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Notes
If Jefferson’s method were the only method ever used for US House seat apportionment, it would have given at least one large state more than the ceiling of its quota in every apportionment since 1860.
Private communication.
We use “state” as a catch-all term that includes DC and territories such as Puerto Rico.
The differences between primary and caucus elections are not important in this article.
Roza also has data for the 2000 and 2012 Democratic primaries, but these were essentially one-candidate contests and thus of no mathematical interest. I was unable to find any usable data for primaries prior to 2000.
Sometimes the CD data must be estimated from county-level information provided by state parties or state secretary of state offices. I use the approximated data because it is the best I can do given the reluctance of state parties and state governments to share data at the district level.
In this context, by “election” I mean a vote distribution along with a number of delegates to apportion. Every CD in every state is a separate election, along with the two statewide elections, one for PLEO and one for at-large apportionment (the number of PLEO delegates does not equal the number of at-large delegates). I created Python code to process the election data.
The size of the House has been fixed at 435 by law since 1913. Prior to 1913, Congress would often change the size of the House after a census to keep pace with an increasing population.
This was the small racket used in early versions of badminton.
I am able to use more elections here because, in contrast to the elimination paradox, I don’t need vote information for eliminated candidates to see whether the Alabama paradox occurred.
One of the reasons this paradox occurs so rarely is that the Democrats use a 15% threshold, which produces elections with fewer candidates, which in turn decreases the likelihood of this paradox being observed. If we go through the Democratic database of elections and look for instances of the paradox when no threshold is used, we find 83 instances of the paradox (including the Iowa at-large example), which is still relatively small. As we increase the threshold from 0% to 15%, the number of instances of the Alabama paradox steadily declines.
Internationally, the population paradox has received a bit more attention. The German Bundestag discarded Hamilton’s method because of its susceptibility to this paradox, for example, and adopted a method that does not suffer from it (see [13]).
See [9] for an interesting investigation of this paradox and some nice ways to visualize it.
I can check more primaries here than the previous 183 because I don’t need any CD data.
References
M. Balinski and H. Young. Fair Representation: Meeting the Ideal of One Man, One Vote, 2nd ed. Brookings Institution Press, 2001.
M. Barrett and B. Popken. How the Iowa caucuses fell apart and tarnished the vote. NBC News, available at https://www.nbcnews.com/politics/2020-election/how-iowa-caucuses-fell-apart-tarnished-vote-n1140346. Accessed September 15, 2021.
Richard E. Berg-Andersson. The math behind the Democratic delegate allocation—2020. The Green Papers, available at https://www.thegreenpapers.com/P20/D-Alloc.phtml. Accessed September 16, 2021.
B. Bradberry. A geometric view of some apportionment paradoxes. Mathematics Magazine 65 (1992), 3–17.
S. Collinson. Vote-reporting mess leaves Iowa with no victor on caucus. CNN, available at https://www.cnn.com/2020/02/03/politics/election-day-in-iowa/index.html. Accessed September 15, 2021.
E. Huntington. The apportionment of representatives in Congress. Transactions of the American Mathematical Society 30:1 (1928), 85–110.
C. Jewitt. The Primary Rules: Parties, Voters, and Presidential Nominations. University of Michigan Press, 2019.
M. Jones, D. McCune, and J. Wilson. The elimination paradox: apportionment in the Democratic Party. Public Choice 178 (2019), 53–65.
M. Jones and J. Wilson. The geometry of adding up votes. Math Horizons 24:1 (2016), 5–9.
E. Lach. On the trail: making sense of the mess in Iowa. New Yorker, available at https://www.newyorker.com/news/campaign-chronicles/on-the-trail-making-sense-of-the-mess-in-iowa. Accessed September 15, 2021.
D. McCune, L. McCune, and D. Nelson. The cutoff paradox in the Kansas presidential caucuses. UMAP Journal 40:1 (2019), 21–45.
A. Prokop. How the Iowa caucus results will actually work—and why 2020’s could be more confusing than ever. Vox, available at https://www.vox.com/2020/1/30/21083701/iowa-caucuses-results-delegates-math. Accessed September 15, 2021.
F. Pukelsheim. Proportional Representation: Apportionment Methods and Their Applications, 2nd ed. Springer, 2017.
A. Robinson and D. Ullman. The Mathematics of Politics, 2nd ed. CRC Press, 2016.
T. Wright. From Cauchy–Schwarz to the House of Representatives: applications of Lagrange’s identity. Mathematics Magazine 94:4 (2021), 244–256.
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McCune, D. The Many Apportionment Paradoxes of the 2020 Iowa Democratic Presidential Caucuses. Math Intelligencer 45, 55–63 (2023). https://doi.org/10.1007/s00283-022-10196-9
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DOI: https://doi.org/10.1007/s00283-022-10196-9