Arrow's Impossibility Theorem as a Generalisation of Condorcet's Paradox

Abstract

Arrow's Impossibility Theorem is a seminal result of Social Choice Theory that demonstrates the impossibility of ranked-choice decision-making processes to jointly satisfy a number of intuitive and seemingly desirable constraints, including that aggregated preferences are transitive. The theorem is often described as a generalisation of Condorcet's Paradox, wherein pairwise majority voting may fail to jointly satisfy the same constraints due to the occurrence of elections that result in intransitive preference cycles. In the strict preference case, i.e., where indifference between alternatives is not allowed, D'Antoni's formulation of Arrow's Impossibility Theorem makes this relationship explicit by constructing aggregations with preference cycles given Arrow's constraints other than transitivity. In this paper, we generalise D'Antoni's methodology to apply to Arrow's Impossibility Theorem in full (i.e., accounting for weak preferences). We additionally leverage this generalisation to strengthen Arrow's Impossibility Theorem by establishing that the joint satisfaction of its constraints other than transitivity yields an aggregation consisting of an intransitive preference cycle that spans all available alternatives. Notably, a preference cycle that spans all alternatives precludes any global ranking information.