popular votes and electoral votes the electoral college as a vote counting mechanism...

POPULAR VOTES AND ELECTORAL VOTES

The Electoral College as a Vote Counting Mechanism

http://www.research.umbc.edu/~nmiller/RESEARCH/PVEV/PVEV.htm

The EC as a Vote Counting Mechanism

• Let’s consider the Electoral College simply as a vote counting mechanism.

• Instead of having a single national election for President (in which votes are added up nationwide, taking no account of state boundaries), we have 51 separate state (and DC) elections for President.

• We determine the national winner by – awarding the plurality winner in each state all the electoral votes

of that state, and then – adding up electoral votes across the nation to determine the

winner, – with an absolute electoral vote majority requirement,– and a House “runoff” in the event no candidate receives the

required majority.

The EC as a Vote Counting Mechanism (cont.)

• We ignore constitutional details, e.g., – states might revert to legislative election of electors;– electors might be elected otherwise than on a statewide general

ticket basis;– electors might violate their pledges, etc.

• Moreover, here we focus for the most part on the case in which there are just two “serious” candidates (who have any chance of carrying states) for President, – so that (excepting a mathematically possible 269-269 electoral

vote tie) one candidate must receive an absolute majority of electoral votes, and the House “runoff” procedure is avoided.

• We also ignore the fact that two small states elect electors by district and thereby allow electoral votes to be divided.

Districted Electoral Systems

• A districted election system is an electoral system in which voters are partitioned into (geographically defined) districts.

• Each district is apportioned a number of seats in a national electoral body.

• Parties or candidates compete for seats district by district.

Districted Electoral Systems (cont.)• Districted electoral system are extremely

widespread. • Most electoral democracies are parliamentary

systems, in which the head of government (prime minister, premier, chancellor, etc.) is selected indirectly, and the general election simply fills the seats in the (lower house of) national legislature.

• All countries except the Netherlands, Israel, and some mini-states have districted elections, though some other countries have “national adjustment” seats that largely counteract the effect of districts.

Districted Electoral Systems (cont.)

• Within each district, some voting rule or electoral formula, i.e., a specification of how voters declare their preferences on ballots and how this information on preferences is aggregated to fill the seats, must be used.

• Clearly, in single-member districts (SMDs), any formula entails a “winner-take-all” district-level outcome (since there is only one seat to take). – As we have seen, several different voting rules can be used,

e.g., Simple Plurality, (Instant) Runoff, Approval Voting, Borda Score, selecting the Condorcet Winner, etc.

• Multi-member districts (MMDs) allow a great variety of possible voting rules, including many that divide seats more or proportionally among competing parties (and well as other that produce “winner-take-all,” or at least “winner-take-most,” outcomes).


• Given an electoral system that applies a (quasi-) proportional electoral formula on large MMDs (or on the [undistricted] nation as a whole), there is an essentially determinate (and proportional) relationship between the popular votes received by a party and the number of seats it wins.

• However, given an electoral system that uses SMDs or applies a “winner-take-all” formulas to MMDs, the relationship between overall seats and popular votes is complex and contingent --- in particular, it depends on how popular votes for a parties or candidates are distributed over the districts.


• The British (and Canadian and other) electoral systems are simple “winner-take-all districted systems, in that all districts are SMDs.

• The U.S. Electoral College system is a more complex “winner-take-all” districted electoral system, in that different “districts” (states) have different numbers of “seats” (electoral votes).

Winner-Take-All Districted Systems

• A winner-take-all districted system tends to produce a two–party system. [Duverger’s Law].

• A winner-take-all districted system tends nationwide to give a disproportionate number of seats to the leading party or candidate and to give few if any seats to trailing (ranked third or lower) parties or candidates. [Exaggeration Effect]

Winner-Take-All Districted Systems (cont.)

• In a two-party system, the exaggeration effect benefits the winning party and penalizing the losing party.

• In winner-take-all districted systems, small parties (or third candidates) with geo-graphically concentrated support do better than small parties (or third candidates) with geographically dispersed support.

The Swing Ratio

• The magnitude of the exaggeration effect is reflected by the swing ratio.– A swing ratio of 3, for example, means that a

party that increases its national vote share by 1% can expect to increase its seat share by about 3%.

– The claim that such systems have exagger-ation effects is simply to say that the swing ratio is greater than one.

Reversal of Winners• Any districted electoral system can

produce a reversal of winners.

• That is, the candidate or party that wins the most popular votes may fail to win the most seats or electoral votes (and there-fore lose the election).

• Such outcomes are actually more common in many parliamentary systems than in U.S. Presidential elections.

“Historical” Overview of EC as a Vote Counting System

• The following chart is a scattergram that plots the relationship between popular votes and electoral vote from 1928 through 2004.– 1948 and 1968 are excluded because third

candidates with concentrated electoral support won substantial electoral votes in those years.

But evidently the translation of popular votes into electoral votes is cannot be entirely “linear.”

Evidently EC has no systematic tendency to produce wrong winners

Limitations of the “Historical” Approach

• Trade-off between number of elections to include (plotted points in scattergram) and uniformity of the political environment.

• In any case, we are limited to about 50 data points.

• We can get as many data points from a single election using a “cross-sectional” approach.

The “Cross-sectional” or “Uniform National Swing” Approach

• This analytical technique allows us to identify the swing ratio, the “wrong winner interval,” and other characteristics of the translation of popular votes into electoral votes in individual elections.

• We use state-by-state popular vote data to profile the “electoral landscape” that character-ized a particular election.

• Then we let the “political tides” in favor of one or other party/candidate rise and fall in a uniform national swing.

Uniform National Swing:1988 as an Example

• 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”).

• Given state-by-state popular vote totals, we can display the relationship between Democratic popular and electoral votes in 1988, if we take the actual state-by-state vote totals as the starting point and then consider how states would tip into or out the Democratic column in the face of a uniform national swing of varying magnitudes for or against the party. – For example, a uniform national swing of 2.50% in favor of the

Democrats would increase their national popular vote percent to 48.60 and would shift every state they lost by less than 2.50% into the Democratic column.

1988 List• The first column lists the states (plus DC) ordered in

terms of the performance of the Democratic ticket in the 1988 Presidential election.

• The second (D2PC) column shows the Democratic percent of the two-party presidential vote (i.e., excluding votes casts for minor parties) in each state.

• The third column (DSWG) is equal to 50 - D2PC. – Each negative entry represents the magnitude of a uniform

national swing against the Democrats that would just cost them the state in question.

• For example, Dukakis carried his home state of MA with 53.98% of the 2-party vote. Thus Dukakis would still carry MA in the face of a uniform national swing against him of up to 3.98% but would lose MA in the face of a larger national swing.

– Each positive entry represents the magnitude of a uniform national swing in favor of the Democrats that would just gain them the state in question.

• For example, Dukakis lost CA with 48.19% of the vote. Thus Dukakis would still lose CA with a uniform national swing in his favor of anything less than 1.81% but would win CA with any larger favorable national swing.

1988 List (cont.)

• The fourth column (DPOP) is equal to 46.10 + DSWG. – It represents the Democratic national popular vote given a

national swing just big enough to tip the state. • For example, the 3.98% national swing against the Democrats just

sufficient to tip MA into the Republican column results in a 42.12% national popular vote for the Democrats;

• For example, the 1.81% swing in favor of the Democrats just sufficient to tip CA into the Democratic column results in a 47.91% national popular vote for the Democrats.

• The fifth column (EVCM) is the total electoral vote for the Democratic ticket cumulating from their strongest to weakest state.

• The fourth and fifth columns together allow us to examine the relationship between popular votes and electoral votes, taking the actual state-by-state 1988 vote as a baseline and considering uniform national swings in both directions from this baseline.

1988 PV=>EV Chart

• A scattergram (with the points connected) plotting EVCM against DPOP produces the monotonically increasing (i.e., never decreasing) PVEV step function shown in the following chart. – The plot is “monotonic” because it assumes the increase in the

Democratic national popular is uniform across states. – It is a step function because electoral votes do not increase

continuously with popular votes but rather in discrete increments (of no less than three votes) whenever another state tips into the Democratic column.

• Dukakis actually won 46.1% of the popular vote, which translated into 112 electoral votes. This is shown in the chart by the dashed green vertical and horizontal reference lines that intersect at the actual election outcome (DPOP = 46.1%, EVCM = 112).

Political “Landscape” vs. Political “Tides”

• The PVEV function may be said to depict the political “landscape” in a given election.

• We then examine what happens as PV goes up or down (as political “tides” wax or wane) as a result of uniform national swings of varying magnitude.

1988 PVEV ChartNote that the general pattern of the 1988 chart

broadly resembles that of the historical chart (especially the curvilinear variant).– However, the slope of the function in the middle of the

chart is considerably steeper.– Indeed, the “tidal” version of the chart shows that,

while Dukakis got 112 electoral votes (and 416 for Bush) with about 46% of votes, with 54% he would have gotten about 390 electoral votes (and 148 for Bush).

• Note that the 1988 PVEV function is not quite symmetric about DPOP = 50%, at least at 50% ± 4%.

– Thus an 8% swing in the popular vote would have gained Dukakis 278 electoral votes (34.75 electoral votes for each 1% of the popular vote, implying a swing ratio of about 6.5.

Wrong Winners in PVEV Charts• Such a chart can partitioned into four equal quadrants by

vertical and horizontal lines located at DPOP = 50% and EVCUM = 269.

• An election outcome located at the intersection of these lines is a perfect tie, with respect to both popular and electoral votes. – An outcome (including the actual 1988 outcome) in the south-

west quadrant (“Rep Winner”) is one in which the Democrats lose both the popular and electoral votes.

– An outcome in the northeast quadrant (“Dem Winner”) is one in which the Democrats win both the popular and electoral vote.

– An outcome in the northwest quadrant entails a Democratic electoral vote victory with less than half of the two-party popular vote, i.e., the Democrat is a “wrong winner.”

– An outcome in the southeast quadrant entail a Democratic electoral vote loss despite a popular vote majority, i.e., the Republican is a wrong winner.

Wrong Winners in PVEV Charts (cont.)

• Assuming uniform national swings from the actual state-by-state popular vote, an Electoral College “wrong winner” (or “reversal of winners” or “misfire”) can occur if and only the PVEV function fails to pass precisely through the perfect tie point at the center of the chart (50.000% => 269).

• It is evident that, given any “landscape,” the electoral vote function almost always fails to pass through the perfect tie point, so the probability of a wrong winner approaches 50% as the popular vote division approaches a perfect tie.

• Because the PVEV step function is monotonic, it can pass through only one of the two wrong winner areas.– So for a given electoral landscape, only one candidate can be a

potential wrong winner.

The “Wrong Winner Interval” in 1988

• The 1988 chart suggests that that the 1988 popular vote split would have had to be a virtually perfect tie in order to produce a wrong winner (and it isn’t evident from the full-sized chart which candidate it might be).

• We can examine this with precision if we go back to the data on which the chart is based.– We see that if there had been a wrong winner, it

would have been Bush (but this outcome would have been very unlikely even if the election had been much closer).


• We see that, if Dukakis had won precisely 50% of the popular vote, he would have lost the election with only 252 electoral votes.

• If Dukakis had won 50.05% of the popular vote, Colorado would have tipped to the Democrats, but he still would have lost with only 260 electoral votes.

• But if Dukakis had reach 50.08% of the popular vote, Michigan would have tipped, giving him an electoral vote majority of 280.

• Thus in 1988 there would have been a “wrong winner” [Bush] if (under the uniform swing assumption) Dukakis had received between 50.0000% and 50.0765% of the popular vote.


• The following chart zooms in on the critical region in the vicinity of DPOP = 50% to show the "wrong winner interval” in 1988, i.e., the DPOP interval from 50.0000% to 50.0765%.

The “Wrong Winner Area” in 1988

• We can also determine the "wrong winner area [rectangle]” of the electoral vote function, i.e., the rectangle with its southwest corner at 50% and 252 and its northwest corner at 50.0765% and 280. – The width of this rectangle is the wrong winner

interval and its height is the Dukakis’s gain in electoral votes over this interval.

– In 1988, this rectangle occupies about 0.000042 of the total area in the full chart (i.e., 100% × 538)

– It occupies 0.000333 of the maximum wrong winner rectangle.

• This maximum in turn is equal to 1/8 of the full chart [ignoring “apportionment effects” – we’ll return to this later].

Symmetry of the PVEV Function

• Call a PVEV (almost) symmetric if it is true that, if the Democratic candidate would win X electoral votes with Y% of the popular vote, then the Republican candidate would likewise win (almost) X electoral with Y% of the popular vote.– Clearly if PVEV is (almost) symmetric, there is

(almost) no possibility of a wrong winner.– We have seen that, while Dukakis got 112 electoral

votes with about 46% of votes, Bush would have gotten 148 electoral votes with 46% of the popular vote, so in this respect the 1988 PVEV function was somewhat asymmetric.

Symmetry of the PVEV Function (cont.)

• We can observe the overall degree of symmetry by superimposing the PVEV function for one party over that for the other.

• The 1988 PVEV function is actually highly symmetric, – except In the vicinity of D2PC = 50% ± 3-4%;– except for the distinctive case of DC.

Two Sources of Wrong Winners• This PVEV visualization makes clear that there are two distinct ways

in which wrong winners may occur. – First, a wrong winner may occur is as a result of the (non-

systematic) “rounding error” (so to speak) necessarily entailed by the fact that the electoral vote function moves up and down in discrete steps.

• In this event, a particular electoral landscape may allow a wrong winner of one party but small perturbations of that landscape allows a wrong winner of the other party.

– Second, a wrong winner may occur as result of (systematic) asymmetry or bias in the general character of the PVEV function.

• In this event, smaller perturbations of the electoral landscape will not change the partisan identity of potential wrong winners.

• Such asymmetry or bias in turn results from two distinct phenomena:

– apportionment effects; and

– distribution effects.

Wrong Winners Produced by “Rounding Error”

• The 1988 landscape provides a clear illustration of a possible wrong winner due to “rounding error” only.

• While the general path of the electoral vote function takes it through the perfect tie point, the stepwise character of the precise path means that it almost certainly misses the perfect tie point.

• Thus a “wrong winner” interval occurs in a narrow popular vote interval on one or other side of the 50% popular vote mark.

Wrong Winners Produced by Asymmetry

• The second source of possible wrong winners is substantial asymmetry or bias in the PVEV function such that its general path clearly misses the perfect tie point and it passes through either through the northwest quadrant or the southeast quadrant.

• In times past (e.g., in the New Deal era and earlier), there was a clear asymmetry in the PVEV function that result primarily from the electoral peculiarities of the old “Solid South”: namely– its overwhelmingly Democratic popular vote

percentages, combined with– its strikingly low voting turnout.

Wrong Winners Produced by Asymmetry (cont.)

• The asymmetry of the PVEV function is most extreme (and favored the Democrats with respect to landslide elections, but this is without consequence for determining the winner.

• What might easily have affect the outcome of elections in this period is the smaller asymmetry in the vicinity of D2PC = 50%.

• This bias against the Democrats was such that the electoral vote function would have regularly produced a “wrong [Republican] winner” if the Democratic ticket received between 50% and about 51.5% of the vote.

• The Democratic Party was dominant in Presidential elections during this period despite this unfavorable electoral landscape because it benefited from consistently favorable high political tides.

• In 1940, the wrong winner interval runs from 50.00% to 51.51%, almost 20 times wider than in 1988– In the 1940, the wrong winner area is 38 times larger than in

1988.

Wrong Winner in 2000• Potential wrong winner outcomes all

recent elections are due to “rounding errors” in essentially symmetric PVEV functions.

• The wrong winner interval extended from D2PC = 50% to D2PC = 50.2716 (about 3.5 times wider than in 1988 but less than 1/5 as wide as in 1940.

• The wrong winner outcome in 2000 occurred because the actual D2PC fell just within this interval, i.e., D2PC = 50.2664%.

Wrong Winner in 1860

• The grand daddy of all “wrong winners” was occurred in 1860. – This electoral landscape exhibits the same kind of bias as 1940

(produced by extreme Republican weakness in the South) but in even more extreme degree.

• It is well known that with slightly less than 40% of the national popular vote, Lincoln won a comfortable electoral vote majority (180 out of 303) against a divided opposition.

• But this victory was quite different from (for example) Wilson’s electoral vote majority (435 out of 531) victory against divided opposition in 1912.

Wrong Winner in 1860 (cont.)• Even if he had confronted a single non-Republican

candidate able to assemble all Douglas, Breckinridge, and Bell votes, Lincoln’s electoral vote total would have been only slightly reduced (whereas Wilson would have lost badly against a similarly united opposition).– The only states that Lincoln actually won but would have lost against

united opposition were California and Oregon (which he won by a pluralities against a divided opposition).

– He would have held every other state that he actually carried, because he carried them with an absolute majority of the popular vote.

– Though Douglas is credited with a popular plurality in New Jersey, Lincoln (for peculiar reasons) won four of its seven electoral votes.

– Even if we shift these four electoral votes out of the Lincoln column along with the seven electoral votes from California and Oregon, Lincoln wins 169 electoral votes (with 39.8% of the popular vote) against 134 electoral votes (with 60.2% of the popular vote) for the united opposition.

– The 1860 PVEV function for this scenario displays extraordinary asymmetry.

Factors Producing PVEV Asymmetry and Systematic Wrong Winners

• Two distinct characteristics of districted electoral systems can produce asymmetry or bias in contribute to reversals of winners: – apportionment effects; and– distribution effects.

• Either effect alone can produce a reversal of winners.

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 if and only if it wins states with X% of the total popular vote.– Note that this say nothing about the popular vote margin by

which the party/candidate wins (or loses) state.– Therefore this does not say that the party wins X% (or any other

specific %) of the popular vote.

• An electoral system can 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.

– Such a system would eliminate apportionment effects from the Electoral College system (while fully retaining its distribution effects).

– Reversal of winners could still occur under Natapoff’s perfectly apportioned system.

– Natapoff’s perfectly apportioned EC system would create perverse turnout incentives in “non-battleground” states.

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, in many ways that we have noted. – House (and electoral vote) apportionments are anywhere from

two (e.g., 1992) to ten years (e.g., 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 (and turnout varies considerably

from state to state).– 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.• Similar imperfections apply (in lesser or greater degree)

in all districted systems.

Perfect Apportionment (cont.)

• With perfect apportionment, the PVEV function looks essentially the same as a typical PVEV function.– It remains a step function and follows the

same S-curve form.

• See following to compare the actual and perfect apportionment PVEV functions for 1988.

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 a reversal of winners independent of apportionment effects.

• Here is the simplest possible example of distribution effects producing a reversal of winners in a simple and perfectly apportioned district system.There are 9 voters partitioned into 3 districts, and candidates D and R win

popular votes as follows: (R,R,D) (R,R,D) (D,D,D):

Popular Votes Electoral Votes

D 5 1

R 4 2

R’s votes are more efficiently distributed, so R wins a majority of electoral votes with a minority of electoral votes.

The 25%-75% Rule• What is the most extreme logically possible example of a

“wrong winner” in perfectly apportioned system?• One candidate or party wins just over 50% of the popular

votes in just over 50% of the (simple) districts or in complex 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+% of 50+%) of the popular vote and the other candidate with almost 75% of the popular vote loses the election.

• If the candidate or party with the favorable vote distri-bution is also favored by imperfect apportionment, a reversal of winners could be even more extreme.

Distribution Effects (cont.)• A proposal to “reform” the Electoral College that

was actually considered seriously in the 1950s was the “Lodge-Gossett Plan.”– The existing apportionment of electoral votes would

be maintained.– The office of elector would be abolished.– In each state, candidates would be awarded electoral

votes “exactly” proportional to their popular vote share in the state.

• Under this plan, the PVEV function would be (essentially) smooth and would generally follow the EV = PV line, but it would wander a bit from side to side.– Reversal of winners could still occur (favoring candi-

dates who do exceptionally well in small and/or low turnout states).

Apportionment vs. Distribution Effects in 1860

• The 1860 election was based on highly imper-fect 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 [the wrong winner interval of about 10% points] would have been even greater in the absence of the countervailing apportionment effects.


• Lincoln was the majority winner in all northern states except NJ, CA, and OR.

• Thus he also would have carried these states against a united opposition.

• These states together held a (modest) majority of the electoral votes.

• Lincoln carried many of these states (especially the more populous ones) by modest margins in the 50%-55% range.

• Lincoln received almost no votes in any southern (slave) states (and literally none in most of them).


• Thus the popular vote distribution closely approximated the 25%-75% pattern.

• Lincoln carried the northern states that held a bit more than half the electoral votes (and a larger majority of the [free] population), generally by modest popular vote margins.

• On the other hand, the anti-Lincoln opposition:– carried the 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.

Sterling Diagrams• 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 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.

• Rescale 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.

• 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 [or 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.