Arrow's Theorem for incomplete relations

Abstract

Two theorems are presented that extend Arrow's Theorem to relations that are not necessarily complete. Let U be a set with three or more elements, let W be the set of weak orderings of U, let L be the set of linear orderings of U, and let f be an n-ary function mapping Wn to W. By Arrow's Impossibility Theorem, if f satisfies Arrow's Condition P (“Pareto”) and Condition 3 (“independence of irrelevant alternatives”) then f violates Arrow's condition of non-dictatorship, which implies that the restriction of f to linear orderings is a projection function, i.e., there is some k∈{1,…,n} such that f(R1,…,Rn)=Rk for all linear orderings R1,…,Rn∈L. Thus Arrow's Theorem provides a characterization of projection functions on linear orderings as those that satisfy two conditions that clearly hold for projection functions, but which are sufficient (by Arrow's Theorem) to imply a function is a projection function. We prove a similar characterization in Theorem 3 and use it to derive Arrow's characterization for projection functions on linear orderings. Arrow's Theorem does not characterize projection functions on weak orderings, but Theorem 3 does characterize projection functions on a strictly larger class than the weak orderings, namely the transitive co-transitive relations. Along the way we prove, as a consequence of Theorem 2, that if a transitive-valued multivariate relational function f satisfies relational versions of Arrow's Conditions P and 3, and maps equivalence relations on U to transitive relations on U, then for some D0⊆{1,…,n}, f(R1,…,Rn) is just the intersection of all Ri for i∈D0.