Abstract
This chapter uses the Banzhaf power measure to calculate the a priori voting power of individual voters under the existing Electoral College system for electing the President of the United States, as well as under variants of this system in which electoral votes are either apportioned among the states in a different manner or cast by the states in a different manner. While the present winner-take-all manner of casting state electoral gives a substantial advantage to voters in the largest states, this advantage is diluted by the small-state advantage in apportionment. Moreover, most of the alternative Electoral College plans that have been proposed to remedy this large-state advantage give an equally substantial voting power advantage to voters in small states. Direct popular election of the President uniquely maximizes and equalizes individual voting power.
An earlier version of this chapter was published in Homo Oeconomicus: 26/3-4, 2009 (Essays in Honor of Hannu Nurmi, Vol. I, edited by Manfred J. Holler and Mika Widgrén).
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- 1.
Cited by Riker (1986).
- 2.
However, Congress has the power to change the size of the House without a constitutional amendment, so in principle it can increase or decrease the small-state advantage in electoral vote apportionment by decreasing or increasing the size of the House. For an analysis of how House size can influence the outcome of Presidential elections, see Neubauer et al. (2003).
- 3.
Computer Algorithms for Voting Power Analysis (http://www.warwick.ac.uk/~ecaae/). The calculations in this chapter used its ipgenf algorithm.
- 4.
This “theorem” is actually a conjecture that has been proved in important special cases and supported by a wider range of simulations; see Lindner and Machover (2004) and Chang et al. (2006). The number of voters need not be very large in order for the theorem statement to be true to good approximation. Indeed, given the provisional apportionment of 65 House seats among only 13 states that was the focus of Luther Martin’s objections, the Penrose Limit Theorem held to reasonable approximation (Virginia’s advantage then was roughly comparable to California’s today), so Martin’s complaint about the disproportionate voting power of large states, while theoretically insightful, was in the circumstances largely off-the-mark (but also see Footnote 8).
- 5.
Specifically, Banzhaf found that voters in New York (the largest state at the time of the 1960 census) had 3.312 times the voting power of voters in the District of Columbia; they had 2.973 times the voting power of voters in the least favored state (Maine). The maximum disparity resulted from the stipulation in the 23rd Amendment that the District cannot have more electoral votes than the least populous state. In the 1960 census, the District not only had a population larger than every state with 3 electoral votes but also larger than several states with 4 electoral votes. The District today has a smaller population than every state except Wyoming.
- 6.
This was taken to be 43.37 %, which is equal to the total popular vote for President in 2004 (122,294,000) as a percent of the U.S. population in 2000. A priori, we have no reason to expect that the percent of the population that is eligible to vote, or of eligible voters who actually do vote, varies by state (though, empirically and a posteriori, we know there is considerable variation in both respects). Using a different (fixed) percent of the population to determine the number of voters in each state would (slightly) affect the following estimates of absolute individual voting power but not comparisons across states or Electoral College variants.
- 7.
This probability can be derived from other values calculated and displayed by the ipgenf algorithm of Computer Algorithms for Voting Power Analysis.
- 8.
Had Luther Martin’s concern been the two-tier voting power of individual members of the House (rather than the voting power of state delegations) under the assumption of bloc voting by state delegations, his complaint that states should not have representation proportional to population would have been strongly supported by the theory of voting power measurement, because large-state members benefit from the (direct) Banzhaf Effect. Using the same example, under the Martin setup the Delaware and Michigan delegations have second-tier voting power of 0.008314 and 0.125606 respectively, so their members have two-tier voting power of 0.008314 and 0.026311 respectively, the latter being more than three times greater than the former.
- 9.
Many of these Electoral College variants have actually been proposed as constitutional amendments, while few if any amendments have proposed changes in the apportionment of electoral votes. For a review of proposed constitutional amendments pertaining to the Electoral College, see Peirce and Longley (1981), especially chapter 6 and Appendix L. However, provided the position of Presidential elector is retained, each state legislature is free to change its mode of casting electoral votes (or, more directly, its mode of selecting Presidential electors) and, as previously noted, Maine and Nebraska actually depart from the winner-take-all arrangement at the present time.
- 10.
Evidently most members of the Constitutional Convention expected that electors would be popularly elected in this manner. However, their Constitution left this matter up to individual states legislatures. Under the original Electoral College system, each elector cast two undifferentiated votes for President. The candidate with the most votes became President (provided he received votes from a majority of electors) and the runner-up became Vice-President. After the first two contested Presidential elections in 1796 and 1800, it was clear that this system could not accommodate elections in which two parties each ran a ticket with both a Presidential and Vice-Presidential nominee. Following the election of 1800, there was considerable consensus to change the manner of casting electoral votes so that each elector would cast one designated vote for President and one designated vote for Vice President, and this was accomplished by the Twelfth Amendment. Though early drafts included the requirement that electors be popularly elected in the manner of the Pure District Plan, this provision was ultimately dropped from the amendment; see Kuroda (1994).
- 11.
This is the system used at present by Maine (since 1972) and Nebraska (since 1992). The 2008 election for the first time produced a split electoral vote in Nebraska, where Obama carried one Congressional District; the Republican-dominated legislature may now switch state law back to winner-take-all. A proposed constitutional amendment (the Mundt-Coudert Plan) in the 1950s would have mandated the Modified District Plan for all states.
- 12.
A proposed constitutional amendment (the Lodge-Gossett Plan) along these lines was seriously considered in Congress in the late 1940s and 1950s. Since fractional electoral votes would be cast, the position of Presidential elector would necessarily be abolished, so this change can be effected only by constitutional amendment. Since minor candidates would presumably win (fractional) electoral votes, it becomes more likely that neither major candidate would win a majority of the electoral votes, so such an amendment would also have to specify what would happen in this event. (The Lodge-Gossett Plan would have elected the electoral-vote plurality winner, unless that candidate failed to receive at least 40 % of the electoral votes, in which case Congress voting by joint ballot would choose between the top two candidates ranked by electoral votes.).
- 13.
Since electoral votes would still be cast in whole numbers, the position of elector can be retained, and a state may use this formula unilaterally. Indeed, such a system was proposed in Colorado as initiative Proposition 36 in 2004. Since third candidates would be likely to win a few electoral votes (especially in large states), this system, if widely adopted, would throw more elections into the House.
- 14.
The principal purpose of such a plan is evidently to reduce the probability of an “election inversion” (Miller 2012) of the sort that occurred in 2000. The larger the national bonus, the more this probability is reduced. A bonus of 102 electoral votes has been most commonly discussed. (It would make sense, however, to make the bonus an odd number so as to preclude electoral vote ties.)
- 15.
Banzhaf III (1968) presented calculations for the Modified District Plan that ignored these interdependencies. Had he displayed the absolute voting power of voters in each state, it would have been evident that mean individual voting power under the district plan (as he calculated it) exceeded that under direct popular vote, which Felsenthal and Machover (1998, pp 58–59) show is an impossibility. However, his rescaled voting power values are quite close to those presented here.
- 16.
The simulation took place at the level of the 436 districts, not individual voters. For each Bernoulli election, the popular vote for the focal 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 0.5\( \sqrt{n} \) (i.e., the normal approximation of the symmetric Bernoulli distribution), where n is the number of voters in the district. (Of course, the other candidate won the residual vote.) The winner in each district was determined, the district votes in each state were added up to determine the state winner, and electoral votes are allocated accordingly.
- 17.
Even with the very large sample, 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, namely those that fell within 0.2 standard deviations of an exact tie. (In a standard normal distribution, the ordinate at ±0.2 × SDs from the mean is about 0.98 times that at the mean.) It needs to be acknowledged that Fig. 12 (and Figs. 15a and 15b for the National Bonus Plan) are not as accurate as other figures, as they entail some sampling error, some other approximations (including the one just noted), and possibly other errors.
- 18.
The Lodge-Gossett Plan proposed in the 1950s specified that candidates would be credited with fractional electoral votes to the nearest one-thousandth of an electoral vote. As proportionality becomes less refined, this system begins to resemble the Whole-Number Proportional System. The Pure Proportional Plan has recently been reinvented as the “Weighted Vote Shares” proposal of Barnett and Kaplan (2007). Combining a precisely proportional method of casting of electoral votes with a precisely proportional apportionment of electoral votes (as discussed earlier) would give every voter equal weight and would be equivalent to direct popular vote.
- 19.
Banzhaf III (1968) presented similar calculations based on similar, though less explicit, assumptions.
- 20.
Mean voting power under the Pure Proportional Plan (as calculated here) is 0.000072150172 versus 0.0000721502396 under direct popular vote.
- 21.
Colorado’s Proposition 36 had no explicit vote threshold but used a distinctly ad hoc apportionment formula that was overtly biased in favor of the leading candidate and against minor candidates.
- 22.
Given three or more candidates, simple rounding does not always work, because the rounded quotas may not add up to the required number of electoral votes—hence the “apportionment problem” definitively treated by Balinski and Young (1982).
- 23.
Similar calculations and chart were independently produced and published by Beisbart and Bovens (2008).
- 24.
Just as a statewide winner under the Modified District Plan must win at least one district, the national popular vote winner must win at least one state with at least 3 electoral votes; 533 is the smallest number B such B + 3 > 538−3.
- 25.
It would appear that Maine and Nebraska have been penalizing themselves in the same fashion for several decades, but the penalty for departing from winner-take-all is much less severe for smaller states. If Maine used the Pure District System instead of winner-take-all, the power of its voters would be cut approximately in half. Since it actually uses the Modified District Plan and is small enough that this system entails “winner-take-almost-all” (i.e., at least three of its four electoral votes), the actual reduction in voting power of Maine voters is less than this. (Another consequence of a Florida switch to districts would have been that—at least considering “mechanical” effects only—Gore would have been elected President in 2000, with no room for dispute and regardless of who won the statewide vote in Florida).
References
Balinski, M. L., & Young, H. P. (1982). Fair representation: meeting the ideal of one man, one vote. New Haven: Yale University Press.
Banzhaf III, J. F. (1968). One man, 3.312 votes: a mathematical analysis of the Electoral College. Villanova Law Review, 13, 304–332.
Barnett, A., & Kaplan, E. H. (2007). A cure for the Electoral College? Chance, 20, 6–9.
Beisbart, C., & Bovens, L. (2008). A power measure analysis of the amendment 36 in Colorado. Public Choice, 134, 231–246.
Chang, P.-L., Chua, C. H. V., & Machover, M. (2006). L.S. Penrose’s limit theorem: tests by simulation. Mathematical Social Sciences, 51, 90–106.
Felsenthal, D. S., & Machover, M. (1998). The measurement of voting power: theory and practice, problems and paradoxes. Cheltenham: Edward Elgar.
Gelman, A., Katz, J. N., & Tuerlinckx, F. (2002). The mathematics and statistics of voting power. Statistical Science, 17, 420–435.
Gelman, A., Katz, J. N., & Bafumi, J. (2004). Standard voting power indexes do not work: an empirical analysis. British Journal of Political Science, 34, 657–674.
Katz, J. N., German, A., & King, G. (2004). Empirically evaluating the Electoral College. In A. N. Crigler, M. R. Just, and E. J. McCaffrey (Eds.), Rethinking the vote. Oxford: Oxford University Press, pp. 75–88.
Kolpin, V. (2003). Voting power under uniform representation. Economics Bulletin, 4, 1–5.
Kuroda, T. (1994). The origins of the Twelfth Amendment: the Electoral College in the early republic, 1787–1804. Westport: Greenwood Press.
Lindner, I., & Machover, M. (2004). L.S. Penrose’s limit theorem: proof of some special cases. Mathematical Social Sciences, 47, 37–49.
Margolis, H. (1983). The Banzhaf fallacy. American Journal of Political Science, 27, 321–326.
Martin, L (1787). “The genuine information delivered to the legislature of the state of maryland relating to the proceeding of the general convention,” in M. Farand (Ed.), The records of the Federal Convention of 1787 (1937 rev. ed.), Vol. 3, New Haven: Yale University Press, pp. 172–232.
Miller, N. R. (2012). Election inversions by the U.S. Electoral college. In D.S. Felsenthal and M. Machover (Eds.), Electoral systems: assumptions, and procedures. Berlin: Springer, pp. 93–127.
Miller, N. R. (2013). “A priori voting power when one vote counts in two ways, with application to two variants of the U.S. Electoral College.” In R. Fara, D. Leech, and M. Salles, (Eds.), Voting power and procedures: essays in honour of Dan S. Felsenthal and Moshé Machover. Berlin: Springer.
Neubauer, M. G., & Zeitlin, J. (2003). Outcomes of presidential elections and House size. PS: Politics and Political Science, 36, 221–224.
Peirce, N. R., & Longley, L. D. (1981). The people’s president: the Electoral College in American history and the direct vote alternative. New Haven: Yale University Press. revised.
Penrose, L. S. (1946). The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109, 53–57.
Riker, W. H. (1986). The first power index. Social Choice and Welfare, 3, 293–295.
Acknowledgments
I thank Dan Felsenthal, Moshe Machover, and especially Claus Beisbart for very helpful criticisms and suggestions.
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Miller, N.R. (2013). A Priori Voting Power and the US Electoral College. In: Holler, M., Nurmi, H. (eds) Power, Voting, and Voting Power: 30 Years After. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35929-3_22
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