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 180Douglas Northern Democrat 29.46 12Breckinridge Southern Democrat 18.09 72Bell Constitutional Union 12.61 39

Total Democratic Popular Vote 47.55Total 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 169Democratic 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 AnalysisOf 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 VotesD 5 1R 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