Exploring Apportionment

Prepared by:

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

email:

malkevitch@york.cuny.edu

web page:

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

The "traditional" apportionment problem arises in two settings of interest to running a country in a democratic manner. In the United States the problem concerns how to translate the different populations of states into a positive integer number of seats for each state in the House of Representatives. In European and other democracies it is concerned with translating different votes in a general election for parties (e.g. social democrats, liberal, green, conservatives) into a positive integer number of seats in parliament. The mathematical essence of the problem is that there is some fixed positive integer number h, the house size, that is to be distributed to "claimants." Each claimant is to be awarded a non-negative (in some cases positive) portion of h where the portions given to all the claimants add to h.

Here are several problem instances with the hope that they will help you discover two appealing methods of apportionment "on your own."

Instance 1

Three states (parties) are vying for portions of a "parliament" of size h equal to 100 based on the "claims" shown. You can think of these claims as populations or votes. What is a fair way of apportioning the 100 seats:

P1 = 436; P2 = 323; P3 = 241

Instance 2

Three states (parties) are vying for portions of a "parliament" of size h equal to 100 based on the "claims" shown. You can think of these claims as populations or votes. What is a fair way of apportioning the 100 seats:

P1 = 456; P2 = 329; P3 = 215

Instance 3

Three states (parties) are vying for portions of a "parliament" of size h equal to 100 based on the "claims" shown. You can think of these claims as populations or votes. What is a fair way of apportioning the 100 seats:

P1 = 453; P2 = 342; P3 = 205

Repeat what you did above but this time do the calculations for a house size h of 10 (in contrast to house size 100).

Note:

If P denotes the sum of the numbers on which the finite number r of claims (P1, P2....,Pi,..,Pr) for the h seats are made, one can compute what might be called a fair share for claimant i as follows:

Fair share for claimant i is given by (Pi/P)h

Fair share for claimant i is given by Pi/(P/h)

Are these two numbers the same? Can you give simple verbal descriptions of Pi/P and P/h?

What skills traditionally taught in K-12 mathematics classes come into play in the concepts and algorithms to apportion h seats to claimants? The calculations above are made easier by the fact that h is either 100 or 10 and that the sum of the claims is a "simple" number to work with. In many situations the calculations are "sloppy" and require a calculator.