Choosing subsets: a size-independent probabilistic model and the quest for a social welfare ordering

``Subset voting'' denotes a choice situation where one ®xed set of choice alternatives (candidates, products) is o€ered to a group of decision makers, each of whom is requested to pick a subset containing any number of alternatives. In the context of subset voting we merge three choice paradigms, ``approval voting`` from political science, the ``weak utility model'' from mathematical psychology, and ``social welfare orderings'' from social choice theory. We use a probabilistic choice model proposed by Falmagne and Regenwetter (1996) built upon the notion that each voter has a personal ranking of the alternatives and chooses a subset at the top of the ranking. Using an extension of Sen's (1966) theorem about value restriction, we provide necessary and sucient conditions for this empirically testable choice model to yield a social welfare ordering. Furthermore, we develop a method to compute Borda scores and Condorcet winners from subset choice probabilities. The technique is illustrated on an election of the Mathematical Association of America (Brams, 1988).