Two approaches to fuzzification of payments in NTU coalitional game

Abstract

There exist several possibilities of fuzzification of a coalitional game. It is quite usual to fuzzify, e.\,g., the concept of coalition, as it was done in [1]. Another possibility is to fuzzify the expected pay-offs, see [3,4]. The latter possibility is dealt even here. We suppose that the coalitional and individual pay-offs are expected only vaguely and this uncertainty on the "input" of the game rules is reflected also by an uncertainty of the derived "output" concept like superadditivity, core, convexity, and others. This method of fuzzification is quite clear in the case of games with transferable utility, see [6,3]. The not transferable utility (NTU) games are mathematically rather more complex structures. The pay-offs of coalitions are not isolated numbers but closed subsets of n-dimensional real space. Then there potentially exist two possible approaches to their fuzzification. Either, it is possible to substitute these sets by fuzzy sets (see, e.g.[3,4]). This approach is, may be, more sophisticated but it leads to some serious difficulties regarding the domination of vectors from fuzzy sets, the concept of superoptimum, and others. Or, it is possible to fuzzify the whole class of (essentially deterministic) NTU games and to represent the vagueness of particular properties or components of NTU game by the vagueness of the choice of the realized game (see [5]). This approach is, perhaps, less sensitive regarding some subtile variations in the the fuzziness of some properties but it enables to transfer the study of fuzzy NTU coalitional games into the analysis of classes of deterministic games. These deterministic games are already well known, which fact significantly simplifies the demanded analytical procedures.

This brief contribution aims to introduce formal specifications of both approaches and to offer at least elementary comparison of their properties.