ANALYZING THE ELECTORAL COLLEGE

1
ANALYZING THE ELECTORAL COLLEGE

• Nicholas R. Miller
• Political Science, UMBC
• INFORMS Meeting
• October 14, 2008
• http//userpages.umbc.edu/nmiller/ELECTCOLLEGE.ht
  ml

2
Preface

• Polsbys Law Whats 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.

3
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.

4
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).

5
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.

6
Chart 1. The Small-State EV Apportionment
Advantage
7
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.

8
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 voters absolute Banzhaf voting power is
• the probability that the voters 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).

9
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 v (2 /
  pn),
• Which implies that that individual voting power
  is inversely proportional to the square root of
  the number of voters.

10
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_L
  ist
• which was used in making most of the
  calculations that follow.

11
A Priori State Voting Power in the Electoral
College (with Winner-Take-All)

• A states a priori voting power is
• the probability that the states 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 Leechs website), the a priori
  voting power of CA (with 55 EV out of 583) .475
  .
• This means if every other states 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 CAs 55 votes will determine
  the outcome.

12
Chart 2. Share of Voting Power by Share of
Electoral Votes
13
Chart 3. Share of Voting Power by Share of
Population
14
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 voters
• state is decisive in the Electoral
  College
• or
  equivalently
• individual voting power in
  the state
• multiplied
  by
• state voting power in the Electoral
  College

15
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.
  

16
Banzhaf Effect in Bernoulli Elections
17
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.

18
Individual Voting Power By State Population
Existing Electoral College
19
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 Californias a priori
  advantage would be reflected in its a posteriori
  voting power as well.

20
Winners Margin by State Size
21
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.

22
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

23
Individual Voting Power by State
PopulationHouse Electoral Votes Only
24
Individual Voting Power by State
PopulationElectoral Votes Precisely
Proportional to Population
25
Individual Voting Power by State
PopulationElectoral Votes Proportional
Population, plus Two
26
Individual Voting Power by State
PopulationElectoral Votes Apportioned Equally
Among States
27
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.

28
Individual Voting Power when States Have Equal
Population (Versus Apportionment Proportional to
Actual Population)
29
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.

30
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).

31
(Almost) Equalized Individual Voting Power
32
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.

33
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

34
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.

35
Pure District System
36
Modified District System (Approximate)
37
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.
• Madisons 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.

38
(No Transcript)
39
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 Prisoners Dilemma.

40
(No Transcript)
41
(Pure) Pure Proportional System
42
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.

43
National Bonus Plan(s)
44
Individual Voting Power Summary Chart
45
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, lets 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 gt
  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

46

• 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.

47
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

48
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
  Winners 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.

49
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.

50
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

51
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).

52
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).

53
The PVEV Step Function for 1988
54
Zoom In on the Reversal Interval
55
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.

56
The PVEV Step Function for 2000
57
The 2000 Reversal Interval
58
Magnitude and Direction of Election Reversal
Intervals
59
Distribution of Reversal Intervals
60
Distribution of Reversal Intervals1952-2004
61
Distribution of Reversal IntervalsAll Scenarios
62
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.

63
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.

64
(No Transcript)
65
(No Transcript)
66
A Sample of 32,000 Simulated Elections Based on
Perturbations of 2004 Electoral Landscape
67
Estimated (Symmetric) Probability of Election
Reversals By Popular Vote (Based on 2004
Landscape)
68
Estimated (Symmetric) Probability of Electoral
Vote Tie By Popular Vote (Based on 2004
Landscape)
69
Another Sample of 32,000 Simulated Elections
Based on Perturbations of 2004 Electoral
Landscape
70
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.

71
(No Transcript)
72
(No Transcript)
73
Highly Asymmetric PVEV Function in 1940
74
(No Transcript)
75
1860 Election
76
Even More Asymmetric PVEV Function in 1860
77
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.

78
Apportionment Effects

• A perfectly apportioned districted electoral
  system is one in which each states 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).

79
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 states
  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
  Natapoffs perfectly apportioned system (due to
  distribution effects).
• Natapoffs 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).

80
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 cant
  be precisely proportional to anything.
• Small states are guaranteed a minimum of three
  electoral votes.

81
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.

82
1988 PVEV Based on Perfect vs. Imperfect
Apportionment
83
1940 PVEV Based on Perfect vs. Imperfect
Apportionment
84
1860 PVEV Based on Perfect vs. Imperfect
Apportionment
85
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 candidates or partys vote may be more
  efficiently distributed than the others,
  causing an election reversal independent of
  apportionment effects.

86
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
• Rs votes are more efficiently distributed, so R
  wins a majority of electoral votes with a
  minority of popular votes.

87
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.

88
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.

89
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).

90
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.

91
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, ones evaluation of the EC may depend
  on whether one thinks Lincoln should have been
  elected President in 1860.

92
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
  states electoral vote and
• the height of each bar is proportional to the
  winning partys percent of the popular vote in
  that state.
• Note this isnt yet a proper Sterling diagram.
• Carleton W. Sterling, Electoral College
  Misrepresentation A Geometric Analysis, Polity,
  Spring 1981.

93
(No Transcript)
94
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 isnt 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 states share of electoral votes, but
• to the states 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.

95
Sterling Diagrams Apportionment Effects
96
Sterling Diagram for 1848
97
Sterling Diagrams The 25-75 Rule (with Perfect
Apportionment)
98
Sterling Diagrams The 25-75 Rule Approximated
99
Sterling Diagram 1860
100
Sterling Diagram 1860
101
Typical Sterling Diagram (50-50 Election)
102
Sterling Diagram1988
103
Sterling Diagram1936
104
Sterling Diagram 2000
105
Sterling Diagram 2000 under Pure District Plan
106
Sterling Diagram 2000 House Seats