It is possible for a candidate to win more total popular votes than another candidate, yet win fewer total electoral votes and thus lose the election. In the 55 presidential elections from 1789 to 2004, this has happened twice — in 1888 and 2000. (In 1824 and 1876, the candidate with the most popular votes did not win, but these elections were eventually decided by select groups of voters.)

What is the minimum percent of the popular vote required to win the election? Using various data and methods, Froehle [4], Ingalls [6], Ruderman [13] and FairVote [2] found that for the 2000 election, the answer was 20 percent to 22 percent. Online, "jacob" [7] posed the problem for the 2004 election; he and various posters pegged it between 21 percent and 23 percent.

Some of the people involved in the preceding references used heuristics. Working separately, Ruderman [13] and Tom Moertel [7] used dynamic programming, and their methods may be non-heuristic. Moertel had a problem formulation very similar to the optimization program that I present later; I did not know about his formulation until July of this year. In 2004, "Wes" [13] considered the converse problem: What is the maximum percent of the popular vote that a candidate can win and lose the election? He notes that this is a knapsack problem and can be solved using dynamic programming.

My results agree with and extend the preceding results. I considered 11 electoral vote allocations, writing a linear optimization program for each one to minimize the number of popular votes required to win the election for all past elections. An advantage of optimization over heuristics is that optimization finds the exact answer, not an approximation.

I end this introduction with a similar problem that some people considered: What is the minimum number of states required to win? In 2004, Grey Hodge [13], after a discussion with Ruderman, said that a candidate needs to win only the 13 largest states. About a week later, "Chris" [6] said that a candidate needs to win the 10 most populous states. FairVote [2] says that 11 states were needed for the 2000 and 2004 elections. The last answer is correct. As Hodge, Ruderman and "Chris" allude, this problem can be solved using a greedy algorithm; a candidate needs to win the states with the most electoral votes. I do not consider this problem further.

Electoral Vote Allocations


Below, I describe the electoral vote allocations, along with a little history. I coded the optimization programs in AMPL ("A Math Programming Language") [3]. I give the program for only the winner-take-all allocation. Some of the other programs are more complicated. If you would like to see them, e-mail me.

I treated the District of Columbia as a state. I assumed that there were two candidates, that the entire population of a state voted and that the population of a state was what it was on the most recent census, in the same year as or before the election (but for the 1789 election, I used the 1790 census). Also, I ignored integrality of integer division (for example, when I divided the population of a state by its number of electors).

Winner-take-all. The candidate who wins the most popular votes in a state wins all the electoral votes for that state. All states except Maine and Nebraska use this allocation.

In the program, win[s] is a binary variable indicating whether the winning candidate wins states s. The constraint says that the number of electoral votes won by the winning candidate should be over half the number of total electoral votes. The objective function says that if the winning candidate wins a state s, he or she should win just over half the popular vote in state s, to minimize his total number of popular votes.

District. Each state has two types of electoral votes: two at-large electoral votes (corresponding to senators) and some non-at-large votes (corresponding to representatives). The two at-large electoral votes go to the candidate who wins the most popular votes. The candidate who wins the most votes in an (non-at-large) electoral district wins the (non-at-large) electoral vote for that district. Maine and Nebraska use this allocation. Turner [17] gives the results of the 1952-1996 elections if all states used the district allocation. Sagarin [14] does the same for the 1968-2000 elections. Madison, Hamilton, and Jefferson favored this allocation, and various states used it early in U.S. history [11, 12, 18]. Congressional proponents included Karl Mundt and Frederic Coudert [11, 5].

Equal-representation district. This allocation is like the district allocation, except that there are no at-large votes.

Sub-state. This allocation is like the district allocation, except that the at-large votes are treated the same as the non-at-large votes. Hamilton considered this allocation [1].

Proportional-real/integer. The electoral votes of a state go to the candidates in proportion to the statewide popular vote. The number of electoral votes a candidate won can be real or integer. Congressional proponents of the real version of this allocation (with a limit of three decimal places for electoral vote allocations) included Henry Cabot Lodge and Ed Gossett [16, 2], and Estes Kefauver and Price Daniel [5, 8].

Bonus. The candidate who wins the most nationwide popular votes wins a bonus number of electoral votes, equal to two times the number of states in the nation, with D.C. A majority of the larger total of electoral votes is required to win. Historian Arthur Schlesinger Jr. proposed this allocation in 1977 [15].

Humphrey-real/integer. This allocation is like the district allocation, except that the non-at-large electoral votes from all states form a set of nationwide electoral votes and go to the candidates in proportion to the nationwide popular vote. Senator Hubert Humphrey proposed this allocation in 1956 [5].

District-and-proportional-real/integer. This allocation is like the district allocation, except that the non-at-large electoral votes in each state go to the candidates in proportion to the statewide popular vote. I created this allocation.

In any election, an important value is 50 percent. Theorem 1 shows that another important value is 25 percent.

Theorem 1 (Benchmark). Consider a U.S.-style presidential election in a nation with a number of states, each with the same population, one electoral vote and a winner-take-all allocation. Assume that everybody votes. Then a candidate must win more than 25 percent of the popular vote to win the election.

Proof: The winning candidate must win more than half the popular votes in over half the states.

Results


Figure 1 shows the minimums of the optimization programs, expressed as a percentage of the total population. The electoral data is from Dave Leip's "Atlas of U.S. Presidential Elections" [9], and the population data is from the "2003 World Almanac" [10].

Figure 1:
(click here to view a larger version in a separate window)

The sub-state, district and winner-take-all allocations have nearly identical graphs. (The numerical data, not shown, for Figure 1 shows that for every year, the sub-state allocation is at most the district allocation, which in turn is at most the winner-take-all allocation.) These three allocations and the equal-representation-district allocation are below the benchmark value of 25 percent in Theorem 1. All other allocations are above 25 percent.

All allocations that involve some proportionality are above 30 percent and often above 35 percent.

For the district-and-proportional and proportional allocations, for every year, the real version has a higher minimum than the integer version. For the Humphrey allocation, the opposite is true.

The proportional-real allocation is the only allocation to always be above 40 percent. Its minimums are somewhat close to 50 percent. Intuitively, an explanation for this is that a direct popular vote is an extreme case of proportional-real allocation in which the entire nation consists of one state.



Winston C. Yang (winston@cs.wisc.edu) is a lecturer in math and computer science at the University of Wisconsin-Stout.

Acknowledgment I thank my former adviser Robert R. Meyer for reading drafts of this paper and for his comments.

1. Richard E. Berg-Andersson, 2001, "To the Spoils Go the Victor: Proposals and Prospects for Electoral College Reform," www.thegreenpapers.com/
   PCom/20010123-0.html, Jan. 23, 2001.
2. FairVote, 2008, "Does your vote matter?," www.fairvote.org/?page=971.
3. Robert Fourer, David M. Gay, and Brian W. Kernighan, 2004, "AMPL: A modeling language for mathematical programming," ampl.com.
4. Craig Froehle, 2008, "Our Electoral College System is Broken: A Numerical Illustration," craigfroehle.com/voting/.
5. Ralph M. Goldman, 1958, "Hubert Humphrey's S.J. 152: A New Proposal for Electoral College Reform," Journal of Political Science, February 1958, Vol. 2, No. 1, pp. 89 — 96.
6. Logan Ingalls, 2004, plutor.org/blog/2004/10/14/08.19.24/#comments, Oct 14.
7. jacob, 2004, "Micro-mini-pfc: Electoral-college-blaster!," www.hulver.com/scoop/story/2004/10/6/112537/693, Oct. 6.
8. George W. King, 1969, "Who Should Elect the President?," Publication #345, League of Women Voters of the United States.
9. David Leip, 2004, "Dave Leip's Atlas of U.S. Presidential Elections," www.uselectionatlas.org.
10. William A. McGeveran, 2003, "The World Almanac and Book of Facts 2003," World Almanac Books, New York, N.Y.
11. Mark McKenzie, 2004, "Will Congress Ever Reform the Alma Mater of Presidents? A Study of Systematic Biases Affecting Congressional Voting on Electoral College Reform," www.allacademic.com/meta/p_mla_apa_
    research_citation/0/8/3/2/3/pages83238/p83238-1.php, April 17.
12. Neal R. Peirce and Lawrence D. Longley, 1981, "The People's President (2nd edition), Simon & Schuster, New York.
13. Jesse Ruderman, 2004, "Winning an election with 22% of the popular vote," www.squarefree.com/archives/000621.html#more, Nov. 1.
14. Jeff Sagarin, 2004, "Maine-Nebraska 'District Plan' Electoral College Vote Method Applied to Entire Country," www.kiva.net/—jsagarin/sports/electoral.htm.
15. Arthur Schlesinger, 2002, "Not the People's Choice," www.prospect.org/cs/articles?article=not the peoples choice, March 25.
16. Ruth C. Silva, 1950, "The Lodge-Gossett Resolution: A Critical Analysis," The American Political Science Review, Vol. 44 (March 1950), No. 1, pp. 86 — 99.
17. Robert C. Turner, 2005, "Do Nebraska and Maine Have the Right Idea? The Political and Partisan Implications of the District System," Presidential Studies Quarterly, Vol. 35 (March 2005), No. 1, pp. 116 — 137.
18. Lucius Wilmerding, 1949, "Reform of the Electoral System," Political Science Quarterly, Vol. 64 (March 1949), No. 1, pp. 1 — 23.

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