Andrzej Wieczorek

The Kakutani property and the fixed point property of topological spaces with abstract convexity

Abstract The paper deals with the usual fixed point property and the following Kakutani property of a space X: for every upper semicontinuous function Φ from X to non-empty closed convex subsets of X, there exists x0 such that x0 ϵ Φ(x0). We derive this property of X from various separation properties of convex subsets of X and a kind of local convexity of X. Convexity in our setup is given in an abstract axiomatic way. Special emphasis is given to the case where X has the form of a product. The obtained results cover several known fixed point theorems: Ky Fan-Glicksberg, Wallace, and special cases of Eilenberg-Montgomery. We also discuss an open problem concerning the fixed point property of finite posets and its role in proving more advanced theorems.

Kakutani property of the polytopes implies Kakutani property of the whole space

Abstract This paper deals with the following property of a space X : for every upper semicontinuous function φ from X to nonempty closed convex subsets of X , there exists x 0 which is in φ ( x 0 ). We derive this property of X from the same property assumed to hold for the polytopes in X . Convexity in our setup is given in an abstract axiomatic way. The results are also reformulated in the order theoretical language of continuous lattices. Our results directly generalize (a) an extension of the Kakutani Fixed Point Theorem to compact convex sets in locally convex spaces, due to Ky Fan and I. L. Glicksberg, and (b) the Fixed Point Theorem of Wallace concerning set-valued functions on trees.

Compressibility: A property of topological spaces related to abstract convexity

Abstract An abstract operation of hull, in a set Y, associates, with every finite set F ⊆ Y, a set hull F ⊆ Y. A topological space X is said to be compressible into a topological space Y, equipped with an operation of hull, if for every sequence (A1, …, An) of open sets covering X and every (y1, …, yn) ϵ Yn there exists a continuous function φ: X → Y such that for every x ϵ X, φ(x) ϵ hull {y i ¦x ϵ A i } . The concept of compressibility is extremely useful when dealing with problems of the existence of fixed points, continuous selections, or with KKM-Property. The obtained results generalize several known theorems; they also provide a broad range of new original results.

Spot functions and peripherals: Krein-Milman type theorems in an abstract setting

Abstract The concepts of a spot function and peripheral relation furnish a framework to formulate and prove a very general theorem of Krein-Milman type. In turn, this result implies several new as well as already existing generalizations of the Krein-Milman Theorem to an abstract convexity setup. The notion of a spot function offers a new approach to axiomatic convexity problems and it is studied in detail.

Fibonacci Numbers and Equilibria in Large "Neighborhood" Games

We deal with a game-theoretic framework involving a finite number of infinite populations, members of which have a finite number of available strategies. The payoff of each individual depends on her own action and distributions of actions of individuals in all populations. A method to find all equilibria is discussed which requires the search through all nonempty subsets of the types’ strategy sets, assigning equilibria to each of them. The method is then used to find equilibria in two types of “neighborhood” games in which there is one type of player who has strategies in V = {1, . . . , k} and payoff functions Ф(j; p) = α · p j−1 + p j + α · p j+1 for j = 2, . . . , k − 1 and: in the case of “chain” games Ф(1; p) = p 1 + α · p 2; Ф(k; p) = α p k−1 + p k; in the case of “circular” games Ф(1; p) = α · p k + p 1 + α · p 2; Ф(k; p) = α · p k−1 + p k + α p 2; (in both cases 0 ≤ α ≤ ½; p is a distribution on V). The Fibonacci numbers are used to determine the coordinates of equilibria in the case α = 1/3, for other values of ? we need to construct numerical Fibonacci-like sequences which determine, in an analogous manner, coordinates of equilibria. An alternative procedure makes use of some numerical Pascal-like triangles, specially constructed for this purpose.

Convergence of mann iteration processes for nonexpansive operators in equi-connected metric spaces

The paper deals with various conditions implying the convergence of a Mann type iteration process constructed for a non-expansive operator in an equi-connected space (i. e. metric space equipped with a connecting function; so the iterates are taken along certain curves). Coefficients of the iterates do not have to be separated from 0 or 1. 1].

Joint pseudo-utility representations

Abstract A pseudo-utility representation of a preference relation P in a set X is a real function defined over X2 whose support coincides with the graph of P. It is proved that, for a parametrized family ( Π t ¦ t ϵ T ) of preference relations in the same domain X, there exists a family of their pseudo-utility representations πt(x, x′), quasi-concave in x′, such that π is continuous over T × X2. The paper justifies the relevance of the problem and presents various special cases, settled in the abstract convexity framework.

A note on the ‘surjectivity property’

Abstract The authors dealing with Nash implementation problems often assume, while formulating and proving their theorems, that the set of alternatives has a ‘surjective property’. In this note it is proved that all sets A have this property and even more: for every set B there exists a function T : A B → A such that for every b ϵ B and every a ϵ A B ⧹{ b } , the function t : A → A defined by t (α):= T ( a ;α) is a bijection. For finite B and A with cardinality not higher than the continuum, an explicit specification of such T is given.

On representations of social preferences - an algebraic approach

The aim of the present paper is to extend some constructions in the field of utility theory, which are usually made for individuals, in the case when a bigger population is considered. Since the situation in this case becomes more complicated, we decided to proceed in a very formal algebraic way.

Coalition games without players an application to Walras equilibria (Announcement of result)

We propose here a general concept of games based on the notion of a coalition. It is assumed that a set of coalitions is given, in which some operations (join, meet and complement) are defined; in general single players (agents) need not exist (formally, coalitions form a Boolean б-algebra). The objects under consideration are states (e.g. states of an economy, like allocations); we restrict ourselves to the case, when the space of states coincides with a set of vector measures; however, further generalizations are possible (Section 4).