Social Choice and Welfare, 48 3 , — Faliszewski, P. Multiwinner voting: A new challenge for social choice theory. Endress Ed.

Kilgour, D. Approval balloting for multi-winner elections.

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Sanver Eds. Multi-winner voting.

## Article I, Section 2, Clause 3: Enumerated Power: Enumerated Clause

Estudios de Economia Applicada, 36 1 , — How to elect a representative committee using approval balloting. Machover Eds. Laslier, J. Handbook on approval voting. Liptak, A. Supreme Court avoids an answer on partisan gerrymandering. New York Times June Monroe, B. Fully proportional representation. American Political Science Review, 89 4 , — Potthoff, R. An underrated relic can modify divisor methods to prevent quota violation in proportional representation and U.

House apportionment. Representation, 50 2 , — Proportional representation: Broadening the options. Journal of Theoretical Politics, 10 2 , — Pukelsheim, F. Proportional representation: Apportionment methods and their applications. Sivarajan, S. A generalization of the minisum and minimax voting methods Preprint. Subiza, B. A consensual committee using approval balloting. Toplak, J. Temple Law Review, 81 1 , — Title Multiwinner approval voting: an apportionment approach. Authors: Steven J. However, there is also the perspective of the size of the difference relative to the size of the populations of the states involved.

## Washington as Land Speculator

Michigan is a much bigger state than Arkansas. In relative terms the Webster method resulted in a relative difference of This example should not mislead one into thinking that Huntington-Hill always does a better job. Measuring the absolute difference in representatives per person would be the basis for defending why the Webster apportionment is better. I use this example only to illustrate the distinction between relative and absolute difference ideas. Here is a summary of what Huntington discovered: a. For relative differences, Huntington-Hill is the optimal method for all the fairness measures about to be listed.

Recall that a i and P i are the number of seats for state i and its population, respectively.

For a fixed divisor method, the formula can be used to compare whether or not giving the next seat to state i or state j is more justified, as measured by the the given fairness formula. Again, it might not be obvious that these seemingly similar measures of absolute pairwise fairness would give rise to such different methods to achieve optimality. Yet Huntington showed that if one is concerned with relative differences, all the criteria formulas are optimal only for Huntington-Hill. In , two mathematicians, Michel Balinski and H. Peyton Young , published the very important book, Fair Representation: Meeting the Ideal of One Man, One Vote , in which they reported in detail on the history of the apportionment problem and described work of their own on the mathematics of the apportionment problem that had appeared in a variety of research papers.

This work built on the earlier work of Huntington but carried the mathematical theory of apportionment much further. In particular, they followed in the footsteps of Kenneth Arrow's work in understanding fairness in voting and elections by looking in detail at fairness issues growing out of apportionment problems. Specifically, they noted the tension between different views of the essential fairness questions.

These fairness questions take the form of stating various axioms or rules that an apportionment method should obey. Many of these issues are quite technical but an intuitive overview follows. There are now many variants of similar sounding axioms which differ in their details. Here are some fairness issues that might be raised: Is an apportionment method house monotone i. Does an apportionment method obey quota? Is an apportionment method biased in the sense that when used to decide many apportionment problems, it tends to be unfair to small or large states in a systematic way?

Is an apportionment method population monotone?

For example, in comparing the results of applying the same apportionment method to two consecutive censuses, could a state whose population went down get more seats than it did previously, while at the same time a state whose population went up lose seats? Does an apportionment avoid the new states paradox? They also examined the consequences of a state splitting into two states to get more seats. This is an important issue for the AP in the European context.

Balinski and Young showed that these fairness conditions do not mix well. Informally their results some of which were known to earlier researchers can be stated: If a method is well behaved with regard to changes in population, then it must be a divisor method rounding rule method. Balinski and Young reject the use of an ingenious method they developed referred to as the quota method.

This method, though it obeys quota and is house monotone, does not avoid the population paradox. In fact, no method which avoids the population paradox guarantees giving every state its lower or upper quota. Divisor methods are house monotone. Divisor methods rounding rule methods avoid paradoxical results when new states are added to the apportionment mix. Balinski and Young also call attention to the issue of bias of an apportionment method which involves the consequences of using this method time after time.

If a method tends to give more seats to large states or more seats to small states this might be deemed a strike against it.

## Marshall vs. Jefferson: Then and Now

The difficulty is arriving at either a theoretical or empirical framework for analyzing bias. The issues involved here are a classic example of the difficulties in the interface between theoretical results in mathematics and how they are applied. It is worthwhile to note that sometimes one can take advantage of the unfairness that mathematics shows is there, either from a theoretical or empirical point of view. For example, Jefferson's method known also as d'Hondt is clearly generous to large states. However, in the European democracy context, if a country uses d'Hondt, then parties which get relatively large votes are likely to get more than their fair share of seats.

This tendency, some believe, means trading stability to some extent for equity. If it is more likely that a single party gets a majority in parliament, or can more easily form a coalition of parties to govern, this may be better for society than having unstable coalitions form.

Coalitions with many partners may result in many changes of government, which may not be healthy in the long term. Political scientists have done a variety of empirical studies related to these issues. Mathematical and empirical studies of apportionment problems continue to receive intensive investigation. Due to perceived unfairness in the way seats have been apportioned or in which districts are created on the basis of census data, American courts continue to have to examine the issues involved.

Even though the Supreme Court upheld the constitutionality of the use of Huntington-Hill relatively recently , this does not mean that if new ideas show that Huntington-Hill is not the best choice available, that the Court will not in some future case conclude that Huntington-Hill is unconstitutional. The apportionment problem and its many relatives will no doubt continue to intrigue mathematicians for a long time to come.

Abeles, F. Dodgson and apportionment for proportional representation, Bull. History of Math. Anderson, M. Fienberg, Who Counts? Baily, W. Balinski, M.

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Young, The quota method of apportionment, Amer. Monthly 82 Applied Math. Monthly 84 Young, Stability, coalitions, and schisms in proportional representation schemes, Amer. Science Rev. Young, Criteria for proportional representation, Operations Research 27 Young, Quotatone apportionment methods, Math. Young, The Webster method of apportionment, Proc. Pollack, M. Rothkopf, and A. Barnett, eds. Young, ed. Bennett, S.