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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 950482, 12 pages
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.
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.
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 and player and it can be denoted by a matrix. Thus, such a game is called a matrix game. More formally, a matrix game is an matrix of real numbers. A (mixed) strategy of player is a probability distribution over the rows of , that is, an element of the set Similarly, a strategy of player is a probability distribution over the columns of , that is, an element of the set A strategy of player is called pure if it does not involve probability, that is, for some and it is denoted by . Similarly, pure strategies of player are denoted by for .
If player plays row (i.e., pure strategy ) and player plays column (i.e., pure strategy ), then player receives payoff and player pays , where is the entry in row and column of matrix . If player plays strategy and player plays strategy , then player receives the expected payoff where denotes the transpose of .
A strategy is called maximin strategy of player in matrix game if for all and a strategy is called minimax strategy of player in matrix game if for all . Therefore, a maximin strategy of player maximizes the minimal payoff of player , and a minimax strategy of player minimizes the maximum that player has to pay to player .
von Neumann and Morgenstern (see ) proved that for every matrix game there is a real number with the following properties. (i)A strategy of player guarantees a payoff of at least to player (i.e., for all strategies of player ) if and only if is a maximin strategy. (ii)A strategy of player guarantees a payment of at most by player to player (i.e., for all strategies of player ) if and only if is a minimax strategy.
Hence, player can obtain a payoff at least by playing a maximin strategy, and player can guarantee to pay not more than by playing a minimax strategy. For these reasons, the number is also called the value of the game .
A position is called a saddle point if for all and for all , that is, if is maximal in its column and minimal in its row . Evidently, if is a saddle point, then 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
A fuzzy set on a set is a function . Generally, the symbol is used for the function and it is said that the fuzzy set is characterized by its membership function which associates with each , a real number . The value of is interpreted as the degree to which belongs to .
Let be a fuzzy set on . The support of is given as and the height of is defined as If , then the fuzzy set is called a normal fuzzy set.
Let be a fuzzy set on and . The -cut (-level set) of the fuzzy set is given by where denotes the closure of sets.
The notion of convexity is extended to fuzzy sets on as follows. A fuzzy set on is called a convex fuzzy set if its -cuts are convex sets for all .
Let be a fuzzy set in , then is called a fuzzy number if (i) is normal, (ii) is convex, (iii) is upper semicontinuous, and(iv)the support of is bounded.
From now on, we will use lowercase letters to denote fuzzy numbers such as and we will denote the set of all fuzzy numbers by the symbol . Generally, some special type of fuzzy numbers, such as trapezoidal and triangular fuzzy numbers, are used for real life applications. We consider here -fuzzy numbers.
The function satisfying the following conditions is called a shape function: (i) is even function, that is, for all , (ii), (iii) is nonincreasing on , (iv)if , then and is called the zero point of .
Let be any number and let be any positive number. Let be any shape function. Then a fuzzy number is called an -fuzzy number if its membership function is given by Here, . Real numbers and are called the center and the deviation parameter of , respectively. In particular, if we get and is called a symmetric triangular fuzzy number.
It is clear that for any shape function , an arbitrary -fuzzy number can be characterized by the its center and the deviation parameter . Therefore, we denote the -fuzzy number by . In particular, if is a symmetric triangular fuzzy number, we write . We also denote the set of all -fuzzy numbers by .
Let be an -fuzzy number then by (2.4) we see that the graph of approaching line as tends to zero from the right. Therefore, we can write that The function in (2.6) is just a characteristic function of the real number . Hence, we get . From now on, we will call a fuzzy number as an -fuzzy number if its membership function is given by (2.4) or (2.6).
Let and be any real number. Then the sum of fuzzy numbers and and the scalar product of and are defined as respectively. In particular, if and are -fuzzy numbers and is any real number, then one can verify that
Let be any -fuzzy number. By the definition of the -cut, is a closed interval for all . Therefore, for all we can denote the -cut of by , where and are end points of the closed interval .
For any symmetric triangular fuzzy numbers Ramík and Římánek (see ) introduced binary relations as follows: 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 ). Let and be any symmetric triangular fuzzy numbers. Then the statements 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 be arbitrary but a fixed real number. For any -fuzzy numbers and we define an order relation with parameter by where is the zero point of . The simple expression of (2.11) is as follows:
It is clear that for any -fuzzy numbers and , if and only if . Therefore, the relation is the order among the centers of -fuzzy numbers. On the other hand, if and only if or they are incomparable and . For , the relation determines the order with respect to their values of center and their size of ambiguity. The smaller is, the larger the possibility of ordering by the value of center is, and the larger is, the larger the possibility of ordering by the size of ambiguity is.
Theorem 2.2 (see ). For every shape function and for each , the relation is a total order relation on .
Let be fixed arbitrarily and let be any -fuzzy vector, that is, all components of are -fuzzy numbers and expressed by a common shape function . Then maximum and minimum of in the sense of the total order are denoted as respectively.
Example 2.3. Let be -fuzzy vector. Then for all .
Let and be any -fuzzy numbers, then the Hausdorff distance between and is defined as that is, is the maximal distance between -cuts of and . In particular, if and are any symmetric triangular fuzzy numbers, then .
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 matrix whose entries are fuzzy numbers. Let be a fuzzy matrix game and , , that is, and are strategies for players and . Then the expected payoff for player is defined by
Example 2.4. Let
be a fuzzy matrix game whose entries are symmetric triangular fuzzy numbers.
For this game, if player plays second row () and player plays third column (), then player receives and correspondingly player pays a payoff . On the other hand, for a pair of strategies and the expected payoff for player is .
Now, we define three types of minimax equilibrium strategies based on the fuzzy max order relation (see ). A point is said to be a minimax equilibrium strategy to game if relations hold.
If is the minimax equilibrium strategy to game , then a point is said to be the (fuzzy) value of game and the triplet is said to be a solution of game under the fuzzy max order “”.
A point is said to be a nondominated minimax equilibrium strategy to game if (i)there is no such that , (ii)there is no such that hold.
A point is said to be a weak nondominated minimax equilibrium strategy to game if (i)there is no such that , (ii)there is no such that hold.
By the above definitions, it is clear that if is a minimax equilibrium strategy to game , it is a nondominated minimax equilibrium strategy, and if is a nondominated minimax equilibrium strategy to game , then it is a weak nondominated minimax strategy.
Furthermore, if is crisp, that is, game 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 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 assuming that the strategies of the other players in step correspond to the frequency with which the various strategies were applied in the previous steps. First, Brown (see ) conjectured and Robinson (see ) proved the convergence of this method for matrix games. This method has also been adapted to interval valued matrix games (see ).
Let be matrix. will denote the th row of and is the th column.
A system consisting of a sequence of -dimensional vectors and a sequence of -dimensional vectors is called a vector system for provided that (i), (ii), , where Here, and denote the th and th components of the vectors and , respectively.
Theorem 3.1 (see ). If is a vector system for and is the value of , then
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 be a fuzzy matrix game whose entries are -fuzzy numbers expressed by a common shape function . Then a vector system for fuzzy matrix is expressed as follows.
Definition 3.2. Let be fixed. Then for all a pair consisting of -dimensional -fuzzy vectors and -dimensional -fuzzy vectors provided that is called a vector system for fuzzy matrix . Here, and satisfy where and denote the th row and th column of , respectively.
Instead of defining and simultaneously, a new vector system can be obtained by changing the condition on as . In numerical calculations, the latter converges more rapidly than the former.
Theorem 3.3. Let be an fuzzy matrix game whose entries are -fuzzy numbers expressed by a common shape function and let be the value of . If is a vector system for and fixed, then Here, the convergence is with respect to the Hausdorff metric on .
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 ). 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:
We choose and we first assume that and . Then .
In the next step (), since all components are the same, we can choose and as any integer from 1 to 3 and from 1 to 4, respectively. If we choose and , then we find
In the second step, we get
Therefore, we obtain , and 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 .
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.
This work is supported by the Anadolu University Research Foundation under Grant no. 1103F059.
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