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Section 12.3 Webster's Method

Daniel Webster (1782-1852) proposed a method similar to Jefferson’s in 1832. It was adopted by Congress in 1842, but replaced by Hamilton’s method in 1852. It was then adopted again in 1901. The difference is that Webster rounds the quotas to the nearest whole number rather than dropping the decimal parts. If that doesn’t produce the desired results at the beginning, he says, like Jefferson, to adjust the divisor until it does. (In Jefferson’s case, at least the first adjustment will always be to make the divisor smaller. That is not always the case with Webster’s method.)

Definition 12.3.1. Webster’s Method.

  1. Determine how many people each representative should represent. Do this by dividing the total population of all the states by the total number of representatives. This answer is called the standard divisor.

  2. Divide each state’s population by the divisor to determine how many representatives it should have. Record this answer to several decimal places. This answer is called the quota.

  3. Round all the quotas to the nearest whole number (but don’t forget what the decimals were). Add up the remaining whole numbers.

  4. If the total from the previous step was less than the total number of representatives, reduce the divisor and recalculate the quota and allocation. If the total from the previous step was larger than the total number of representatives, increase the divisor and recalculate the quota and allocation. Continue doing this until the total in the previous step is equal to the total number of representatives. The divisor we end up using is called the modified divisor or adjusted divisor.

Example 12.3.2.

Again, consider Delaware, with an initial divisor of \(21,900.82927\text{.}\) Use Webster's method to appotion the 41 state representatives.

Table 12.3.3.
County Population
Kent 162,310
New Castle 538,479
Sussex 197,145
Total 897,934
Solution.
Table 12.3.4.
County Population Quota Initial
Kent 162,310 7.4111 7
New Castle 538,479 24.5872 25
Sussex 197,145 9.0017 9
Total 897,934 41

This gives the required total, so we’re done.

Example 12.3.5.

Again, consider Rhode Island, with an initial divisor of \(14,034.22667\text{.}\) Use Webster's method to apportion the 75 representatives.

Table 12.3.6.
County Population
Bristol 49,875
Kent 166,158
Newport 82,888
Providence 626,667
Washington 126,979
Total 1,052,567
Solution.
Table 12.3.7.
County Population Quota Initial
Bristol 49,875 3.5538 4
Kent 166,158 11.8395 12
Newport 82,888 5.9061 6
Providence 626,667 44.6528 45
Washington 126,979 9.0478 9
Total 1,052,567 76

This is too many, so we need to increase the divisor. Let’s try \(14,100\text{.}\)

Table 12.3.8.
County Population Quota Initial
Bristol 49,875 3.5372 4
Kent 166,158 11.7843 12
Newport 82,888 5.8786 6
Providence 626,667 5.8786 44
Washington 126,979 9.0056 9
Total 1,052,567 75

This works, so we’re done.

Like Jefferson’s method, Webster’s method carries a bias in favor of states with large populations, but rounding the quotas to the nearest whole number greatly reduces this bias. (Notice that Providence County, the largest, is the one that gets a representative trimmed because of the increased quota.) Also like Jefferson’s method, Webster’s method does not always follow the quota rule, but it follows the quota rule much more often than Jefferson’s method does. (In fact, if Webster’s method had been applied to every apportionment of Congress in all of American history, it would have followed the quota rule every single time.)

In 1980, two mathematicians, Peyton Young and Mike Balinski, proved what we now call the Balinski-Young Impossibility Theorem.

Definition 12.3.9. Balinski-Young Impossibility Theorem.

The Balinski-Young Impossibility Theorem shows that any apportionment method which always follows the quota rule will be subject to the possibility of paradoxes like the Alabama, New States, or Population paradoxes. In other words, we can choose a method that avoids those paradoxes, but only if we are willing to give up the guarantee of following the quota rule.