Resolving the Topological Social Choice Paradox

1.1. Topological Social Choice Theory. Social choice theory aims to understand the nature of, and provide methods for, the aggregation of individual preferences to yield a social preference which is fair and satisfactory to individual voters on an axiomatic ground. An important example of this is in popular elections. In the 1980s, Chichilnisky and colleagues developed a topological approach pioneered by Eckmann in the 1950s in [2] to model this situation. Here, the space of preferences may be modeled as a parafinite CW complex. These constitute a very large collection of spaces whose topology is very well-understood, and for example includes all manifolds (among many other classes of examples). In this framework, Chichilnisky and Heal proved in [1] (following Eckmann in [2]) a topological analogue of Arrow’s result, known as the Resolution Theorem. It was shown that contractibility of P , a very strong topological condition, is necessary and sufficient for there to exist a social aggregation rule P → P which is continuous and respects anonymity and unanimity (called a Chichilnisky map). The contractibility restriction is central to the topological theory of social aggregation, so we will explore and interpret contractibility in this context. The main idea is that it is impossible for a non-contractible preference spaces to have a tied (inconclusive) vote. The Resolution Theorem may be viewed as requiring that no ties are possible, where the definition of “tie” depends on the context. In particular, every possible preference profile has an aggregated social outcome.