Tomáš Kroupa

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

Abstract In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n -person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.

Enlarged cores and bargaining schemes in games with fuzzy coalitions

In this paper we introduce a new concept of solution for games with fuzzy coalitions, which we call an enlarged core. The enlarged core captures an idea that various groups of fuzzy coalitions can have different bargaining power or influence on the final distribution of wealth resulting from the cooperation process. We study a bargaining scheme for the enlarged core, which is an iterative procedure for generating sequences converging to elements of the enlarged core. It is shown that the enlarged core coincides with Aubins core for a specific class of games with fuzzy coalitions.

Application of the choquet integral to measures of information in possibility theory

Possibility measures are analyzed from the information‐theoretic point of view. The significant role of the Choquet integral is argued in this context. The main result of the article is a representation theorem for the nonspecificity of possibility distribution and a new definition of divergence for possibility measures. An application of the divergence to construction of possibilistic models is outlined.

Extension of belief functions to infinite-valued events

We generalise belief functions to many-valued events which are represented by elements of Lindenbaum algebra of infinite-valued Łukasiewicz propositional logic. Our approach is based on mass assignments used in the Dempster–Shafer theory of evidence. A generalised belief function is totally monotone and it has Choquet integral representation with respect to a unique belief measure on Boolean events.

Filters in fuzzy class theory

A concept of filter is introduced in fuzzy class theory. Graded properties of filters, prime filters, and related constructions are investigated. In particular, the properties of filters relativized with respect to the three basic t-norms (Godel, product, and Lukasiewicz) are studied. A naturally arising example of a graded filter is constructed and it is shown that the models of filters include well-known families of set functions.

Topology in Fuzzy Class Theory: Basic Notions

In the formal and fully graded setting of Fuzzy Class Theory (or higher-order fuzzy logic) we make an initial investigation into basic notions of fuzzy topology. In particular we study graded notions of fuzzy topology regarded as a fuzzy system of open or closed fuzzy sets and as a fuzzy system of fuzzy neighborhoods. We show their basic graded properties and mutual relationships provable in Fuzzy Class Theory and give some links to the traditional notions of fuzzy topology.

Core of Coalition Games on MV-algebras1

Coalition games are generalized to semisimple MV-algebras. Coalitions are viewed as [0, 1]-valued functions on a set of players, which enables to express a degree of membership of a player in a coalition. Every game is a real-valued mapping on a semisimple MV-algebra. The goal is to recover the so-called core: a set of final distributions of payoffs, which are represented by measures on the MV-algebra. A class of sublinear games are shown to have a non-empty core and the core is completely characterized in certain special cases. The interpretation of games on propositional formulas in Łukasiewicz logic is introduced.

States in Łukasiewicz logic correspond to probabilities of rational polyhedra

It will be shown that probabilities of infinite-valued events represented by formulas in Lukasiewicz propositional logic are in one-to-one correspondence with tight probability measures over rational polyhedra in the unit hypercube. This result generalizes a recent work on rational measures of polyhedra and provides an elementary geometric approach to reasoning under uncertainty with states in Lukasiewicz logic.

Geometry of possibility measures on finite sets

The paper presents a purely geometrical characterization of the convex set of probabilities dominated by a possibility measure on a finite set. It is demonstrated that the set of dominated probabilities can be represented as a very special kind of convex polyhedral set, the so-called simple polytope, which enhances performance of computational methods. A lower bound and a new upper bound for the number of extreme points are established. It is shown that the upper bound leads in some cases to a better estimate than the exponential bound appearing in the literature.

Modal extensions of Łukasiewicz logic for modelling coalitional power

Modal logics for reasoning about the power of coalitions capture the notion of effectivity functions associated with game forms. The main goal of coalition logics is to provide formal tools for modeling the dynamics of a game frame whose states may correspond to different game forms. The two classes of effectivity functions studied are the families of playable and truly playable effectivity functions, respectively. In this paper we generalize the concept of effectivity function beyond the yes/no truth scale. This enables us to describe the situations in which the coalitions assess their effectivity in degrees, based on functions over the outcomes taking values in a finite Łukasiewicz chain. Then we introduce two modal extensions of Łukasiewicz finite-valued logic together with many-valued neighborhood semantics in order to encode the properties of many-valued effectivity functions associated with game forms. As our main results we prove completeness theorems for the two newly introduced modal logics.