Abstract
The President of the United States is elected, not by a direct national popular vote, but by a two-tier Electoral College system in which (in almost universal practice since the 1830s) separate state popular votes are aggregated by adding up state electoral votes awarded, on a winner-take-all basis, to the plurality winner in each state. Each state has electoral votes equal in number to its total representation in Congress and since 1964 the District of Columbia has three electoral votes. At the present time, there are 435 members of the House of Representatives and 100 Senators, so the total number of electoral votes is 538, with 270 required for election (with a 269–269 tie possible). The U.S. Electoral College is therefore a two-tier electoral system: individual voters cast votes in the lower-tier to choose between rival slates of ‘Presidential electors’ pledged to one or other Presidential candidate, and the winning elector slates then cast blocs of electoral votes for the candidate to whom they are pledged in the upper tier. The Electoral College therefore generates the kind of weighted voting system that invites analysis using one of the several measures of a priori voting power. With such a measure, we can determine whether and how much the power of voters may vary from state to state and how individual voting power may change under different variants of the Electoral College system.
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Notes
- 1.
The website may be found at http://www.warwick.ac.uk/~ecaae/.
- 2.
Such calculations reveal that the Modified District and Pure Proportional plans both give a substantial advantage to voters in small states (due to their advantage in the apportionment of electoral votes). The voting power implications of the Whole Number Proportional Plan are truly bizarre: voters in states with an even number of electoral votes have (essentially) zero voting power, while voters in states with an odd number of electoral votes have voting power as if electoral votes were equally apportioned among these states (Beisbart and Bovens 2008). All these findings are presented in Miller (2009) using charts similar to Fig. 1.
- 3.
Recalculation of Banzhaf’s (1968) results shows that the same anomaly existed under the 1960 apportionment of electoral votes.
- 4.
If the number of voters n is even (e.g., n = 100), the interpretation of a decisive vote differs somewhat according to whether the voting context is parliamentary or electoral. Under usual parliamentary rules, a tie vote defeats a motion, so voter i is decisive in any voting combination in which 50 other voters vote ‘yes’ and 49 vote ‘no,’ as the motion passes or fails depending on whether i votes ‘yes’ or ‘no.’ However, in elections between two candidates (our present concern), voting rules are typically neutral between the candidates, so a tie outcome might be decided by the flip of a coin. In this event, a voter i is “half decisive” in any voting combination in which 50 other voters vote for A and 49 for B (A wins if i votes for A and each candidate wins with 0.5 probability if i votes for B) and also in any voting combination in which 49 other voters vote for A and 50 for B. The upshot is that voter i’s total Banzhaf score (and voting power) is the same under either interpretation. Thus we can (and will) speak loosely “the probability of a tie vote” even when the number of other voters is even. More obviously, we can (and will) speak interchangeably between “the probability of voter i breaking what would otherwise be a tie vote” and “the probability of a tie vote” when the number of voters is large.
- 5.
However, with only nine voters, the Large At-Large System with a single vote that counts in two ways is effectively equivalent to the Pure At-Large System, because the candidate who wins the at-large vote must win at least one district and thus three out of five electoral votes.
- 6.
Taking the sum of the voting powers associated with each of the voter’s (district and at-large) votes may appear to double-count those voting combinations in Contingency 1 in which both of i’s two votes are doubly decisive, but at the same time it misses voting combinations in Contingency 1 in which neither vote by itself is doubly decisive but the two votes together are, and it turns out that these combinations exactly balance out (Beisbart 2007).
- 7.
The correlation between the number of uniform districts carried by a candidate and the candidate’s national popular vote is about +0.784. This degree of associations appears to be essentially constant regardless of the number of voters or districts, provided the latter is more than about 20 and the former is more than a thousand or so per district.
- 8.
As before, each of the 45 districts has 2,223 voters, a number selected so that both district and at-large vote ties may occur before focal voter i (in District 1) casts his vote and so that no ties occur after i has voted. The simulations, which are generated by SPSS syntax files, operate at the level of the district: the vote for candidate A in each district is a number drawn randomly from a normal distribution with a mean of 2,223/2 = 1,111.5 and a standard deviation of \( \sqrt{.25 \times 2223} \), i.e., the normal approximation to the binomial distribution with p = 0.5, and then rounded to the nearest integer.
- 9.
The vertical axis in Figs. 7 and 9a, b must show actual, rather than rescaled, voting power, because the voting power of the least favored voter varies as the bonus (at-large) component varies.
- 10.
Though this idea had been around earlier, it was most notably proposed by Schlesinger (2000) following the 2000 election. He proposed a national bonus of 102 electoral votes—two for each state plus the District of Columbia. However, given an even number (538) of ‘regular’ electoral votes, it would seem sensible to make the bonus an odd number in order to definitively eliminate the possibility of electoral vote ties (though ties would be far less likely than at present given any substantial nation bonus). It is clear that the motivation for a national bonus is to reduce the probability of election inversions, not to redistribute voting power.
- 11.
Given such a large electorate size, few if any elections were tied at the state or national level, so electoral vote distributions were taken from a somewhat wider band of elections, namely those that fell within 0.2 standard deviations of an precise tie in the state or national popular vote. (Random elections with many voters are very close, so the standard deviation is very small. Moreover, the ordinate of a normal curve at a standard score of ±0.2 is about 0.98 times that at a standard score of zero, so the density of elections is essentially constant in the neighborhood of a tie.)
- 12.
- 13.
With each vote counting the same way at the state and national levels, the national popular vote winner must win at least one state with at least three electoral votes, and 533 is the smallest number B such B + 3 > 0.5(538 + B).
- 14.
This system is used at present by Maine (since 1972) and Nebraska (since 1992). The 2008 election for the first time produced a split electoral vote in one of these states, namely Nebraska, where Obama carried one Congressional District. A proposed constitutional amendment (the Mundt-Coudert Plan) in the 1950s would have mandated the Modified District Plan for all states.
- 15.
Again these simulations were generated at the level of the 436 districts, not individual voters. For each random election, the popular vote for one candidate was generated in each Congressional District by drawing a random number from a normal distribution with a mean of n/2 and a standard deviation of \( \sqrt{.125n} \), where n is the number of voters in the district, i.e., the normal approximation to the Bernoulli distribution with p = 0.5. The winner in each district was determined, the district votes in each state were added up to determine the state winner, and electoral votes were allocated accordingly.
- 16.
Even given this very large sample of elections, the large electorate size meant that few elections were tied at the district or state level, so the relevant electoral vote distributions were taken from a somewhat wider band of elections, in this case those falling within about 0.1 standard deviations of an exact tie.
- 17.
Unlike those in Fig. 10b, the plotted points in Fig. 10a are subject to some sampling error (though its effects are probably almost invisible), as well as errors due to other approximations noted in the text. However, the most prominent apparent anomalies in Fig. 10a, where voters in a slightly more populous state (e.g., Rhode Island or Iowa) may have somewhat greater voting power than voters is slightly less populous states (e.g., Montana or Kansas) primarily reflect real discrepancies affecting voters in states with approximately similar populations that happen to fall on opposite sides of a threshold in the (whole-number) apportionment of electoral votes. For example, Rhode Island is the smallest state with four electoral votes, while Montana is the largest state with three electoral votes. (Such discrepancies are found in all Electoral College variants that apportion electoral votes into whole numbers.)
References
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Acknowledgements
This is a revised and expanded version of a paper on “Voting Power with District Plus At-Large Representation” presented at the 2008 Annual Meeting of the (U.S.) Public Choice Society, San Antonio, March 6–9, 2008. It also draws on material from “A Priori Voting Power and the U.S. Electoral College” Homo Oeconomicus, 26 (3/4, 2009): 341–380. I thank Dan Felsenthal, Moshé Machover, and especially Claus Beisbart for very helpful criticisms and suggestions on early stages of this work.
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Miller, N.R. (2014). A Priori Voting Power When One Vote Counts in Two Ways, with Application to Two Variants of the U.S. Electoral College. In: Fara, R., Leech, D., Salles, M. (eds) Voting Power and Procedures. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-319-05158-1_11
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