A Generalization of Arrow's Impossibility Theorem Through Combinatorial Topology

Abstract: To the best of our knowledge, a complete characterization of the domains that escape the famous Arrow's impossibility theorem remains an open question. We believe that different ways of proving Arrovian theorems illuminate this problem. This paper presents a new combinatorial topology proof of Arrow's theorem. In PODC 2022, Rajsbaum and Ravent\'os-Pujol proved this theorem using a combinatorial topology approach. This approach uses simplicial complexes to represent the sets of profiles of preferences and that of single preferences. These complexes grow in dimension with the number of alternatives. This makes it difficult to think about the geometry of Arrow's theorem when there are (any) finite number of voters and alternatives. Rajsbaum and Ravent\'os-Pujol (2022) use their combinatorial topology approach only for the base case of two voters and three alternatives and then proceed by induction to prove the general version. The problem with this strategy is that it is unclear how to study domain restrictions in the general case by focusing on the base case and then using induction. Instead, the present article uses the two-dimensional structure of the high-dimensional simplicial complexes (formally, the $2$$\unicode{x2013}$skeleton), yielding a new combinatorial topology proof of this theorem. Moreover, we do not assume the unrestricted domain, but a domain restriction that we call the class of polarization and diversity over triples, which includes the unrestricted domain. By doing so, we obtain a new generalization of Arrow's theorem. This shows that the combinatorial topology approach can be used to study domain restrictions in high dimensions through the $2$$\unicode{x2013}$skeleton.