Charles Horvath

Maximal elements and fixed points for binary relations on topological ordered spaces

Abstract Topological semilattices are partially ordered topological spaces X in which each pair of elements x , x ′ ∈ X has a least upper bound x V x ′ and the function ( x , x ′)↦ x V x ′ is continuous. We establish in such a context an order theoretical version of the classical result of Knaster-Kuratowski-Mazurkiewicz, as well as fixed point theorems for multivalued mappings. One can then, as in the context of topological vector spaces, obtain existence results for the largest elements of a weak preference relation or maximal elements for a strict preference relation. Beyond these particular results, we wish to attract attention to path-connected topological semilattices, examples of which will be found in the introduction, and their rich geometric structure — a geometric structure rich enough to provide order theoretical versions of some of the basic tools from mathematical economics and, therefore, also an alternative to the usual convexity assumptions.

Network games; adaptations to Nash-Cournot equilibrium

We consider nonlinear flow problems involving noncooperative agents, all active in the same network. To find Nash equilibria, we develop an algorithm that lends itself to decentralized computation and parallel processing. The algorithm, which proceeds in terms of iterative strategy adjustments, is, in essence, of subgradient type. One advantage of that type is the ease with which stochastic and nonsmooth data can be accommodated.

On the existence of constant selections

Abstract Given a relation Ω :X→Y between topological spaces, we inquire whether it has a constant selection. This problem has been investigated from different points of view, purely topological or convex. We present here a synthesis of some of the most interesting results, with some generalizations and new insights.

Toward A geometric theory of minimax equalities

We present new ideas and concepts in minimax equalities. Two important classes of multifunctions will be singled out, the Weak Passy-Prisman multifunctions and multifunctions possessing the finite simplex property. To each class of multifunctions corresponds a class of functions. We obtain necessary and sufficient conditions for a multifunction to have the finite intersection property, and necessary and sufficient conditions for a function to be a minimax function. All our results specialize to sharp improvements of known theorems, Sion, Tuy, Passy-Prisman, Flåm-Greco. One feature of our approach is that no topology is required on the space of the maximization variable. In a previous paper [6] we presented a “method of reconstruction of polytopes” from a given family of subsets, this in turn lead to a “principle of reconstruction of convex sets” Theorem 3, which plays a major role in this paper. Our intersection theorems bear no obvious relationship to other results of the same kind, like K.K.M. or other more elementary approaches based on connectedness. We conclude our work with a remark on the role of upper and lower semicontinuous regularization in mimmax equalities

Reconstruction of polytopes by convex pastings

Consider a convex polytope X and a family F of convex sets, satisfying a given property P. Moreover, assume that F is closed under operations of cutting and convex pasting along hyperplanes. Necessary and sufficient conditions are given to have X∈ F. As a consequence, it follows that, if all simplices or small enough simplices have the property in question, then X also has that property.

A Topological Investigation of the Finite Intersection Property

When does a finite family of sets {R 0,...,R n } have a nonempty intersection? From the topology of the unions formed from these sets we obtain sufficient conditions. Homotopy or homology properties will guarantee that the intersection is not empty.

Topological versions of passy– prisman’S minimax theorem

A topological version of Passy–Prisman’s minimax Theorem is proved. We introduce to pological cones and we prove our results under connectedness assumptions. We give examples of cones in spaces without any linear structure. Even when interpreted in a linear framework our results are new and improve Passy–Prisman’s minimax Theorem, and consequently Sion’s minimax Theorem

Stochastic mean values, rational expectations, and price movements

Numerous economic problems assume the form of finding a fixed point of a continuous self-mapping on a compact interval. We consider instances where the mapping is a parametrized expected value, and we offer an iterative scheme for locating a fixed point. The proposed method can be seen as an adaptive learning method, akin to stochastic approximation.

Efficient Nash equilibria on semilattices

This contribution introduces the so-called quasi-Leontief functions. In the framework and the language of tropical algebras, our quasi-Leontief functions are the additive functions defined on a semimodule with values in the semiring of scalars. This class of functions encompasses as a special case the usual Leontief utility function. We establish the existence of efficient Nash equilibria when the strategy spaces are compact and pathconnected topological semilattices.