Notes on Modeling: Session VIII: Apportionment

Prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

email: 

malkevitch@york.cuny.edu

web page: 

http://york.cuny.edu/~malk/



Although the apportionment problem from a certain point of view involves only lots of arithmetic, the details of the different algorithms and other aspects of this problem are very "technical."

As we have seen there are some over arching methods but the details of what is done differ a fair amount depending on whether one is working in the situation:

a. Every claimant no matter how small gets at least k seats (in the US, for apportionment after the census held every 10 years, k = 1; for some situations in Mexico, k is 2)

b. Claimants may not get any seats depending on the size of the claims involved.

My apologies for making a bit of a muddle about introducing names for the methods that address the complexities here.

Let us try to use this terminology in the future. D'Hondt is the divisor method (where the rounding rule is always round down) which proceeds on the assumption that a claimant need not be guaranteed a seat. I will try to use Jefferson to be the name of the method where we use the always round down divisor approach but must meet the restriction of giving a claimant at least k (usually 1) seat. Sometime the terminology on the class web page does not always adhere to this convention. When one applies the rounding down method (rather than the table method to implement either D'Hondt or Jefferson) the number of seats given out may not give one seat to each claimant. In this case the "Jefferson method" algorithm must be modified to deal with this situation. Rather than learn how to deal with modifying the approach we took to D'Hondt to solve the Jefferson version, we will note that in principle, when the rounding down rule approach does not yield each claimant one seat, one can always use the table approach to D'Hondt/Jefferson and get the desired apportionment (obeying the condition of giving each claimant one seat if that is required).

The table method and the rounding rule method (use a rounding rule, modifying the district size to allocate the proper number of seats if necessary) are not equally convenient, depending on the house size. For h small, the table method is usually more convenient, while for large values of h the rounding rule approach usually works better. However, one can always use the table rule if one wants to. When the Census Bureau has reported the apportionment results using Huntington-Hill they usually have shown the table method approach to the find the correct apportionment.