Null preference and the resolution of the topological social choice paradox

We investigate the role of contractibility in topological social choice theory. The Resolution Theorem states that there exists an aggregation map that is anonymous, unanimous, and continuous if and only if the space of individual preferences is contractible. Here, we turn a non-contractible space of social preferences (modeled as a CW complex) into a contractible space by adding the null preference which models full indifference of society, following the possibility results of Jones et al. (2003) which is based on a topology first considered but rejected by Le Breton and Uriarte (1990). We prove the corresponding extension of the Resolution Theorem by showing that the null preference as a social outcome precisely captures those voter profiles that represent a “tie” under a Chichilnisky map. Further, the space of these tie profiles is shown to have measure zero in the case that the space of individual preferences is a sphere in any dimension.