Chapter 15: Apportionment Part 6: Huntington-Hill Method


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Chapter 15: Apportionment


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



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.



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.



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.



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



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.



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.



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.



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.



Example: Huntington-Hill Method

  • Here, we are calculating q*, which is the geometric mean of the upper and lower quota for each state.



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.



Example: Huntington-Hill Method

  • Finally, we round based on the following rule:



Example: Huntington-Hill Method

  • We must add a seat because the initial apportionment sums to 74 when the total house size is 75.



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…



Example: Huntington-Hill Method

  • 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.




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