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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 950482, 12 pages
http://dx.doi.org/10.1155/2012/950482
Research Article

A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

Department of Mathematics, Faculty of Science, Anadolu University, 26470 Eskişehir, Turkey

Received 2 February 2012; Accepted 6 August 2012

Academic Editor: Allan Peterson

Copyright © 2012 Emrah Akyar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Fuzzy matrix games, specifically two-person zero-sum games with fuzzy payoffs, are considered. In view of the parametric fuzzy max order relation, a fictitious play algorithm for finding the value of the game is presented. A numerical example to demonstrate the presented algorithm is also given.

1. Introduction

Game theory is a mathematical discipline which studies situations of competition and cooperation between several involved parties, and it has many applications in broad areas, such as strategic warfare, economic or social problems, animal behaviour, and political voting systems.

The simplest game is a finite, two-person, zero-sum game. There are only two players, player I and player II and it can be denoted by a matrix. Thus, such a game is called a matrix game. More formally, a matrix game is an 𝑚×𝑛 matrix 𝐺 of real numbers. A (mixed) strategy of player I is a probability distribution 𝑥 over the rows of 𝐺, that is, an element of the set 𝑋𝑚=𝑥𝑥=1,,𝑥𝑚𝑚𝑥𝑖0𝑖=1,,𝑚,𝑚𝑖=1𝑥𝑖.=1(1.1) Similarly, a strategy of player II is a probability distribution 𝑦 over the columns of 𝐺, that is, an element of the set 𝑌𝑛=𝑦𝑦=1,,𝑦𝑛𝑛𝑦𝑖0𝑖=1,,𝑛,𝑛𝑖=1𝑦𝑖.=1(1.2) A strategy 𝑥 of player I is called pure if it does not involve probability, that is, 𝑥𝑖=1 for some 𝑖=1,,𝑚 and it is denoted by I𝑖. Similarly, pure strategies of player II are denoted by II𝑗 for 𝑗=1,,𝑛.

If player I plays row 𝑖 (i.e., pure strategy 𝑥=(0,0,,𝑥𝑖=1,0,,0)) and player II plays column 𝑗 (i.e., pure strategy 𝑦=(0,0,,𝑦𝑗=1,0,,0)), then player I receives payoff 𝑔𝑖𝑗 and player II pays 𝑔𝑖𝑗, where 𝑔𝑖𝑗 is the entry in row 𝑖 and column 𝑗 of matrix 𝐺. If player I plays strategy 𝑥 and player II plays strategy 𝑦, then player I receives the expected payoff 𝑔(𝑥,𝑦)=𝑥T𝐺𝑦,(1.3) where 𝑥T denotes the transpose of 𝑥.

A strategy 𝑥 is called maximin strategy of player I in matrix game 𝐺 if 𝑥minT𝐺𝑦,𝑦𝑌𝑛𝑥minT𝐺𝑦,𝑦𝑌𝑛,(1.4) for all 𝑥𝑋𝑚 and a strategy 𝑦 is called minimax strategy of player II in matrix game 𝐺 if 𝑥maxT𝐺𝑦,𝑥𝑋𝑚𝑥maxT𝐺𝑦,𝑥𝑋𝑚(1.5) for all 𝑦𝑌𝑛. Therefore, a maximin strategy of player I maximizes the minimal payoff of player I, and a minimax strategy of player II minimizes the maximum that player II has to pay to player I.

von Neumann and Morgenstern (see [1]) proved that for every matrix game 𝐺 there is a real number 𝜈 with the following properties. (i)A strategy 𝑥 of player I guarantees a payoff of at least 𝜈 to player I (i.e., 𝑥T𝐺𝑦𝜈 for all strategies 𝑦 of player II) if and only if 𝑥 is a maximin strategy. (ii)A strategy 𝑦 of player II guarantees a payment of at most 𝜈 by player II to player I (i.e., 𝑥T𝐺𝑦𝜈 for all strategies 𝑥 of player I) if and only if 𝑦 is a minimax strategy.

Hence, player I can obtain a payoff at least 𝜈 by playing a maximin strategy, and player II can guarantee to pay not more than 𝜈 by playing a minimax strategy. For these reasons, the number 𝜈 is also called the value of the game 𝐺.

A position (𝑖,𝑗) is called a saddle point if 𝑔𝑖𝑗𝑔𝑘𝑗 for all 𝑘=1,,𝑚 and 𝑔𝑖𝑗𝑔𝑖𝑙 for all 𝑙=1,,𝑛, that is, if 𝑔𝑖𝑗 is maximal in its column 𝑗 and minimal in its row 𝑖. Evidently, if (𝑖,𝑗) is a saddle point, then 𝑔𝑖𝑗 must be the value of the game.

2. Fuzzy Numbers and a Two-Person Zero-Sum Game with Fuzzy Payoffs

In the classical theory of zero sum games the payoffs are known with certainty. However, in the real world the certainty assumption is not realistic on many occasions. This lack of precision may be modeled via fuzzy logic. In this case, payoffs are presented by fuzzy numbers.

2.1. Fuzzy Numbers

In this section, we give certain essential concepts of fuzzy numbers and their basic properties. For further information see [2, 3].

A fuzzy set 𝐴 on a set 𝑋 is a function 𝐴𝑋[0,1]. Generally, the symbol 𝜇𝐴 is used for the function 𝐴 and it is said that the fuzzy set 𝐴 is characterized by its membership function 𝜇𝐴𝑋[0,1] which associates with each 𝑥𝑋, a real number 𝜇𝐴(𝑥)[0,1]. The value of 𝜇𝐴(𝑥) is interpreted as the degree to which 𝑥 belongs to 𝐴.

Let 𝐴 be a fuzzy set on 𝑋. The support of 𝐴 is given as 𝑆𝐴=𝑥𝑋𝜇𝐴(𝑥)>0,(2.1) and the height h(𝐴) of 𝐴 is defined as h𝐴=sup𝑥𝑋𝜇𝐴(𝑥).(2.2) If h(𝐴)=1, then the fuzzy set 𝐴 is called a normal fuzzy set.

Let 𝐴 be a fuzzy set on 𝑋 and 𝛼[0,1]. The 𝛼-cut (𝛼-level set) of the fuzzy set 𝐴 is given by 𝐴𝛼=𝑥𝑋𝜇𝐴]𝐴(𝑥)𝛼,if𝛼(0,1cl𝑆,if𝛼=0,(2.3) where cl denotes the closure of sets.

The notion of convexity is extended to fuzzy sets on 𝑛 as follows. A fuzzy set 𝐴 on 𝑛 is called a convex fuzzy set if its 𝛼-cuts 𝐴𝛼 are convex sets for all 𝛼[0,1].

Let 𝐴 be a fuzzy set in , then 𝐴 is called a fuzzy number if (i)𝐴 is normal, (ii)𝐴 is convex, (iii)𝜇𝐴 is upper semicontinuous, and(iv)the support of 𝐴 is bounded.

From now on, we will use lowercase letters to denote fuzzy numbers such as ̃𝑎 and we will denote the set of all fuzzy numbers by the symbol 𝔽. Generally, some special type of fuzzy numbers, such as trapezoidal and triangular fuzzy numbers, are used for real life applications. We consider here 𝐿-fuzzy numbers.

The function 𝐿 satisfying the following conditions is called a shape function: (i)𝐿 is even function, that is, 𝐿(𝑥)=𝐿(𝑥) for all 𝑥, (ii)𝐿(𝑥)=1𝑥=0, (iii)𝐿() is nonincreasing on [0,+), (iv)if 𝑥0=inf{𝑥>0|𝐿(𝑥)=0}, then 0<𝑥0<+ and 𝑥0 is called the zero point of 𝐿.

Let 𝑎 be any number and let 𝛿 be any positive number. Let 𝐿 be any shape function. Then a fuzzy number ̃𝑎 is called an 𝐿-fuzzy number if its membership function is given by 𝜇̃𝑎(𝑥)=𝐿𝑥𝑎𝛿0,𝑥.(2.4) Here, 𝑥𝑦=max{𝑥,𝑦}. Real numbers 𝑎 and 𝛿 are called the center and the deviation parameter of ̃𝑎, respectively. In particular, if 𝐿(𝑥)=1|𝑥| we get 𝜇̃𝑎1(𝑥)=1𝛿[]|𝑥𝑎|,𝑥𝑎𝛿,𝑎+𝛿0,otherwise(2.5) and ̃𝑎 is called a symmetric triangular fuzzy number.

It is clear that for any shape function 𝐿, an arbitrary 𝐿-fuzzy number ̃𝑎 can be characterized by the its center 𝑎 and the deviation parameter 𝛿. Therefore, we denote the 𝐿-fuzzy number ̃𝑎 by ̃𝑎(𝑎,𝛿)𝐿. In particular, if ̃𝑎 is a symmetric triangular fuzzy number, we write ̃𝑎(𝑎,𝛿)𝑇. We also denote the set of all 𝐿-fuzzy numbers by 𝔽𝐿.

Let ̃𝑎(𝑎,𝛿)𝐿 be an 𝐿-fuzzy number then by (2.4) we see that the graph of 𝜇̃𝑎(𝑥) approaching line 𝑥=𝑎 as 𝛿 tends to zero from the right. Therefore, we can write that 𝜇̃𝑎(𝑥)=1,𝑥=𝑎0,𝑥𝑎.(2.6) The function in (2.6) is just a characteristic function of the real number 𝑎. Hence, we get 𝔽𝐿. From now on, we will call a fuzzy number ̃𝑎 as an 𝐿-fuzzy number if its membership function is given by (2.4) or (2.6).

Let ̃̃𝑎,𝑏𝔽 and 𝑘 be any real number. Then the sum of fuzzy numbers ̃𝑎 and ̃𝑏 and the scalar product of 𝑘 and ̃𝑎 are defined as 𝜇̃𝑎+̃𝑏(𝑧)=sup𝑧=𝑥+𝑦𝜇miñ𝑎(𝑥),𝜇̃𝑏,𝜇(𝑦)𝑘̃𝑎(𝑧)=max0,sup𝑧=𝑘𝑥𝜇̃𝑎,(𝑥)(2.7) respectively. In particular, if ̃𝑎(𝑎,𝛿1)𝐿 and ̃𝑏(𝑏,𝛿2)𝐿 are 𝐿-fuzzy numbers and 𝑘 is any real number, then one can verify that ̃̃𝑎+𝑏𝑎+𝑏,𝛿1+𝛿2𝐿,||𝑘||𝛿𝑘̃𝑎𝑘𝑎,1𝐿.(2.8)

Let ̃𝑎 be any 𝐿-fuzzy number. By the definition of the 𝛼-cut, [̃𝑎]𝛼 is a closed interval for all 𝛼[0,1]. Therefore, for all 𝛼[0,1] we can denote the 𝛼-cut of ̃𝑎 by [𝑎𝐿𝛼,𝑎𝑅𝛼], where 𝑎𝐿𝛼 and 𝑎𝑅𝛼 are end points of the closed interval [̃𝑎]𝛼.

For any symmetric triangular fuzzy numbers ̃𝑏̃𝑎, Ramík and Římánek (see [4]) introduced binary relations as follows:̃̃𝑎𝑞𝑏𝑎𝐿𝛼𝑏𝐿𝛼,𝑎𝑅𝛼𝑏𝑅𝛼[]̃̃̃[]̃𝛼0,1(fuzzymaxorder),̃𝑎𝑏̃𝑎𝑞𝑏,̃𝑎𝑏𝛼0,1(strictfuzzymaxorder),̃𝑎𝑏𝑎𝐿𝛼>𝑏𝐿𝛼,𝑎𝑅𝛼>𝑏𝑅𝛼[]𝛼0,1(strongfuzzymaxorder).(2.9) Following theorem is a useful tool to check fuzzy max order and strong fuzzy max order relations between symmetric triangular fuzzy numbers.

Theorem 2.1 (see [5]). Let ̃𝑎(𝑎,𝛿1) and ̃𝑏(𝑏,𝛿2) be any symmetric triangular fuzzy numbers. Then the statements ̃||𝛿̃𝑎𝑞𝑏1𝛿2||̃||𝛿𝑎𝑏,̃𝑎𝑏1𝛿2||<𝑎𝑏(2.10) hold.

It is not difficult to check that the fuzzy max order is a partial order. Then we may have many minimal and maximal points with respect to fuzzy max order. Therefore, use of the fuzzy max order is not so efficient in computer algorithms. Furukawa introduced a total order relation which is a modification of the fuzzy max order with a parameter (see [5, 6]).

Let 0𝜆1 be arbitrary but a fixed real number. For any 𝐿-fuzzy numbers ̃𝑎(𝑎,𝛿1)𝐿 and ̃𝑏(𝑏,𝛿2)𝐿 we define an order relation with parameter 𝜆 by ̃𝑎𝜆̃𝑏(i)𝑥0||𝛿1𝛿2||𝑏𝑎,or(ii)𝜆𝑥0||𝛿1𝛿2||𝑏𝑎<𝑥0||𝛿1𝛿2||,||||or(ii)𝑎𝑏<𝜆𝑥0||𝛿1𝛿2||,𝛿2>𝛿1,(2.11) where 𝑥0 is the zero point of 𝐿. The simple expression of (2.11) is as follows: ̃𝑎𝜆̃𝑏(i)𝜆𝑥0𝛿1+𝑎<𝜆𝑥0𝛿2+𝑏,or(ii)𝜆𝑥0𝛿1+𝑎=𝜆𝑥0𝛿2+𝑏,𝛿2𝛿1.(2.12)

It is clear that for any 𝐿-fuzzy numbers ̃𝑎(𝑎,𝛿1)𝐿 and ̃𝑏(𝑏,𝛿2)𝐿, ̃𝑎0̃𝑏 if and only if 𝑎𝑏. Therefore, the relation 0 is the order among the centers of 𝐿-fuzzy numbers. On the other hand, ̃𝑎1̃𝑏 if and only if ̃𝑏̃𝑎 or they are incomparable and 𝛿2>𝛿1. For 0<𝜆<1, the relation 𝜆 determines the order with respect to their values of center and their size of ambiguity. The smaller 𝜆 is, the larger the possibility of ordering by the value of center is, and the larger 𝜆 is, the larger the possibility of ordering by the size of ambiguity is.

Theorem 2.2 (see [5]). For every shape function 𝐿 and for each 𝜆[0,1], the relation 𝜆 is a total order relation on 𝔽𝐿.

Let 𝜆[0,1] be fixed arbitrarily and let ̃𝑣𝑉=(1,̃𝑣2̃𝑣,,𝑛) be any 𝐿-fuzzy vector, that is, all components of 𝑉 are 𝐿-fuzzy numbers and expressed by a common shape function 𝐿. Then maximum and minimum of 𝑉 in the sense of the total order 𝜆 are denoted as Max𝜆𝑉,Min𝜆𝑉,(2.13) respectively.

Example 2.3. Let 𝑉=((2,0.1)𝐿,(0,0.1)𝐿,(3,0.3)𝐿,(1,0.4)𝐿) be 𝐿-fuzzy vector. Then Max𝜆𝑉=(0,0.1)𝐿,Min𝜆𝑉=(3,0.1)𝐿,(2.14) for all 𝜆[0,1].

Let ̃𝑎 and ̃𝑏 be any 𝐿-fuzzy numbers, then the Hausdorff distance between ̃𝑎 and ̃𝑏 is defined as 𝑑̃𝑏̃𝑎,=sup[]𝛼0,1||𝑎max𝐿𝛼𝑏𝐿𝛼||,||𝑎𝑅𝛼𝑏𝑅𝛼||,(2.15) that is, ̃𝑑(̃𝑎,𝑏) is the maximal distance between 𝛼-cuts of ̃𝑎 and ̃𝑏. In particular, if ̃𝑎(𝑎,𝛿1) and ̃𝑏(𝑏,𝛿2) are any symmetric triangular fuzzy numbers, then ̃𝑑(̃𝑎,𝑏)=|𝑎𝑏|.

2.2. Two-Person Zero-Sum Game with Fuzzy Payoffs and Its Equilibrium Strategy

In this section, we consider zero-sum games with fuzzy payoffs with two players, and we assume that player I tries to maximize the profit and player II tries to minimize the costs.

The two-person zero-sum game with fuzzy payoffs is defined by 𝑚×𝑛 matrix 𝐺 whose entries are fuzzy numbers. Let 𝐺 be a fuzzy matrix game 𝐺=II1II2II𝑛I1I2I𝑚̃𝑔11̃𝑔12̃𝑔1𝑛̃𝑔21̃𝑔22̃𝑔2𝑛̃𝑔𝑚1̃𝑔𝑚2̃𝑔𝑚𝑛(2.16) and 𝑥𝑋𝑚, 𝑦𝑌𝑛, that is, 𝑥 and 𝑦 are strategies for players I and II. Then the expected payoff for player I is defined by ̃𝑔(𝑥,𝑦)=𝑥T𝐺𝑦=𝑖𝑗𝑥𝑖𝑦𝑗̃𝑔𝑖𝑗.(2.17)

Example 2.4. Let 𝐺=II1II2II3II4I1I2I3(10,0.1)𝑇(8,0.7)𝑇(8,0.7)𝑇(6,0.7)𝑇(9,0.8)𝑇(1,0.7)𝑇(3,0.8)𝑇(8,0.9)𝑇(3,0.2)𝑇(1,0.5)𝑇(2,0.2)𝑇(3,0.7)𝑇(2.18) be a fuzzy matrix game whose entries are symmetric triangular fuzzy numbers.
For this game, if player I plays second row (𝑥=(0,1,0)) and player II plays third column (𝑦=(0,0,1,0)), then player I receives and correspondingly player II pays a payoff ̃𝑔(I2,II3)(3,0.8)𝑇. On the other hand, for a pair of strategies 𝑥=(1/2,1/2,0) and 𝑦=(0,1/3,1/3,1/3) the expected payoff for player I is ̃𝑔(𝑥,𝑦)=(2,3/4)𝑇.

Now, we define three types of minimax equilibrium strategies based on the fuzzy max order relation (see [7]). A point (𝑥,𝑦)𝑋𝑚×𝑌𝑛 is said to be a minimax equilibrium strategy to game 𝐺 if relations 𝑥T𝐴𝑦𝑞𝑥T𝐴𝑦,𝑥𝑋𝑚,𝑥T𝐴𝑦𝑞𝑥T𝐴𝑦,𝑦𝑌𝑛(2.19) hold.

If (𝑥,𝑦)𝑋𝑚×𝑌𝑛 is the minimax equilibrium strategy to game 𝐺, then a point ̃𝜈=𝑥T𝐴𝑦 is said to be the (fuzzy) value of game 𝐺 and the triplet (𝑥,𝑦,̃𝜈) is said to be a solution of game 𝐺 under the fuzzy max order “”.

A point (𝑥,𝑦)𝑋𝑚×𝑌𝑛 is said to be a nondominated minimax equilibrium strategy to game 𝐺 if (i)there is no 𝑥𝑋𝑚 such that 𝑥T𝐴𝑦𝑥T𝐴𝑦, (ii)there is no 𝑦𝑌𝑛 such that 𝑥T𝐴𝑦𝑥T𝐴𝑦hold.

A point (𝑥,𝑦)𝑋𝑚×𝑌𝑛 is said to be a weak nondominated minimax equilibrium strategy to game 𝐺 if (i)there is no 𝑥𝑋𝑚 such that 𝑥T𝐴𝑦𝑥T𝐴𝑦, (ii)there is no 𝑦𝑌𝑛 such that 𝑥T𝐴𝑦𝑥T𝐴𝑦hold.

By the above definitions, it is clear that if (𝑥,𝑦)𝑋𝑚×𝑌𝑛 is a minimax equilibrium strategy to game 𝐺, it is a nondominated minimax equilibrium strategy, and if (𝑥,𝑦)𝑋𝑚×𝑌𝑛 is a nondominated minimax equilibrium strategy to game 𝐺, then it is a weak nondominated minimax strategy.

Furthermore, if 𝐺 is crisp, that is, game 𝐺 is a two-person zero-sum matrix game, then these definitions coincide and become the definition of the saddle point.

3. The Fictitious Play Algorithm

The solution of matrix games with fuzzy payoffs has been studied by many authors. Most solution techniques are based on linear programming methods (see [3, 811] and references therein).

The Fictitious Play Algorithm is a common technique to approximate calculations for the value of a two-person zero-sum game. In this algorithm, the players choose their strategies in each step 𝑘 assuming that the strategies of the other players in step 𝑘 correspond to the frequency with which the various strategies were applied in the previous 𝑘1 steps. First, Brown (see [12]) conjectured and Robinson (see [13]) proved the convergence of this method for matrix games. This method has also been adapted to interval valued matrix games (see [14]).

Let 𝐺=(𝑔𝑖𝑗) be 𝑚×𝑛 matrix. 𝑔𝑟𝑖 will denote the 𝑖th row of 𝐺 and 𝑔𝑐𝑗 is the 𝑗th column.

A system (𝑈,𝑉) consisting of a sequence of 𝑛-dimensional vectors 𝑈0,𝑈1, and a sequence of 𝑚-dimensional vectors 𝑉0,𝑉1, is called a vector system for 𝐺 provided that (i)min𝑈0=max𝑉0, (ii)𝑈𝑘+1=𝑈𝑘+𝑔𝑟𝑖(𝑘), 𝑉𝑘+1=𝑉𝑘+𝑔𝑐𝑗(𝑘), where 𝑣𝑖(𝑘)=max𝑉𝑘,𝑢𝑗(𝑘)=min𝑈𝑘.(3.1) Here, 𝑣𝑖(𝑘) and 𝑢𝑗(𝑘) denote the 𝑖(𝑘)th and 𝑗(𝑘)th components of the vectors 𝑉 and 𝑈, respectively.

Theorem 3.1 (see [13]). If (𝑈,𝑉) is a vector system for 𝐺 and 𝜈 is the value of 𝐺, then lim𝑘min𝑈𝑘𝑘=lim𝑘max𝑉𝑘𝑘=𝜈.(3.2)

Now, in view of Furukawa's parametric total order relation we will adapt this method for two-person zero-sum games with fuzzy payoffs.

Let 𝐺=II1II2II𝑛I1I2I𝑚̃𝑔11̃𝑔12̃𝑔1𝑛̃𝑔21̃𝑔22̃𝑔2𝑛̃𝑔𝑚1̃𝑔𝑚2̃𝑔𝑚𝑛(3.3) be a fuzzy matrix game whose entries are 𝐿-fuzzy numbers expressed by a common shape function 𝐿. Then a vector system (𝑈,𝑉) for fuzzy matrix 𝐺 is expressed as follows.

Definition 3.2. Let 𝜆[0,1] be fixed. Then for all 𝑘 a pair (𝑈,𝑉) consisting of 𝑛-dimensional 𝐿-fuzzy vectors 𝑈𝑘=(̃𝑢1,̃𝑢2,,̃𝑢𝑛) and 𝑚-dimensional 𝐿-fuzzy vectors 𝑉𝑘̃𝑣=(1,̃𝑣2̃𝑣,,𝑚) provided that Min𝜆𝑈0=Max𝜆𝑉0,𝑈𝑘+1=𝑈𝑘+̃𝑔𝑟𝑖(𝑘),𝑉𝑘+1=𝑉𝑘+̃𝑔𝑐𝑗(𝑘),(3.4) is called a vector system for fuzzy matrix 𝐺. Here, 𝑖(𝑘) and 𝑗(𝑘) satisfy ̃𝑣𝑖(𝑘)=Max𝜆𝑉𝑘,̃𝑢𝑗(𝑘)=Min𝜆𝑈𝑘,(3.5) where ̃𝑔𝑟𝑖 and ̃𝑔𝑐𝑗 denote the 𝑖th row and 𝑗th column of 𝐺, respectively.

Instead of defining 𝑈𝑘 and 𝑉𝑘 simultaneously, a new vector system can be obtained by changing the condition on 𝑗 as ̃𝑢𝑗(𝑘+1)=Min𝜆(𝑈𝑘+1). In numerical calculations, the latter converges more rapidly than the former.

Now, we can state our main theorem, the proof of which resembles the proof of the theorem given in [13, 14].

Theorem 3.3. Let 𝐺 be an 𝑚×𝑛 fuzzy matrix game whose entries are 𝐿-fuzzy numbers expressed by a common shape function 𝐿 and let ̃𝜈 be the value of 𝐺. If (𝑈,𝑉) is a vector system for 𝐺 and 𝜆[0,1] fixed, then lim𝑘Max𝜆𝑉𝑘𝑘=lim𝑘Min𝜆𝑈𝑘𝑘=̃𝜈.(3.6) Here, the convergence is with respect to the Hausdorff metric on 𝔽.

4. A Numerical Example

The best way to demonstrate the Brown-Robinson method for fuzzy matrix games is by means of an example.

Let us consider the modified example of Collins and Hu (see [15]). This shows an investor making a decision as to how to invest a nondivisible sum of money when the economic environment may be categorized into a finite number of states. There is no guarantee that any single value (return on the investment) can adequately model the payoff for any one of the economic states. Hence, it is more realistic to assume that each payoff is a fuzzy number. For this example, it is assumed that the decision of such an investor can be modeled under the assumption that the economic environment (or nature) is, in fact, a rational “player” that will choose an optimal strategy. Suppose that the options for this player are strong economic growth, moderate economic growth, no growth or shrinkage, and negative growth. For the investor player the options are to invest in bonds, invest in stocks, and invest in a guaranteed fixed return account. In this case, clearly a single value for the payoff of either investment in bonds or stock cannot be realistically modeled by a single value representing the percent of return. Hence, a game matrix with fuzzy payoffs better represents the view of the game from both players' perspectives. Consider then the following fuzzy matrix game for this scenario, where the percentage of return is represented in decimal form: (𝐺=BondsStocksFixedStrongModerateNoneNegative0.1230,0.1300)𝑇(0.1420,0.1700)𝑇(0.0450,0)𝑇(0.1025,0.1950)𝑇(0.0310,0.0110)𝑇(0.0450,0)𝑇(0.0555,0.0065)𝑇(0.0310,0.0110)𝑇(0.0450,0)𝑇(0.0260,0.0040)𝑇(0.1250,0.0275)𝑇(0.0450,0)𝑇.(4.1)

We choose 𝜆=0.5 and we first assume that 𝑈0=((0,0.1)𝑇,(0,0.1)𝑇,(0,0.1)𝑇) and 𝑉0=((0,0.1)𝑇,(0,0.1)𝑇,(0,0.1)𝑇,(0,0.1)𝑇). Then Min𝜆(𝑈0)=Max𝜆(𝑉0)=(0,0.1)𝑇.

In the next step (𝑘=1), since all components are the same, we can choose 𝑖(1) and 𝑗(1) as any integer from 1 to 3 and from 1 to 4, respectively. If we choose 𝑖(1)=1 and 𝑗(1)=1, then we find 𝑈1=𝑈0+̃𝑔𝑟𝑖(1)=(0,0.1)𝑇,(0,0.1)𝑇,(0,0.1)𝑇+(0.1230,0.1300)𝑇,(0.1420,0.1700)𝑇,(0.0450,0)𝑇=(0.1230,0.2300)𝑇,(0.1420,0.2700)𝑇,(0.0450,0.1)𝑇,𝑉1=𝑉0+̃𝑔𝑐𝑗(1)=(0,0.1)𝑇,(0,0.1)𝑇,(0,0.1)𝑇,(0,0.1)𝑇+(0.1230,0.1300)𝑇,(0.1025,0.1950)𝑇,(0.0555,0.0065)𝑇,(0.0260,0.0040)𝑇=(0.1230,0.2300)𝑇,(0.1025,0.2950)𝑇,(0.0555,0.1065)𝑇,(0.0260,0.1040)𝑇.(4.2)

In the second step, we get Min𝜆𝑈1=Min𝜆(0.1230,0.2300)𝑇,(0.1420,0.2700)𝑇,(0.0450,0.1)𝑇=(0.0450,0.1)𝑇,Max𝜆𝑉1=Max𝜆(0.1230,0.2300)𝑇,(0.1025,0.2950)𝑇,(0.0555,0.1065)𝑇,(0.0260,0.1040)𝑇=(0.1230,0.2300)𝑇.(4.3)

Therefore, we obtain 𝑖(2)=1, 𝑗(2)=3 and 𝑈2=𝑈1+̃𝑔𝑟i(2)=(0.1230,0.2300)𝑇,(0.1420,0.2700)𝑇,(0.0450,0.1)𝑇+(0.1230,0.1300)𝑇,(0.1420,0.1700)𝑇,(0.0450,0)𝑇=(0.2460,0.3600)𝑇,(0.2840,0.4400)𝑇,(0.0900,0.1000)𝑇,𝑉2=𝑉1+̃𝑔𝑐𝑗(2)=(0.1230,0.2300)𝑇,(0.1025,0.2950)𝑇,(0.0555,0.1065)𝑇,(0.0260,0.1040)𝑇+(0.1420,0.1700)𝑇,(0.0310,0.0110)𝑇,(0.0310,0.0110)𝑇,(0.1250,0.0275)𝑇=(0.1680,0.2300)𝑇,(0.1485,0.2950)𝑇,(0.1025,0.1065)𝑇,(0.0740,0.1040)𝑇.(4.4) Continuing in this way, and using the Maple computer algebra system, we build up Table 1. By virtue of Table 1, we see that the value of the game approaches the crisp number (0.045,0)𝑇.

tab1
Table 1: The Brown-Robinson method for solving the example.

5. Conclusion

In this paper, we have adapted the Brown-Robinson method to fuzzy matrix games. It is shown that by means of this method, the value of fuzzy matrix games can be easily calculated. Although the method is no way as useful as linear programming for calculating the solution of the game exactly, it is an interesting result and the method can be easily programmed by novice programmers. In addition, linear programming methods are not efficient enough for high dimensional fuzzy matrix games, but the Brown-Robinson method can be used even if the matrix dimension is too high.

Acknowledgment

This work is supported by the Anadolu University Research Foundation under Grant no. 1103F059.

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