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.

Here are some fairness issues that might be raised: Is an apportionment method house monotone (i.e. avoids giving fewer seats to a state in a larger house)? 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.:

** No divisor methods guarantee giving each state its lower or upper quota. (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. 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.

1. Apportionment Systems
2. Fairness and Apportionment
3. Balinski and Young's Contribution
4. Where to Next?
5. References