Chapter 15: Apportionment

1. Chapter 15: Apportionment Part 6: Huntington-Hill Method

2. Huntington-Hill Method • This method is similar to both the Jefferson and Webster Methods. The Huntington-Hill, Webster and Jefferson methods are all called “divisor methods” because of the way in which a critical divisor is used to determine the apportionment. • Like the other divisor methods, the Huntington-Hill method begins by determining a standard divisor and then calculating a quota for each state. • Next, in the Huntington-Hill method, instead of rounding the quota in the usual way, we round to get the initial apportionments in a way that is based on a calculation involving the geometric mean of two numbers. • Given two numbers a and b, the geometric mean of these numbers is

3. Geometric Mean • We can visualize the geometric mean in two ways. Here is the first way: • Suppose we have two numbers a and b. What is the geometric mean of these two numbers ? • The geometric mean of a and b is the length of the side of a square who area is equal to the area of a rectangle with sides a and b. a b

4. Geometric Mean • Here is another way to visualize the geometric mean of two numbers. • Suppose we have two numbers a and b. What is the geometric mean of these two numbers ? • The geometric mean of a and b is the length of the perpendicular segment connecting a point on the diameter of a circle with diameter length a + b to a point on that circle. a b

5. Geometric Mean • Here is another way to visualize the geometric mean of two numbers. • Suppose we have two numbers a and b. What is the geometric mean of these two numbers ? • The geometric mean of a and b is the length of the perpendicular segment connecting a point on the diameter of a circle with diameter length a + b to a point on that circle. Length is geometric mean of a and b a b

6. Huntington-Hill Method • To determine how to round q, we must calculate the geometric mean of each state’s upper and lower quota. If q is less than this geometric mean, we round down. If q is equal or greater than the geometric mean, we round up. • Let q* represent the geometric mean of the upper and lower quota of q. That is, • We define << q >> to be the result of rounding q using the geometric mean. • Thus

7. Huntington-Hill Method • Back to the Huntington-Hill Method … • First, we calculate the standard divisor. Then we calculate q, the initial apportionment for each state. • Next, we round q using the geometric mean method. • Then, we determine if seats must be added or removed to result in the desired apportionment. • If seats must be added or removed, we must choose a modified divisor, as in the Jefferson and Webster method, so that rounding the resulting quotas by the geometric-mean method will produce the required total.

8. Example: Huntington-Hill Method • Let’s use the fictional country from a previous example. Suppose this country has states A, B and C with populations as given in the table below. Suppose the house size is 75 seats.

9. Example: Huntington-Hill Method • Let’s use the fictional country from a previous example. Suppose this country has states A, B and C with populations as given in the table below. Suppose the house size is 75 seats. We calculate the standard divisor first… In this case, s = p/h = 1517/75 = 20.2267

10. Example: Huntington-Hill Method • Let’s use the fictional country from a previous example. Suppose this country has states A, B and C with populations as given in the table below. Suppose the house size is 75 seats. We must calculate the standard divisor first. In this case, s = p/h=1517/75 = 20.2267 Now we can calculate the quota for each state … q = pi/s

11. Example: Huntington-Hill Method • Here, we are calculating q*, which is the geometric mean of the upper and lower quota for each state.

12. Example: Huntington-Hill Method • Let’s use the fictional country from a previous example. Suppose this country has states A, B and C with populations as given in the table below. Suppose the house size is 75 seats.

13. Example: Huntington-Hill Method • Finally, we round based on the following rule: • If the quota q is greater or equal to q* then we round up • If the quota q is less than q* then we round down in each case we round down because in each case q < q*

14. Example: Huntington-Hill Method • We must add a seat because the initial apportionment sums to 74 when the total house size is 75.

15. Example: Huntington-Hill Method • We must add a seat because the initial apportionment sums to 74 when the total house size is 75. • With some experimentation, we find that a modified divisor of 20.15 will work…

16. Example: Huntington-Hill Method Answer • Here we use a modified divisor of md = 20.15. That produces modified quotas as shown above. These are compared with the geometric mean of the upper and lower modified quotas. • Notice that with state C we round up because q is larger than q*. • By chance, it happened that we got the same apportionment as we had using Webster’s method. Often, Webster’s method and the Huntington-Hill method will give the same result – but not always.