Fair sharing under dichotomous preferences

In an important article on collective choice with dichotomous preferences, Bogomolnaia et al. (2005) propose a fairness criterion called fair welfare share. We argue that this criterion permits mechanisms that are not fair and we propose an alternative fairness concept called proportional sharing. It guarantees to each subgroup, such as a social class, gender or ethnic group, a fair probability that at least one of its members will like the final outcome. It is compatible with anonymity, neutrality, ex ante efficiency and strategy-proofness if and only if there are at most four agents or at most three possible outcomes.