3. Balinski and Young's Contribution
In 1982, 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.
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