Acyclic Collective Choice Rules

Abstract

This paper establishes a natural and satisfying characterization of the class of collective choice rules which are acyclic and satisfy the Arrow axioms (unrestricted domain, independence of irrelevant alternatives, and the weak Pareto principle). We show that, when the number of alternatives is larger than the number of individuals, there must exist an individual who can at least some critical number of pairwise decisions. This critical number of veto pairs depends on the number of alternatives and individuals, and, as the number of alternatives increases without limit, the fraction of all pairs which some individual can veto approaches unity. We also present a global veto theorem and an axiomatic characterization of the Pareto extension rule which utilizes acyclicity rather than quasi-transitivity. ARROW [1] SHOWED that the only collective choice rules that yield weak order social preference relations and satisfy unrestricted domain, independence of irrelevant alternatives, and the weak Pareto principle are dictatorial. Gibbard [9] demonstrated that by relaxing the rationality requirement from transitivity to quasi-transitivity (i.e., transitivity of the strict preference relation) we can evade the letter though not the spirit of the Arrow dictatorship result: oligarchy, a weaker form of dictatorship, still obtains when the other three axioms are imposed. In this paper we prove a theorem parallel to those of Arrow and Gibbard for the weaker rationality requirement of acyclicity (i.e., the absence of cycles of strict preference). Since acyclicity is a necessary and sufficient condition for the existence of a nonempty set of maximal elements in every finite feasible set, there are powerful reasons for imposing it. Moreover, as we argue in Blair and Pollak [2], it is difficult to justify any stronger rationality property such as quasi-transitivity without at the same time justifying some even stronger rationality condition which implies dictatorship. Our principal result shows that, when the number of alternatives is larger than the number of individuals, there must exist an individual who can at least some critical number of pairwise decisions. (We say that individual i has a veto over the ordered pair (y, x) if he is weakly decisive for x against y-that is, if his strict preference for x over y implies weak social preference for x over y, regardless of the preferences of other individuals.) This critical number of veto pairs depends on the number of alternatives and the number of individuals. As the number of alternatives increases without limit, the fraction of all pairs that some individual can veto approaches unity. There may be more than one individual who can veto at least the critical number of pairs; indeed, it is possible for every individual to have a veto over every ordered pair of alternatives.