Revistes Catalanes amb Accés Obert (RACO)

Two approaches to fuzzification of payments in NTU coalitional game

Milan Mareš


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

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

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