Abstract
When considering the probability of voters being decisive in presidential elections, public choice economists typically proceed as if the probability can reasonably be approximated by assuming that the winner is determined by simple majority vote. It is well known the Electoral College can cause presidential candidates to lose elections despite winning the popular vote (an election inversion). But the Electoral College’s ability to increase the probability of some voters being decisive effectively has been ignored, despite such increases having occurred in 10 of the last 49 presidential elections. By examining the influence of the Electoral College in most of the presidential elections from 1824 to 2016, I explain and give examples of how the probabilities of some voters being decisive were elevated above what they would have been under majority rule, with the increase being truly astounding in several cases. The examples do not weaken the importance of expressive voting, but our understanding of such voting is improved by considering the probability effects of the Electoral College on voter decisiveness.
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Notes
The voter typically wants the outcome he or she votes for, but the only benefit realized with certainty from voting is the emotional satisfaction of expressing support for or against a candidate and/or the policies he or she advocates. However, expressive voting also can explain why voters sometimes vote for outcomes they would not choose if their decisions at the polls were decisive. See Tullock (1971).
Technically, the conclusion requires an odd number of voters. Elections featuring an even number of voters obviously can end in ties, and in that event cannot be decided by one vote and no voter can be individually decisive in determining the winner. I shall ignore that problem, as is typically done in expressive voting models.
For an early discussion of the use of the binomial distribution in voting models, see Beck (1975). Because of the limited number of probabilities in Table 1, throughout the paper I shall use a spreadsheet containing the formula (4–6) in Brennan and Lomasky (1993, p. 56) to calculate the probabilities of tied elections. See the “Appendix”.
Brennan and Lomasky adopt the lower turnout number because they are considering presidential elections with voter turnouts ranging from 47.5 million to 74.1 million.
There was at least one decisive state in 16 of the 49 presidential elections from 1824 to 2016, with multiple decisive states in some of them. I consider only elections from 1824 to 2016 because in the first nine presidential elections few states chose electors by popular vote, and totals are difficult to obtain even when they did. See McLaughlin (2019). I read the McLaughlin paper after I had struggled with some frustrating attempts to get started on this paper. His emphasis on the importance of decisive states in understanding the Electoral College convinced me to take an ex post approach that started with those states. McLaughlin does not consider the influence of the Electoral College on the probability of decisive voters, but without his paper I doubt I would have finished this paper.
This is a common situation despite fewer voters in state than national elections. The reason is that party preferences are commonly more evenly spread over the entire country than over many states, with the vote gap likely being much smaller in national than in state elections. .
Even though the Electoral College increases the probability of decisiveness of voters in only some states in only some presidential elections, the increases can be, and have been, so large that they increased significantly the average probability of voter decisiveness. See footnotes 11 and 14.
The magnitude of the increase is not always clear, even approximately, because moderately sized vote gaps can prevent the calculation of numerical estimates of tied elections.
No one seriously doubts that Gore won the popular vote in the national election. But, given how close the election was, it is useful to ask the counter-factual question: Would Bush have won more national votes than Gore if the Electoral College had not existed? That question cannot be answered, but it makes us consider the fact that the Electoral College affects how presidential candidates campaign. Without the Electoral College Bush would have campaigned differently, as Gore also would have, by concentrating on getting more popular votes instead of more electoral votes. See Silver (2011) and Stromberg (2008).
I occasionally cite the probability of a tie for the largest number of voters for which a numerical approximation is provided by a spreadsheet I relied on, which in this case is only 50 million for the national election instead of the actual number of 100 million. That difference obviously results in a large overstatement of the probability of the national election being tied.
The number is almost exactly four times larger than the probability of a tied election with 100 million voters turning out and a vote gap of zero, which is 1/(12,500). Also, since Florida’s voters represented approximately 5% of the voters nationwide, its added decisiveness alone (ignoring the increase in New Hampshire) would have raised the average probability of decisiveness for all votes to roughly 1/(63,000). The largest probability of a tied national presidential election was 1/320,000 in 1880, when the number of national voters was only 9 million. This election is discussed in Sect. 5. I am grateful to an anonymous referee for suggesting those comparisons.
I ignore the 1824 election as irrelevant in that regard because the voting pattern that led to 1824’s outcome contrasts starkly with the voting patterns that led to the election inversions in 1876, 1888, 2000 and 2016. Miller (2012, p. 16) doesn’t include the 1824 election in his list of inverted elections. The 2016 election didn’t result in the Electoral College increasing the probability of decisive voters, at least not that I could be sure of, but, as I shall argue, it came very close to doing so.
See Sect. 5 for examples of national elections being too one-sided or too close for the Electoral College to increase the probability of decisive voters.
The probability 1/(2500) is significantly larger than 1/(4000), which is the probability of the national election being tied with a vote gap of zero and 10 million voters. Also, since New York was home to slightly more than 10% of the national voters, the state’s 1/(2500) probability of voter decisiveness would have increased the average probability of decisiveness of all votes in the country to approximately 1/(25,000).
See Table 7.1 and Brennan and Lomasky’s (1993, p. 119) discussion.
Gelman et al. (2012) consider the Electoral College as an influence on the probability of voter decisiveness in an article on a prediction they made about the 2008 US presidential election. That article predicted far smaller differences between groups of states in the probability of voter decisiveness than I find from ex post evidence. In summarizing their forecast, Gelman et al. (2012, pp. 323–324) state that “we estimate the probability of a single vote being decisive as, at most, about 1 in 10 million in a few states near the national median. Averaging these probabilities over all the states and weighing by turnout yield a 1 in 60 million chance that a randomly selected voter would be decisive.” While recognizing that theirs was a forecast, it is worth pointing out that no states were decisive in the 2008 election, so the Electoral College could not have affected the probability of decisive voters. In addition, the popular vote gap between Barack Obama and John McCain was 7.2% out of more than 129 million votes cast, driving the probability of a tie, given the lack of any influence by the Electoral College, deep into the 1/∞ category. Although the authors mention the importance of decisive states in their general discussion, nowhere do they mention that the probability of a tie in a decisive state election must exceed the probability of a tie in the national election if the Electoral College is to have any effect on voter decisiveness. This is probably all explained by the fact that they were making a forecast, a very different exercise than I have pursued here.
Consider that the eight states and the District of Columbia with three electoral votes can largely cancel each other out, and commonly do.
For example, we hear and obsess about income differences because a few people are many times wealthier than most of us, with little consideration given to the general advantages we all realize from the process that makes those differences possible.
Reconstruction was soon ended anyway because of an agreement between Republicans and Democrats that led to Hayes’ election and to the Jim Crow era in the South.
See Visser (2020).
References
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Acknowledgements
I want to thank Danny Hughes, Professor of Economics at the Georgia Institute of Technology, for programming a spreadsheet to approximate the probabilities of elections being tied and instructing me on its use. Also, an anonymous referee deserves my gratitude for helpful comments.
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Appendix
Appendix
The formula used to compute the probabilities and programmed into the spreadsheet is given by
where
t = The decimal version of the vote gap divided by 2.
For example, if the vote gap is 0.02 then t is equal to 0.0002/2 or 0.0001.
See formula (4.6) in Brennan and Lomasky (1993, p. 56).
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Lee, D.R. Ignoring the Electoral College: why public choice economists understate the probability of decisive voters. Public Choice 187, 439–454 (2021). https://doi.org/10.1007/s11127-020-00787-5
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DOI: https://doi.org/10.1007/s11127-020-00787-5