dorsal/arxiv
View SchemaCondorcet's Paradox as Non-Orientability
| Authors | Ori Livson, Siddharth Pritam, Mikhail Prokopenko |
|---|---|
| Categories | |
| ArXiv ID | 2601.07283vv1 |
| URL | https://arxiv.org/abs/2601.07283 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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.
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"abstract": "Preference cycles are prevalent in problems of decision-making, and are contradictory when preferences are assumed to be transitive. This contradiction underlies Condorcet\u0027s 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\u0027s existing topological model of strict, ordinal preferences on 3 alternatives. In our framework, the contradiction underlying Condorcet\u0027s 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\u0027s 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.",
"arxiv_id": "2601.07283",
"authors": [
"Ori Livson",
"Siddharth Pritam",
"Mikhail Prokopenko"
],
"categories": [
"math.AT",
"cs.GT",
"econ.TH"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Condorcet\u0027s Paradox as Non-Orientability",
"url": "https://arxiv.org/abs/2601.07283",
"version": "v1"
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