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. 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 contradictory preference cycles. However, a formal proof of this relationship has been limited to D'Antoni's work, which applies only to the strict preference case, i.e., where indifference between alternatives is not allowed. In this paper, we generalise D'Antoni's methodology to prove in full (i.e., accounting for weak preferences) that Arrow's Impossibility Theorem can be equivalently stated in terms of contradictory preference cycles. This methodology involves explicitly constructing profiles that lead to preference cycles. Using this framework, we also prove a number of additional facts regarding social welfare functions. As a result, this methodology may yield further insights into the nature of preference cycles in other domains e.g., Money Pumps, Dutch Books, Intransitive Games, etc.