Since the Constitution indicates that each state should get at least one seat the authors of that document worried to some extent about state equity. Balinski and Young's book talk in detail about the trade-offs between different points of view about equity. In Europe, where there is an apportionment problem for allocating seats to parties in elections there are similar noisy details of "applied mathematics" mirroring the American problem of having to give each state one seat. This difficulty is that many countries will not give a party a seat unless the party gets at least a certain percentage of the popular vote. In Europe more attention seems to have been given than in this country to global discrete optimization approaches to solving apportionment problems.
A good reference for this point of view is: P. di Cortona, C. Manzi, A. Pennisi, F. Ricca, and B. Simeone, Evaluation and Optimization of Electoral Systems.
I agree with you that the concern for house monotonicity seems beside the point since by law the house size in the US has been fixed for a long time. There are a wide variety of population monotonicity assumptions and some of them seem reasonable. Thus, in deciding between a divisor method and largest remainders (Hamilton) one must decide which is more bothersome: violating quota or violating population monotonicity.
Cheers,
Joe
Many thanks to Joe Malkevich (Archive Message 10835) for the web reference (http://www.aps.org/apsnews/0401/040117.html) to Youngs very readable and useful summary paper on apportionment methods.Again - and as a caveat to some conclusions one might draw from the paper - there are various viewpoints on just which criteria and measures thereof are most important to optimize. For some of us, what counts is fairness to and among persons, more than to and among states. For me, the preferred apportionment should maximize, for ones chosen convex utility function, the sum over all persons of each persons utility value for her per-cap representation level. So, other things being equal, it is likely better to under-represent a few people (at a given level of per-cap representation) than to under-represent (at the same level) many people. Conventional criteria featured in Youngs paper, however, directly address the issue of fairness to and among states rather than to and among persons. (Typically, each state, or each pair of states, gets equal weight in an objective function to be maximized or minimized.) These criteria include house or population monotonicities, and lack of bias as between small vs large states.
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Joseph Malkevitch
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Phone: 718-262-2551
Web page: http://www.york.cuny.edu/~malk
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