Locally complete path independent choice functions and their lattices

Abstract

The concept of path independence (PI) was first introduced by Arrow [Social Choice and Individual Values, Yale University Press, Hew Haven, 1963] as a defense of his requirement that collective choices be rationalized by a weak ordering. Plott [Econometrica 41 (1973) 1075–1091] highlighted the dynamic aspects of PI implicit in Arrow’s initial discussion. Throughout these investigations, two questions, both initially raised by Plott, remained unanswered. What are the precise mathematical foundations for path independence? How can PI choice functions be constructed? We give complete answers to both these questions for finite domains and provide necessary conditions for infinite domains. We introduce a lattice associated with each PI function. For finite domains these lattices coincide with locally lower distributive or meet-distributive lattices and uniquely characterize PI functions. We also present an algorithm, effective and exhaustive for finite domains, for the construction of PI choice functions and hence for all finite locally lower distributive lattices. For finite domains, a PI function is rationalizable if and only if the lattice is distributive. The lattices associated with PI functions that satisfy the stronger condition of the weak axiom of revealed preference are chains of Boolean algebras and conversely. Those that satisfy the strong axiom of revealed preference are chains and conversely.