Condorcet's Paradox as Non-Orientability

TL;DR

This work reframes Condorcet's Paradox as a non-orientability problem in a topological model of three-alternative preferences, extending Baryshnikov's nerve-complex framework to capture strict orders and preference cycles. By constructing four canonical topological models corresponding to the four combinations of cycle validity and realisation, the authors show that contradictory cycles yield non-orientable surfaces, specifically the Klein Bottle or Real Projective Plane, while valid cycles correspond to orientable surfaces. They then reduce Arrow's Impossibility Theorem to an orientability question, linking social-choice impossibilities to topological properties. The approach illuminates deep connections between non-orientability, economic impossibilities, and logical paradoxes, and opens paths to extending the framework to more alternatives and indifference structures, as well as to related paradoxes in economics and logic.

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 reduce Arrow's Impossibility Theorem to a statement about the orientability of a surface. Furthermore, these results contribute to existing wide-ranging interest in the relationship between non-orientability, impossibility phenomena in Economics, and logical paradoxes more broadly.

Figures (14)

• Figure 1: Example of a simplicial complex
• Figure 2: Nerve complexes for families of sets $\{U_i\}_{i \in \{1,2,3\}}$ and $\{V_i\}_{i \in \{1,2,3\}}$ with different intersection patterns.
• Figure 3: The impossibility of orienting the Möbius Strip (left) and the Klein Bottle (right) non-orientability-normals
• Figure 4: Fundamental Polygons for various surfaces.
• Figure 5: Two surfaces homeomorphic to $S^1 \times [0,1]$, i.e., the product of a hollow circle and unit interval: the Annulus (left) annulus, and hollow Cylinder (right) cylinder

Theorems & Definitions (41)

• Definition 2.1.1
• Theorem 2.1.2: Arrow's Impossibility Theorem
• Definition 2.2.1
• Theorem 2.2.2
• Definition 3.1.1
• Proposition 3.1.2
• proof
• Definition 3.2.1
• Proposition 3.2.3
• proof