Application of Soft Set in Game Theory

Application of Soft Set in Game Theory

B. K. Tripathy (VIT University, India), Sooraj T. R. (VIT University, India) and Radhakrishna N. Mohanty (VIT University, India)
DOI: 10.4018/978-1-5225-7368-5.ch031

Abstract

In recent years, most of the applications in game theory have been developed based on the theory of fuzzy sets. But the inadequacy of the parameterization tool in fuzzy set theory leads to difficulties for decision making in the game theory. Soft sets were introduced by Molodtsov to overcome this problem in fuzzy sets and it was illustrated by him. Choice functions play an important role in game theory. Soft set theory gives an opportunity to construct new mathematical tool which keeps all good sides of choice function and eliminates its drawbacks. Also, decision making is an integral part of games and many researchers have applied soft set theory in decision making. In this chapter, the authors describe all these and propose some important improvements leading to better deals in game environments.
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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 eE, 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.

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