Transcript ELECTION INVERSIONS UNDER THE ELECTORAL COLLEGE N. R. Miller Election Inversions • An election inversion occurs when the candidate (or party) that wins.

ELECTION INVERSIONS UNDER
THE ELECTORAL COLLEGE
N. R. Miller
Election Inversions
• 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, etc.) and therefore loses the election.
• The 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 also.
It has also produced one possible inversion and one massive but “latent”
and usually unrecognized election inversion.
Manifest EC Election Inversions
Election
2000
1888
1876*
1960
EC Winner
271 [Bush (R)]
233 [Harrison (R)]
185 [Hayes (R)]
303 [Kennedy (D)]
EC Loser
267 [Gore (D)]
168 [Cleveland (D)]
184 [Tilden (D)]
220 [Nixon (R)]
EC Loser’s 2-P PV%
50.27%
50.41%
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.
** No good way exists to determine Kennedy’s popular vote in AL because electors were elected individually
and some Democratic electors were not pledged to the Democratic ticket.
Electoral Map:1860
The 1860 Election: A Latent But Massive Inversion
Candidate
Party
Lincoln
Douglas
Breckinridge
Bell
Republican
Northern Democrat
Southern Democrat
Constitutional Union
Total Democratic Popular Vote
Total anti-Lincoln Popular Vote
Pop. Vote
EV
39.82%
29.46%
18.09%
12.61%
180
12
72
39
47.55%
60.16%
• Two inconsequential inversions (between Douglas and Breckinridge and
between Douglas and Bell) are manifest.
• Suppose that, had one Democratic candidate withdrawn, the other would
have inherited all or most of the his support. Then 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.
– But in fact, 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 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
Republican
Anti-Republican
Pop. Vote
39.82%
60.16%
EV
169
134
Counterfactual 1860 Electoral Map
Scattergram:
DEV by DPV:
1928-2012
Excludes 1948 and 1968
Total number of EV
varies slightly
Bound is ±65 EV
Does this indicate
the propensity of
the EC to produce
election inversions?
The PVEV Step Function
• The problem with such a historical scattergram is that the
structure of party competition has changed vastly over the 85
years of elections depicted,
– especially with respect to the disappearance of the old
“solid South.”
• We turn to a more informative historical analysis of election
inversions, by using state-by-state popular vote percentages in
a given year to produce what may be called the Popular VoteElectoral Vote (or PVEV) function for each election.
The PVEV 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 Very
Small Pro-Republican Inversion Interval
Inversion Intervals:1988 vs. 2000
• The 2000 election produced an actual (pro-Republican)
election inversion.
– However, the (pro-Republican) 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% (vs. the actual 60.16%).
Inversion Intervals (cont.)
• The following chart shows the magnitude and direction of the
inversion intervals in all ‘straight-fight’ elections since 1828
(not including the counter-factual 1860 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, 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 exhibit 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 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 informative 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 inversions obviously
depends on the
closeness of the
election (in terms
of popular votes).
• This chart stacks
all (pro-Dem and
pro-Rep) inversion intervals in
order of magnitude.
• It makes the
historical proRepublican advantage evident.
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 twoparty 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.
The Probability of Election
Inversions (cont.)
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 (nonsystematic) “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 Jim Crow
“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.
Hypothetical Apportionment Effects
State
A
B
C
Total
Vote
100
100
300
500
EV
3
3
5
8
DPV RPV DEV REV
40
60
0
3
40
60
0
3
180 120
5
0
260 240
5
6
– Note: each state is carried by the same margin (60%-40%), so there are no “distribution
effects.”
• Malapportionment need not cause inversions (or party bias),
– specifically if both parties do equally well in small (favored) and large
(disfavored) states.
Apportionment Effects
• We start with the benchmark of a perfectly apportioned twotier electoral system, in which apportionment effects are
eliminated because
– electoral votes are (hypothetically) apportioned 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 remaining bias in the PVEV function, and any remaining
inversion interval, 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 counterbalancing 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 winnertake-all at the state (or district) level character of these 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.
• Here is the simplest possible example of a distribution effect producing an
election inversion in a small, uniform, 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).
D
R
Popular Votes
5
4
Electoral Votes
1
2
– R’s votes are more “efficiently” distributed [or the system is
“gerrymandered” in R’s favor], so R wins a majority of electoral votes
with a minority of popular votes.
The 25% vs. 75% Rule
• If the number of districts or states 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 districts or states that collectively have just over 50% of the
electoral votes.
– These districts or states 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 resulting 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.
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 free [northern] states except
NJ, CA, and OR
• which collectively held a bit more than half the electoral votes
(and a larger majority of the [free] population),
• generally by modest (typically 51-55% and never exceeding about
60%) popular vote margins.
– The anti-Lincoln opposition would have
• carried all southern and border [slave] 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.
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.
Election Inversions [and EV Ties] under EC
Variants
• Would election inversions be more or less likely (would average
inversion increases become smaller or larger and more pro-Rep
or pro-Dem) under other Electoral College Variants
• Random Elections: each voter votes as if independently flipping a
fair coin.
– Such elections are extraordinarily close with respect to popular votes (but
less close with respect to electoral votes).
– They are free of any systematic distribution effects.
• Random elections based on some historical “landscape.”
– 2000s
– New Deal era (1936-1944)