Fuzzy Measures and integrals for evaluating strategies Yasuo Narukawa Toho Gakuen, 3-1-10 Naka, Kunitachi, Tokyo, 186-0004 Japan E-mail: narukawa@d4.dion.ne.jp Vicenç Torra IIIA-CSIC, Campus UAB s/n 08193 Bellaterra, Catalonia, Spain E-mail: vtorra@iiia.csic.es Abstract AI techniques have been applied in games since the beginning of the field. Fuzzy measures and fuzzy integrals, a powerful tool for decision making, have potentiality in games for the evaluation of strategies. In this paper we will explore the use of such soft computing techniques and show their interest in decision making and modeling auctions. The paper describes the role of fuzzy measures and integrals in decision making and reviews some computational aspects that are important for the application of such methods in real applications. The paper also gives a short overview on the methods for learning the parameters for these integrals from examples. Keywords: Games, decision making, auctions, soft computing, fuzzy sets, fuzzy theory, model building, Sugeno and Choquet integrals. 1 Introduction Games have been a testbed for artificial intelligence since its inception, being Kasparov defeat against Deep Blue the most famous achievement. Nevertheless, although computer progams have already proven [14] their superiority in several other games (e.g. Othello, checkers) there are some games where further research is needed. Probably is Go [2] one of the current major goals. The current state of computer technology has increased even more the interest of AI techniques in gaming. While previous AI work was mostly oriented on solving the game and, thus, on finding the appropriate decisions so that the computer, as a player, wins the game (See [2] for details on present AI techniques for games following this approach); current trends in the field consider a completely different perspective. It is well known that, nowadays, computer games heavily rely on virtual worlds and synthetic characters. Due to this, ongoing research is on the development of synthetic characters that behave in an intelligent way and with a credible/plausible behavior. An important aspect for demonstrating a credible behavior of the characters is their ability in making decisions. Making an appropriate and on time decision is essential for the credibility of such syntetic characters. In recent years, several tools have been developed in the areas of soft computing and decision theory for decision making. In Section 2 we will review some of these tools, focusing on fuzzy measures and fuzzy integrals. Fuzzy measures and fuzzy integrals are versatile operators that can be used in other contexts than decision making. For example, they have a broad use in aspects related with information fusion. In games, as well as in electronic auctions, they can be used to model coalitions, or participants. In Section 3 we will review this application. An important aspect for building real applications in the context of games and auctions is the computational cost of the algorithms required for making a decision. Due to this, in Section 4 we study the computational cost of two different fuzzy integrals: the Choquet and the Sugeno ones. We will see that their cost is acceptable for most applications as it is just slightly worse than the weighted mean. 2 Decision making Decision making problems are defined in terms of a set of alternatives (say ). Then, a decision maker has to select one alternative from the set (one Ü from ). The simplest case corresponds to the case in which there exists a univariate function that measures the utility of selecting a particular alternative. Then, selection is done so that this function is maximized (select Ü such that ܵ Ñ Ü Ü¾ ܵ, or, equivalently, Ü Ö Ñ Ü Üµ). Alternatively, a preference relation can be built from the function (Ü ½ is preferred to Ü ¾ if when Ü ½ µ Ü ¾ µ) and the most preferred alternative is selected (Ü is selected if Ü is preferred to all Ý in Ò Ü). A multicriteria decision corresponds to the case that instead of a univariate function a multi-valuated one is considered. I.e., ½ ¾ Å µ. In this situation, there are cases in which there is not a unique maximizing solu-
¾ 2.1 Aggregation of ¾ Ü ¾µ Ü ¾ In this case, a value is defined in terms of the other values. This is, in terms of ½ Æ, or with the condition of independence of irrelevant alternatives: Ü µ ½ Ü µ Æ Ü µµ ¾ Ü ½µ Ü ½ ½ Ü ½µ ½ Ü ¾µ Figure 1. Pareto efficient solutions: ܽ, ܾ tion. For example, for ½ ¾µ we have a non optimal case when ܽ ܾ and ½ ܽµ ½ ܾµ but ¾ ܽµ ¾ ܾµ. Figure 1 illustrates this example. This situation raises the concept of Pareto efficiency: Definition 1 An alternative is Pareto efficient if and only if there is no other alternative that is better for all criteria. Definition 2 The Pareto set is the set of alternatives that are Pareto efficient. In the example above, ܽ (and also ܾ) is Pareto efficient because there is no other alternative that is better for all criteria. Naturally, the set ܽ ܾ define the Pareto set. Multicriteria decision problems have been studied by two major schools. On the one hand, the researchers under the umbrella of multicriteria decision aid (MCDA) are mainly devoted to the development of tools that help users to understand their preferences. On the other hand, researchers in multicriteria decision making (MCDM) are more interested on building descriptive approaches. In this latter case, the goal is to build a model of the behavior of the decision makers and, then, apply this model to new problems. According to this distinction, the second approach is more suitable in games as the main concern is the construction of software for making actual decisions. Therefore, model building is the main interest. Two main approaches can be distinguished for multicriteria decision problems: Aggregate the functions ½ Æ into a new function and then use this aggregated function for selecting the best alternative (or for building its corresponding preference relation and then selecting the alternative). Compute a preference relation Ê for each and then aggregate the preference relations into a Ê. Select the best alternative according to the aggregated preference relation. ½ Recall that the condition of independence of irrelevant alternatives states that the aggregation of several functions for a particular object Ó can be directly expressed in terms of an aggregation of the values of the functions for that object. In our context this corresponds to the following equation (where ¼ is the function to aggregate the functions): ¼ ½ Æ µ Ü µ ½ Ü µ Æ Ü µµ Here, is a function that combines Æ values in a given range (or domain following AI and Machine Learning terminology) and yields to another value in the same range. Usually, is one of the so-called aggregation operators. This is, is a function that satisfies (for simplicity we consider the range of equal to the unit interval) the following conditions: 1. Ü Üµ Ü (unanimity) 2. Ñ Ò Ü Ü ½ Ü Æ µ Ñ Ü Ü (compensation) 3. Ü ½ Ü Æ µ Ü ¼ ½ Ü Æ µ when Ü Ü ¼ (monotonicity) Arithmetic mean and weighted mean are well-known examples of aggregation operators. Choquet and Sugeno integrals are more complex and powerful aggregation operators. Their main characteristic is that they can be used to model situations in which the values Ü are not independent but there are dependences among them. I.e., some of the values are redundant or correlated with other of the values. Dependences between the values are expressed in terms of the so-called fuzzy measures. We define them below (see e.g. [9] for details): Definition 3 [8] A set function Ð µ ¼ ½ is a fuzzy measure (or a game) if it satisfies the following axioms: 1. µ ¼, µ ½ (boundary conditions) 2. implies µ µ (monotonicity) In this paper, we assume that is a finite set, that is, ½. Choquet and Sugeno integrals ( Á and ËÁ)aredefined as an integral of a function ¼ ½ with respect to a fuzzy measure on. Their definition are as follows: Definition 4 [3] Let be a fuzzy measure on Ð µµ then the Choquet integral of a function Ê with respect to the fuzzy measure is defined by: Á µ Æ ½ Þ µ µ Þ ½µµ µ µ (1)
where Þ µµ indicates that the indices have been permuted so that ¼ Þ ½µµ Þ Òµµ ½, µ Þ µ Þ Æµ and Þ ¼µµ ¼. ¼ ½ possibility fuzzy measures are an interesting family of fuzzy measures for modeling auctions. We define them below and we will show their interest in Section 3. Definition 5 Let A be a non empty subset of. A set function ÈÓ defined as: ÈÓ µ ½if ÈÓ µ ¼if is called the 0-1 possibility measure focused on. A set function Æ defined as: Æ µ ½if Æ µ ¼if is called the 0-1 necessity measure focused on. The next proposition follows from the definition of Choquet integral. Proposition 1 Let be a non-empty subset of and a function on. 1. Á ÈÓ µ Ñ Ü Ü¾ ܵ 2. Á Æ µ Ñ Ò Ü¾ ܵ 3. For every fuzzy measure, Ñ Ò Ü¾ ܵ Á Ñ Ü Üµ ܾ Definition 6 [8] The Sugeno integral Ë µ of a function ¼ ½ with respect to is defined by Ë µ Æ ½ Þ µ µ µ µ (2) where Þ µµ indicates that the indices have been permuted so that ¼ Þ ½µµ Þ Òµµ ½, µ Þ µ Þ Æµ, Æ ½µ. The next proposition follows from the definition above and Proposition 1. Proposition 2 Let be a fuzzy measure and ¼ ½. 1. If is 0-1 valued, then Á µ Ë µ 2. For every f.m., Ñ Ò Ü¾ ܵ Ë µ Ñ Ü Ü¾ ܵ In our context, the set corresponds to the set of criteria (or functions ½ Æ ). Accordingly, the fuzzy measure can be defined over the set ½ Æ (i.e., ½ Æ ). Then, for each Ü we define the function Ü as Ü µ Ü µ. Therefore, is computed as: or: Ü µ ½ Ü µ Æ Ü µµ Ü µ Á Ü ½µ Ü Æµµ Ü µ Ë Ü ½µ Ü Æµµ 2.2 Aggregation of preference relations The general approach is to build a new preference relation from the preference relations corresponding to each attribute. This is, for each a relation Ê is built and, then, from all these relations a relation Ê is constructed. Again, the condition of independence of irrelevant alternatives described in Section 2.1 can take a role. Here, such condition stands that the relation Ê for two objects Ü Ü does only depend on the values Ê for these same objects. This can be formulated as follows: ¼ ʽ Æ µ Ü Ü µ ʽ Ü Ü µ Ê Æ Ü Ü µµ Methods for aggregation of preference relations have been studied for a long time (e.g. by Ramon Llull [15, 5] (S. XIII) or, more recently, Borda and Condorcet (S. XVIII)). A well known result is Arrow s impossibility theorem [1]. See e.g. [4] for a state-of-the art description of the field. 3 Modeling auctions In this section we review the use of fuzzy measures and integrals for modeling auctions. We will present first the general theory and then its application to auctions. 3.1 Fuzzy measure with parameter: General theory Let Ø be a real number. We consider a family of fuzzy measures Ø Ø¾Ì with parameter Ø where Ì Ê. Denote Ø Ð µµ as Ø «if µ Ð Ñ Ø «Ø µ for all ¾ Ð µ and Ø Áµ as Ø «if Á µ Ð Ñ Ø «Á Ø µ for all ¼ ½. We have the next fundamental theorem for convergence. The detailed proof is in [6]. Theorem 1 Let Ø be a family of fuzzy measures with parameter Ø on a finite set Ð µµ and is a fuzzy measure on Ð µµ. Ø Áµ if and only if Ø Ð µµ A concrete example of a fuzzy measure with parameter can be obtained by using distorted probabilities. We say that a fuzzy measure on Ð µµ is a distorted probability if there exist a probability È on Ð µµ and a function ¼ ½ ¼ ½ such that Æ È. Definition 7 Let Ø be a positive real number, and a family of functions Ø ¼ ½ ¼ ½ defined as Ø Üµ Ü Ø for Ü ¾ ¼ ½. We say that a fuzzy measure Ø is a basic distorted probability generated by a probability È if Ø Ø Æ È
The next lemma follows immediately from the definition above. Lemma 1 Let È be a probability on Ð µµ satisfying È µ ½only if and È µ ¼only if and Ø a basic distorted probability generated by È. Then we have: 1. Ø Æ Ð µµ as Ø ½, 2. Ø ÈÓ Ð µµ as Ø ¼. The next proposition follows from Theorem 1. Proposition 3 Let È be a probability on Ð µµ satisfying È µ ½only if and È µ ¼only if and Ø a basic distorted probability generated by È. Then we have: 1. Ð Ñ Ø ½ Á Ø µ Ñ Ò Ü¾ ܵ, 2. Ð Ñ Ø ¼ Á Ø µ Ñ Ü Ü¾ ܵ, for every real-valued function on. According to the previous results, a single parameter permits to cover the whole range between the minimum and the maximum. Therefore, changing the parameter Ø, we can obtain a suitable fuzzy measure. 3.2 Application Fuzzy measures and integrals can be used for modeling auctions. In this case, we have that the reference set corresponds to the list of participants. This is, is the set of individuals that participate in a particular auction. Then, fuzzy measures can be used to model whether a certain participant is participating or not in a particular time instant. In this case, modelization is based on 0-1 possibility measures (i.e., measures that take values in ¼ ½ instead on ¼ ½ ). Then, we define Ø over as a function of Ø, where Ø is a variable used to denote time instants. In this case, Ø is to express which members of take part in the auction. Formally speaking, when Å Ø is the set of participants in the auction at time Ø, then Ø µ ½if Å Ø Ø µ ¼if Å Ø. Therefore, when the participant in participates in the auction in the instant Ø ½ then, ¾ Å Ø and the 0-1 possibility measure is such that ؽ Å Ø µ ½. Then, we consider a function Ø Ê that modelizes the price that participants have attached to instant Ø. Under this interpretation, the Choquet integral Á Ø Ø µ is equivalent to Ñ Ü Ü¾Å Ø Ø Üµµ, that is the highest price that could be achieved in instant Ø. Then, ؽ corresponds to the termination instant. When Ê is the amount of money that participants can pay, then Á ؽ µ Ð Ñ Ø ½ Á Ø µ becomes the price of the final bid accepted. Using this construction, it is clear that various auctions models can be defined in terms of Ø and Ø. 4 Computational aspects An important element for the use of Choquet and Sugeno integrals in real applications is their computational requirements. In this section we evaluate the computational cost of Choquet and Sugeno integrals. It is easy to see that the order of the computational cost of applying a Choquet integral to a function on with respect to a fuzzy measure is equivalent to the cost of ordering the elements of in terms of ܵ. This is, the cost of applying the Choquet integral is Æ ÐÓ Æ where Æ. We give below an algorithm with this cost. In the first step, the function is ordered and at the same time the permutation is constructed. The cost of this step is Æ ÐÓ Æ where Æ as said before. Then, if the fuzzy measure is stored in a look-up table (¾ Æ positions are needed) we have that we can use a binary representation for the sets µ. In this case, removing an element Ü µ from the set is just changing the µ bit from ½ into ¼ (assigning the corresponding bit to zero). Under this representation accessing µµ requires a constant time. Therefore, the whole Step 3 process takes Ç Æ µ steps. For all this, the whole process is bounded by the time Step 1 takes. Step 1: order and define accordingly Step 2: ØÓØ Ð ¼ Step 3: Ø ½ ½µ Step 3: for ½ to Æ Step 3.1: Ü µµ Ü ½µµ Step 3.2: ÑÑ Ø Øµ Step 3.3: ØÓØ Ð ØÓØ Ð ÑÑ Step 3.4: Ø Ø Ò µ Step 4: return ØÓØ Ð Note that this computational time is the same than the time required by the OWA operator (this operator also requires the ordering of the elements) and only slightly worse than the time required by the weighted mean (recall that the weighted mean is linear Ç Æ µ). Therefore, with respect to the computational cost, the Choquet integral is a competitive alternative to simpler aggregation methods. Similarly, the computational cost of the Sugeno integral is also Ç Æ ÐÓ Æ µ. Note that in this case, the outline of the previous algorithm is also appropriate with some minor modifications. In particular, in this case, equals to Ü µµ and for computing the new ØÓØ Ð instead of multiplying by ÑÑ and adding it to previous ØÓØ Ð the max-min composition should be applied. I.e., ØÓØ Ð Ñ Ü ØÓØ Ð Ñ Ò Ñѵµ. This is illustrated below:
Step 1: order and define accordingly Step 2: ØÓØ Ð ¼ Step 3: Ø ½ ½µ Step 3: for ½ to Æ Step 3.1: Ü µµ Step 3.2: ÑÑ Ø Øµ Step 3.3: ØÓØ Ð Ñ Ü ØÓØ Ð Ñ Ò Ñѵµ Step 3.4: Ø Ø Ò µ Step 4: return ØÓØ Ð As said, the cost of this algorithm is also Ç Æ ÐÓ Æ µ but as now addition and multiplication is replaced by maximum and minimum the effective cost is decreased. 4.1 Fuzzy measures The calculation of the computational cost of the Choquet and Sugeno integrals above is based on the fact that the determination of the fuzzy measure for a particular set can be done using a look-up table. Therefore, an initialization step is required for establishing such look-up table. By the way, for real applications, the definition of a suitable fuzzy measure is the cornerstone of the system. As can be deduced from its definition, a fuzzy measure requires ¾ Æ values. Therefore, for situations in which the cardinality of the reference set is large, such definition becames awkward. To avoid such general definition, fuzzy measures with reduced complexity have been defined so that less than ¾ Æ parameters are required. Decomposable fuzzy measures, k- additive and Ò-dimensional distorted probabilities are some of the existing fuzzy measures with reduced complexity. See [13] for detailed references. In applications, fuzzy measures are usually defined by the system developer or by a domain expert. An alternative is to obtain (learn/determine) from examples. The first study of this subject, for fuzzy integrals, that included some empirical results were obtained by Tanaka and Murofushi [12] in 1989 for the Choquet integral. Since then several algorithms (e.g. based on operation research algorithms and genetic algorithms) have been proposed. Chapter 2 (p. 22) in [13] briefly reviews main results on model building for fuzzy integrals. Two chapters also in [13] (Grabisch, and Imai, Asano and Sato) review and describe state-of-the-art methods for this purpose. 5 Conclusion In this paper we have reviewed the use of fuzzy measures and integrals (Choquet and Sugeno integrals) and their application in state-of-the-art applications. We have also considered their computational cost showing that it is not much larger than the straightforward information fusion methods (e.g. the arithmetic mean and the weighted mean) usually used in most applications. In particular, we have shown that the computational cost of the integrals is Ç Æ ÐÓ Æ µ instead of being Ç Æ µ in the weighted mean. Acknowledgments This work was partly supported by the Spanish Ministry of Science and Technology Fund through project no. TIC- 2001-0633-C03-02 STREAMOBILE and by Generalitat de Catalunya AGAUR, 2002XT 00111. References [1] K. J. Arrow, Social choice and individual values, Wiley, 1963. [2] B. Bouzy, T. Cazenave, Computer Go: An AI oriented survey, Artif. Intel. 132 (2001) 39-103. [3] G. Choquet, Theory of Capacities, Ann. Inst. Fourier 5 (1954) 131-296. [4] J. Fodor, M. Roubens, Fuzzy preference modelling and multicriteria decision support, Kluwer Academic Publishers, 1994. [5] G. Hägele, F. Pukelsheim, Llull s writings on electoral systems,studia Lulliana 41 (2001) 3-38 [6] Y. Narukawa, T. Murofushi, and M. Sugeno: Space of fuzzy measures and convergence, Fuzzy Sets and Systems, vol. 138, no. 3 (2003) pp. 497-506. [7] E. Pap, (Ed.), Handbook of Measure Theory, North-Holland, 2002. [8] M. Sugeno, Theory of fuzzy integrals and its application, Thesis, Tokyo Institute of Technology, 1974. [9] M. Sugeno, T. Murofushi, Fuzzy Measure, Tokyo, Nikkan Kogyo Shinbunsha (in Japanese), 1993. [10] J. Schaeffer, H. J. van den Herik, Games, computers, and artificial intelligence, Artif. Intel. 134 (2002) 1-7. [11] M. Sugeno, K. Fujimoto, and T. Murofushi, Hierarchical decomposition of Choquet integral models, Intl. J. Unc., Fuzz. and KBS, 3, (1995) 1 15. [12] A. Tanaka, T. Murofushi, A learning model using fuzzy measures and the Choquet integral, 5th Fuzzy System Symposium, Kobe, Japan, 213 218 (in japanese), 1989. [13] V. Torra, On some aggregation operators for numerical information, in V. Torra (Ed.), Information fusion in data mining, Springer, 2003, 9 26. [14] H. J. van den Herik, J. W. H. M. Uiterwijk, J. van Rijswijck, Games solved: Now and in the future, Artif. Intel. 134 (2002) 277-311. [15] http://www.math.uni-augsburg.de/stochastik/lull/