A Fictitious Play Algorithm for Matrix Games with Fuzzy Payoffs

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 𝑋_𝑚=𝑥=𝑥₁,…,𝑥_𝑚∈ℝ^𝑚∶𝑥_𝑖≥0∀𝑖=1,…,𝑚,^𝑚_𝑖=1𝑥_𝑖=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 𝑌_𝑛=𝑦=𝑦₁,…,𝑦_𝑛∈ℝ^𝑛∶𝑦_𝑖≥0∀𝑖=1,…,𝑛,^𝑛_𝑖=1𝑦_𝑖=1.(1.2) A strategy 𝑥 of player I is called pure if it does not involve probability, that is, 𝑥_𝑖=1 for some 𝑖=1,…,𝑚 and it is denoted by I_𝑖. Similarly, pure strategies of player II are denoted by II_𝑗 for 𝑗=1,…,𝑛.

If player I plays row 𝑖 (i.e., pure strategy 𝑥=(0,0,…,𝑥_𝑖=1,0,…,0)) and player II plays column 𝑗 (i.e., pure strategy 𝑦=(0,0,…,𝑦_𝑗=1,0,…,0)), then player I receives payoff 𝑔_𝑖𝑗 and player II pays 𝑔_𝑖𝑗, where 𝑔_𝑖𝑗 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)=𝑥^T𝐺𝑦,(1.3) where 𝑥^T denotes the transpose of 𝑥.

A strategy 𝑥^∗ is called maximin strategy of player I in matrix game 𝐺 if min𝑥^∗^T𝐺𝑦,𝑦∈𝑌_𝑛≥min𝑥^T𝐺𝑦,𝑦∈𝑌_𝑛,(1.4) for all 𝑥∈𝑋_𝑚 and a strategy 𝑦^∗ is called minimax strategy of player II in matrix game 𝐺 if max𝑥^T𝐺𝑦^∗,𝑥∈𝑋_𝑚≤max𝑥^T𝐺𝑦,𝑥∈𝑋_𝑚(1.5) for all 𝑦∈𝑌_𝑛. 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., 𝑥^T𝐺𝑦≥𝜈 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., 𝑥^T𝐺𝑦≤𝜈 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 𝑔_𝑖𝑗≥𝑔_𝑘𝑗 for all 𝑘=1,…,𝑚 and 𝑔_𝑖𝑗≤𝑔_𝑖𝑙 for all 𝑙=1,…,𝑛, that is, if 𝑔_𝑖𝑗 is maximal in its column 𝑗 and minimal in its row 𝑖. Evidently, if (𝑖,𝑗) is a saddle point, then 𝑔_𝑖𝑗 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 𝐴∶𝑋→[0,1]. Generally, the symbol 𝜇_𝐴 is used for the function 𝐴 and it is said that the fuzzy set 𝐴 is characterized by its membership function 𝜇_𝐴∶𝑋→[0,1] which associates with each 𝑥∈𝑋, a real number 𝜇_𝐴(𝑥)∈[0,1]. The value of 𝜇_𝐴(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 𝑆𝐴=𝑥∈𝑋∶𝜇_𝐴(𝑥)>0,(2.1) and the height h(𝐴) of 𝐴 is defined as h𝐴=sup_𝑥∈𝑋𝜇_𝐴(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 𝐴^𝛼=𝑥∈𝑋∶𝜇^𝐴cl𝑆𝐴,(𝑥)≥𝛼,iiff𝛼𝛼∈=0(,0,1](2.3) where cl denotes the closure of sets.

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

Let 𝐴 be a fuzzy set in ℝ, then 𝐴 is called a fuzzy number if (i)𝐴 is normal, (ii)𝐴 is convex, (iii)𝜇_𝐴 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 𝑎̃ 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∶ℝ→ℝ satisfying the following conditions is called a shape function: (i)𝐿 is even function, that is, 𝐿(𝑥)=𝐿(−𝑥) for all 𝑥∈ℝ, (ii)𝐿(𝑥)=1⇔𝑥=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 𝑎̃ is called an 𝐿-fuzzy number if its membership function is given by 𝜇_̃𝑎(𝑥)=𝐿𝑥−𝑎𝛿∨0,𝑥∈ℝ.(2.4) Here, 𝑥∨𝑦=max{𝑥,𝑦}. Real numbers 𝑎 and 𝛿 are called the center and the deviation parameter of 𝑎̃, respectively. In particular, if 𝐿(𝑥)=1−|𝑥| we get 𝜇_̃𝑎(𝑥)=10,−𝛿1|𝑥−𝑎|,𝑥oth∈er[w𝑎i−se𝛿,𝑎+𝛿](2.5) and 𝑎̃ is called a symmetric triangular fuzzy number.

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

Let 𝑎̃≡(𝑎,𝛿)_𝐿 be an 𝐿-fuzzy number then by (2.4) we see that the graph of 𝜇_̃𝑎(𝑥) approaching line 𝑥=𝑎 as 𝛿 tends to zero from the right. Therefore, we can write that 𝜇_̃𝑎(𝑥)=10,,𝑥𝑥≠=𝑎𝑎.(2.6) The function in (2.6) is just a characteristic function of the real number 𝑎. Hence, we get ℝ⊂𝔽_𝐿. From now on, we will call a fuzzy number 𝑎̃ as an 𝐿-fuzzy number if its membership function is given by (2.4) or (2.6).

Let 𝑎,̃𝑏̃∈𝔽 and 𝑘 be any real number. Then the sum of fuzzy numbers 𝑎̃ and 𝑏̃ and the scalar product of 𝑘 and 𝑎̃ are defined as 𝜇𝜇_̃𝑎+𝑏̃(𝑧)=sup_{𝑧=𝑥+𝑦}min𝜇_̃𝑎(𝑥),𝜇_̃𝑏(𝑦)_𝑘,𝑎̃(𝑧)=max0,sup_{𝑧=𝑘𝑥}𝜇_̃𝑎(𝑥),(2.7) respectively. In particular, if 𝑎̃≡(𝑎,𝛿₁)_𝐿 and 𝑏̃≡(𝑏,𝛿₂)_𝐿 are 𝐿-fuzzy numbers and 𝑘 is any real number, then one can verify that 𝑎̃+𝑘𝑏𝑎̃≡̃≡𝑎𝑘+𝑎,𝑏||𝑘,𝛿||₁𝛿₁+𝛿₂_𝐿,_𝐿.(2.8)

Let 𝑎̃ 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 𝑎̃ by [𝑎^𝐿𝛼,𝑎^𝑅𝛼], where 𝑎^𝐿𝛼 and 𝑎^𝑅𝛼 are end points of the closed interval [𝑎]̃^𝛼.

For any symmetric triangular fuzzy numbers 𝑎,̃𝑏̃ Ramík and Římánek (see [4]) introduced binary relations as follows:𝑎̃≻𝑎𝑎̃̃≽𝑏⪴̃⟺𝑏̃𝑞⟺𝑏̃𝑎⟺_𝐿𝛼𝑎̃⪴𝑎^𝐿𝛼𝑞≧𝑏,̃𝑏𝑎^𝐿𝛼̃,≠𝑎𝑏^𝑅𝛼̃≧∀𝑏𝛼^𝑅𝛼∈∀[0𝛼,∈1][(0s,tr1i]ct(ffuuzzzzyymmaaxxoorrddeerr)),,>𝑏^𝐿𝛼,𝑎^𝑅𝛼>𝑏^𝑅𝛼∀𝛼∈[0,1](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 𝑎̃≡(𝑎,𝛿₁) and 𝑏̃≡(𝑏,𝛿₂) be any symmetric triangular fuzzy numbers. Then the statements 𝑎̃𝑎⪴̃≻𝑞𝑏𝑏̃⟺̃⟺||𝛿||𝛿₁₁−𝛿₂||≦𝑎−𝑏,−𝛿₂||<𝑎−𝑏(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 0≤𝜆≤1 be arbitrary but a fixed real number. For any 𝐿-fuzzy numbers 𝑎̃≡(𝑎,𝛿₁)_𝐿 and 𝑏̃≡(𝑏,𝛿₂)_𝐿 we define an order relation with parameter 𝜆 by 𝑎≤̃_𝜆𝑏̃⟺⎧⎪⎪⎨⎪⎪⎩(i)𝑥⁰o(iri)𝜆𝑥||𝛿₁₀−𝛿₂||≤𝑏−𝑎,o(iri)||𝑎−||𝑏𝛿||₁<−𝜆𝛿𝑥₂||₀≤𝑏−𝑎<𝑥₀||𝛿₁−𝛿₂||,||𝛿₁−𝛿₂||,𝛿₂>𝛿₁,(2.11) where 𝑥₀ is the zero point of 𝐿. The simple expression of (2.11) is as follows: 𝑎≤̃_𝜆𝑏̃⟺⎧⎪⎨⎪⎩(i)𝜆𝑥⁰o(iri)𝜆𝑥₀𝛿₁+𝑎<𝜆𝑥₀𝛿₂+𝑏,𝛿₁+𝑎=𝜆𝑥₀𝛿₂+𝑏,𝛿₂≤𝛿₁.(2.12)

It is clear that for any 𝐿-fuzzy numbers 𝑎̃≡(𝑎,𝛿₁)_𝐿 and 𝑏̃≡(𝑏,𝛿₂)_𝐿, 𝑎≤̃₀𝑏̃ if and only if 𝑎≤𝑏. Therefore, the relation ≤₀ is the order among the centers of 𝐿-fuzzy numbers. On the other hand, 𝑎≤̃₁𝑏̃ if and only if 𝑏̃⪴𝑎̃ 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 𝔽_𝐿.

Let 𝜆∈[0,1] be fixed arbitrarily and let 𝑉=(𝑣₁̃,𝑣₂̃,…,𝑣_𝑛̃) 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 Max_𝜆𝑉,Min_𝜆𝑉,(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 Max_𝜆𝑉=(0,0.1)^𝐿,Min_𝜆𝑉=(−3,0.1)^𝐿,(2.14) for all 𝜆∈[0,1].

Let 𝑎̃ and 𝑏̃ be any 𝐿-fuzzy numbers, then the Hausdorff distance between 𝑎̃ and 𝑏̃ is defined as 𝑑𝑎,̃𝑏̃=sup_[]𝛼∈0,1max||𝑎^𝐿𝛼−𝑏^𝐿𝛼||,||𝑎^𝑅𝛼−𝑏^𝑅𝛼||,(2.15) that is, 𝑑(𝑎,̃𝑏)̃ is the maximal distance between 𝛼-cuts of 𝑎̃ and 𝑏̃. In particular, if 𝑎̃≡(𝑎,𝛿₁) and 𝑏̃≡(𝑏,𝛿₂) are any symmetric triangular fuzzy numbers, then 𝑑(𝑎,̃𝑏)̃=|𝑎−𝑏|.

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=II¹II₂⋯II_𝑛I₁I₂I⋮_𝑚⎛⎜⎜⎝𝑔̃¹¹𝑔̃₁₂⋯𝑔̃_1𝑛𝑔̃₂₁𝑔̃₂₂⋯𝑔̃_2𝑛𝑔⋮̃_𝑚1⋮⋱⋮𝑔̃_𝑚2⋯𝑔̃_𝑚𝑛⎞⎟⎟⎠(2.16) and 𝑥∈𝑋_𝑚, 𝑦∈𝑌_𝑛, that is, 𝑥 and 𝑦 are strategies for players I and II. Then the expected payoff for player I is defined by 𝑔̃(𝑥,𝑦)=𝑥^T𝐺𝑦=_𝑖_𝑗𝑥_𝑖𝑦_𝑗𝑔̃_𝑖𝑗.(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 𝑔̃(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 𝑔̃(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^∗)∈𝑋_𝑚×𝑌_𝑛 is said to be a minimax equilibrium strategy to game 𝐺 if relations 𝑥𝑥^∗T_∗T𝐴𝑦^∗⪴𝑞𝑥^T𝐴𝑦^∗,∀𝑥∈𝑋_𝑚,𝐴𝑦⪴𝑞𝑥^∗T𝐴𝑦^∗,∀𝑦∈𝑌_𝑛(2.19) hold.

If (𝑥^∗,𝑦^∗)∈𝑋_𝑚×𝑌_𝑛 is the minimax equilibrium strategy to game 𝐺, then a point 𝜈̃=𝑥^∗T𝐴𝑦^∗ is said to be the (fuzzy) value of game 𝐺 and the triplet (𝑥^∗,𝑦^∗,𝜈̃) is said to be a solution of game 𝐺 under the fuzzy max order “⪴”.

A point (𝑥^∗,𝑦^∗)∈𝑋_𝑚×𝑌_𝑛 is said to be a nondominated minimax equilibrium strategy to game 𝐺 if (i)there is no 𝑥∈𝑋_𝑚 such that 𝑥^T𝐴𝑦^∗≽𝑥^∗T𝐴𝑦^∗, (ii)there is no 𝑦∈𝑌_𝑛 such that 𝑥^∗T𝐴𝑦^∗≽𝑥^∗T𝐴𝑦hold.

A point (𝑥^∗,𝑦^∗)∈𝑋_𝑚×𝑌_𝑛 is said to be a weak nondominated minimax equilibrium strategy to game 𝐺 if (i)there is no 𝑥∈𝑋_𝑚 such that 𝑥^T𝐴𝑦^∗≻𝑥^∗T𝐴𝑦^∗, (ii)there is no 𝑦∈𝑌_𝑛 such that 𝑥^∗T𝐴𝑦^∗≻𝑥^∗T𝐴𝑦hold.

By the above definitions, it is clear that if (𝑥^∗,𝑦^∗)∈𝑋_𝑚×𝑌_𝑛 is a minimax equilibrium strategy to game 𝐺, it is a nondominated minimax equilibrium strategy, and if (𝑥^∗,𝑦^∗)∈𝑋_𝑚×𝑌_𝑛 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, 8–11] 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 𝑘−1 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=(𝑔_𝑖𝑗) be 𝑚×𝑛 matrix. 𝑔^𝑟𝑖 will denote the 𝑖th row of 𝐺 and 𝑔^𝑐𝑗 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)𝑈_𝑘+1=𝑈_𝑘+𝑔^{𝑟𝑖(𝑘)}, 𝑉_𝑘+1=𝑉_𝑘+𝑔^{𝑐𝑗(𝑘)}, where 𝑣_𝑖(𝑘)=max𝑉_𝑘,𝑢_𝑗(𝑘)=min𝑈_𝑘.(3.1) Here, 𝑣_𝑖(𝑘) and 𝑢_𝑗(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 lim_𝑘→∞min𝑈_𝑘𝑘=lim^𝑘→∞max𝑉_𝑘𝑘=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=II¹II₂⋯II_𝑛I₁I₂I⋮_𝑚⎛⎜⎜⎝𝑔̃¹¹𝑔̃₁₂⋯𝑔̃_1𝑛𝑔̃₂₁𝑔̃₂₂⋯𝑔̃_2𝑛𝑔⋮̃_𝑚1⋮⋱⋮𝑔̃_𝑚2⋯𝑔̃_𝑚𝑛⎞⎟⎟⎠(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∈ℕ a pair (𝑈,𝑉) consisting of 𝑛-dimensional 𝐿-fuzzy vectors 𝑈_𝑘=(𝑢₁̃,𝑢₂̃,…,𝑢_𝑛̃) and 𝑚-dimensional 𝐿-fuzzy vectors 𝑉_𝑘=(𝑣₁̃,𝑣₂̃,…,𝑣_𝑚̃) provided that 𝑈_𝑘+1Min_𝜆𝑈⁰=Max_𝜆𝑉⁰,=𝑈_𝑘+𝑔̃^{𝑟𝑖(𝑘)},𝑉_𝑘+1=𝑉_𝑘+𝑔̃^{𝑐𝑗(𝑘)},(3.4) is called a vector system for fuzzy matrix 𝐺. Here, 𝑖(𝑘) and 𝑗(𝑘) satisfy 𝑣̃_𝑖(𝑘)=Max_𝜆𝑉^𝑘,𝑢̃^𝑗(𝑘)=Min_𝜆𝑈^𝑘,(3.5) where 𝑔̃^𝑟𝑖 and 𝑔̃^𝑐𝑗 denote the 𝑖th row and 𝑗th column of 𝐺, respectively.

Instead of defining 𝑈_𝑘 and 𝑉_𝑘 simultaneously, a new vector system can be obtained by changing the condition on 𝑗 as 𝑢̃_𝑗(𝑘+1)=Min_𝜆(𝑈_𝑘+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 𝜈̃ be the value of 𝐺. If (𝑈,𝑉) is a vector system for 𝐺 and 𝜆∈[0,1] fixed, then lim_𝑘→∞Max_𝜆𝑉^𝑘𝑘=lim^𝑘→∞Min_𝜆𝑈^𝑘𝑘=𝜈̃.(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=MNSNeotgdrooaentnrieagvtee⎛⎜⎜⎝(0.1230,0B.1o3n0d0s)^𝑇StocksFixed(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 𝑈₀=((0,0.1)_𝑇,(0,0.1)_𝑇,(0,0.1)_𝑇) and 𝑉₀=((0,0.1)_𝑇,(0,0.1)_𝑇,(0,0.1)_𝑇,(0,0.1)_𝑇). Then Min_𝜆(𝑈₀)=Max_𝜆(𝑉₀)=(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 𝑈₁=𝑈₀+𝑔̃^𝑟𝑖(1)=+(0,(00..11)2_𝑇3,0(,00,.01.310)_𝑇0,)_𝑇(0,0.1)_𝑇=(0.1230,0.2300)_𝑇,(0.1420,0.1700)_𝑇,(0.0450,0)_𝑇𝑉₁,(0.1420,0.2700)_𝑇,(0.0450,0.1)_𝑇,=𝑉₀+𝑔̃^𝑐𝑗(1)=+(0,(00..11)2_𝑇3,0(,00,.01.310)_𝑇0,)_𝑇(0,0.1)_𝑇,(0,0.1)_𝑇=(0.1230,0.2300)_𝑇,(0.1025,0.1950)_𝑇,(0.0555,0.0065)_𝑇,(0.0260,0.0040)_𝑇,(0.1025,0.2950)_𝑇,(0.0555,0.1065)_𝑇,(0.0260,0.1040)_𝑇.(4.2)

In the second step, we get Min_𝜆𝑈¹==M(0.i0n_𝜆450(,00.1.12)3_𝑇0,0.2300)_𝑇,(0.1420,0.2700)_𝑇,(0.0450,0.1)_𝑇Max_𝜆,𝑉¹==M(0.a1x_𝜆230(,00..12233000,)0_𝑇.2300)_𝑇,(0.1025,0.2950)_𝑇,(0.0555,0.1065)_𝑇,(0.0260,0.1040)_𝑇.(4.3)

Therefore, we obtain 𝑖(2)=1, 𝑗(2)=3 and 𝑈₂=𝑈₁+𝑔̃^𝑟_i(2)=+(0.(102.13203,00.,203.10300)_𝑇0,)(_𝑇0.1420,0.2700)_𝑇,(0.0450,0.1)_𝑇=(0.2460,0.3600)_𝑇,(0.1420,0.1700)_𝑇,(0.0450,0)_𝑇𝑉₂,(0.2840,0.4400)_𝑇,(0.0900,0.1000)_𝑇,=𝑉₁+𝑔̃^𝑐𝑗(2)=+(0.(102.13402,00.,203.10700)_𝑇0,)(_𝑇0.1025,0.2950)_𝑇,(0.0555,0.1065)_𝑇,(0.0260,0.1040)_𝑇=(0.1680,0.2300)_𝑇,(0.0310,0.0110)_𝑇,(0.0310,0.0110)_𝑇,(−0.1250,0.0275)_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.

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.