### Two approaches to fuzzification of payments in NTU coalitional game

#### Resum

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.

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.