A Statistics-Related Approach to the Legislative Apportionment Problem…and also, a Second One

Summary

We investigate global optimization methods in the Apportionment Problem, using objective functions which arise rather naturally from the field of Probability and Statistics. Minimizing the Chi-Square statistic (and also, minimizing the variance in seat-share per person) is equivalent to using Webster’s method. Meanwhile, finding a mode of the multinomial distribution (hence maximizing the P-value in an Exact Test) is equivalent to using Jefferson’s method.

Acknowledgments

Many thanks to the anonymous referees, and to the Editor, for all their work and careful attention in helping me to improve this article. Also, thanks to Karen Saxe and to James Swenson, who read some early drafts and kindly shared some very helpful comments. Finally, thank you to my colleague Pam Carriveau, for many enlightening conversations on the intersection of political science and mathematics, and also for a truly wonderful experience co-teaching an Honors Colloquium in Fall 2020, on the topic of Gerrymandering and Redistricting.

Notes

1 The reader is invited to consider [4] as an elegant and very promising approach to evaluating fairness.

2 Victor D’Hondt (1841–1902) studied parliamentary elections in which each party “should” win a number of seats proportional to the number of votes it receives. Replacing “party” with “state,” and “votes” with “residents,” this is exactly the Apportionment Problem—except we do not require that each party must win at least one seat.

Additional information

Notes on contributors

Dan Swenson

Dan Swenson ([email protected]) received a Ph.D. in mathematics from the University of Minnesota in 2009, and he is currently a Professor of Mathematics at Black Hills State University. He enjoys coffee and the occasional candy bar, and he thinks that voting in elections is a pretty good idea. Dan lives in Spearfish, South Dakota, with his wife Becky and their dog Bowie.