Wise man. Daniel Webster's method of apportioning seats was the best, says mathematician Peyton Young.

A House Divided

SAN FRANCISCO--Of all the uses of the U.S. census, the most important--to politicians--is the apportionment of congressional seats. Since 1941, Congress has assigned seats by a complicated formula that replaced a simpler approach designed in 1832 by the American statesman Daniel Webster. But Peyton Young, a senior fellow at the Brookings Institution in Washington, D.C., says Webster's system was actually better. The current method is biased, giving less populous states 3% to 4% more seats than they deserve, Young reported here on 17 February at the annual meeting of the American Association for the Advancement of Science (publisher of ScienceNOW).

Assigning House seats might seem simple: Calculate each state's fraction of the population, multiply by 435, round up the largest fractions to achieve a total of 435, and round down the rest. But no matter how you do the rounding, the simple arithmetic approach is susceptible to outcomes that mathematicians call paradoxical and politicians consider unacceptable. The most famous case occurred in 1880, when a decision to expand Congress from 299 to 300 seats caused the state of Alabama to lose a seat. Simple rounding can also cause states to lose seats when they gain population or gain seats by failing to count some of their citizens.

Webster's method avoids these paradoxes with a mathematical formula that involves multiplying the fractions by a tiny bit more than 435 and rounding to the nearest whole number. Its 1941 successor was supposed to be even better, because it minimizes differences in states' relative per capita representation. Many mathematicians, including the famous John von Neumann, endorsed the method.

But it introduced a bias, says Young, because it results in small numbers being rounded up more often than large numbers. The number 1.45 gets rounded up, for example, whereas 54.45 gets rounded down. Because every state gets two senators regardless of population, "the whole system is now rigged toward small states," Young says. He thinks Congress should switch back to Webster's method, and now is the time, he says, because the population happens to be aligned so that the new and old methods yield the same results.

Steven Brams, an expert on voting systems at New York University, doesn't foresee any surge of support for a change. But he agrees that Webster's scheme--or even the older, simple rounding method--"would be superior" to what's in place today.

Related sites
U.S. map showing the changes in apportionment based on the 2000 census
The Census Bureau's role in the apportionment process
Census Bureau information about computing apportionment
Peyton Young's home page at the Brookings Institution