ELECTION INVERSIONS BY THE U.S. ELECTORAL COLLEGE

Nicholas R. MillerDepartment of Political Science, UMBC

nmiller@umbc.eduhttp://userpages.umbc.edu/~nmiller/index.htm

For presentation at the 2011 Annual Meeting of the Public Choice SocietySan Antonio, TexasMarch 10-13, 2011

Overview• An election inversion occurs when the candidate (or party) that wins the most votes

from the nationwide electorate fails to win the most electoral votes (or parliamentary seats) and therefore loses the election.– Other names for such an event include ‘election reversal,’ ‘reversal of winners,’ ‘wrong

winner,’ ‘representative inconsistency,’ ‘compound majority paradox,’ and ‘referendum paradox.’

• An election inversion can occur under U.S. Electoral College or any other districted electoral system.– Such an event actually occurred in the 2000 Presidential election.

• I first note the three historical manifestations of election inversions by the U.S. Electoral College

• I also discuss one massive but ‘latent’ inversion in more detail. • I then use uniform swing analysis based on historical state-by-state election data in

order to estimate the probability, magnitude, and direction of potential election inversions.

• Along the way, I identify three distinct sources of election inversions – rounding effects, – apportionment effects, and – distribution effects.

• Finally I examine their separate impacts on the likelihood of election inversions.

Examples of Election Inversions• Election inversions are at least as common in Westminster parliamentary

systems (with uniform districts) as in U.S. Presidential elections.• Most such elections are very close with respects to both seat and votes,

• but there are exceptions, e.g., Canada, 1979.• However, many such inversions result from votes for (and at least a few

seats won by) regional or other third parties,• a factor not considered in the following US-focused analysis.

Election Leading Parties Pop. Vote SeatsUK 1929 Conservatives 38.1% 260

Labour 37.1% 287UK 1951 Labour 48.78% 297

Conservatives 47.97% 302UK 1974 (Feb.) Conservatives 37.2% 297

Labour 37.9% 301NZ 1978 Labour 40.4% 40

National 39.8% 51NZ 1981 Labour 39.0% 43

National 38.8% 47CA 1979 Liberals 40.1% 114

Conservatives 35.1% 136

Electoral College Election Inversions• The U.S. Electoral College has produced three manifest

election inversions,– all of which were very close with respect to popular votes, and – two of which were very close with respect to electoral votes,

plus one massive but “latent” and usually unrecognized election inversion.

Manifest EC Election Inversions

Election EC Winner EC Loser EC Loser’s 2-P PV%

2000 271 [Bush (R)] 267 [Gore (D)]50.27%1888 233 [Harrison (R)] 168 [Cleveland (D)] 50.41% 1876* 185 [Hayes (R)] 184 [Tilden (D)] 51.53%

*The 1876 election was decided (just before the inauguration) by an Electoral Commission that, by a bare majority and straight party-line vote, awarded all of 20 disputed electoral votes to Hayes.

The 1860 Election: A Latent But Massive InversionCandidate 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 47.55%Total anti-Lincoln Popular Vote 60.16%

• Two inconsequential inversions (between Douglas and Breckinridge and between Douglas and Bell) are manifest.

• It may appear that Douglas and Breckinridge were spoilers against each -----other – that is, that either Democrat could have won if the other had not been in the race.– Under a direct popular vote system, this would have been true (if we suppose that, had

one Democratic candidate withdrawn, the other would have inherited all or most of the his support).

– But in fact, under the Electoral College system, Douglas and Breckinridge were not spoilers against each other (even if we suppose that, had one Democratic candidate withdrawn, the other would have inherited all or most of the his support).

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 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 this counterfactual 1860 election:

Party Pop. Vote EVRepublican 39.82% 169Anti-Republican 60.16% 134

The Probability of Election Inversions: First Cut – Historical Estimate

• Number of Inversions/Number of elections (since 1828) 3/46 = .06522 4/46 = .08696

• Clearly an important determinant of the probability of an election inversion is the probability of a close division of the popular vote.

• Considering only elections in which the winner’s popular vote margin was no greater than about 3 percentage points, the frequency of inversions is considerably higher, namely

3/12 = .25.

The PVEV Step Function• We turn to a more informative historical analysis of election

inversions, by using state-by-state popular vote percentages to produce what may be called the Popular Vote-Electoral Vote (or PVEV) step function for each election.– The PVEV is based on the kind of uniform swing analysis pioneered by

Butler (1951), which has also be called • hypothetical (single-year) swing analysis (Niemi and Fett, 1986),• the Bischoff method (Peirce and Longley, 1981),• and has also been employed by Nelson (1974), Garand and Parent (1991),

and others.

• The PVEV function – is a cumulative distribution function and is therefore (weakly)

monotonic, and– is a step function because, while the independent variable (PV) is

essentially continuous, the dependent variable (EV) is discrete (taking on only whole number values and jumping up in discrete steps).

The PVEV Step-Function: 1988 as an Example

In 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 was lost to a “faithless elector”).

1988 Example (cont.)Of all the states that Dukakis carried,

he carried Washington (10 EV) by the smallest margin of 50.81%.

If the Dukakis national popular vote of 46.10% were (hypothetically) to decline by 0.81 percentage points 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 percentage points uniformly across all states (to 47.15%), IL would tip into his column (increasing his EV to 136).

The Full PVEV for 1988 Appears To Go Through the Perfect-Tie Point

But If We Zoom in on PV ≈ 50%, We Find a Small Pro-Republican Inversion Interval

Inversion Intervals:1988 vs. 2000

• The 2000 election produced an actual (pro-Republican) election inversion.– However, the election inversion interval was (in absolute terms) only

slightly larger in 2000 than in 1988:• DPV 50.00% to 50.08% in 1988• DPV 50.00% to 50.27% in 2000

– The key difference between the 2000 and 1988 elections was not the magnitude of the inversion interval but that 2000 was much closer.• The actual D2PV was 50.267%, putting it (just) within the inversion

interval,• though if the 2000 inversion interval had been as small as the

1988 interval (but still pro-Republican) Gore would have won.• In the counterfactual 1860 election, the (pro-Republican)

inversion interval was– DPV 50.00% to 61.26%.

Inversion Intervals (cont.)• The following chart shows the magnitude and direction of the inversion

interval in all ‘straight-fight’ elections since 1828 (excluding the counter-factual 1869 straight fight).– When the number of electoral votes is even (as at present), a PVEV

may also have a tie interval.– A tie interval may span the PV = 50% mark, in which case there is no

inversion interval at all.– However, no such spanning tie interval appears in any historical PVEV.

• Apart from aside the counterfactual 1860 election (not shown in the chart), inversion intervals are typically quite small, rarely exceeding two percentage points.– The mean of the absolute intervals (i.e., ignoring whether they are

pro-Republican or pro-Democratic) over the entire period is about 0.76 percentage points, and• They exhibited an overall pro-Republican bias more often than

not. – However, considering only elections since the mid-twentieth century,

inversion intervals have been smaller, rarely exceeding one percentage point and averaging about 0.5, and • they exhibit no clear party bias.

– The 1988 election turns out to have the smallest inversion interval on record.

Historical Magnitude and Direction of Election Inversion Intervals

• Note: The scale on the right gives the percent of the popular vote required for a Democratic electoral vote majority . On average, the Democrat needed 50.30% of the popular vote to win an electoral vote majority.

The Probability of Election Inversions Based on Historical Inversion Intervals

• Based as they are on state-by-state data for all ‘straight-fight’ Presidential elections, these results provide a more informa-tive basis for estimating the likelihood election inversion by the Electoral College.

• If we assume that the Democratic Two-Party Popular Vote % (DPV%) is (more or less) normally distributed with a mean of 50%, the estimated the probability of election inversions is as follows.

The Probability of Election Inversions (cont.)• The probability of

an election inver-sions obviously depends on the closeness of the election (in terms of popular votes).

• This chart stacks all (pro-Dem and pro-Rep) inver-sion intervals in order of magni-tude.

• It makes the historical pro-Republican advan-tage evident.

The Probability of Election Inversions (cont.)

This chart stacks all absolute (tie or) inversion interval

It can be interpreted as indicating the approximate probability of an election inversion as a function of the popular vote winner’s margin above 50% of the two-party vote.

• If that margin is barely more than 50%, the a priori probability of an inversion is just about 0.5;

• If it exceeds 52%, the probability is almost zero (in the absence of extreme sectional conflict like 1860).

• By interpolation,– if it is about 50.5%, the

probability is about .25; – if it is about 51%, the

probability is about 0.125; and

Two Distinct Sources of Possible Election Inversions

• The PVEV visualization makes it evident that there are two distinct ways in which election inversions may occur.

• The first source of possible election inversions 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. – That is to say, as the Democratic vote swings upwards, almost certainly

the pivotal state (that gives the Democratic candidate 270 or more electoral votes) will almost never tip into the Democratic column precisely as the Democratic popular vote crosses the 50% mark.• In any event, a specific PVEV allows (in a sufficiently close election) a

“wrong winner” of one party only (e.g., Republican in 1860, 1988, and 2000).

• But small perturbations in some (but not all) PVEVs allow a “wrong winner” of either party.

First Source (cont.)

More Generally, the Democratic and Republican PVEVs in 1988 are Almost Identical

The Second Source of Possible Election Inversions

• Second, an election reversal may occur as result of a (systematic) asymmetry or (partisan) bias in the general character of the PVEV function,– particularly in the vicinity of PV = 50%.– In this event, small perturbations of the PVEV cannot change the

partisan identity of potential wrong winners. • In times past, there was a clear (pro-Republican) asymmetry

in the PVEV function in the vicinity of PV = 50% 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.

An Asymmetric PVEV (1940) with a Highly Evident Inversion Interval (No Zoom Needed)

Democratic vs. Republican PVEVs in 1940 are Quite Distinct

Generalized PVEV Function in 1940

The Non-Republican PVEV in 1860

Non-Republican and Republican PVEVs in 1860 are Entirely Distinct

Two Sources of Asymmetry or Bias in the PVEV

• Asymmetry or bias in a PVEV can result from either or both of two distinct phenomena:– apportionment effects and– distribution effects.

• Either effect alone can produce election inversions.

• In combination, they can either reinforce or counterbalance each other.

Apportionment Effects• We start with the benchmark of a perfectly apportioned two-

tier electoral system, in which apportionment effects are eliminated because – electoral votes are reapportioned among the states in a way that is

precisely proportional to the total popular vote cast within each state.• It follows that, in a perfectly apportioned system, a candidate

who wins X% of the electoral vote also carries states that collectively cast X% of the total popular vote.

• The concept of a perfectly apportioned electoral system is an analytical tool.– As a practical matter, an electoral system can be perfectly apportioned

only retroactively, i.e., only after the popular vote in each state is known.

• Apportionment effects refer to the (net) effects on the inversion interval of any deviations from perfect apportionment.

Imperfect Apportionment of Electoral Votes

• The U.S. Electoral College system is (very) imperfectly apportioned, for at least six reasons. – House seats (and therefore electoral votes) must be apportioned in

(relatively small) whole numbers, and therefore cannot be precisely proportional to anything.

– There are many different methods of apportioning whole numbers of seats on the basis of population, none of which is uniquely best.

– House (and therefore electoral vote) apportionments are anywhere from two to ten years out-of-date at the time of a Presidential election.

– The apportionment of electoral votes is skewed in favor of smaller states, as they are guaranteed a minimum of three electoral votes (due to their guaranteed one House seat and two Senate seats) and (approximate) proportionality begins only after that.

– The size of the House is not fixed by the Constitution and can be changed by law (as it was regularly in the nineteenth century), so the magnitude of the small-state bias can be reduced (or enhanced) by law, by increasing (or reducing) the size of the House.

Imperfect Apportionment of Electoral Votes (cont.)• House seats (and therefore electoral votes) are apportioned

to states on the basis of their total population and not on the basis of their – voting age population, or – voting eligible population (excluding non-citizens, etc.), or– number of registered voters, or – number of actual voters in a given election (or typical levels of

turnout).– In particular,

• For several elections prior to the ratification of the Nineteenth Amendment, women could vote in some states but not others; and

• more generally, in early years the qualifications for (adult male) voting varied significantly from state to state.

• In addition, prior to the 13th Amendment, House seats were apportioned on the basis of the total free population plus three fifths of ‘all other persons’ (who certainly could not vote).

Imperfect Apportionment (cont.)• Imperfect apportionment may or may not create bias in the

PVEV function.– This depends on the extent to which state advantages with

respect to apportionment effects is correlated with their support for one or other candidate or party.

• We can separate apportionment effects from distribution effects by recalculating the PVEV function– with electoral votes retroactively and precisely (and therefore

fractionally) reapportioned on the basis of the total (two-party) popular vote in each state.

• Any residual bias in the PVEV function must be due to distribution effects.

In 1988 Apportionment Effects Were Minimal

In 1940 Apportionment Effects Were Quite Substantial

Apportionment Effects in 1860 • As noted before, the grand daddy of all election inversions

occurred in the (counterfactual version of) the 1860 election.– The 1860 electoral landscape exhibited the same kind of bias as 1940

(reflecting low turnout and Republican weakness in the South) but to an even more extreme degree, especially in the second respect.

• 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.– Southern states had on average smaller populations than 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 Effects in 1860 (cont.)

• But all of these apportionment effects favored the south and therefore the Democrats.

• Thus the huge pro-Republican inversion interval was (more than) entirely due to distribution effects.

• The magnitude of the inversion interval in 1860 would have been even (slightly) greater in the absence of the counter-balancing apportionment effects.

1860 PVEV Based on Perfect vs. Imperfect Apportionment

Apportionment Effects (cont.)

• We might expect that perfect apportionment would greatly reduce the frequency and magnitude of election inversions.

• In fact, with respect to actual election results, perfect apportionment would have – “corrected” the election inversions of 1876 and 2000, but at the same

time– failed to “correct” the election inversion of 1888, and moreover– created a new election inversion in 1916.

• Perhaps surprisingly, perfect apportionment would have actually increased the degree of Republican bias, and – as a consequence of this, considerably increased the average

magnitude of absolute inversion intervals.

Magnitude and Direction of Election Inversion Intervals Under Perfect Apportionment

Note: The scale on the right gives the percent of the popular vote required for a Democratic electoral vote majority . On average, the Democrats needed 50.62% of the popular vote to win an electoral vote majority under perfect apportionment.

• The inversion intervals shown above, with apportionment effects removed, must reflect distribution effects, to which we now turn.

Distribution Effects• Distribution effects in districted electoral system result from

the winner-take-all at the state (or district) level character of these systems.– Such effects can be powerful even under

• uniformly 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 inversion independent of apportionment effects.

• Footnote: Consistent with the strongly two-party nature of U.S. Presidential (and other) elections, everything in this presentation – and the discussion of distribution effects in particular – supposes that minor parties play no significant role in elections.– Presidential elections for which this supposition is inappropriate have been excluded

from analysis.– The present analysis therefore is not adequate for a discussion of potential election

inversions in the UK and Canada in the present era, where support for minor or regional parties plays an important role in the translation of votes into seats for the two major parties.

Distribution Effects (cont.)• Here is the simplest possible example of a distribution effect

producing an election inversion in a small, uniform, and perfectly apportioned district system.

• Nine voters are perfectly apportioned into three uniform districts.

• The individual votes for candidates D and R in each district are 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% vs. 75% Rule• Suppose we have k uniform districts each with n voters (both

odd numbers). • A candidate can win by carrying

• a bare majority of (k + 1) / 2 districts • each with a bare majority of (n + 1) / 2 votes.

– Thus a candidate can win with as few as [(k + 1) / 2] × [(n + 1) / 2] = (nk + n + k +1) / 4

efficiently distributed total votes. • With n = 3 and k = 3, the last expression is 4/9 = 44.4%, but – as n and k become large, the last expression approaches a

limit of nk / 4, i.e., 25% of the total popular vote.

The 25% vs. 75% Rule (cont.)• Stated more intuitively, if the number of districts is fairly large and the

number of voters is very large, the most extreme logically possible example of an election inversion in a 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 voting weight.

– 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% of the popular vote and the other candidate (with just under 75% of the popular vote) loses the election.

– The election inversion interval is (just short of) 25 percentage points wide.– However, if the candidate or party with the favorable vote distribution is also

favored by imperfect apportionment, the inversion interval could be could be even greater.

E. E. Schattschneider, Party Government (1942), p. 70Kenneth May, “Probabilities of Certain Election Results,” American Mathematical Monthly (1948)Gilbert Laffond and Jean Laine, “Representation in Majority Tournaments,” Mathematical Social

Sciences (2000)

The 25%-75% Rule in 1860 (cont.)• In the 1860 Lincoln vs. anti-Lincoln scenario, the popular vote

distribution over the states approximated the logically extreme 25%-75% pattern, but in the presence of relatively extreme and adverse (to Lincoln) apportionment effects.– Lincoln would have carried all 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 (50-60%) 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 almost (in many cases literally) 100% margins; and• lost all other states other than NJ, CA, and OR by relatively narrow

margins.

Accounting for both Apportionment and Distribution Effects

• By definition, in the absence of both apportionment effects and distribution effects, the PVEV function would show that the PV% required to give the Democrats an electoral vote majority is 50% (plus or minus a small amount due to rounding effects). – Since these rounding effects are small, they may be ignored to good

approximation, so we would expect the inversion interval to be zero under these circumstances.

• For each election, we have determined the minimum percent of the popular vote that gives the Democrats an electoral vote victory, i.e., 50% plus or minus the overall (pro-Republican or pro-Democratic) inversion interval.

• Call this quantity Dad, as it reflects both apportionment effects and distribution effects.

Accounting for Both Effects (cont.)

• We have also determined the minimum PV% that would give the Democrats an electoral vote majority in the absence of apportionment effects (or apportionment effects that work to the net advantage of either party, – i.e., 50% plus or minus the (pro-Republican or pro-Democratic)

inversion interval under perfect apportionment. • Call this quantity Dd, since it reflects distribution effects only.

– Thus (50% − Dd) is the portion of the Democratic inversion interval due to distribution effects only.

• What we have not yet determined is the minimum PV% that would give the Democrats an electoral vote majority in the event that distribution effects did not work to the advantage of either party.

• Call this quantity Da, since it reflects apportionment effects only.

– Thus (50% − Da) is the Democratic inversion interval due to apportionment effects only.

Accounting for Both Effects (cont.)• The magnitude of Da can be inferred on the basis of the following

accounting identity:Dad = 50% + (50% − Dd) + (50% − Da).

• ThusDa = 50% + 50% − Dd + 50% − Dad = 150% − (Dd + Dad).

• For example, in 1988Da = 150% − (50.05% + 50.08%) = 49.87%,

so the inversion interval resulting from apportionment effects only is 0.13%, giving the accounting identity 50.08% = 50% + 0.13% −0.05%.

• For the counterfactual 1860 election Da = 150% – (62.51 + 61.26) = 26.23%

so the inversion interval resulting from apportionment effects only is 23.77, giving the accounting identity 61.26% = 50% + 23.77% – 12.51%

• The following chart shows the magnitude and direction of apportionment and distribution effects in every election and it therefore summarizes the main results of this paper.

Summary Chart and Conclusions

• Over the entire period, apportionment effects have generally favored Democrats and distribution have generally favored Republicans, with latter effects being somewhat stronger than the former, producing the slight pro-Republican bias.– In this respect, the counterfactual 1860 election can be characterized as a

massive exaggeration of the typical pattern.

Summary Chart and Conclusions (cont.)

• More specifically, throughout the 19th Century, there are no consistent patterns, – evidently reflecting relatively loose party ties in the early

party systems coupled with the disruptive events leading to and coming out of the Civil War.

Summary Chart and Conclusions (cont.)

• The overall pattern is especially clear from 1908 through 1956 (with the interesting exception of 1928).– This pattern reflects the special character of the “Solid South,”– where the Democrats won overwhelming (and thus “inefficient”)

popular vote margins on the basis of very low turnout.

Summary Chart and Conclusions (cont.)

• In the next two elections (1964 and 1972), the “old” South begins to “switches sides,” so the partisan impact of the two effects is reversed.

• But as the Voting Rights Act takes full effect, (black) turnout increases in the South and also provides the basis for a substantial (but rarely winning DPV% in Southern states.)– Thus both effects become relatively small.