# 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: 10.4018/978-1-5225-2255-3.ch281

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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.

## Key Terms in this Chapter

Game Theory: The study of mathematical models of conflict and cooperation between intelligent rational decision-makers.

Choice Function: Choice function is a mapping which associates a set of strategies with a given situation.

Payoff Function: Payoff function is used for modeling human behavior. Payoff function for a player is a mapping from the cross-product of players strategy spaces to the players set of payoffs.

Fuzzy Set: Fuzzy sets are an extension of the classical notion of sets whose elements have degrees of membership.

S-Function (Soft Function): It is a payoff function defined using notion of soft set.

Soft Game: A game using soft payoff functions.

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

Nash Equilibrium: It is a concept to find out the solution of a non cooperative game involving two or more players in game theory.

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