Combinatorial topology and Arrovian model

One of the main problems in Social Choice and decision-making is finding good rules aggregating individual preferences into a single social preference representing the whole society.
Arrow in 1951 [1] proved that it is not possible to make such good aggregations in the universal domain of rankings satisfying the Paretian property, the independence of irrelevant alternatives and non-dictatorial when there are at least three alternatives.

There is much literature studying how we can weak the required properties to obtain good rules such as restricting domains or using other types of orders as preferences.

However, there is a parallel approach consisting on looking for alternative models resembling the Arrovian one to get the desired rules. One of them are the fuzzy Arrovian models treated in a subsequent section.

My contributions on the study of the classic Arrovian model relies on the study of the domain restrictions. That is, for some non-universal domains of individual preferences, aggregation rules satisfying the remaining properties exist. In spite of the large literature studying some of these domains, there is not still a characterization of the domains allowing aggregation neither a deep understanding about which characterizes of the domains are related to the solution.

Sergio Rajsbaum and I though that a geometric or topological study could shed some light on this problem. We took the approach developed by Baryshnikov [2], but we did not apply homology techniques as Baryshnikov, instead, we applied combinatorial topology.

Baryshnikov approach consists on translating the space of profiles and social preferences to simplicial complexes (a type of topological space build by vertices, edges, triangles, tetrahedron, etc), and aggregation functions to simplicial maps. We used simple techniques as the \emph{index lemma}, or simple topological properties as connectedness, to characterize the domains in which is possible to aggregate (in the base case $n=2$ and $|X|=3$), and give new original proofs of Arrow’s theorem.

The importance of this research line relies on using combinatorial topology in social choice, as far as I know, for first time. It allows us to use geometry as well as the usual combinatorics on solving these models: A good example is the first sistematic domain characterization in therms of aggregation. Even if it is a result for the first case, it gives intuition on possible generalizations. Moreover, these type of topological of techniques are more intuitive than the homological ones as well as they allow computational implementations that in the future will be useful to practitioners.