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Article Excerpt
The Adams apportionment method is the only divisor method that consistently favors small districts relative to the Hamilton method. The Jefferson method is the only divisor method that favors large districts relative to the Hamilton method. These statements hold for the Balinski and Young [3] interpretation of "favoring small/large districts," and for the partial "majorization ranking" introduced by Marshall et al. [7]. As such, Hamilton is positioned in between Adams and Jefferson.
Key words: apportionment; proportional representation; discrete allocation; divisor methods; majorization
MSC2000 subject classification: Primary: 90B80, 91B12; secondary: 91B14, 91F10
OR/MS subject classification: Primary: government; secondary: elections
1. Introduction. Apportionment methods are used to round the population proportions of electoral districts (or the vote proportions of parties) to integer numbers of seats in a representative body. The translation of almost continuous proportions into natural numbers nearly always involves adjusting. As the gain or loss of a single seat is usually considered important, the effect of the rounding process is a central issue in the theory of apportionment. In particular, different apportionment methods are tested on whether or not they systematically favor the smaller districts over the larger districts (or the larger over the smaller). There are at least two ways to study this phenomenon. First, one can study the seat bias, either by estimates from empirical data or by exact formulae derived from a probability model. Second, one can try to rank different methods according to the preferential treatment of smaller versus larger districts. Before we recall some of the literature on both tracks, we illustrate the following apportionment methods:
* Hamilton: a quota method with residual fit by the largest remainders
* Adams: the divisor method with rounding up
* Jefferson: the divisor method with rounding down
* Webster: the divisor method with standard rounding
These methods are properly defined below. For now, it suffices to see how they allocate 10 seats over the three-tuple (603, 249, 148). The total population amounts to 1,000. We list the outcomes in Table 1. Moving in this table from the left to the right, the small districts 2 and 3 each hand in one seat to the large district. The focus is on whether this example exposes fundamental properties of the apportionment methods.
Balinski and Young [3, Appendix A, [section] 5] use historical data to calculate the average percentage bias of some divisor methods. They observe that (i) the method of Adams favors the smaller districts the most; (ii) the Webster method is close to unbiased; and (iii) the Jefferson method has a plain bias toward the larger districts. Recently, Schuster et al. [8], Schwingenschlogl and Drton [9], and Drton and Schwingenschlogl [5] confirm and extend these results on the basis of exact formulae for the expected bias. With respect to the Hamilton method, Schuster et al. [8] expose that its seat bias is practically zero.
Concerning the second track in the literature, Balinski and Young [3, p. 118] formalize the idea of favoring small districts as follows. Method M' favors small districts relative to method M if moving from an M'-allocation toward an M-allocation; it cannot happen that simultaneously a smaller district loses seats and a larger district gains seats. In view of the results on seat bias, one might expect that Adams favors small districts relative to Webster and Jefferson, and that Webster favors small districts relative to Jefferson. As a matter of fact, these results appear in Balinski and Young [3, Proposition 5.1, p. 119].
Marshall et al. [7] observe that, in general, the Balinski-Young relation fails transitivity. They propose a weaker (and transitive) relation as follows. Consider an apportionment problem. First, order the districts from large to small. Denote a seat allocation proposed by Adams by ([a.sub.1], [a.sub.2], ..., [a.sub.s]) and an allocation proposed by Jefferson by ([b.sub.1], [b.sub.2], ..., [b.sub.s]). Then, they show that the following holds:
[a.sub.1] [less than or equal to] [b.sub.1], [a.sub.1] + [a.sub.2] [less than or equal to] [b.sub.1] + [b.sub.2], ... and [a.sub.1] + [a.sub.2] + ... + [a.sub.s-1] [less than or equal to] [b.sub.1] + [b.sub.2] + ... + [b.sub.s-1],
where s is the number of districts. The totals [a.sub.1] + [a.sub.2] + ... + [a.sub.s] and [b.sub.1] + [b.sub.2] + ... + [b.sub.s], of course, coincide. Marshall et al. [7] thus show that the Adams method is majorized by the Jefferson method. They use this majorization criterion to rank a number of divisor methods. The relationships between Adams, Webster, Jefferson, and other divisor methods obtained by Balinski and Young are recovered. (1)
Because the Hamilton apportionment method is not a divisor method, the above results are all silent about the positioning of the Hamilton method (and other quota methods) in these rankings. Still, the Hamilton method was and is today widely used. This paper attempts to fill the gap. Our first result reads:
PROPOSITION 1.1. (i) The method of Adams favors small districts relative to the method of Hamilton. (ii) The method of Hamilton favors small districts relative to the method of Jefferson.
Hence, in...
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