EXAMINING THE U'S' ELECTORAL COLLEGE: VARIANTS AND PROBLEMS

1
EXAMINING THE U.S. ELECTORAL COLLEGE VARIANTS
AND PROBLEMS

• Nicholas R. Miller
• UMBC
• LSE/VPP Talk
• May 6, 2008
• http//userpages.umbc.edu/nmiller/ELECTCOL.html

2
Origins of the Electoral College

• The mode of appointment of the Chief Magistrate
  of the United States is almost the only part of
  the system, of any consequence, which has escaped
  without severe censure, or which has received the
  slightest mark of approbation from its opponents.
  . . . I venture somewhat further, and hesitate
  not to affirm that if the manner of it be not
  perfect, it is at least excellent. (Publius
  Alexander Hamilton, Federalist 68)
• Many subsequent evaluations (and the many
  proposed constitutional amendments) suggest a
  less favorable assessment of the mode of
  appointment of the Chief Magistrate, which has
  been variously viewed as
• part of a generally elitist and anti-democratic
  constitution or
• a last-minute jerry-built compromise or
• a well designed compromise among diverse
  considerations, or possibly
• the embodiment of well-thought selection criteria.

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Origins of the Electoral College (cont.)

• The Electoral College the term is not used in
  the Constitution was a compromise between
• legislative election of the President (which
  might make the President subservient to Congress)
  and
• national popular election (which presented
  formidable practical difficulties at the time).
• The perceived advantage of the Electoral College
  was that (unlike Congress) it would perform a
  single task i.e., cast votes for President --
  and would then disband.
• A new Electoral College of unknown composition
  would re-elect (or not) the President four years
  later.

4
The Original Electoral College

• Each state selects a number of intermediate
  electors equal in number to its total (House
  Senate) representation in Congress (H 2).
• The legislature of each state determines the mode
  of selection of the electors from that state, the
  most likely alternatives being
• election by the legislature itself,
• popular election from districts, and
• popular election on a state-wide general
  ticket.
• Electors were originally required to
• cast two votes for two different candidates,
• at least one of whom had to be a resident of
  another state.
• To be elected by the Electoral College, a
  candidate was required to receive
• votes from a majority of electors and
• more votes than any other candidate.
• Given the double vote system, these are logically
  distinct requirements.

5
The Original Electoral College (cont.)

• If no candidate met the above requirement, the
  election would be thrown into the House of
  Representatives.
• The House could choose
• between two (or more) tied candidates, in the
  event both (or all) receive votes from a majority
  of electors, or
• among the top five candidates, in the event no
  candidate receives votes from a majority of
  electors.
• Voting in the House is by state delegation, each
  delegation casting one vote.
• Balloting continues until some candidate is
  supported by a majority of state delegations.
• In any event, the runner-up Presidential
  candidate becomes Vice President.

6
The Original Electoral College (cont.)

• This original Electoral College system was
  designed to operate in a non-partisan
  environment.
• It was generally hoped and expected that electors
  would typically be
• popularly elected
• from single-member districts (like most state
  legislators, delegates to the state ratifying
  conventions, members of the British House of
  Commons, and as was expected also for members of
  the new U.S. House and
• that they would be well-informed local notables
  who would act as representative trustees of their
  states and districts.
• It was further expected that
• there would typically be multiple potential
  Presidential candidates, and
• electors would choose among these candidates on
  the basis of their character and connections, not
  party affiliation or policy promises.
• It was therefore expected that the House
  contingent procedure would be needed 19 times
  out of 20, so
• big states would have the dominant role in
  screening/nominating candidates, while
• small states would have equal role in most final
  elections.

7
Duvergers Law and Crackup of the Original
Electoral College

• These expectations did not anticipate the
  development of a national two-party system.
• Duvergers Law Given politically ambitious
  candidates, single-winner elections produce (in
  equilibrium) two-candidate (or two-ticket
  Pres.VP) contests (the wasted vote argument
  and all that) and then sustain a two-party
  system.
• Given the development of a two-party Federalist
  vs. Republican system, the double-vote
  runner-up-is-VP provisions of the original
  Electoral College turned out to be a fatal flaw.
  
• In 1796 the Presidential candidate (Jefferson) of
  the losing (Republican) party became Vice
  President.
• In 1800 an electoral vote tie between the two
  (Presidential and Vice Presidential) candidates
  of the winning (Republican) party was broken by a
  House of Representatives controlled by the losing
  (Federalist) party.

8
The 12th Amendment

• After the 1800 fiasco, Congress proposed, and the
  states quickly ratified (in time for 1804
  election), the 12th Amendment to the
  Constitution.
• Electors now cast separate votes for President
  and Vice President.
• The required electoral vote majority for
  President (and for Vice President) is a simple
  majority of votes cast ( number of electors).
• If no candidate receives the required simple
  majority for President, the House (still voting
  by state delegations) chooses from among the top
  three vs. top five candidates.
• If no candidate receives the required majority
  for Vice President, the Senate (voting
  individually) chooses from among the top two
  candidates.

9
The Transformation of the Electoral College

• By the 1830s, the Electoral College as formally
  modified by the 12th Amendment had been further
  transformed into the kind of (essentially)
  automatic popular vote counting system that
  exists today.
• This transformation
• was driven largely by the development of a
  two-party system, and
• was brought about without any further
  constitutional amendments or (with one minor
  exception) any change in federal law.

10
Elements of the Transformation

• Beginning with the first contested Presidential
  election in 1796, elector candidates almost
  invariably party men, pledged (and faithful) to
  the (Presidential and Vice Presidential) nominees
  of their party.
• Put otherwise, electors became party delegates
  rather that trustees of their states or districts
  cf. regular delegates vs. superdelegates
• Pledged electors were almost universal as early
  as 1796.
• Notably, Samuel Miles (Fed., PA) violated his
  pledge and voted for Jefferson rather than Adams.
• An angry Federalist supporter complained What,
  do I choose Samuel Miles to determine for me
  whether John Adams or Thomas shall be President?
  No! I choose him to act, not to think.
• Once pledged and faithful electors have been
  selected, the prospective electoral vote for
  Presidential candidates is also known.
• The 12th Amendment took effect in 1804.
• A nationwide Presidential election day (i.e., day
  for selecting Presidential electors) was
  established by Congress in 1845.

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Elements of the Transformation (cont.)

• In early elections, the mode of selecting
  Presidential electors was regularly manipulated
  by party politicians in each state, on the basis
  of partisan calculations.
• 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 so the
  dominant party wins all of a states electoral
  votes, it is folly or worse for the other six
  not to follow.
• By 1832, Presidential electors were almost
  universally selected by popular (vs. legislative)
  vote (and by much expanded electorates).
• With popular election of pledged electors,
  American voters may be forgiven for thinking they
  are actually voting for Presidential candidates
  on election day.
• By 1836, the mode of popular election in every
  state is (following Madisons strategic analysis
  above) the general ticket (or party slate),
  rather than election from districts (or by any
  kind of propor-tional representation).
• This induces winner-take-all at the state
  level.
• However, at the present time two small states (ME
  and NE) use the Modified District Plan (discussed
  later).

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Elements of the Transformation (cont.)

• Moreover, the two-party system has effectively
  bypassed the House contingent procedure.
• On this point, the election of 1824 (the second
  and last time an election was thrown into the
  House) was the exception that proved the rule.
• The Federalist Party had collapsed and the
  Democratic-Republican Party was unchallenged.
• Consequently there was no longer pressure for
  Democratic-Republicans to unite behind a single
  candidate (or Presidential-Vice Presidential
  ticket).
• Four candidates, all nominally belonging to the
  same party, sought the Presidency.
• Unsurprisingly, no candidate received a majority
  of the electoral votes and the election was
  thrown into the House (for the second and last
  time).
• However, whenever there is a serious
  third-party ticket (especially one with a
  geographical base of support, e.g., George
  Wallace vs. Ross Perot), fears of the election
  being thrown into the House arise.
• Moreover, since the 23rd Amendment, giving the
  District of Columbia three electoral votes, was
  ratified in 1961, the total number of electoral
  votes has been an even number (538),
• so an electoral vote tie (269-269) is possible,
  and
• an election might be thrown into the House even
  in the absence of a third-party candidate winning
  election votes.

13
The Electoral College as a Vote-Counting Mechanism

• On election day (November 4), voters in each
  state vote for either the Democratic or
  Republican (or possibly other) slates of elector
  candidates, who are pledged to their partys
  (Pres. VP) nominees.
• The elector slate receiving the most votes wins.
• The electors meet in their state capital in
  mid-December and cast their electoral votes (as
  pledged).
• Electoral vote tallies are transmitted from each
  state capital to Congress and counted before a
  joint session on January 5.
• The President of the Senate the incumbent Vice
  President of the U.S. announces the vote and
  proclaims the President-elect.
• So in (almost invariable) practice, everything is
  determined on election night in November, and the
  remaining steps are merely ceremonial that is,
  TV prognosticators can
• report the popular vote winner in each state,
• add up the corresponding electoral votes, and
• declare a President-elect.

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(Alleged) Problems with the Electoral College as
a Vote-Counting Mechanism

• The Voting Power Problem. Do voters in different
  states have unequal voting power?
• If so, the EC violates the criterion of One
  Person, One Vote (OPOV).
• The Election Reversal Problem. The candidate who
  wins the most popular votes nationally may fail
  to be elected.
• The election 2000 provides an example (provided
  we take the official popular vote in FL at face
  value).
• The Partisan Bias Problem. Does the Electoral
  College as a vote counting system favor one party
  over the other?
• This is essentially a generalization of the
  Election Reversal Problem.
• The Battleground States Problem. The Electoral
  College focuses the Presidential election contest
  on a few battleground states, which get very
  disproportionate attention from the candidates
  and parties.

15
A Priori Voting Power

• The Electoral College itself generates a
  51-player weighted voting game susceptible a
  priori voting power analysis using an appropriate
  voting power measure.
• In addition, there is an simple majority voting
  game within each state to select members of the
  Electoral College.
• So the overall Presidential election is a
  two-tier voting game.
• The measurement and interpretation of a priori
  voting power has been a major focus of the LSE
  program on Voting Power and Procedures.
• Though VPP research has focused primarily on
  voting power with respect to the EU Council of
  Ministers (and other EU institutions).
• Measures of a priori voting power applied to the
  Electoral College take account of the its formal
  rules plus the population of each state and the
  mathematical formula used to apportion House
  seats among the states,
• but not demographics, historic voting patterns,
  ideology, polling results, election forecasts,
  etc.

16
Voting Power (cont.)

• With a measure of voting power, we can address
  questions that arise with respect to voting power
  in the Electoral College, in particular
• How much (if any) does individual voting power
  vary from state to state?
• In so far as it does vary, are voters in larger
  or smaller states favored? (A priori, population
  is the only way states differ.)
• How would voting power change under various
  proposed (and other) alternatives to the existing
  Electoral College?

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Voting Power (cont.)

• With respect to the second question, directly
  contra-dictory claims are commonly made.
• The Electoral College . . . gives greater voting
  power to voters in states with less than average
  population and less to those with greater than
  average population. (Gary Parish, The Electoral
  College Source of Inequality and Social
  Injustice in America)
• According to a frequently published argument . .
  . , there is an intrinsic large-state advantage
  in Presidential elections. (Howard Margolis,
  The Banzhaf Fallacy)
• Partly because the Electoral College is viewed by
  some as favoring small states and by others as
  favoring large states, it is commonly asserted
  that a constitutional amending modifying or
  abolishing the Electoral College can never by
  ratified by the required 38 states.
• Hence the National Popular Vote Plan (to use an
  interstate compact to bypass the constitutional
  amendment process).

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Voting Power (cont.)

• First, we need to define and distinguish between
• voting weight and
• voting power.
• Second and specifically with respect to the
  Electoral College, we also need to distinguish
  between
• how electoral votes are apportioned among the
  states (which determines voting weight), and
• how electoral votes are cast within states
  (which, in conjunction with voting weight,
  determines voting power).

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The Apportionment of Electoral Votes

• Remember that each state has electoral votes
  equal to its total (House Senate)
  representation in Congress,
• which guarantees every state at least three
  electoral votes and
• more generally produces a systematic and
  substantial small-state advantage in the
  apportionment of electoral votes (relative to the
  apportionment population).
• This is the basis of the argument that the
  Electoral College advantages voters in smaller
  (rural, etc.) states.
• The magnitude of this small-state apportionment
  advantage is not fixed in the Constitution.
• It varies (inversely) with the size of the House
  (relative to the Senate), which is determined by
  Congress.
• If the House had been sufficiently larger than
  435, Gore would have won the 2000 election (even
  while losing Florida).

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The Small-State Apportionment Advantage
21
Another View of the Small-State Advantage
22
The Casting of Electoral Votes

• Remember that state choice of the general ticket
  for electing electors produces the
  winner-take-all practice that produces the
  weighted voting game noted at the outset.
• Many have believed that this practice produces a
  large-state advantage in voting power that
  counteracts (in some degree) the small-state
  advantage in apportion-ment.
• This is the basis for the argument that the
  Electoral College gives disproportionate voting
  power to voters in larger (urban, etc.) states.

23
Voting Weight vs. Voting Power

• Suppose that four parties receive these vote
  shares Party A, 27 Party B, 25 Party C, 24
  Party 24.
• Seats in a 100-seat parliament are proportionally
  awarded, so
• Party A 27 seats Party C 24 seats
• Party B 25 seats Party D 24 seats
• While seats (voting weights) have been
  apportioned in a way that is precisely
  proportional to vote support, voting power has
  not been proportionally apportioned (and cannot
  be).
• Since no party controls a majority of 51 seats, a
  governing coalition of two or more parties must
  be formed.
• A partys voting power is reflects its
  opportunity to create (or destroy) winning
  (governing) coalitions.
• But, with a small number of parties, coalition
  possibilities -- and therefore different patterns
  in the distribution of voting power -- are highly
  restricted.

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Voting Weight vs. Voting Power (cont.)

• A 27 seats B 25 seats C 24 seats
  D 24 seats
• Once the parties start negotiating, they will
  find that Party A has voting power that greatly
  exceeds its slight advantage in seats. This is
  because
• Party A can form a winning coalition with any one
  of the other parties and
• the only way to exclude Party A from a winning
  coalition is for Parties B, C, and D to form a
  three-party coalition.
• The seat allocation above (totaling 100 seats) is
  strategically equivalent to this much simpler
  allocation (totaling 5 seats)
• Party A 2 seats
• Parties B, C, and D 1 seat each
• Total of 5 seats, so a winning coalition requires
  3 seats, i.e., (32,1,1,1)
• So the original seat allocation is strategically
  equivalent to one in which Party A has twice the
  weight of each of the other parties (which is not
  proportional to their vote shares).

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Voting Weight vs. Voting Power (cont.)

• Suppose at the next election the vote and seat
  shares change a bit
• Before Now
• Party A 27 Party A 30
• Party B 25 Party B 29
• Party C 24 Party C 22
• Party D 24 Party D 19
• While seats shares have changed only slightly,
  the strategic situation has changed
  fundamentally.
• Party A can no longer form a winning coalition
  with Party D.
• Parties B and C can now form a winning coalition
  by themselves.
• The seat allocation is equivalent to this much
  simpler allocation
• Parties A, B, and D 1 seat each
• Party D 0 seats
• Total of 3 seats, so a winning coalition requires
  2 seats, i.e., (21,1,1,0)
• Party A has lost voting power, despite gaining
  seats.
• Party C has gained voting power, despite losing
  seats.
• Party D has become powerless (becoming a
  so-called dummy), despite retaining a substantial
  number of seats.

26
Felsenthal and Machover, The Measurement of
Voting Power

• In this book, Dan Felsenthal and Moshé Machover
  (both members of the VPP Board of Directors)
  present the most conclusive study of voting power
  measures.
• They conclude that the so-called Absolute Banzhaf
  Measure is the appropriate measure for evaluating
  power in typical voting situations, including the
  Electoral College.
• They further show that this measure has the
  following a probabilistic interpretation
• A voters absolute Banzhaf voting power is the
  voters a priori probability of casting a
  decisive vote, i.e., one that determines the
  outcome of an election (for example, breaking
  what otherwise would be a tie).
• In this context, a priori probability means, in
  effect, given that all voters vote randomly,
  i.e., vote for either candidate as if they
  independently flip fair coins), so that every
  point in the Bernoulli space (every combination
  of voters) is equally likely to occur.
• I call such a two-candidate elections Bernoulli
  elections

27
Calculating Power Index Values

• There are mathematical formulas and algorithms
  can calculate or estimate voting power values.
• 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/
• which was used in making the calculations that
  follow.
• It is apparent that
• with 51 states (as opposed to 4 parties in the
  earlier example) voting power is in fact closely
  proportional to voting weight (electoral votes),
  though
• the largest states (especially California) are
  somewhat advantaged.
• Remember that (for example) Californias value of
  .475 means that, if all states other than CA
  repeatedly cast their electoral votes by
  independently flipping coins, almost half of the
  time they would split their 483 votes
  sufficiently equally that Californias 55 votes
  determine the winner).

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State Voting Power in the Existing EC (cont.)

• It is evident from the following charts that
• only Californias share of voting power
  substantially deviates from (and exceeds) its
  share of electoral votes
• the modest large-state advantage in voting power
  (relative to voting weight) is not sufficient to
  balance out the small-state advantage in
  apportionment indeed,
• even Californias distinctive advantage in terms
  of voting power (relative to voting weight) is
  not sufficient to give it voting power
  proportional to its population.

30
Share of Voting Power by Share of Electoral Votes
31
Share of Voting Power by Share of Population
32
The 125 Million Two-Tier Voting Game

• But as previously noted, a U.S. Presidential
  election really is a two-tier voting system, in
  which the casting of electoral votes is
  determined by a simple majority voting games
  within each state.
• In such a two-tier system, individual a priori
  voting power is the probability of double
  decisiveness, i.e.,
• that a voter casts a decisive vote within his or
  her state (i.e., that there is tie among the
  other voters in the state),
• and
• that his or her state casts a decisive bloc of
  electoral votes (i.e., that neither candidate
  wins 270 electoral votes from the other states).
• Put otherwise, individual voting power in the
  two-tier system is equal to
• individual voting power in the unweighted (but
  large number) majority voting game within the
  state
• times
• the states voting power in the 51-state weighted
  voting game.
• We have already determined the second term in
  this product as shown in the last column of the
  previous table.

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The Two-Tier Voting Game (cont.)

• Probability theory and the properties of Banzhaf
  measure tell us that the individual voting power
  in the first tier voting game is inversely
  proportional (to excellent approximation once n
  gt 25) to the square root of the number of voters
  in the state.
• We previously saw that a states voting power in
  the second tier voting game is approximately
  proportional to its electoral vote (and
  therefore, apart from the small state
  apportionment advantage, roughly proportional to
  its population.
• Putting these two considerations together, it
  follows that individual a priori voting power in
  the two-tier system
• increases with the population of the voters
  state, and
• is approximately proportional to the square root
  of the number of voters in the voters state.

34
The Two-Tier Voting Game (cont.)

• This effect (first noted with explicit reference
  to the Electoral College by Banzhaf) may be
  dubbed the Banzhaf effect.
• It had been noted in a more general context
  twenty years earlier by Penrose, and it is a
  consequence of what Felsenthal and Machover call
  the Penrose square-root rule.
• With specific respect to the Electoral College,
  the Banzhaf effect
• is in some degree counterbalanced by the (quite
  substantial) small-state advantage in
  apportionment,
• in small degree reinforced by the (quite weak)
  large-state advantage in voting power, and
• among the smallest states is largely hidden by
  the unavoidable rounding error in apportioning
  House seats into small whole numbers.

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Individual Voting Power Under the Existing
Electoral College
37
Individual Voting Power Under the Existing
Electoral College (cont.)

• The chart also shows mean individual voting power
  nationwide.
• Note that the individual voting power values
  plotted in the chart must be weighted by ELECT
  SIZE in order to determine mean voting power
  nationwide.
• The chart also shows individual voting power
  under direct popular vote (calculated in the same
  manner as individual voting power within a
  state).
• This is necessarily uniform over the nation.
• 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.

38
Alternatives to the Existing Electoral College

• We now consider two categories of alternatives to
  the existing Electoral College
• those that keep the state-level winner-take all
  practice but use a different formula for
  apportioning electoral votes among states, and
• those that keep the existing apportionment of
  electoral votes but use something other than
  winner-take-all for the casting of state
  electoral votes.
• Almost all Electoral College reform plans (and
  certainly all reforms that can be implemented at
  the state level) fall in the second category.
• Methodological note in these each 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.

39
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 House seats 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

40
Individual Voting Power by State
PopulationHouse Electoral Votes Only
41
Individual Voting Power by State
PopulationElectoral Votes Precisely
Proportional to Population
42
Individual Voting Power by State
PopulationElectoral Votes Proportional
Population, plus Two
43
Individual Voting Power by State
PopulationElectoral Votes Apportioned Equally
Among States
44
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 bloc of electoral votes e.g., 100
  are awarded on a winner-take-all basis to the
  national popular vote winner. Bush 271, Gore
  367 Bush 386, Kerry 252

45
Alternative Rules for Casting Electoral Votes
(cont.)

• Calculations for the Proportional Plan, the
  Whole-Number Proportional Plan, and the Pure
  District Plans are straightforward.
• Under the Modified District Plan (and the
  National Bonus Plans), each voter casts a single
  vote that counts both within a district (or
  state) and also at-large (state or nation).
• Calculating individual voting power in such
  systems is far from straightforward.
• I have made approximations by simulating
  Bernoulli elections.

46
The (Pure) Proportional Plan

• Electoral votes are retained but the office of
  elector is abolished.
• A states electoral votes are cast fractionally
  in precise proportion to the candidates shares
  of the popular vote in the state (which is why
  the office of elector must be abolished).
• If proportionality is sufficiently refined, such
  a system becomes a 122-million single-tier
  weighted voting game,
• where the weights applied to individual votes
  reflect their states electoral votes per voter.
• As proportionality becomes less refined, such a
  system begins to resemble a whole-number
  proportional system (considered next).

47
The (Pure) Proportional Plan (cont.)

• The constitutional amendment for a proportional
  system Lodge-Gossett Plan proposed in the
  1950s specified that candidates would be credited
  with fractional electoral votes calculated out to
  four decimal places.
• The following chart assumes that proportionality
  is sufficiently refined to create a single-tier
  weighted voting game.
• We can invoke the Penrose Limit Theorem to
  justify the assumption that voting weight
  voting power in this single-tier very large-n
  weighted voting game.

48
Individual Voting Power by State
Population(Pure) Pure Proportional System
49
(Pure) Proportional Plan (cont.)

• The small-state apportionment advantage carries
  through very strongly to voting power.
• Voters in Wyoming in have almost four times the
  voting power of voters in California.
• Nevertheless, voters in Montana (the largest
  state with only three electoral voters) have less
  voting power than voters in Rhode Island (the
  smallest state with four electoral votes).
• Close inspection of the chart shows that similar
  but less striking inversions exist with respect
  to a number of larger states.
• Since sufficiently refined proportionality
  creates what is effectively a (weighted)
  single-tier voting game, mean individual voting
  power is equal to individual voting power under a
  direct (unweighted) popular vote.

50
The Whole-Number Proportional Plan

• Such a plan was proposed by Colorados
  Proposition 36 in 2004.
• This plan divides a states electoral votes
  between (or among) the candidates in a way that
  is as close to proportional to the candidates
  state popular vote shares as possible, given that
  the apportionment must be in whole numbers.
• Unlike the (Pure) Proportional Plan, whole-number
  proportionality allows for the retention of
  electors.
• Accordingly, it is the only proportional plan
  that can be implemented at the state level (as
  Colorado Prop. 36 proposed).

51
The Whole-Number Proportional Plan (cont.)

• In principle, there are as many such plans as
  there are apportionment formulas.
• In addition, candidates might be required to meet
  some vote threshold in order to win any electoral
  votes.
• Colorado Proposition 36
• had no explicit vote threshold, and
• used a distinctly ad hoc apportionment formula
• that was overtly biased toward the strongest
  candidate and against the weakest candidates.
• But, in the event there are just two candidates
  (as we assume here), all apportionment formulas
  work in the same straightforward way
• multiply each candidates share of the popular
  vote by the states number of electoral votes and
  then round off in the normal manner (to the
  nearest whole number of electoral votes).

52
Whole-Number Proportional Plan (cont.)

• In this two-tier system, individual a priori
  voting power is the probability that
• the voter casts a decisive vote within his or her
  state in the sense that
• other votes are so divided that the individuals
  vote determines whether a candidate gets k or k1
  electoral votes from the state,
• and
• this single electoral vote is decisive in the
  Electoral College.
• where, as usual, these probabilities result from
  the Bernoulli elections.
• The following chart shows that this plan produces
  a distinctly odd distribution of a priori
  individual voting power.
• Note. I have discovered that similar
  calculations and chart have been produced,
  independently and earlier, by Claus Beisbart and
  Luc Bovens, A Power Analysis of the Amendment 36
  in Colorado, University of Konstanz, May 2005,
  and Public Choice, forthcoming.

53
Individual Voting Power by State
PopulationWhole-Number Proportional Plan
54
Whole-Number Proportional Plan (cont.)

• As can be seen, this plan produces a truly
  bizarre allocation of voting power.
• Voters in a large number (17) of states are
  rendered (essentially) powerless.
• These are voters in states with an even number of
  electoral votes.
• Heres why this happens.
• Given Bernoulli elections and any fairly large
  number of voters, the vote essentially always is
  almost equally divided between the two
  candidates.
• As previously noted, the expected (i.e., the mean
  over repeated coin flipping elections) vote share
  for each candidate is .5, and
• the standard deviation of the distribution of
  vote shares is .5vn .

55
Equal Apportionment vs. Whole-Number Proportional
56
The Pure District Plan

• All electors are elected by popular vote in
  single-member districts, so winner-take-all is
  applied at the district, rather than state,
  level.
• Historical Note 1 this is how the most of the
  framers thought electors would (and should) be
  selected.
• Historical Note 2 this mode of selection was
  prescribed in early drafts of what became the
  12th Amendment.
• Accordingly, there are 538 electoral districts
  and the popular vote winner in each district is
  awarded one electoral vote.
• All districts are subdivisions of states.
• All districts in the same state have equal
  numbers of voters.
• Districts in smaller states have smaller numbers
  of voters than those in larger states, because of
  the small-state advantage in the apportionment of
  electoral votes.

57
The Pure District Plan (cont.)

• Individual voting power within states is equal,
• because the number of voters in each district is
  equal.
• All districts have equal voting power in the
  Electoral College,
• because they have equal weight, i.e., 1 EV, and
  the second tier voting game entails 538 equally
  weighted voters.
• Individual voting power across states is not
  equal,
• because districts in different states have
  unequal numbers of voters.

58
Individual Voting Power by State PopulationPure
District System
59
The Pure District Plan (cont.)

• We see that the small-state advantage in
  apportionment carries through in terms of voting
  power.
• Voters in Wyoming have an almost 2 to 1 voting
  power advantage over voters in California.
• While Wyoming districts have only about ¼ as many
  voters as California districts, the Banzhaf
  effect means that California voters have about ½
  the voting power as Wyoming voters.
• Nevertheless, voters in Montana (the largest
  state with only three electoral voters) have less
  voting power than voters in Rhode Island (the
  smallest state with four electoral votes).
• Close inspection of the chart shows that similar
  but less striking inversions exist with respect
  to a number of larger states.

60
Supplementary Slides
61
Can Electoral Vote Apportionment 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.

62
Individual Voting Power when States Have Equal
Population (Versus Apportionment Proportional to
Actual Population)
63
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.

64
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 (at least to close approximation), electoral
  votes can be apportioned by applying the Penrose
  Square Root Rule in reverse (as an engineering
  principle, rather than as a descriptive law)
• Individual voting power is equalized when
  electoral votes are apportioned so that state
  voting power is proportional to the square root
  of state population.
• But such Penrose Apportionment is tricky, because
  what must be made proportional to population is
  not electoral votes (what we directly apportion)
  but state voting power (a consequence of the
  apportionment of electoral votes).

65
(Approximately) Equalized Individual Voting Power
66
EV Apportionment to Equalize VP (cont.)

• These two methods of apportionment that equalize
  individual voting power equalize it at
  (essentially) the same level, namely
• 0.00005785 (vs. 0.00007215 for direct popular
  vote).
• If the Penrose square root rule is used but
  apportion-ment of electoral votes must be in
  whole numbers,
• individual voting power is imperfectly equalized
  (especially among small state voters), and
• mean individual voting power is reduced ever so
  slightly (to 0.00005784).

67
State Voting Power When EVs are Apportioned on
the Basis of SQRT of Population (cont.)

• Under such square-root apportionment rules, the
  outcome of the 2004 Presidential elections 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.