2. Excursions in Modern Mathematics, 7e: 4.6 - 2 Copyright © 2010 Pearson Education, Inc. 4 The Mathematics of Apportionment 4.1Apportionment Problems 4.2Hamilton’s Method and the Quota Rule 4.3 The Alabama and Other Paradoxes 4.4Jefferson’s Method 4.5Adam’s Method 4.6Webster’s Method

3. Excursions in Modern Mathematics, 7e: 4.6 - 3 Copyright © 2010 Pearson Education, Inc. What is the obvious compromise between rounding all the quotas down (like Jefferson) and rounding all the quotas up (like Adams)? What about conventional rounding? (round quotas down when fractional part is below 0.5 and up otherwise) Since we’re allowing modified divisors to manipulate the quotas, it’s possible to find a suitable divisor that will make conventional rounding work. This is the idea behind Webster’s method. Webster’s Method

4. Excursions in Modern Mathematics, 7e: 4.6 - 4 Copyright © 2010 Pearson Education, Inc. Step 1Find a “suitable” divisor D. Step 2Using D as the divisor, compute each state’s modified quota modified quota = state population/D Step 3Find the apportionment by rounding each modified quota the conventional way. WEBSTER’S METHOD

5. Excursions in Modern Mathematics, 7e: 4.6 - 5 Copyright © 2010 Pearson Education, Inc. Our first task is to find a suitable divisor D: Should it be more than the SD, or less? Use the standard quotas as a starting point. Rounding the standard quotas to the nearest integer yields a total of 251 (row 4 Table 4-16). This is too high (by one seat), which tells us that we should try a divisor D a bit larger than the standard divisor. We try D = 50,100. Example 4.10Parador’s Congress (Webster’s Method)

6. Excursions in Modern Mathematics, 7e: 4.6 - 6 Copyright © 2010 Pearson Education, Inc. Row 5 shows the modified quotas, and the last row shows these quotas rounded to the nearest integer. Now we have a valid apportionment! The last row shows the final apportionment under Webster’s method. Example 4.9Parador’s Congress (Webster’s Method)

7. Excursions in Modern Mathematics, 7e: 4.6 - 7 Copyright © 2010 Pearson Education, Inc. A flowchart illustrating how to find a suitable divisor D for Webster’s method using trial- and-error is on the next slide. The most significant difference when we use trial-and-error to implement Webster’s method as opposed to Jefferson’s method is the choice of the starting value of D. With Webster’s method we always start with the standard divisor SD. If we are lucky and SD happens to work, we are done! Webster’s Method

8. Excursions in Modern Mathematics, 7e: 4.6 - 8 Copyright © 2010 Pearson Education, Inc. Webster’s Method

9. Excursions in Modern Mathematics, 7e: 4.6 - 9 Copyright © 2010 Pearson Education, Inc. When the standard divisor works as a suitable divisor for Webster’s method, every state gets an apportionment that is within 0.5 of its standard quota. This is as good an apportionment as one can hope for. If the standard divisor doesn’t quite do the job, there will be at least one state with an apportionment that differs by more than 0.5 from its standard quota. Webster’s Method

10. Excursions in Modern Mathematics, 7e: 4.6 - 10 Copyright © 2010 Pearson Education, Inc. In general, Webster’s method tends to produce apportionments that don’t stray too far from the standard quotas. Occasional violations of quota rules occur but such violations are rare in real-life apportionments. Webster’s method has a lot going for it – it does not suffer from any paradoxes, and it shows no bias between small and large states. Webster’s Method

11. Excursions in Modern Mathematics, 7e: 4.6 - 11 Copyright © 2010 Pearson Education, Inc. Surprisingly, Webster’s method had a rather short tenure in the U.S.House of Representatives. It was used for the apportionment of 1842, then replaced by Hamilton’s method, then reintroduced for the apportionments of 1901, 1911, and 1931, and then replaced again by the (currently used) Huntington-Hill method. Webster’s Method