Published online 8 January 2008 | Nature | doi:10.1038/news.2008.419

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As the US campaign revs up, mathematicians debate how states should be represented.

How many seats should each US state get? It isn't simple... GETTY

With 53 seats in the US House of Representatives, California has long dominated congressional and electoral politics. Now mathematician Paul Edelman says that a much-needed rehaul of the way these seats are assigned would knock them down three notches.

Edelman, who is also a law professor at Vanderbilt University in Tennessee, is proposing a different method for apportionment, the process that divvies up the 435 congressional representatives based on state populations. He claims his method is fairer than the existing one because it comes closest to the "one person, one vote" ideal set forth by the US Supreme Court. The current method "takes no account of what the law has to say", argues Edelman, who outlined his method in a talk on 6 January at the Joint Mathematics Meetings in San Diego, California.

The electoral map is basically constructed by looking at population numbers that are tallied every decade by the US Census Bureau. A state's percentage of the national population should be reflected in its percentage of seats in the House. But this results in a lot of fractional numbers of seats that need to be cleaned up. Over US history, four main mathematical methods have been used to do this, all with various strengths and weaknesses.

One for you, one for me

The Hamilton method, used from 1850 until 1900, is the simplest. In this method, an 'ideal' district size is determined by dividing the US population by 435 (the number of seats). The state populations are then divided by this ideal size to find their deserved fraction of seats. In 2000, for example, California was entitled to a quota of 52.44 seats. The states are then ordered by the size of their fractional remainders. Those with the biggest remainder are the first to be rounded up and given an extra representative. Remaining seats are distributed, down the list, until all 435 seats are meted out.

The other methods round up or down without regard to rank. But this can easily result in a total of more or less than 435 seats. So then the 'ideal' district size is adjusted and the numbers re-crunched until the right number of seats comes out of the mix.

These methods — Jefferson, Webster and the current one, Huntington-Hill, which has been in effect since the 1940 census — use different rounding points. For example, the Huntington-Hill method rounds up or down from the geometric mean of the nearest integers (so if California deserves 52.44 seats it is rounded down, as the geometric mean of 52 and 53 is 52.4976). Since the geometric mean is proportionally larger for higher numbers, the current method has an inherent bias towards giving small states a boost up — something Edelman and others have criticized.

Minimally unfair

Edelman's method is instead designed to minimize the difference between the most over-represented state and the most under-represented one, in terms of the difference between the actual number of people per representative and the ideal number. This is done through an iterative process that evaluates 385 scenarios to find this minimum total deviation. He argues that this comes closest to matching the ideal of “one person, one vote”.

Using his method for populations in 2000, Montana, Delaware, South Dakota, Utah and Mississippi would each gain one seat; Texas, New York, Florida, Ohio and North Carolina would lose one; and California would lose three. “That could very well freak people out,” says Edelman.

The method is “interesting” but has a few problems, says Don Saari, director of the Institute for Mathematical Behavioral Sciences at the University of California at Irvine. The method has even more of a bias towards small states than the existing method, he argues. And it doesn't necessarily come up with unique solutions — there could be many ways to achieve equal 'unfairness'. And, like the current method, it can overshoot and adjust a state's seat up or down by more than one vote.

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Saari prefers the simplicity of the Hamilton method, which is the only one that ensures a state won’t be rounded up or down past the nearest integer. As the populations of big states grow and those of small ones shrink, Saari says, the over- and undershooting of the current method will become more pronounced — and certain states will probably complain. “I expect we’ll see challenge after challenge,” he says.

That probably won’t happen until the next census in 2010 – when the winners and losers under the new population counts become apparent. Then the bickering will begin, Edelman says. It's good to be prepared for that, he adds, with a good analysis of the various options available.