Condorcet's Paradox as Non-Orientability
For social choice theorists, this provides a novel topological characterization of preference cycles and a restatement of Arrow's Theorem, though the results are incremental extensions of existing topological models.
This paper introduces a topological framework for modeling preference cycles, showing that Condorcet's Paradox corresponds to the non-orientability of surfaces like the Klein Bottle or Real Projective Plane, and restates Arrow's Impossibility Theorem in terms of orientability.
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.