Chapter 15: Apportionment

1. Chapter 15: Apportionment Part 5: Webster’s Method

2. Webster’s Method • Webster’s method is similar to Jefferson’s method. The first step is the same in both methods. • To understand the differences, we’ll need some notation: • Let <q> represent the integer obtained by rounding the real number q in “the usual way.” That is, if the decimal part of q is greater or equal to .5, then we round q up to the next greatest integer – otherwise, we round it down. • For example, <6.8> = 7 and <3.4446> = 3.

3. Webster’s Method • We begin Webster’s method by calculating the standard divisor, s, just as with Jefferson’s method. • Remember s = p/h where p = total population and h = house size. • Then we calculate each state’s quota: where pi = population of state i.

4. Webster’s Method • Once we find each state’s quota, we give that state an initial apportionment equal to <q>. That is, we’ll round the quota q to the integer <q>. • At this point, if the total apportionment we’ve assigned equals the house size then we are done. • However, we may have already assigned more than the available seats or there may be extra seats available that have not yet been assigned. • In either case (too many seats assigned or not enough) we will determine a modified divisor that yield the required total apportionment when rounding the normal way.

5. Example 1: Webster’s Method • Let’s determine the apportionment by the Webster Method for the fictional country introduced in a previous example: Suppose a country has 6 states with populations as given in the table. Also, suppose there are 250 seats in the house of representatives for this country. In this case, the standard divisor is 12500/250 = 50.

6. Example 1: Webster’s Method • Calculate each state’s quota, q. • Notice that rounding the normal way produces a total apportionment that is actually more than the available seats. • Therefore, we must find a modified divisor that will yield the desired total when rounding the normal way…

7. Example 1: Webster’s Method After some experimentation… We discover that a modified divisor of d=50.1 will work. Notice that to make the total a little lower, we needed to find a modified divisor a little larger than the standard divisor.

8. Example 1: Webster’s Method

9. Final answer Example 1: Webster’s Method

10. Example 2: Webster’s Method For another example, let’s consider a country with 3 states, named A, B and C. Suppose the populations of each state are as given below. Suppose that this country has a house of representatives with 75 seats. What is the standard divisor? The standard divisor is s = p/h = 1517/75 = 20.2267 q = pi/s that is, (state pop.)/(std. divisor) What is the quota for each state ?

11. Example 2: Webster’s Method For another example, let’s consider a country with 3 states, named A, B and C. Suppose the populations of each state are as given below. Notice the total sum of all values of q is the house size, but of course, we expect each state to get an integer number of seats. We now will use Webster’s Method to find the final apportionment.

12. Example 2: Webster’s Method For another example, let’s consider a country with 3 states, named A, B and C. Suppose the populations of each state are as given below. The initial apportionments are found by rounding the quota for each state in the usual way. However, as a result, we have one seat still to be assigned. Note that in the previous example we had to take a seat away. In this example, we will add a seat.

13. Example 2: Webster’s Method For another example, let’s consider a country with 3 states, named A, B and C. Suppose the populations of each state are as given below. Because we need to add a seat, we will search for a slightly smaller modified divisor. We were using a standard divisor of 1517/75 = 20.2267. After some experimentation, we discover that using a modified divisor of 20.15 will produce the desired total.

14. Example 2: Webster’s Method Final answer … The last column is the final result. The answer is A gets 22 seats, B gets 18 and C gets 35 seats.