“So how do we elect our president?” Karen Saxe asked her audience at the MAA Carriage House on November 1. Not by popular vote, the Macalester College professor noted, but through a “rather eccentric Electoral College.”

In a timely talk titled “A Mathematical Adventure through the Census, Reapportionment, and Redistribution,” Saxe discussed how math and politics intersect as seats in the House of Representatives—and thus votes in the Electoral College—are allocated across and within states.

This allocation occurs via a three-step process. First, every 10 years, the census determines the populations of the states. Next, the 435 House seats are divvied up among the states based on these population figures. “And then after each state gets its seats,” Saxe explained, “the states have to be carved up into geographic areas, one for each seat.”

Although math is certainly involved in census taking, Saxe did not dwell on it. She mentioned that the U.S. Census is relatively costly to conduct—between $40 and $50 per person as opposed to about $1 per person in China and $0.40 in India—but quickly moved on to “the reapportionment problem.”

Each state has at least one seat in the House, with additional seats awarded in proportion to population. The exact number of seats that a state deserves based on its population is called its quota. In the original House of Representatives, for example, Virginia was entitled to 18.31 of the 105-seat total.

But, as Saxe observed, since you can’t have 18.31 people, “the question becomes: How should we round this number?”

It’s a tricky question. Since 1790, four different reapportionment methods have been used, and none has functioned in a way that quite squares with common sense.

The Hamilton Method, for instance, which Saxe said her students are often able to devise on their own, falls prey to the so-called Alabama paradox. A situation could—and did, in 1880—arise in which a state loses a seat, even though the only change is an increase in the number of seats to be apportioned.

The current reapportionment method, named after statistician Joseph Hill and mathematician Edward Huntington, uses geometric means and iterative calculation of priority numbers. Though the Hill-Huntington method is, according to Saxe, “considered the fairest method,” it can in extreme circumstances violate the quota rule. It is possible, in other words, for Hill-Huntington to award a state a number of seats other than what you’d get if you rounded its quota up or down.

“That’s a little bit upsetting,” Saxe admitted.

So why has an upsetting means of apportioning House seats been in use since 1941? Because, as Michel Balinksi and H. Peyton Young proved in 1982, no apportionment method can both observe the quota rule and steer clear of paradoxes.

Of course, even if a state has not gained or lost House seats, changes in population distribution within the state necessitate redistricting. In redrawing their congressional districts, states must observe three criteria. Districts must have equal populations, for one thing, and also contiguity.

“To mathematicians that means path-connectness,” Saxe said. Basically the district “isn’t in two bits.”

The final criterion—compactness—is less well defined, with different states employing different “compactness measures.” The Roeck measure of a district, for example, is equal to the area of the district divided by the area of the smallest enclosing circle, while the Polsby-Popper measure is the area of the district divided by the area of the circle with circumference equal to the district’s perimeter.

A good measure of compactness, Saxe argued, should take the shape of the state into consideration. It’s not fair to compare compactness measures between states with vastly different configurations, she said, pointing out that while Cape Cod pretty much necessitates a C-shaped district, it’s hard to imagine why Illinois District 4 should look like two lumps connected by a string.

In collaboration with her student Carl Corcoran, Saxe developed a compactness measure that considers both a state’s shape and the distribution of population within districts. It also discriminates in intuitively satisfying ways between districts awarded identical scores by other compactness measures.

Unfortunately, though, it’s not up to mathematicians to draw district maps. Legislatures do it in some states, while in others the task is left to independent commissions. Saxe herself served on a citizens’ commission that aimed to devise district maps through a “fair, open, and participatory process.”

Though the plan carefully developed by Saxe and 14 fellow Minnesotans was ultimately vetoed by the state’s governor, Saxe retains an interest in and sense of humor about the redistricting process. Her presentation’s final slide provided onlookers with links to a redistricting song and game, sure to both educate and amuse. *—Katharine Merow*

Listen to the full lecture (mp3)

Listen to an interview with Karen Saxe and MAA Director of Publications Ivars Peterson (mp3)

Watch a video of this lecture on MAA's YouTube Channel

Apportioning Representatives in the United States Congress

This MAA Distinguished Lecture was funded by the National Security Agency.