This is the same problem used to introduce the idea of Fair Division. The reason that the same example is appropriate here is that apportionment is a special type of fair division problem. In an apportionment problem the items to be divided are identical and indivisible but (and here's the twist) each participant may be entitled to a different percentage of the items (reminiscent of weighted voting). In fact, the most common (and perhaps most important) apportionment problems relate to government; and, in particular, deal with deciding on the number of representatives a state (or county, etc.) is entitled to in a legislative body. For a problem like this, a participant would be a state (or county, etc.) and the identical and indivisible items to be divided among the participants would be the seats in the legislative body.

For instance, the U.S. House of Representatives has 435 seats (the identical objects to be divided among the states) and states (the participants) are entitled to various numbers of these seats. (Alabama currently has 7 of the 435 seats.) The proof that this is, in fact, not a simple problem is found in the long and continual disputes over apportionment that have occurred in the United States since the first census in 1790.

A little History.

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