Condorcet's Paradox as Non-Orientability

TL;DR

This paper models preference cycles in social choice using topology, showing that Condorcet's Paradox relates to non-orientability of certain surfaces, offering a new geometric perspective on decision-making contradictions.

Contribution

It introduces a topological framework that generalizes existing models and links preference cycles to surface non-orientability, extending the understanding of social choice paradoxes.

Findings

Preference cycles correspond to non-orientable surfaces like the Klein Bottle.

Restates Arrow's Impossibility Theorem in terms of surface orientability.

Provides a topological interpretation of Condorcet's Paradox.

Abstract

Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet's Paradox, a pioneering result of Social Choice Theory, wherein intuitive and seemingly desirable constraints on decision-making necessarily lead to contradictory preference cycles. Topological methods have since broadened Social Choice Theory and elucidated existing results. However, characterisations of preference cycles in Topological Social Choice Theory are lacking. In this paper, we address this gap by introducing a framework for topologically modelling preference cycles that generalises Baryshnikov's existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet's Paradox topologically corresponds to the non-orientability of a surface homeomorphic to either the Klein Bottle or Real Projective Plane, depending on how preference cycles are represented. These findings allow us to restate Arrow's Impossibility Theorem in terms of the orientability of a surface as well.