Path Independent Choice Functions Research Articles

A hyper-relation characterization of weak pseudo-rationalizability

I provide a characterization of weakly pseudo-rationalizable choice functions – that is, choice functions rationalizable by a set of acyclic relations – in terms of hyper-relations satisfying certain properties. For those hyper-relations Nehring calls extended preference relations, the central characterizing condition is weaker than (hyper-relation) transitivity but stronger than (hyper-relation) acyclicity. Furthermore, the relevant type of hyper-relation can be represented as the intersection of a certain class of its extensions. These results generalize known, analogous results for path independent choice functions.

A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries

In this paper we present a duality between nonmonotonic consequence relations and well-founded convex geometries. On one side of the duality we consider nonmonotonic consequence relations satisfying the axioms of an infinitary variant of System P, which is one of the most studied axiomatic systems for nonmonotonic reasoning, conditional logic and belief revision. On the other side of the duality we consider well-founded convex geometries, which are infinite convex geometries that generalize well-founded posets. Since there is a close correspondence between nonmonotonic consequence relations and path independent choice functions one can view our duality as an extension of an existing duality between path independent choice functions and convex geometries that has been developed independently by Koshevoy and by Johnson and Dean.

Closure and Preferences

We investigate the results of Kreps (1979), dropping his completeness axiom. As an added generalization, we work on arbitrary lattices, rather than a lattice of sets. We show that one of the properties of Kreps is intimately tied with representation via a closure operator. That is, a preference satisfies Kreps' axiom (and a few other mild conditions) if and only if there is a closure operator on the lattice, such that preferences over elements of the lattice coincide with dominance of their closures. We tie the work to recent literature by Richter and Rubinstein (2015). Finally, we carry the concept to the theory of path-independent choice functions.

Hyper-relations, choice functions, and orderings of opportunity sets

We prove a coincidence of the class of multi-preference hyper-relations and the class of decent hyper-relations (DHR), that is the class of binary relations on opportunity sets satisfying monotonicity, no-dummy, stability with respect to contraction and extension, and the union property. We study subclasses of DHR. In order to pursue our analysis, we establish a canonical bijection between DHR and the class of no-dummy heritage choice functions. From this we obtain that the no-dummy heritage choice functions have multi-criteria rationalizations with reflexive binary relations. We also prove that the restriction of this bijection to two subclasses of DHR, namely the transitive decent hyper-relations, and the ample hyper-relations, is a bijection between these subclasses and the classes of closed no-dummy choice functions and no-dummy path-independent choice functions (Plott functions), respectively.

Orders on Subsets Rationalised by Abstract Convex Geometries

We find conditions on the order on subsets of a finite set which are necessary and sufficient for the relative ranking of any two subsets in this order to be determined by their extreme elements relative to an abstract convex geometry. It turns out that this question is closely related to the rationalisability of path independent choice functions by hyper-relations.

Orders on Subsets Rationalised by Abstract Convex Geometries

We find conditions on the order on subsets of a finite set which are necessary and sufficient for the relative ranking of any two subsets in this order to be determined by their extreme elements relative to an abstract convex geometry. It turns out that this question is closely related to the rationalisability of path independent choice functions by hyper-relations.

A new characterization of the path independent choice functions

In this paper we introduce a new axiom for choice functions, equivalent to the path independence axiom. We call it the congruence condition. This axiom allows to construct directly the anti-exchange closure operators and the convex geometries associated to path independent choice functions.

Designer path independent choice functions

This paper provides an algorithm for the construction of all PICFs on a finite set of alternatives, V, designed by an a priori given set I of initial choices as well as the determination of whether the initial set I is consistent with path independence. The algorithm is based on a new characterization result for path independent choice functions (PICF) on finite domains and uses that characterization as the basis of the algorithm. The characterization result identifies two properties of a partition of the Boolean algebra as necessary and sufficient for a choice function C to be a PICF: (i): For every subset A of V the set ${\rm arc}(A) = {\{}B: C (B) = C(A){\}}$ is an interval in the Boolean algebra 2 V . (ii): If A/B is an interval in the Boolean algebra such that C(A) = C(B) and if M/N is an upper transpose of A/B then C(M) = C(N). The algorithm proceeds by expanding on the implications of these two properties.

Path-independence and closure operators with the anti-exchange property

Recently in Koshevoy (Math. Social Sci. 38 (1999) 35), it was established a connection between the theory of choice functions satisfying path-independence condition and closure operators with the anti-exchange property. Closure operators with the anti-exchange property are a combinatorial abstraction of usual convex hull closure in Euclidean spaces. Interest in these structures has its sources in different fields of mathematics. We demonstrate that path-independent choice functions provide another source for this structure. Specifically, we associate to a choice function f a collection of expanding maps. We prove that a function f is path-independent if and only if all the maps of this collection are coinciding anti-exchange closure operators. Consequences of such a characterization are demonstrated.

A class of multipartner matching markets with a strong lattice structure

For a two-sided multipartner matching model where agents are given by path-independent choice functions and no quota restrictions, Blair [7] had shown that stable matchings always exist and form a lattice. However, the lattice operations were not simple and not distributive. Recently Alkan [3] showed that if one introduces quotas together with a monotonicity condition then the set of stable matchings is a distributive lattice under a natural definition of supremum and infimum for matchings. In this study we show that the quota restriction can be removed and replaced by a more general condition named cardinal monotonicity and all the structural properties derived in [3] still hold. In particular, although there are no exogenous quotas in the model there is endogenously a sort of quota; more precisely, each agent has the same number of partners in every stable matching. Stable matchings also have the polarity property (supremum with respect to one side is identical to infimum with respect to the other side) and a property we call {\it complementarity}.

Locally complete path independent choice functions and their lattices

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.

The duality between the anti-exchange closure operators and the path independent choice operators on a finite set

In this paper, we show that the correspondence discovered by Koshevoy [Math. Soc. Sci. 38 (1999) 35] and Johnson and Dean [An algebraic characterization of path independent choice functions, Third International Meeting of the Society for Social Choice and Welfare, Maastricht, The Netherlands, 1996; Locally complete path independent choice functions and their lattices. Preprint, 1998] between anti-exchange closure operators and path independent choice operators is a duality between two semilattices of such operators. Then we use this duality to obtain results concerning the ‘ordinal’ representations of path independent choice functions from the theory of anti-exchange closure operators.

Choice functions and abstract convex geometries

A main aim of this paper is to make connections between two well – but up to now independently – developed theories, the theory of choice functions and the theory of closure operators. It is shown that the classes of ordinally rationalizable and path independent choice functions are related to the classes of distributive and anti-exchange closures.

Ideal Structures of Path Independent Choice Functions

Ideals are used to provide necessary and sufficient conditions for classes of choice functions satisfying the Strong Axiom of Preference, the Weak Axiom of Revealed Preference, or Path Independence. Semigroups whose choice functions are not path independent are demonstrated to lack proper ideal structures. A subsemigroup which is a monoid is identified. A motivation for interest in these structures is that the semigroup property is required of all finite state automata. Search-process foundations to choice functions facilitate applying automata results to economics and direct tests of whether an automaton is economic. Journal of Economic Literature Classification Numbers: C69, D00, D83.