Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 37

Tab l e 3 . 5 Movie Viewing History of A 4
Director Title Type Quality

Spielberg Indiana Jones S.F. Average

King Just Call Me Bob Comedy Average

Mora Kung Fung Hero Action Bad

Spielberg E.T. the Extra-Terrestrial S.F. Good

where the universe U is a non-empty set of objects. Each object corre-
sponds to a row in the decision table. A is a non-empty set of attributes
such that each attribute a ∈ A maps U to a set of values V a .Inpar-
ticular, d

A is special attribute called the decision variable, which
represents the classification results of the objects, such that each ob-
ject in U belongs to a one of the decision classes represented by a value
in V d , the set of all decision classes.

∈

Example 3.3 In the Tables 3.2 to 3.5, the universe U is the set
of movies, and the attributes are Director , Type and Quality ,among
which Quality is the decision variable. We consider each of the at-
tribute a function. For example, Quality is a function that maps U to
V Quality ,

Quality : U → V Quality

where V Quality = { Good , Average , Bad } .

We have to note that, however, even with such extra information, it
is still hard to apply the traditional core-based game theoretic analysis
to this game because the outcome (whether the new movie will be
good) will still depend on how the agents' experiences are interpreted
by the individual agents to predict the quality of the current movie.
Let us first consider a two agents movie watching coalition

{ A 1 ,

A 2 } . This case is straightforward enough: both agents will have the

opinion that the movie should be good, since the qualities of all pre-
vious Spielberg's S.F. movies in their experiences have been good. If
these two agents think like common people, they would expect the