―veil of ignorance‖ concerning future political trends.
• Accordingly they should — and to some extent must —
be concerned with how the institutions they are
designing allocate a priori, rather than empirical, voting
power.
– The framers of the U.S. Constitution did not require or expect
electoral votes to be cast en bloc by states.
– However, at least one delegate [Luther Martin] expected that
state delegations in the House of Representatives would vote en
bloc, which he thought would give large states a Banzhaf-like
advantage.
William H. Riker, “The First Power Index.” Social Choice and Welfare, 1986.
Alternative EV Apportionment Rules
• Keep the winner-take all practice [in 2000, Bush 271,
Gore 267; in 2004, Bush 286, Kerry, 252] but use a
different formula for apportioning electoral votes among
states.
– Apportion electoral votes [in whole numbers] on basis of
population only [―House‖ electoral votes only] [Bush 211, Gore
225; Bush 224, Kerry 212]
• Apportion electoral votes [fractionally] to be precisely
proportional to population [Bush 268.96092, Gore 269.03908;
Bush 275.67188, Kerry 262.32812]
• Apportion electoral votes [fractionally] to be precisely
proportional to population but then add back the ―constant two‖
[Bush 277.968, Gore 260.032; Bush 285.40695, Kerry
252.59305]
• Apportion electoral votes equally among the states [in the
manner of the House contingent procedure] [Bush 30, Gore 21;
Bush 31, Kerry 20]
Individual Voting Power by State Population:
―House Electoral Votes‖ Only
Individual Voting Power by State Population:
Electoral Votes Precisely Proportional to Population
Individual Voting Power by State Population:
Electoral Votes Proportional Population, plus Two
Individual Voting Power by State Population:
Electoral Votes Apportioned Equally Among States
Can Electoral Votes Be Apportioned So As
To Equalize Individual Voting Power?
• The question arises of whether electoral votes can be
apportioned so that (even while retaining the winner-
take-all practice) the voting power of individuals is
equalized across states?
• One obvious (but constitutionally impermissible)
possibility is to redraw state boundaries so that all states
have the same number of voters (and electoral votes).
– This creates a system of uniform representation.
Methodological Note: since the following chart compares voting power under
different apportionments, voting power must be expressed in absolute (rather
than rescaled) terms.
Individual Voting Power when States Have Equal Population
(Versus Apportionment Proportional to Actual Population)
Uniform Representation
• Note that equalizing state populations not only:
– equalizes individual voting power across states, but also
– raises mean individual voting power, relative to that under
apportionment based on the actual unequal populations.
• While this pattern appears to be typically true, it is not
invariably true,
– e.g., if state populations are uniformly distributed over a wide
range.
• However, individual voting power still falls below that
under direct popular vote.
– So the fact that mean individual voting power under the Electoral
College falls below that under direct popular vote is
• not due to the fact that states are unequal in population and
electoral votes, and
• is evidently intrinsic to a two-tier system.
Van Kolpin, “Voting Power Under Uniform Representation,” Economics
Bulletin, 2003.
Electoral Vote Apportionment to Equalize
Individual Voting Power (cont.)
• Given that state boundaries are immutable, can we
apportion electoral votes so that (without changing state
populations and with the winner-take-all practice
preserved) the voting power of individuals is equalized
across states?
• Yes, individual voting power can be equalized by
apportioning electoral votes so that state voting power is
proportional to the square root of state population.
– But such apportionment is tricky, because what must be made
proportional to population is
• not electoral votes (which is what we directly apportion) but
• state voting power (which is a consequence of the
apportionment of electoral votes).
(Almost) Equalized Individual Voting Power
Electoral Vote Apportionment to Equalize
Individual Voting Power (cont.)
• Under such square-root apportionment rules, the
outcome of the 2004 Presidential election would be
– Fractional Apportionment: Bush 307.688, Kerry
230.312.
– Whole-Number Apportionment: Bush 307, Kerry 231
– Actual Apportionment: Bush 286, Kerry 252
– Electoral Votes proportional to popular vote: Bush
275.695, Kerry 262.305
• Clearly equalizing individual voting power is not the
same thing as making the electoral vote (more)
proportional to the popular vote.
Alternative Rules for Casting Electoral Votes
• Apportion electoral votes as at present but use
something other than winner-take-all for casting state
electoral votes.
– (Pure) Proportional Plan: electoral votes are cast [fractionally] in
precise proportion to state popular vote. [Bush 259.2868, Gore
258.3364, Nader 14.8100, Buchanan 2.4563, Other 3.1105;
Bush 277.857, Kerry 260.143]
– Whole Number Proportional Plan [e.g., Colorado Prop. 36]:
electoral votes are cast in whole numbers on basis of some
apportionment formula applied to state popular vote. [Bush 263,
Gore 269, Nader 6, or Bush 269, Gore 269; Bush 280, Kerry
258]
– Pure District Plan: electoral votes cast by single-vote districts.
– Modified District Plan: two electoral votes cast for statewide
winner, others by district [present NE and ME practice]. [Bush
289, Gore 249, if CDs are used; no data for 2004]
– National Bonus Plan: 538 electoral votes are apportioned and
cast as at present but an additional 100 electoral votes are
awarded on a winner-take-all basis to the national popular vote
winner. [Bush 271, Gore 367; Bush 386, Kerry 252]
Individual Voting Power under Alternative
Rules for Casting Electoral Votes
• Calculations for the Pure District Plan, Pure Proportional
Plan, and the Whole-Number Proportional Plan are
straightforward.
• Under the Modified District Plan and the National Bonus
Plan, each voter casts a single vote that counts two
ways:
• within the district (or state) and
• ―at-large‖ (i.e., within the state or nation).
– Calculating individual voting power in such systems is far from
straightforward.
– I am in the process of working out approximations based on very
large samples of Bernoulli elections.
Pure District System
Modified District System (Approximate)
District System Is ―Out of Equilibrium‖
• Given a district system, any state can gain power by
unilaterally switching to winner-take-all.
– Madison to Monroe (1800): “All agree that an election by districts
would be best if it could be general, but while ten states choose
either by their legislatures or by a general ticket [i.e., winner-
take-all], it is folly or worse for the other six not to follow.”
– Virginia switched from districts to winner-take-all in 1800.
• If it had not, the Jeffersonian Republicans would almost
certainly lost the 1800 election.
• Madison’s strategic advice is powerfully confirmed in
terms of individual voting power,
– though the voting-power rationale for winner-take-all is logically
distinct from the party-advantage rationale.
Winner-Take-All Is ―In Equilibrium‖
• In the mid-1990s, the Florida state legislature seriously
considered switching to the Modified District Plan.
• The effect of such a switch on the individual voting
power is shown in the following chart.
– However, I assume a switch to the Pure District Plan, because
this can be directly calculated.
• Considering ―mechanical‖ effects only, if Florida had made the
switch, Gore would have been elected President (regardless of the
statewide vote in Florida).
• Although small states are penalizing by the winner-
take-all system, they are further penalized if the
unilaterally switch to districts.
• So even if a district system is universally agreed to be
socially superior (as Madison considered it to be),
states will not voluntary choose to move that direction.
– States are caught in a Prisoner’s Dilemma.
(Pure) Pure Proportional System
Whole-Number Proportional Plan
Similar
calculations
and chart were
produced,
independently
and earlier, by
Claus Beisbart
and Luc
Bovens, ―A
Power Analysis
of the Amend-
ment 36 in
Colorado,‖
University of
Konstanz, May
2005, and
Public Choice,
March 2008.
National Bonus Plan(s)
Individual Voting Power: Summary Chart
The Probability of Election Reversals
• Any districted electoral system can produce an election
reversal.
– That is, the candidate or party that wins the most popular votes
nationwide may fail to win the most ―districts‖ (e.g., parliamentary
seats or electoral votes) and thereby lose the election).
– Such outcomes are actually more common in some
parliamentary systems than in U.S. Presidential elections.
• First, let’s examine the probability that a two-tier Bernoulli election
(i.e., given the probability model used in voting power calculations)
results in an election reversal, i.e.,
– that a majority of individuals voters vote ―heads‖ but the winner
based on ―electoral votes‖ is ―tails‖ or vice versa?
• Based on very large-scale (n = 1,000,000) simulations, if the number
of equally populated districts/states is modestly large (e.g., k > 20),
about 20.5% of such elections produce reversals.
Feix, Lepelley, Merlin, and Rouet, “The Probability of Conflicts in a U.S.
Presidential Type Election,” Economic Theory, 2004
30,000
Bernoulli
elections
with 45
districts
each with
2223 voters
(n = 100,035)
In a more inclusive
sample of 120,000
such elections,
20.36% were
reversals.
Probability of Election Reversals (cont.)
• If the districts are non-uniform (as in the Electoral
College), the probability of an election reversal is
evidently slightly greater.
• Simulations of 32,000 Bernoulli elections for each of
three EC variants:
The Election Reversal Problem
• The U.S. Electoral College has produced three manifest
election reversals (though all were very close),
– plus one massive election reversal that is not usually recognized
as such.
Election Winner Runner-up Winner’s 2-P PV
2000 271 [Bush (R)] 267 [Gore (D)] 49.73%
1888 233 [Harrison (R)] 168 [Cleveland (D)] 49.59%
1876 185 [Hayes (R)] 184 [Tilden (D)] 48.47%
• The 1876 election was decided (on inauguration eve) by a Electoral
Commission that, by a bare majority and on a straight party line
vote, awarded all of 20 disputed electoral votes to Hayes.
– Unlike Gore and Cleveland, Tilden won an absolute majority (51%) of
the total popular vote.
The 1860 Election
Candidate Party Pop. Vote % EV
Lincoln Republican 39.82 180
Douglas Northern Democrat 29.46 12
Breckinridge Southern Democrat 18.09 72
Bell Constitutional Union 12.61 39
Total Democratic Popular Vote 47.55
Total anti-Lincoln Popular Vote 60.16
• Two inconsequential reversals (between Douglas and Breckinridge
and between Douglas and Bell) are manifest.
• It may appear that Douglas and Breckinridge were spoilers against
each other.
– Under a direct popular vote system, this would have been true.
– But under the Electoral College system, Douglas and Breckinridge were
not spoilers against each other.
A Counterfactual 1860 Election
• Suppose the Democrats could have held their Northern
and Southern wings together and won all the votes
captured by each wing separately.
– Suppose further that it had been a Democratic vs. Republican
straight fight and that the Democrats had also won all the votes
that went to Constitutional Union party.
– And, for good measure, suppose that the Democrats had won all
NJ electoral votes (which for peculiar reasons were actually split
between Lincoln and Douglas).
• Here is the outcome of the counterfactual 1860 election:
Party Pop. Vote % EV
Republican 39.82 169
Democratic 60.16 134
An Empirical Approach to the Analysis
of Election Reversals
In the 1988, the
Democratic ticket
of Dukakis and
Bentsen received
46.10% of the
two-party
national popular
vote and won
112 electoral
votes (though
one of these was
lost to a
―faithless
elector‖).
Uniform Swing Analysis
Of all the states that Dukakis carried,
he carried Washington (10 EV) by
the smallest margin of 50.81%.
If the Dukakis popular vote of
46.10% were (hypothetically)
to decline by 0.81% uniformly
across all states (to 45.29%),
WA would tip out of his
column (reducing his EV to
102).
Of all the states that Dukakis failed
carry, he came closest to carrying
Illinois (24 EV) with 48.95%.
If the Dukakis popular vote of
46.10% were (hypothetically)
to increase by 1.05%
uniformly across all states (to
47.15%), IL would tip into his
column (increasing his EV to
136).
The PVEV Step Function for 1988
Zoom In on the Reversal Interval
2000 vs. 1988
• The key difference between the 2000 and 1988 (or 2004
and other recent) elections is that 2000 was much
closer.
– The election reversal interval was (in absolute terms)
hardly larger in 2000 than in 1988:
• DPV 50.00% to 50.08% in 1988
• DPV 50.00% to 50.27% in 2000
– But the actual DPV was 50.267%, i.e., (just) within the
reversal interval.
The PVEV Step Function for 2000
The 2000 Reversal Interval
Magnitude and Direction of Election
Reversal Intervals
Distribution of Reversal Intervals
Distribution of Reversal Intervals:
1952-2004
Distribution of Reversal Intervals:
All Scenarios
Two Distinct Sources of
Possible Election Reversals
• The PVEV step-function defines a particular ―electoral
landscape,‖ i.e., an interval scale on which all states are
placed with respect to the relative partisan composition
of their electorates,
– for example, in 1988 WA was 1.86% more
Democratic than Illinois.
• The PVEV visualization makes it evident that there are
two distinct ways in which election reversals may occur.
First Source of
Possible Election Reversals
• The first source of possible election reversals is
invariably present.
• An election reversal may occur as a result of the (non-
systematic) ―rounding error‖ (so to speak) necessarily
entailed by the fact that the PVEV function moves up in
discrete steps.
– In any event, a given electoral landscape allows (in a sufficiently
close election) a ―wrong winner‖ of one party only.
– But small perturbations of such a landscape allow a ―wrong
winner‖ of the other party.
• The 1988 chart (and similar charts for all recent elections
[including 2000]) provide a clear illustration of election
reversals due to ―rounding error‖ only.
– So if the election had been much closer (in popular votes) and
the electoral landscape slightly perturbed, Dukakis might have
been a wrong winner instead of Bush.
A Sample of 32,000 Simulated Elections Based on
Perturbations of 2004 Electoral Landscape
Estimated (Symmetric) Probability of Election Reversals By
Popular Vote (Based on 2004 Landscape)
Estimated (Symmetric) Probability of Electoral Vote Tie
By Popular Vote (Based on 2004 Landscape)
Another Sample of 32,000 Simulated Elections Based
on Perturbations of 2004 Electoral Landscape
Second Source
of Possible Election Reversals
• Second, an election reversal may occur as result of
(systematic) asymmetry or bias in the general character
of the PVEV function.
– In this event, small perturbations of the electoral
landscape will not change the partisan identity of
potential wrong winners.
• In times past (e.g., in the New Deal era and earlier),
there was a clear asymmetry in the PVEV function that
resulted largely from the electoral peculiarities of the old
―Solid South,‖ in particular,
– its overwhelmingly Democratic popular vote
percentages, combined with
– its strikingly low voting turnout.
Highly Asymmetric PVEV Function in 1940
1860 Election
Even More Asymmetric PVEV Function in 1860
Two Distinct Sources of
Bias in the PVEV
• Asymmetry or bias in the PVEV function can result either
or both from two distinct phenomena:
– distribution effects.
– apportionment effects; and
• Either effect alone can produce a reversal of winners,
and
– they can either reinforce or counterbalance each
other.
Apportionment Effects
• A perfectly apportioned districted electoral system is one
in which each state’s electoral vote is precisely
proportional to its popular vote in every election (and
apportionment effects are thereby eliminated).
• It follows that, in a perfectly apportioned system, a party
(or candidate) wins X% of the electoral vote if and only if
it wins states with X% of the total popular vote.
– Note that this says nothing about the popular vote margin by
which the party/candidate wins (or loses) states.
– Therefore this does not say that the party wins X% (or any other
specific %) of the popular vote.
• An electoral system cannot be perfectly apportioned in
advance of the election (in advance of knowing the
popular vote in each state).
Apportionment Effects (cont.)
• In highly abstract analysis of its workings, Alan Natapoff (an MIT
physicist) largely endorsed the workings Electoral College
(particularly its within-state winner-take-all feature) as a vote
counting mechanism but proposed that each state’s electoral vote
be made precisely proportional to its share of the national popular
vote.
– This implies that
• electoral votes would not be apportioned until after the election, and
• would not be apportioned in whole numbers.
• Actually Natapoff proposes perfect apportionment of ―House‖
electoral votes while retaining ―Senatorial‖ electoral effects
– in order to counteract the ―Lion [Banzhaf] Effect.‖
– Such a system would eliminate apportionment effects from the Electoral
College system (while fully retaining its distribution effects).
– Reversal of winners can still occur under Natapoff’s perfectly
apportioned system (due to distribution effects).
– Natapoff’s perfectly apportioned EC system would create seemingly
perverse turnout incentives in ―non-battleground‖ states,
• though he views this as a further advantage of his proposed.
Alan Natapoff, “A Mathematical One-Man One-Vote Rationale for Madisonian
Presidential Voting Based on Maximum Individual Voting Power,” Public
Choice, 88/3-4 (1996).
Imperfect Apportionment
• The U.S. Electoral College system is (substantially)
imperfectly apportioned, for many reasons.
– House (and electoral vote) apportionments are anywhere from
two (e.g., in 1992) to ten years (e.g., in 2000) out of date.
– House seats (and electoral votes) are apportioned on the basis
of total population, not on the basis of
• the voting age population, or
• the voting eligible population, or
• registered voters, or
• actual voters in a given election.
• All these factors vary considerably from state to state (and district to
district).
– House seats (and electoral votes) must be apportioned in whole
numbers and therefore can’t be precisely proportional to
anything.
– Small states are guaranteed a minimum of three electoral votes.
Imperfect Apportionment (cont.)
• Similar imperfections apply (in lesser or greater degree)
in all districted systems.
• Imperfect apportionment may or may not bring about
bias in the PVEV function.
– This depends on the extent to which states (districts)
having greater or lesser weight than they would have
under perfect apportionment is correlated with their
support for one or other candidate or party.
1988 PVEV Based on Perfect vs. Imperfect
Apportionment
1940 PVEV Based on Perfect vs. Imperfect
Apportionment
1860 PVEV Based on Perfect vs. Imperfect
Apportionment
Distribution Effects
• Distribution effects in districted electoral system result
from the winner-take-all at the district/state level
character of these systems.
• Such effects can be powerful even in
– simple districted (one district-one seat/electoral vote)
systems, and
– perfectly apportioned systems.
• One candidate’s or party’s vote may be more ―efficiently‖
distributed than the other’s, causing an election reversal
independent of apportionment effects.
Distribution Effects (cont.)
• Here is the simplest possible example of distribution
effects producing a reversal of winners in a simple and
perfectly apportioned district system.
• There are 9 voters partitioned into 3 districts, and
candidates D and R win popular votes as follows:
(R,R,D) (R,R,D) (D,D,D):
Popular Votes Electoral Votes
D 5 1
R 4 2
R’s votes are more efficiently distributed, so R wins a majority of
electoral votes with a minority of popular votes.
The 25%-75% Rule
• The most extreme logically possible example of an
election reversal in perfectly apportioned system results
when
– one candidate or party wins just over 50% of the popular votes in
just over 50% of the (uniform) districts or in non-uniform districts
that collectively have just over 50% of the electoral votes.
– These districts also have just over 50% of the popular vote
(because apportionment is perfect).
– The winning candidate or party therefore wins just over 50% of
the electoral votes with just over 25% (50+% x 50+%) of the
popular vote and the other candidate with almost 75% of the
popular vote loses the election.
– The election reversal interval is (just short of) 25 percentage
points wide.
– If the candidate or party with the favorable vote distribution is
also favored by imperfect apportionment, the reversal interval
could be winners could be even more extreme.
The 25%-75% Rule in 1860 (cont.)
• In the 1860 Lincoln vs. anti-Lincoln scenario, the popular
vote distribution approximated the 25%-75% pattern
quite well.
– Lincoln would have carried all the northern states
except NJ, CA, and OR
• which held a bit more than half the electoral votes (and a
larger majority of the [free] population),
• generally by modest popular vote margins.
– The anti-Lincoln opposition would have
• carried all southern states with a bit less than half of the
electoral votes (and substantially less than half of the [free]
population)
• by essentially 100% margins; and
• lost all other states other than NJ, CA, and OR by relatively
narrow margins.
Distribution Effects (cont.)
• The Pure Proportional Plan for casting electoral votes eliminates
distribution effects entirely.
– The Whole Number Proportional and Districts Plans do not
eliminate distribution effects, and so
• they permit election reversals (even with perfect apportionment);
indeed
• the District Plans permit election reversals at the state as well as
national levels.
• But election reversals could still occur under the Pure Proportional Plan
due to apportionment effects.
– The reversals would favor candidates who do exceptionally well in
small and/or low turnout states).
• However, the Pure Proportional Plan combined with perfect
apportionment would be equivalent to direct national popular vote,
– so election reversals could not occur, and
– individual voting power would be equalized (and maximized).
Apportionment vs. Distribution Effects
in 1860
• The 1860 election was based on highly imperfect
apportionment.
– The southern states (for the last time) benefited from
the 3/5 compromise pertaining to apportionment.
– The southern states had on average smaller popula-
tions than the northern states and therefore benefited
disproportionately from the small-state guarantee.
– Even within the free population, suffrage was more
restricted in the south than in the north.
– Turnout among eligible voters was lower in the south
than the north.
Apportionment vs. Distribution
Effects in 1860 (cont.)
• But all these apportionment effects favored the south
and therefore the Democrats.
• Thus the pro-Republican reversal of winners was entirely
due to distribution effects.
– The magnitude of the reversal of winners in 1860
would have been even greater in the absence of the
countervailing apportionment effects.
• If the most salient characteristic of the Electoral College
is that it may produce election reversals, one’s
evaluation of the EC may depend on whether one thinks
Lincoln should have been elected President in 1860.
Sterling Diagrams: Visualizing Apportionment
and Distribution Effects Together
• First, we construct a bar graph of state-by-state popular
and electoral vote totals, set up in the following manner.
– The horizontal axis represents all states:
• ranked from the strongest to weakest for the winning party;
where
• the thickness of each bar is proportional to the state’s
electoral vote; and
• the height of each bar is proportional to the winning party’s
percent of the popular vote in that state.
[Note: this isn’t yet a proper Sterling diagram.]
Carleton W. Sterling, “Electoral College Misrepresentation: A Geometric Analysis, Polity,”
Spring 1981.
Sterling Diagrams (cont.)
• It is tempting to think that the shaded and unshaded
areas of the diagram represent the proportions of the
popular vote won by the winning and losing parties
respectively.
• But this isn’t true until we make one adjustment and
thereby create a Sterling diagram.
• Adjust the width of each bar so it is proportional,
– not to the state’s share of electoral votes, but
– to the state’s share of the popular national popular vote.
– If electoral votes were perfectly apportioned, no adjustment
would be required.
• Draw a vertical line at the point on the horizontal axis
where a cumulative electoral vote majority is achieved.
• In a perfectly apportioned system, this would be at just above the
50% mark.
• If there is no systematic apportionment bias in the particular
election, this will also be [just about] at the 50% mark.
Sterling Diagrams : Apportionment Effects
Sterling Diagram for 1848
Sterling Diagrams: The 25%-75% Rule (with
Perfect Apportionment)
Sterling Diagrams: The 25%-75% Rule
Approximated
Sterling Diagram: 1860
Sterling Diagram: 1860
Typical Sterling Diagram
(50%-50% Election)
Sterling Diagram:1988
Sterling Diagram:1936
Sterling Diagram: 2000
Sterling Diagram:
2000 under Pure District Plan
Sterling Diagram: 2000 House Seats