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)
Copyright: © 2018 |Pages: 11
DOI: 10.4018/978-1-5225-2255-3.ch281

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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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