A fuzzy approach to strategic games - Semantic Scholar

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 6, DECEMBER 1999

A Fuzzy Approach to Strategic Games Qian Song and Abraham Kandel Fellow, IEEE

Abstract— A game is a decision-making situation with many players, each having objectives that conflict with each other. The players involved in the game usually make their decisions under conditions of risk or uncertainty. In this paper, a fuzzy approach is proposed to solve the strategic game problem in which the pure strategy set for each player is already defined. Based on the concepts of fuzzy set theory, this approach will use a multicriteria decision-making method to obtain the optimal strategy in the game, a method which shows more advantages than the classical game methods. Moreover, with this approach, some useful conclusions are reached concerning the famous “prisoner’s dilemma” problem in game theory. Index Terms— Fuzzy set, game, membership function, prisoner’s dilemma, strategic form.

I. INTRODUCTION

A

game is a description of a decision-making situation involving more than one decision maker. Game theory is generally considered to have begun with the publication of von Neumann and Morgenstern’s The Theory of Games and Economic Behavior [1] in 1944. The development of game theory accelerated in papers by Nash [2]–[4], Shapley [5], [6], and Gillies [7]. The behavior of players in a game is assumed to be rational and influenced by other rational players’ behaviors, which distinguishes a game from the general decision-making problem. The decision makers, in general decision-making problems, face only an anonymous nature, which is a set of relevant states. Certainly, this nature is not rational. There are three ways to represent the interaction of players in a game. First, extensive form is the most complete description of the situation in a game. This form not only details the information available to and motivation of players, but it also shows the various stages of the interaction and conditions for a player to move. Second, strategic form is more abstract than extensive form; it only gives all possible strategies of each player, along with the payoffs that result from the strategy choices of players. Finally, characteristic function form describes the social interactions in which players can make agreements with each other. This description is usually used for representing coalitions in cooperative games [8]. In this paper, we study games in strategic form, which may generally correspond to a number of different extensive form games. In order to analyze the behaviors of players and construct a method for each player to choose his action, a strategic Manuscript received November 11, 1998; revised June 15, 1999. The authors are with the Department of Computer Science and Engineering, University of South Florida, Tampa, FL 33620-5350 USA. Publisher Item Identifier S 1063-6706(99)09881-1.

game first defines each individual’s alternative actions. The combination of all the players’ strategies will determine a unique outcome to the game and the payoffs to all players. A solution for a player in a game should allow that player to win or satisfy his objectives for the game. For example, he can maximize his own payoff and/or minimize his opponent’s payoff. Formally, the solution of a game is a situation in which each player plays a best response to the other players’ actual strategy choices. This is the concept of equilibrium. There are several methods for obtaining the equilibria of some special kinds of games such as the dominant strategy for dominant strategy equilibrium or a mixed strategy for mixed strategy equilibrium. A linear programming method is used in matrix games. Most of these methods are based on the maximin principle for selecting optimal strategies. But in 1978, Butnariu [9] pointed out that one of the major assumptions of classical game theory is that all possible choosing strategies are equally possible choices for a player. In reality, there are other rules for selecting optimal strategies and considerations (such as moral, aesthetic, or philosophical motives) that may render some strategies more or less acceptable to a player. Furthermore, all classical methods are only used in special kinds of games and are only based on the crisp set (real number set) of solution space. In fact, most of the problems in practice contain vagueness, imprecision, and uncertainty that cannot be expressed efficiently in crisp numbers. The notion of fuzzy sets first appeared in the papers written by Zadeh [10]–[12]. This notion tries to show that an object more or less corresponds to a particular category. The degree to which an element belongs to a category is an element of the rather than the Boolean pair continuous interval [13]. Using the notion of fuzzy sets, the payoff function in a game can be fuzzified. Furthermore, the solution to the game may also be a fuzzy set. In this paper, we introduce the concept of conditional fuzzy sets, which is similar to the conditional distribution in probability and statistics. The optimal solution to the game is found using this fuzzy model. At the same time, this concept allows us to obtain the solution for a player in a game. Compared with the models in classical game theory, this is a more appropriate method for representing the vagueness of knowledge in a game. At the same time, it does not introduce too many solutions. For example, in the Nash equilibrium solution method, we may have too many equilibria and, hence, no way for the player to decide on an action. Another advantage of conditional fuzzy sets is that the knowledge of a player’s payoff value need not be a precise number. Hence, the fuzzy model is more appropriate to represent the real situation. This fuzzy approach combines the multiple goals of a player

1063–6706/99$10.00  1999 IEEE

SONG AND KANDEL: FUZZY APPROACH TO STRATEGIC GAMES

into one fuzzy model by using the multicriteria method. Using the weight vector to represent the philosophical motives or moral characteristics of a player makes it more general than the maximin principle in classical game theory. By applying this method to the prisoner’s dilemma problem, we obtain new results in choosing an optimal strategy given the payoff matrix and weight vector of prisoner. Relative fuzzy concepts will be given at the beginning of Section II, along with our fuzzy approach. In order to explain the approach clearly, we will give two examples. In Section III, the fuzzy approach is used for prisoner’s dilemma problem. From the results, some useful conclusions can be drawn. Section IV ends the paper with some conclusions on the fuzzy approach, pointing out some open questions for future researchers.

II. FUZZY APPROACH

TO

STRATEGIC GAMES

A strategy is a complete plan for a player to decide how to play the game [8]. Without detailed knowledge of the sequence of moves in a strategic game, we can still analyze the behaviors of players; all that is needed for a solution to a game is to indicate what the player would do in the situation when he must make a move. So it is sufficient if we list each individual’s strategies, the combination of which will determine a unique outcome (payoff to each player) of a game. A game is in strategic form if only the set of players I, the set and the payoff functions for each player of strategies are given, where . The concepts of games in strategic form and mixed strategies used in this paper, as well as the notations, are as defined and described in [8]. It is clear from [8] that the concept of Nash equilibrium creates problems when there are too many equilibria in one game. Some reason for this difficulty is that we do not consider other features of the players, such as moral or philosophical motives, etc. We assume that all possible strategies are equally possible to each player, which is not very appropriate in the classical game. Another disadvantage of the classical game method is the maximin principle. In 1971, Rawls assumed [15] that each participant in the original position would choose among alternative concepts of justice on the basis of the highly irrational maximin principle, which requires every player to act in such a way as if he were absolutely sure that whatever he did, the worst possible outcome of his action would be obtained. This is not very realistic because Rawls is supposedly looking for that concept of justice that rational individuals would choose in the original position. In this section, we do recommend a fuzzy approach, which reduces the shortcomings of classical game methods. Before the first publications of fuzzy set theory by Zadeh [10] in 1965, probability had been the only means for expressing uncertainty. L. Zadeh proposed the concept of a fuzzy set that focuses not only on the uncertainty property, but also on the imprecision that is intrinsic to natural language, imprecision that is assumed to be possiblistic rather than probabilistic [18]. In classical game theory such as in a mixed strategy game, a mixed strategy of player is a vector with and .

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The meaning of this mixed strategy is a representation of the probability of choosing each pure strategy. So it can be noted that all elements of are equally probable to the player , which, in the real world, is not always true. A player’s moral, aesthetic, or philosophical motives should also be considered in deciding the optimal strategy. This may result in some strategy being more or less likely to be chosen by a player. Probability and statistics now are insufficient to represent the ambiguity and vagueness in such models, especially for linguistically represented imprecision. Fortunately, we have another utility to measure the uncertainty: the fuzzy set theory. Indeed, imprecision is used in the sense of vagueness rather than vague conceptual phenomena that can be precisely and rigorously studied. A classical (crisp) set is defined as a collection of elements . The number of elements in the set can or objects be finite, countable, or uncountable. We note that each single . element can either belong to or not belong to a set Such a classical set can be described by using the characteristic function. For instance, by stating a condition for membership , we define a membership such as . means membership (characteristic) function in and indicates nonmembership. For a of fuzzy set, the characteristic function allows various degrees of membership for the elements of a given set. in terms of a relevant universal set For a fuzzy set there is a membership function

represents the degree to which belongs to . Compared with a crisp subset of , the range of the membership , rather than function of fuzzy set is a continuous interval . is the membership function the two-element set to the membership space and the value of that maps is the grade of membership. is also the degree of compatibility or degree of truth of in . When contains is identical only two points 0 and 1, is nonfuzzy and to the characteristic function of a nonfuzzy set. In other words, a crisp set (a nonfuzzy set) is a special case of fuzzy sets. A game contains conflict. There are at least two rational players whose decisions influence each other’s payoffs. Usually their objective functions (payoff functions) conflict with each other. The players are regarded as rational if they try to maximize their objective functions based on their belief about their environment. This rationality, as well as moral and philosophical motives, usually affect the player’s decisions. For example, if the player is cooperative, he may want to maximize his own objective functions as well as maximize other players’ payoffs. If he is noncooperative, even malicious, he would not only like to maximize his own payoff, but at the same time minimize his opponents’ payoffs. The attitude of players cannot be precisely represented in classical game theory. Here fuzzy set theory is used to construct a model that will overcome this shortcoming in the classical game model. be the strategy set Let of player . Assume that the game situation considered is symmetrical with respect to the uncertainty about the other players’ at-

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Fig. 1. Membership function of


F1k .


titudes about strategies. So we only need concern ourselves with one player’s view, for instance player one’s view. The others’ actions can be similarly analyzed. In order to represent more than one motive for the player, the multicriteria decision-making method is used. For example, the first goal of player 1 is to maximize the expected value of his own payoff. The second goal is to maximize his opponent’s expected payoff, etc. This kind of model can represent the cooperative game in classical game theory. If the second goal is not to maximize his opponent’s payoff, but to minimize his opponent’s outcome, the model corresponds is used to represent the th to the noncooperative game. goal function of player , which is a function that measures the attainment of the th goal for the player. Each goal function where is assumed to be a function of is the set of players. Obviously, each other player’s reaction can affect the player’s payoff. in Now player 1 constructs a conditional fuzzy set , which corresponds to the goal on the condition that the strategies of other players are . When the other players play the strategy combination , player 1 can react to a change in his goal is denoted as structure. The membership function of

F1k .

So the membership function of for goal is linear. At the maximum value of the objective function, it equals ; at it will be (see Fig. 2). the minimum of If his goal is to minimize a value, the membership function can be (2) , ; , . Note that player 1 is still uncertain about the strategies that in the other players employ. We need another fuzzy set to represent the possibility that strategy is chosen. The membership function of fuzzy combination is denoted as . In fact, player 1 can use his set experience or knowledge of the raw score outcomes of the (Fig. 3). game to construct the membership function for Generally, if player 1 does not have any knowledge of what the other players will choose or if he has little experience, he will assume the average value has mostly happened. For example, if there are only two players in the game and player 2 only has two pure strategies in the strategy set , the membership is assumed to be function for fuzzy set When

if

if if This membership function represents the degree of player and usually we choose the 1 maximizing his th goal function shown in Fig. 1 to define this membership function. is So

For example, player one’s first goal is to maximize his payoff. We simply assume the membership function to be (1) and are the minimum and maximum value of where the objective function. If one of them is infinite, we will use a sufficiently large number to represent it. This is reasonable since in reality the objective function usually has a finite range.

Player 1 has no bias toward any particular strategy which player 2 will choose because he does not have enough experience or he does not wish to do so. In order to obtain the optimal strategy for player 1, we as the first need to construct an unconditional fuzzy set representation of the attainable goal corresponding to the goal The membership function of is (3) accounts for the possibility of two The multiplication events happening at the same time. The maximum operator on will choose the most possible situstrategy combination ation for player 1 on all strategies of the other players. Thus, that correspond to the kth goal of player based on all 1, weight vector


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The conditional membership function will be

The second goal of player 1 is to minimize the value of

The conditional membership function is


F1 .

Let

is introduced to represent player 1’s attitude about the structure of his goals. Then all fuzzy sets are aggregated by the weight vector to their membership functions. The result will be

and from (3) (6)

If

,

(4) The optimal strategy mined as

for player 1 is eventually deter-

(5)

Thus, is monotonically increasing when inand it will reach its maximum creases on the interval . value when If

The optimal strategies for the other players are similarly obtained. Following this section, two examples will be shown to illustrate the above fuzzy approach. Example 1: ’s strategies

Thus, is monotonically decreasing when inand it will reach its maximum creases on the interval , so from (6), is point where

’s .

Similarly,

strategies

can be determined as

Player 1’s first goal is to maximize his own expected payoff; his second goal is to minimize his opponent’s payoff If the weight vetor is

Let player 1 have the following triangular membership for in : function if if

Depending on the weight vector such that optimal strategy will be

from (4)

from (5), the

The expected payoff value of player 1 is Because is monotonically increasing along with if , finally , the optimal strategy for player 1 is . If , the optimal strategy of player 1 is .

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Example 2: (Prisoner’s dilemma [20]) ’s strategies

For the first case, when the prisoner is noncooperative and selfish, he would like to minimize the other prisoner’s payoff as well as maximize his own. Then his second goal is to minimize the value of

’s .

So is monotonically decreasing when increases and it will reach its maximum value at on the interval , so is

strategies

Let and be the strategy combination for prisoner 1 and prisoner 2, respectively. From the view of player 1, let the membership function for player 1’s fuzzy set in be the triangular shape function The conditional membership function for fuzzy set

if if To generalize the model and represent the subjective attitude of prisoners, we assume that prisoner 1 is (1) noncooperative and selfish or (2) cooperative and devoted. For the first case, the prisoner is likely to maximize his own outcome and minimize the other prisoner’s payoff. The structure of the two goals is determined by the weight vector . If the first goal is to maximize his own outcome it is represented by

From (1), the conditional membership function will be

Let

Let

from (3)

If

Thus, is monotonically increasing on the interval and the maximum value is reached when . , , then If

from (3) So

If

is

is monotonically decreasing on the interval can be determined as

.

Then, if the weight vector is (7)

and , increases on the interval increasing if . reach its maximum when If

is monotonically , and it will

From the above equation, it is easy to see that , meaning that whatever the weight vector is, the prisoner will choose the second strategy. So the . optimal strategy combination for prisoner 1 is If the prisoner is cooperative and devoted and his second goal is to maximize the value of

then the conditional membership function will be


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Similar to the first case

1) Prisoner 1 wants to maximize his payoffand minimize the other prisoner’s outcome

the maximum value happens at So for this case

.

(8) We see that the weight vector will determine what kind of strategy combination is optimal for the prisoner. If the , will increase when weight vector satisfies increases so the maximum value is . The prisoner . This will choose the optimal strategy combination means that if the prisoner is not too selfish and he would like to make his partner get more or equal payoff than he himself gets, he will play the strategy not to confess. Otherwise, if and is large enough, the prisoner will choose as his optimal strategy since he cares too much about his own outcome. In some sense, he tends to be selfish so the result will tend to be the first case. From this special case of the prisoner’s dilemma, we see that the fuzzy model can easily represent the game situation. Moreover, it is possible to describe the prisoner’s subjective attitudes or moral characteristics that are uncertain in the game and cannot be described by probability and statistics in classical game, although they are certain to affect the strategy chosen by players. This is only a special case of prisoner’s dilemma, but we hope this model is general enough to represent all prisoner’s dilemma problems, whatever the payoff matrix might be. The relationship between the payoff matrix and the weight vector is worthy of further study to help the prisoner decide his optimal strategy.

From (1), the conditional membership function will be

Let

based on (3)

If

because of and . monotonically increasing on the interval . If will reach its maximum at

is and it

where III. FUZZY MODEL OF THE PRISONER’S DILEMMA Assume the general case of prisoner’s dilemma payoff table is ’s strategies

’s .

So at , the function maximum value

reaches its

The prisoner’s second goal is to minimize the value of

strategies

The four parameters should satisfy the inequality . Also assume the strategy combination chosen by since there are only two pure strategies prisoner 1 is for him. Similarly, the strategy combination for prisoner 2 . Let prisoner 1 have the following triangular is for the fuzzy set : membership function if if

The conditional membership function is

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If

If

then

Fig. 4. Function

where

f (p1 ; p2 )

.

where

is monotonically decreasing on the interval If

which is the value at

. If the weight vector is

So the optimal strategy combination is and the equilibrium is (C, C). This means that whatever the prisoners’ different payoffs, if they are noncooperative and selfish, they will always choose the strategy to confess. 2) The prisoner is devoted and cooperative. He would like to maximize the other one’s payoff as well as his own. The second goal is to maximize the value

Because of , we cannot decide whether is monotonically increasing or decreasing on the ; but from the following: interval

From Fig. 4, we know the maximum value will happen at and the maximum value is . , then If

We are not certain if the function is monotonical. However, because

or The conditional membership function is

From Fig. 5, the maximum value will happen at ; because , the maximum value will be yielding


Fig. 5. Function

f (p1 ; p2 )

vague and uncertain in practice and not easy to represent in the models of classical game theory. Moreover, this approach combines multiple goals of one player in the game into one model. This makes the model simpler. Compared with prisoner’s dilemma problems in the classical game theory and given the weight vector information that represents the prisoner’s subjective characteristics, the optimal strategy can be easily determined. This model is also more general since the maximin principle is a special case of it. Finally, we point out the relationship between the payoff matrix and the weight vector of the player’s characteristics for choosing the optimal strategy. It will be easier for the decision maker to choose his action based upon the above knowledge. From the examples, we see the solution will be difficult to compute for -person games , although this approach can be extended to -person games. The membership functions for the fuzzy sets in the game model should have more appropriate forms, depending on the knowledge of the characteristics of players in the game, which are assumed to be linear in this paper.

.

If the weight vector is

Let us consider the scalar of

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only. This equals

ACKNOWLEDGMENT The authors would like to thank S. Dick for his advice and helpful suggestions. REFERENCES

Let . Several special cases need to be discussed here. ; that is . In this case, 1) whatever the strategy combination of prisoner 1, the payoff value to him will be the same. , . The prisoner will choose 2) the strategy combination (1, 0); when the prisoner is not very selfish, he would like to maximize his own payoff as well as maximize the other prisoner’s outcome. He is not so selfish as to consider only himself and he will choose the strategy not to confess. , , which means the 3) prisoner is too selfish. He cares too much about his own well being. Although he would like to maximize the other prisoner’s payoff, he is too selfish to care about others and he will choose the strategy to confess. Finally, we see that the fuzzy model can solve the conflict situation in the prisoner’s dilemma, including the prisoner’s attitude and moral features, which is not easily done by probability in classical game theory. Also, from the result, the relationship between the attitude of a prisoner (as represented by the weight vector) and the optimal strategy for the prisoners, given the payoff matrix, are demonstrated. From this, a decision can be made more easily. This result also corresponds to reality. IV. CONCLUSION In this paper we recommend a fuzzy approach to strategic games. Using this method, a fuzzy model is constructed to represent the conflict situation in strategic games. This model uses the fuzzy set to represent the philosophical motives or moral characteristics of players in the game, which are usually

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Abraham Kandel (S’68–M’74–SM’79–F’92) received the B.Sc. (electrical engineering) degree from the Technion—Israel Institute of Technology, Haifa, the M.S. (electrical engineering) degree from the University of California, and the Ph.D. (electrical engineering and computer science) degree from the University of New Mexico, Albuquerque. He is a Professor and the Endowed Eminent Scholar in computer science and engineering and the Chairman of the Department of Computer Science and Engineering, University of South Florida, Tampa. Previously, he was Professor and Founding Chairman of the Computer Science Department, Florida State University (FSU), Tallahassee, as well as the Director of the Institute of Expert Systems and Robotics, FSU, and the Director of the State University System Center for Artificial Intelligence, FSU. He is a member of the editorial board of Fuzzy Sets and Systems, Information Sciences, Expert Systems, Engineering Apllications of Artificial Intelligence, The Hournal of Grey Systems, Control Engineering Practice, Fuzzy Systems—Reports and Letters, Book Series on Studies in Fuzzy Decision and Control, Applied Computing Review Journal, Journal of Neural Network World, The Journal of Fuzzy Mathematics, and BUSEFAL—Bulletin for Studies and Exchange of Fuzziness and its Applications. He has published over 350 research papers for numerous professional publications in computer science and engineering. He is coauthor of Fuzzy Switching and Automata: Theory and Applications (1979), Discrete Mathematics for Computer Scientists (1983), Fuzzy Relational Databases—A Key to Expert Systems (1984), Elements of Computer Organization (1989), Real-Time Expert Systems Computer Architecture (1991), Verification and Validation of Rule-Based Fuzzy Control Systems (1993), Fundamentals of Computer Numerical Analysis (1994), and Fuzzy Expert Tools (1996). He is author of Fuzzy Techniques in Pattern Recognition (1982) and Fuzzy Mathematical Techniques with Applications (1986), He is coeditor of Approximate Reasoning in Expert Systems (1985), Engineering Risk and Hazard Assessment (1988), Hybrid Architectures for Intelligent Systems (1992), Fuzzy Control Systems (1994), and Fuzzy Hardware (1998). He is editor of Fuzzy Expert Systems (1992). Dr. Kandel is a Fellow of the ACM, New York Academy of Sciences, and AAAS. He is a member of the NAFIPS, IFSA, ASEE, and Sigma Xi. He is editor of IEEE MICRO—Fuzzy Track, an associate editor of IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, and is on the editorial board of IEEE TRANSACTIONS ON FUZZY SYSTEMS. He has received the College of Engineering Outstanding Researcher Award, USF 1993–1994, Sigma Xi Outstanding Faculty Researcher Award (1995), the Theodore and Venette–Askounes Ashford Distinguieshed Scholar Award, USF (1995), MOISIL International Foundation Gold Medal for Lifetime Achievements (1996), and the Distinguished Researcher Award, USF (1997).
