Application of Soft Set in Game Theory

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Background

Soft set is a parameterized family of subsets defined over a universe associated with a set of parameters.

The definition of soft set is given below.

• Definition 1(Molodtsov, 1999): A pair (F, E) is called a soft set over U iff F is a mapping of E into the set of all subsets of the universal set U; i.e.F: E → P(U)(1)

where U is the universal set, E is the parameter set and P(U) is the power set U.

In other words, a soft set over U is a parameterised family of subsets of the universe U. For e ∈ E, F(e) can be called as the set of e-approximate elements of the soft set (F, E). So, a soft set can be represented as a collection of approximations. The parameter part of the approximation is called as predicate and for each parameter in E and the set containing all the elements of F(e) is called the value set of e in (F, E).

The pair (U, E) is often regarded as a soft universe. A parameter can be anything adverbial for the elements, such as a number, word, phrase or a sentence which can describe the value set more appropriately.