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On solving matrix games with pay-offs of triangular fuzzy numbers: Certain observations and generalizations

https://doi.org/10.1016/j.ejor.2015.05.011Get rights and content

Highlights

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    This paper reports on a recent publication in EJOR on matrix games with payoffs of triangular fuzzy numbers.

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    It is concluded that the value of the game for each player will not be a common triangular fuzzy number.

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    To solve the game, players need to solve their respective multi-objective linear programming problem.

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    These modifications lead to an algorithm to solve matrix games with payoffs of general piecewise linear fuzzy numbers.

Abstract

The purpose of this paper is to highlight a serious omission in the recent work of Li (2012) for solving the two person zero-sum matrix games with pay-offs of triangular fuzzy numbers (TFNs) and propose a new methodology for solving such games. Li (2012) proposed a method which always assures that the max player gain-floor and min player loss-ceiling have a common TFN value. The present paper exhibits a flaw in this claim of Li (2012). The flaw arises on account of Li (2012) not explaining the meaning of solution of game under consideration. The present paper attempts to provide certain appropriate modifications in Li’s model to take care of this serious omission. These modifications in conjunction with the results of Clemente, Fernandez, and Puerto (2011) lead to an algorithm to solve matrix games with pay-offs of general piecewise linear fuzzy numbers.

Introduction

Li (2012) in his recent paper proposed a new method to solve two person zero-sum matrix games with pay-offs of TFN’s. He emphasized, and in fact illustrated, that the method proposed by him always assures a common TFN-type fuzzy value for both Player I gain-floor and Player II loss-ceiling functions. Therefore, he concluded that any matrix game with pay-offs of TFNs has a TFN-type fuzzy value. He further emphasized that the conclusion (i.e., a common TFN value for both players) is rational because the underlying fuzzy matrix game is a zero-sum game. He also argued that his proposed method results in superior performance as compared to the existing methods in literature, more specifically, Campos (1989), Bector, Chandra, and Vidyottama (2004), Li (1999); Li and Yang (2004) and Li (2008). This is because the existing methods either do not provide a TFN value for the game or even if they do so the value is not common for the two players.

In this paper, we demonstrate that although Li’s first conclusion that the value of the game is a TFN is correct but his other conclusion that both players have a common TFN value is flawed. The mistake in Li’s work is observed due to omission of an essential concept of ‘solution of a game’. In this paper, we proceed with a thorough investigation of Li’s work and highlight a serious omission in an effort to alert future adoption of Li’s approach in fuzzy matrix games. We also suggest appropriate modifications to take care of this omission. These modifications in conjunction with the results of Clemente, Fernandez, and Puerto (2011) lead to an algorithm to solve matrix games with pay-offs of general piecewise linear fuzzy numbers.

The remainder of the paper is organized as follows. Section 2 reviews and points out a serious omission in the method of Li (2012). Section 3 provides our revision in order to resolve the mistakes in Li’s research. A simple numerical example is presented in Section 4. Section 5 describes an algorithm to solve matrix games with pay-offs of general piecewise linear fuzzy numbers. This algorithm is based on our revision of Li’s work as suggested in Section 3. Some concluding remarks are furnished in Section 6.

Section snippets

Review of Li’s method

Rather than presenting the mathematical details of Li’s method to solve a two person zero-sum matrix game with pay-offs of TFNs we consider only the numerical example presented in his work. This has been done to keep the presentation short and also to clearly illustrate how and why we differ from his point of view. The example is cited from Campos (1989) and it has now become almost a bench mark example in the area of fuzzy matrix games (see, Bector & Chandra, 2005). Let the fuzzy matrix game

Our revision on Li (2012) model

With regard to fuzzy matrix games and related topics we shall follow the notations and terminologies of Bector and Chandra (2005). We hope there will be no confusion even if they are somewhat different from Li (2012).

Let Rn denote the n-dimensional Euclidean space and R+n be its non-negative orthant. Let eT=(1,…,1) be a vector of ‘ones’ whose dimension is specified as per the specific context. By a two person zero-sum fuzzy matrix game FG, we mean the triplet FG=(Sp,Sn,A˜) where Sp={x∈R+p,eTx=1}

The numerical example of Li (2012)

Consider the game FG=(S2,S2,A˜) where A˜ is the pay-off matrix taken by Li (2012) and already described in Section 2.

To solve FG for Player I, we need to solve the following multiobjective linear programming problem (MOP-I) Max(vl,vm,vr)subjectto175x1+80x2≥vl,150x1+175x2≥vl,190x1+100x2≥vr,158x1+190x2≥vr,180x1+90x2≥vm,156x1+180x2≥vm,x1+x2=1,x1,x2≥0.

We use the GAMS software on Window 64 bits Intel Core Duo2 platform to solve the problem (Mavrotas, 2007) and get one of the Pareto (or efficient)

Solving games with pay-offs of piecewise linear fuzzy numbers

In this section we show that the modifications as suggested in Section 3 provide a methodology to solve matrix games with pay-offs of piecewise linear fuzzy numbers. Our presentation here is highly motivated by Clemente et al. (2011).

Let FG=(S2,S2,A˜) be the two person zero-sum fuzzy matrix game as introduced in Section 3. But we now assume that elements aij(i=1,2,…,p,j=1,2,…,n) of the matrix A˜ are piecewise linear fuzzy numbers. By a piecewise linear fuzzy number we mean a fuzzy number having

Concluding remarks

In this study, we have described a critical omission in the recent work of Li (2012) for solving zero-sum fuzzy matrix games with pay-offs of TFNs. We described the flaw in Li’s conversion from fuzzy game problem to deterministic linear programming problems. We have provided a revision in Li’s approach suitably and drawn valid conclusions with regard to the solution of such games. Specifically we have concluded that each player have a TFN-type optimal value but it does not make sense to have

Acknowledgment

The authors are thankful to the referees for their valuable suggestions and extensive comments which have helped to improve the paper.

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