Multi-Attribute Proportional Representation

We consider the following problem in which a given number of items has to be chosen from a predefined set. Each item is described by a vector of attributes and for each attribute there is a desired distribution that the selected set should fit. We look for a set that fits as much as possible the desired distributions on all attributes. Examples of applications include choosing members of a representative committee, where candidates are described by attributes such as sex, age and profession, and where we look for a committee that for each attribute offers a certain representation, i.e., a single committee that contains a certain number of young and old people, certain number of men and women, certain number of people with different professions, etc. With a single attribute the problem boils down to the apportionment problem for party-list proportional representation systems (in such case the value of the single attribute is the political affiliation of a candidate). We study some properties of the associated subset selection rules, and address their computation.