A majorization comparison of apportionment methods in proportional representation

Friedrich Pukelsheim (), Albert W. Marshall and Ingram Olkin
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Friedrich Pukelsheim: Institut für Mathematik, Universität Augsburg, D-86315 Augsburg, Germany
Albert W. Marshall: Statistics Department, University of British Columbia, Vancouver BG, V6T 1Z2, Canada
Ingram Olkin: Department of Statistics, Stanford University, Stanford CA, 94305-4065, USA

Social Choice and Welfare, 2002, vol. 19, issue 4, 885-900

Abstract: From the inception of the proportional representation movement it has been an issue whether larger parties are favored at the expense of smaller parties in one apportionment of seats as compared to another apportionment. A number of methods have been proposed and are used in countries with a proportional representation system. These apportionment methods exhibit a regularity of order, as discussed in the present paper, that captures the preferential treatment of larger versus smaller parties. This order, namely majorization, permits the comparison of seat allocations in two apportionments. For divisor methods, we show that one method is majorized by another method if and only if their signpost ratios are increasing. This criterion is satisfied for the divisor methods with power-mean rounding, and for the divisor methods with stationary rounding. Majorization places the five traditional apportionment methods in the order as they are known to favor larger parties over smaller parties: Adams, Dean, Hill, Webster, and Jefferson.

Date: 2002-10-09
Note: Received: 5 August 2000/Accepted: 24 October 2001
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