On Two-Player Interval-Valued Fuzzy Bayesian Games

On Two-Player Interval-Valued Fuzzy Bayesian Games Tiago C. Asmus,1,† Graçaliz P. Dimuro,2,3,∗ Benjamı́n Bedregal4,‡ 1 Instituto de Matemática, Estatı́stica e Fı́sica, Universidade Federal do Rio Grande, Rio Grande, Brazil 2 Centro de Ciência Computacionais, Universidade Federal do Rio Grande, Rio Grande, Brazil 3 Department of Automatic and Computation, Public University of Navarra, Pamplona, Spain 4 Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal, Brazil Game theory is an important basis to simulate several situations where multiple agents interact strategically for decision making and support. In many applications, such as auctions, frequently used for resource management involving two or more agents competing for the resources, the interacting agents only know their own characteristics and must make decisions while having to estimate the characteristics of the others. When probabilities are assigned for the different types of the interacting agents, this kind of strategic interaction constitutes a Bayesian game. In cases in which it is very difficult to characterize the private information of each agent, the payoffs can be given by approximate (not probabilistic) values, but the concept of Bayesian Nash equilibrium cannot be applied in this context. Fuzzy set theory is an excellent basis for studying this type of game, where the payoffs are represented by fuzzy numbers. When it is the case that there is also uncertainty about such fuzzy numbers, the use of interval fuzzy numbers appears as a good modeling alternative. This paper introduces an approach for interval-based fuzzy Bayesian games, based on interval-valued fuzzy probabilities for modeling the types of agents involved in the interaction. We present two different case studies, namely the (Interval) Fuzzy Bayesian Hiring Game and (Interval) Fuzzy Bayesian Prisoner’s Dilemma with Moral Standards, comparing the results obtained with the crisp, fuzzy and interval fuzzy approaches, highlighting a particular case C 2016 in which the interval fuzzy approach presents a solution although the two other do not.  Wiley Periodicals, Inc. 1. INTRODUCTION Game theory1 is a well-known important basis to simulate several situations where multiple agents (also named players) interact strategically for decision ∗ Author to whom all correspondence should be addressed; e-mail: [email protected], [email protected]. † e-mail: [email protected]. ‡ e-mail: [email protected]. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 00, 1–40 (2016) 2016 Wiley Periodicals, Inc. View this article online at wileyonlinelibrary.com. • DOI 10.1002/int.21857  C 2 ASMUS, DIMURO, AND BEDREGAL making and support. Game theory provides a useful framework for modeling/analyze cooperation, negotiation, and coalition in multiagent systems (MAS)2–5 and social simulation.6–9 There are several applications in which the interacting agents only know their own characteristics and must make decisions while having to estimate the characteristics of the other participants of the interaction, which configures a game of incomplete information, as, e.g., the case of auctions in the context of MAS, which is frequently used for resource management involving two or more agents competing for the resources,2 or the case of Cliff–Edge problems, where maximizing profits while preventing the entire deal from falling through is a great challenge for the agents.10 Whenever probabilities are assigned for the different types of the interacting agents, this kind of strategic interaction constitutes a Bayesian game, which is a game of imperfect information,11 the main concern of this paper. In the case of Bayesian games, sometimes it is very difficult to characterize the private information of each agent (e.g., ability, level of effort, influence, personality, interest, strategy), to establish the probabilities of the types that each player may assume. In these situations, the payoffs may be given by approximate (not probabilistic) values, and the concept of Bayesian Nash equilibrium cannot be applied in this context. Fuzzy set theory, which was introduced by Zadeh,12 is an excellent basis for studying this type of game in which the payoffs are represented by fuzzy numbers that can be modeled in different ways. In the literature, one may find a large amount of research on fuzzy games, as discussed by Dang and Hong,13 where two main lines of research on fuzzy games were identified, namely, matrix fuzzy games and noncooperative fuzzy games, the majority focused on fuzzy zero-sum games,14 which are strictly competitive. In the line of the research of noncooperative fuzzy games, see, e.g., (i) the survey by Larbani,15 (ii) the work by Chandra and Aggarwal,16 which proposed an algorithm to solve matrix games with payoffs of general piecewise linear fuzzy numbers, (iii) the proposal of Liu and Kao17 of the application of the extension principle, and (iv) the analyses of the existence of equilibrium solution for a noncooperative game with fuzzy goals and parameters of Kacher and Larbani.18 On the other hand, cooperative game theory is concerned with the situation where the agents have only two alternative possibilities, namely, to join or not a coalition, without any option for the degree of commitment. Cooperative fuzzy games19,20 allow for the partial participation of agents in coalitions, where the attainable outcomes of a game depend on the degree of commitment of the agents, so modeling ambiguous decision making as observed in most group behaviors. Some interesting contributions for the research on fuzzy cooperative games are due to, e.g., (i) Yu and Zhang,21 who extended the notion of fuzzy cooperative fuzzy games to obtain a more general definition, (ii) Monroy et al.,22 with a study of cooperative games with fuzzy payoffs, and, more recently, (iii) Sagara,23 who investigated fuzzy extensions of cooperative games and the coincidence of the solutions for fuzzy and crisp games, and (iv) Borkotokey et al.,24 who proposed a bicooperative game with fuzzy bicoalitions in multilinear extension form. In general, whatever the kind of fuzzy game we adopt, we may face another challenge, whenever there is also uncertainty in the modeling of the fuzzy numbers (e.g., when they are given by a group of different domain experts, or when a single International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 3 expert is in doubt). In this case, interval fuzzy numbers can be used to represent a range of fuzzy numbers. Observe that interval-valued fuzzy sets allow to deal not only with vagueness (lack of sharp class boundaries) but also with uncertainty (lack of information).25–27 In the literature, one can find some works combining intervals and fuzzy games in different ways. For example, cooperative games where the knowledge about the worth of coalitions is described by fuzzy intervals have been introduced by Mares,20 considering the core as a fuzzy set. Mallozzi et al.28 proposed a different approach, in which the core is a subset of vectors whose components are fuzzy intervals. On the other hand, Collins and Hu29 studied interval-valued matrix games with fuzzy logic, extending the results of classical matrix games into interval-valued games, and defining fuzzy relational operators for intervals to compare every pair of possible interval payoffs. Meng et al.30 proposed a generalized form of fuzzy games with interval characteristic functions. Brião et al.31 introduced two approaches for the solution of interval-valued fuzzy zero-sum games, based on interval fuzzy linear programming problems. This paper introduces the concept of interval-based fuzzy Bayesian games, based on interval-valued fuzzy probabilities for modeling the types of agents involved in the interaction. The interval-valued fuzzy probabilities are given by symmetric triangular interval fuzzy numbers,a inspired by Buckley,32 who considered an arithmetic restriction on the interval [0,1] in their representation of uncertain probabilities by fuzzy numbers. This approach allows a simpler formulation for the fuzzy and interval-valued fuzzy Bayesian game models than those based on credibility theory.33 To investigate the solution of fuzzy and interval-valued fuzzy Bayesian games, we adopt a total order relation in the space of symmetric triangular (interval) fuzzy numbers, namely, the (interval) AD-order,34 to guarantee the existence of equilibrium solutions. We present two different case studies, introducing game models to be used in MAS applications, namely (i) the (Interval) Fuzzy Bayesian Hiring Game (FBHG), for decision making in agent negotiation35 (e.g., in resource or team allocation to accomplish a task) and (ii) the (Interval) Fuzzy Bayesian Prisoner’s Dilemma (FBPD) with Moral Standards, an extension of the Prisoner’s Dilemma, whose iterated/evolutionary approaches are commonly used to promote cooperation/coalition among agents.36 In particular, in (ii), the consideration of moral standards37 is an attempt to reflect the influence of human behaviors (e.g., egoism, altruism) in the agents’ payoffs, which may allow the model to predict the outcomes more adequately, as properly suggested by Zhang et al.38 The paper is organized as follows: Sections 2 presents some preliminary concepts that are necessary for the development of the paper. Some introductory concepts of interval-valued fuzzy numbers are presented in Section 3. The adopted approach for interval-valued fuzzy probabilities is presented in Section 4. Section 5 a We decided the use of (interval) symmetric triangular fuzzy numbers as they are sufficient for the desired applications in this paper, and, in general, for other several applications, as discussed by Buckley.32 International Journal of Intelligent Systems DOI 10.1002/int 4 ASMUS, DIMURO, AND BEDREGAL introduces the proposed approaches for the two-player Fuzzy Bayesian Game and Interval Fuzzy Bayesian Game. Section 6 introduces the (Interval) FBHG. Section 7 introduces the (Interval) FBPD with Moral Standards. Section 8 is the Conclusion.b 2. PRELIMINARY CONCEPTS A fuzzy set is defined by means of a graded membership function, which is an extension of the characteristic function of the classical set theory. Given a universe X and a fuzzy subset F of X, the membership function ϕF : X → [0, 1] provides the grade ϕF (x) with which an element x ∈ X belongs to F , where ϕF (x) = 0 and ϕF (x) = 1 represent, respectively, the nonpertinence and the complete pertinence of x to the fuzzy subset F . A fuzzy subset F of X can be then represented by a set of ordered pairs, given by F = {(x, ϕF (x)) | x ∈ X}. The support of F is defined as the set suppF = {x ∈ X|ϕF (x) > 0}. (1) The core of F is defined as the set coreF = {x ∈ X|ϕF (x) = 1}. For 0 < α ≤ 1, the α-cuts of F are defined as the classic subsets of X defined by F [α] = {x ∈ X|ϕF (x) ≥ α}, (2) where suppF is the support of F defined in Equation 1. A fuzzy set is completely defined by its α-cuts.27 A fuzzy set N is called a fuzzy number whenever ϕN is defined on the set of real numbers R, and the following conditions hold:c (i) All α-cuts of N are nonempty closed intervals in R; (ii) The support of N is bounded. In this case, one defines the notion of α-cut of a fuzzy number N for α = 0 as N .d the closure of the support of N, that is, N[0] = supp To operate with fuzzy numbers, we can represent them through their α-cuts and apply interval arithmetic.46 A very preliminary version of this work was presented by Asmus et al.39 There may be found in the literature different and nonequivalent definitions of fuzzy numbers (e.g., by Bojadziev and Bojadziev,40 by Hanss,41 and by Stupnanová42 ). We adopted the definition by Buckley and Eslami43 (the same definition was given by Klir and Yuan,44 and Nguyen and Walker45 ), which is sufficient to develop this work. d The closure of an open real interval ]a1 , a2 [ is defined as the closed real interval [a1 , a2 ]. b c International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 5 Figure 1. The membership function ϕN of a symmetrical triangular fuzzy number N = (a/m/b). A fuzzy number is called triangular if its membership function is defined as ⎧ 0, if x ≤ a ⎪ ⎪ ⎪ ⎪ ⎨ x−a , if a < x ≤ m m−a (3) ϕN (x) = x−b ⎪ , if m < x ≤ b ⎪ m−b ⎪ ⎪ ⎩ 0, if x ≥ b, where a, m, b ∈ R, a ≤ m ≤ b. A triangular fuzzy number N is determined by the real numbers a, m, and b and is denoted by the ordered tuple (a/m/b), where m is the unique element of its core coreN = {m}. The support of N is an open real interval suppN =]a, b[, and its representation through its α-cuts is given by N[α] = [(m − a)α + a, (m − b)α + b], (4) for 0 ≤ α ≤ 1. Of particular interest for this work are the symmetric triangular fuzzy numbers (see Figure 1), where b − m = m − a, since they are used to model the expression “around,” commonly used for modeling fuzzy probabilities32,47 and certain fuzzy games.33 Given two fuzzy numbers N1 and N2 , there are several ordering methods for analyzing if N1 ≤ N2 , some of them given by partial orders and other not, some are adequate for just particular shapes of membership functions (see, e.g., the discussions presented by Buckley,32 Asmus and Dimuro,34 Bustince et al.,48 and, more recently, by Wang and Wang49 ). The choice of which method to use depends on the membership functions of N1 and N2 and the problem being modeled. We decided to adopt the AD-order,34 which is efficient and practical to work with symmetrical triangular fuzzy numbers and, at the same time, takes into consideration the accuracy and quality of the information contained in the fuzzy number, in the sense of an information order,50,51 providing meaningful results. International Journal of Intelligent Systems DOI 10.1002/int 6 ASMUS, DIMURO, AND BEDREGAL DEFINITION 2.1.34 Consider α ∈ [0, 1] and a degree of imprecisione ρ ∈ (0, 1], and let N1 = (a1 /m1 /b1 ) and N2 = (a2 /m2 /b2 ) be symmetric triangular fuzzy numbers. Then, N1 < N2 if and only if one of the following conditions hold: (1) (a1 < a2 ) ∧ (b1 ≤ b2 ) (5) (2) (a1 < a2 ) ∧ (b2 < b1 ) ∧ (m1 ≤ m2 ) (3) (a1 < a2 ) ∧ (b2 < b1 ) ∧ (m2 < m1 ) ∧ [∀α : (0 ≤ α ≤ ρ) ⇒ (m1 − a1 )α + a1 ≤ (m2 − a2 )α + a2 ] (4) (a2 ≤ a1 ) ∧ (b1 < b2 ) ∧ (m1 < m2 ) ∧ [∃α : (0 ≤ α ≤ ρ) ∧ (m1 − a1 )α + a1 < (m2 − a2 )α + a2 ] and N1 ≤ N2 ⇔ N1 = N2 ∨ N1 < N2 . (6) THEOREM 2.1.34 The relation ≤ is a total order relation in the set of symmetric triangular fuzzy numbers. 3. INTERVAL-VALUED FUZZY NUMBERS Consider the set of real intervals IR and let U = {[a, b] ∈ IR | 0 ≤ a ≤ b ≤ 1} be the set of subintervals of the unit interval [0, 1]. An interval fuzzy subset F of a universe X is defined as the set of ordered pairs F = {(x, νF (x)) | x ∈ X}, where νF : X → U is the interval-valued membership function of F, which provides the interval membership grade νF (x) containing the uncertain membership grade with which an element x ∈ X belongs to F. Whenever the interval membership function νF is continuous,f then there exist continuous functions ϕFl , ϕFu : X → [0, 1], called, respectively, the lower membership function (LMF) and the upper membership function (UMF), such that, for every x ∈ X, it holds that27 νF (x) = [ϕFl (x), ϕFu (x)], (7) e The degree of imprecision ρ indicates in which extension the cores’ values of the compared symmetric triangular fuzzy numbers are relevant for ordering fuzzy numbers that are too close. When ρ assumes values approaching zero, the quality of the information provided by the fuzzy number is more considered than the numeric quantity that this fuzzy number represents.34,50,51 f The continuity of interval functions was defined by Moore as an extension of the continuity of real functions.50,51 International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 7 where ϕFl (x) ≤ ϕFu (x). The inner and outer supports of an interval fuzzy set F of X are defined, respectively, by lsuppF = {x ∈ X | ϕFl (x) > 0} and usuppF = {x ∈ X | ϕFu (x) > 0}, and its core is defined by coreF = {x ∈ X | νF (x) = [1, 1]} = {x ∈ X | ϕFl (x) = ϕFu (x) = 1}. For [α1 , α2 ] ∈ U, with α1 > 0, the [α1 , α2 ]-cuts of F are defined by27 F[α1 , α2 ] = {x ∈ X | νF (x) ≥K [α1 , α2 ]}, (8) where ≥K is the Kulisch–Miranker interval order relation.50,51 As in the classical fuzzy theory, an interval fuzzy set is completely determined by its [α1 , α2 ]-cuts.27  is defined as an interval fuzzy set of R satisfying An interval fuzzy number N the following properties:27  are real intervals of IR; (a) the [α1 , α2 ]-cuts and the core of N (b) lsuppN and usuppN are bounded. In this case, one defines the notion of [α1 , α2 ]-cuts of an interval fuzzy number  for α1 = 0 as N [0, α2 ] = N ⎧ l [0] ∩ {x ∈ X | νN (x) ≥K [0, α2 ]}, ⎨N l [0] ∩ N u [0], ⎩N if α2 = 0 if α2 = 0, (9) l [0] and N u [0] are the closures of the supports lsuppN and usuppN , rewhere N spectively.  is called triangular if both LMF and UMF are An interval fuzzy number N defined as in Equation 3, where the lower and upper parameters are al , m, bl and au , m, bu , respectively. An interval fuzzy number is denoted by  = ([au , al ]/[m, m]/[bl , bu ]), N  is a and whenever it holds that bl − m = m − al and bu − m = m − au , then N symmetric triangular interval fuzzy number. One can observe that the functions ϕN l : R → [0, 1] and ϕN u R → [0, 1] define, respectively, the membership functions of fuzzy numbers N l and N u , which define  Thus, an interval fuzzy number may be represented as an interval fuzzy number N. an ordered pair of fuzzy numbers:  = (N l , N u ), N International Journal of Intelligent Systems DOI 10.1002/int 8 ASMUS, DIMURO, AND BEDREGAL = Figure 2. The membership function νN of a symmetric triangular interval fuzzy number N (N l , N u ). where N l and N u are called the lower generator fuzzy number and the upper , respectively. generator fuzzy number of N In fact, assuming that N l = (al /m/bl ) and N u = (au /m/bu ) are symmetric triangular fuzzy numbers defined, respectively, by the membership functions ϕN l  = ([au , al ]/[m, m]]/[bl , bu ]) is a symmetric triangular interval and ϕN u , then N fuzzy number defined by the interval membership function νN : R → U, such that νN (x) = [ϕN l (x), ϕN u (x)], for all x ∈ R.27 (See the representation of the membership function νN of a sym = (N l , N u ) in Figure 2.) metric triangular interval fuzzy number N DEFINITION 3.1.34 Let N1l and N1u be, respectively, the lower and the upper generator 1 = (N1l , N1u ). Let N2l and N2u be, fuzzy numbers of an interval fuzzy number N respectively, the lower and the upper generator fuzzy numbers of an interval fuzzy 2 = (N2l , N2u ). Then we define number N and 1 < 2 ⇔ (N1u < N2u ) ∨ (N1u = N2u ∧ N1l < N2l ) N N (10) 1 ≤ 1 = N 2 ∨ N 1 < 2 , 2 ⇔ N N N N (11) where ≤ is the AD-order relation given in Definition 2.1.  is a total order relation in the set of symmetric THEOREM 3.1.34 The relation ≤ triangular interval fuzzy numbers. 4. INTERVAL-VALUED FUZZY PROBABILITIES To define interval-valued fuzzy probabilities, first we present the definition of fuzzy probabilities proposed by Buckley,32 based on stochastic vectors under an arithmetic restriction, not using a standard fuzzy probability theory.52 International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 9 Observe that Costa et al.53 have introduced an approach of fuzzy intuitionist probabilities that does not use an arithmetic restriction. Although this approach facilitates the calculation of each fuzzy intuitionist probability, avoiding the interdependence between them, it produces, in general, fuzzy numbers with greater dispersion than the ones produced by our approach. As the interval-valued fuzzy probabilities defined in this paper are calculated to be used in problems of decision making, we decided to work with fuzzy numbers with the least possible dispersion to avoid ambiguous cases in the ordering of imprecise values. Let X = {x1 , . . . , xn } be a finite set and P : ℘(X) → [0, 1] a probability function defined for all subsets of X, with P ({xi }) = φi , 1 ≤ i ≤ n, 0 ≤ φi ≤ 1 and n i=1 φi = 1, characterizing a finite and discrete probability distribution, represented by  = {φ1 , . . . , φn }. We observe that the elements of  are often obtained by experts’ opinions, not always representing precise values or consensus. To model this uncertainty, one may use fuzzy numbers φi , generating a new set  = {φ1 , . . . , φn }. Consider that the elements of a fuzzy probability function P are symmetric triangular fuzzy numbers P ({xi }) = φi = (ai /mi /bi ), (12) for 1 ≤ i ≤ n, where ai , bi , and mi represent the pessimistic, optimistic, and “most likely” estimate of occurrence of xi , respectively.g For (X, ) to represent a finite and discrete distribution of probabilities, it is necessary to make an arithmetic restriction, since the interval sum of the α-cuts of in 1.32,54 Then, for all 0 ≤ α ≤ 1, we choose an ei from each all φi hardly results α-cut φi [α] so that ni=1 ei = 1. To define the arithmetic restriction, consider the following set of stochastic vectors: n n E = (e1 , . . . , en ) ∈ [0, 1] | ei = 1 (13) i=1 and define the sets as follows: Aα = φ 1 [α] × · · · × φ n [α]. (14) Sα = Aα ∩ E. (15) Consider a subset A ⊆ X, and let IAX = {i ∈ {1, . . . , n} | xi ∈ A} (16) g See Refs. 32 and 47 for the explanation on the method for constructing symmetric triangular fuzzy numbers based on the variance of all the pessimistic and optimistic estimates provided by several experts. International Journal of Intelligent Systems DOI 10.1002/int 10 ASMUS, DIMURO, AND BEDREGAL be the set of indexes of the elements of A related to the indexes of X = {x1 , . . . , xn }.h A The function f,α : Sα → [0, 1], defined by A f,α (e1 , . . . , en ) = ei , (17) i∈IA where 0 ≤ α ≤ 1, and Sα is given in Equation 15, guarantees the desired arithmetic restriction. The fuzzy probability of a subset A ⊆ X is obtained through its α-cuts by A P (A)[α] = f,α (e) | e = (e1 , . . . , en ) ∈ Sα , (18) A : Sα → [0, 1] is defined in Equation 17. where f,α PROPOSITION 4.1.47 P (A) is a symmetric triangular fuzzy number. The properties of fuzzy probabilities were analyzed by Asmus et al.,47 considering the AD-order relation (Definition 2.1). Finally, we show how to calculate the fuzzy mean μr through the multiplication of each  = {φ1 , . . . , φn } by the real values in r = (r1 , . . . , rn ) ∈ Rn . To this end, r the function g,α : Sα → R, defined by n r (e1 , . . . , en ) = g,α ei · ri , (19) i=1 where 0 ≤ α ≤ 1 and Sα is given by Equation 15, ensures that the arithmetic restriction is respected. The fuzzy mean for r = (r1 , . . . , rn ) ∈ Rn is defined through its α-cuts as r μr [α] = g,α (e) | e = (e1 , . . . , en ) ∈ Sα (20) r for 0 ≤ α ≤ 1, and g,α : Sα → R defined by Equation 19. PROPOSITION 4.2.47 μr is a triangular symmetric fuzzy number. Now, we present the approach for interval fuzzy probabilities, which was first introduced by Asmus et al.47 Analogously to which was presented in the beginning of this section, let  = {φ1 , . . . , φn } be a finite probability distribution and consider that some of these φi , with 1 ≤ i ≤ n, represent uncertain values, in a way that even its fuzzy modeling is not trivial (e.g., there is no consensus related to the fuzzy numbers to be adopted in the modeling). In this case, one may use interval fuzzy numbers, substituting all crisp φj for an interval fuzzy number φj , constituting a h When X can be understood by context, IAX is written simply as IA . International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 11 , in a way that the . Then, we have an interval fuzzy probability function P new set   elements of  are represented by symmetric triangular interval fuzzy number as ({xi }) = φ i = (φi l , φi u ), P (21) for 1 ≤ i ≤ n, where φi l represents the fuzzy modeling with the least imprecision and φi u represents the fuzzy modeling with the greater imprecision, according to the experts’ opinions. ) to represent a finite (and, therefore, discrete) probability function, it For (X,  is necessary to define an arithmetic restriction. For 0 ≤ α1 ≤ α2 ≤ 1, we choose from i [α1 , α2 ] an ei such that ni=1 ei = 1. To define the arithmetic each [α1 , α2 ]-cut φ restriction, consider the set E of stochastic vectors, given in Equation 13, and define the sets: 1 ,α2 1 [α1 , α2 ] × · · · × φ n [α1 , α2 ]. =φ Aα  α1 ,α2 1 ,α2 = Aα ∩ E. S   (22) (23) Consider a subset A ⊆ X and the set IAX of indexes of A given in Equation 16. The α1 ,α2 A → [0, 1], defined by function f ,[α ,α ] : S  1 2 A f ,[α1 ,α2 ] (e1 , . . . , en ) = ei , (24) i∈IA α1 ,α2 is given in Equation 23, defines the arithmetic where 0 ≤ α1 ≤ α2 ≤ 1 and S  restriction. The interval fuzzy probability of a subset A ⊆ X is defined through its [α1 , α2 ]-cuts as α1 ,α2 (A)[α1 , α2 ] = fA , (e) | e = (e1 , . . . , en ) ∈ S P  ,[α1 ,α2 ] (25) α A where f ,[α ,α ] : S  → [0, 1] is defined in Equation 24. 1 2 (A) is a symmetric triangular interval fuzzy number. PROPOSITION 4.3.47 P (A) may be obtained through its PROPOSITION 4.4.47 The interval fuzzy probability P generator fuzzy probabilities P (A)l and P (A)u , as (A)[α1 , α2 ] = P (A)l [α1 ] ∩ P (A)u [α2 ], P (26) (A) = (P (A)l , P (A)u ). P (27) for 0 ≤ α1 ≤ α2 ≤ 1. Thus, interval fuzzy probabilities can be given through their generator fuzzy probabilities, as follows: International Journal of Intelligent Systems DOI 10.1002/int 12 ASMUS, DIMURO, AND BEDREGAL The properties of interval fuzzy probabilities were analyzed in Ref. 47, considering the interval AD-order relation (Definition 10). α1 ,α2 r → R, defined by Now, consider the auxiliary function q ,[α ,α ] : S  1 2 n r q ,[α1 ,α2 ] (e1 , . . . , en ) ei · ri , = (28) i=1 α1 ,α2 is given in Equation 23, guarantees the arithwhere 0 ≤ α1 ≤ α2 ≤ 1 and S  metical restriction. For an r = (r1 , . . . , rn ) ∈ Rn , we define the interval fuzzy mean through its [α1 , α2 ]-cuts, as α1 ,α2 r  μr .  [α1 , α2 ] = q ,[α1 ,α2 ] (e) | e = (e1 , . . . , en ) ∈ S  (29) μr PROPOSITION 4.5.47   is a symmetric triangular interval fuzzy number. μr PROPOSITION 4.6.47 The interval fuzzy mean   may be calculated through the r r generator fuzzy means μ and μ , as l for 0 ≤ α1 ≤ α2 ≤ 1. 5. u r r  μr  [α1 , α2 ] = μl [α1 ] ∩ μu [α2 ], (30) THE TWO-PLAYER FUZZY AND INTERVAL FUZZY BAYESIAN GAMES A simultaneous game in which one or more players are unaware of the type of some other player or even his own type defines a game of incomplete information, or a Bayesian game. The player who does not know the type of the other must assign probabilities to each possible type that this agent can take. Once these probabilities have been assigned (possibly through experts’ opinions on the subject), then the payoff functions can be calculated and the game becomes a game of imperfect information.1,11,33 Let L = {1, . . . , l} and M = {1, . . . , m} be finite sets of possible types that players JA and JB can assume, respectively. The players JA and JB of specific types ′ t ∈ L and t ′ ∈ M are denoted by JAt and JBt , respectively. We denote by JAL and JBM the players JA and JB whose types vary in L and M, respectively. ′ The players JAt and JBt can choose strategies from the action sets A = {A1 , . . . , An } and B = {B1 , . . . , Bn′ }, respectively. The payoffs that the players JAL and JBM receive for adopting certain strategies depend on the types assumed by them and are determined by the respective payoff functions ′ httA : A × B → R International Journal of Intelligent Systems (31) DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES ′ httB : A × B → R, 13 (32) for each combination of types t ∈ L and t ′ ∈ M. We build a game for each combination of types in L and M, obtaining an L × M game matrix  tt ′  GLM AB = RAB t∈L,t ′ ∈M , ′ (33) tt where each entry RAB is the payoff matrix of the game between the players JAt and ′ ′ ′ JBt . Considering the payoff functions httA and httB , specified in Equations 31 and 32, respectively, we have that ′ tt RAB =   ′ ′ httA (Ai , Bj ), httB (Ai , Bj ) Ai ∈A,Bj ∈B . (34) Let P and P ′ be known probabilities functions, with P ({t}) = φt , for t ∈ L, and P ′ ({t ′ }) = φt′′ , for t ′ ∈ M. Then,  = {φ1 , . . . , φl } and ′ = {φ1′ , . . . , φm′ } are the sought probability distributions. These probabilities are used to build a single probabilistic game from a matrix GLM AB of deterministic games, as given in Equation 33. In this game matrix, each cell is a combination of strategies chosen by the ′ players JAt and JBt , whose payoff value is given by combining the probabilities φt and φt ′ assigned to each player type with the values of the payoffs contained in the tt ′ (Equation 34). respective matrix RAB Thus, the payoffs of the Bayesian game are obtained through the functions l m hLA : Al × Bm → R and hM B : A × B → R, defined by l hLA (A∗ , B ∗ ) m ′ ′ ′ ′ φt · φt′′ · httA (At , B t ) = (35) t=1 t ′ =1 m l ∗ ∗ hM B (A , B ) = φt · φt′′ · httB (At , B t ), (36) t ′ =1 t=1 in which A∗ = (A1 , . . . , Al ) and B ∗ = (B 1 , . . . , B m ) represent vectors that have in each position one of the actions of the players JAL and JBM , respectively, so that A1 , . . . , Al ∈ A = {A1 , . . . , An } and B 1 , . . . , B m ∈ B = {B1 , . . . , Bn′ }. We obtain a matrix ZAB with order {nl } × {n′ m }, which covers all the combinations of strategies and types of each player. Then, considering the discussion in the previous paragraphs, the definition of a two-player Bayesian game can be stated as follows, where we adopt the same notation used by Zhang et al.:38 International Journal of Intelligent Systems DOI 10.1002/int 14 ASMUS, DIMURO, AND BEDREGAL DEFINITION 5.1. A two-player Bayesian Game is defined as a eight-tuple  2−BG = L, M, A, B,  ′ httA , t∈L,t ′ ∈M   ′ httB , , ′ . t∈L,t ′ ∈M (37) Considering the payoff functions defined in Equations 35 and 36, Definition 5.1 can be given equivalently by the following definition: DEFINITION 5.2. A two-player Bayesian game is defined as a eight-tuple 2−BG′ = (L, M, A, B, hLA , hM B ). (38) A combination of strategies is said to be a Nash equilibrium (NE) when each strategy is the best response to the strategies of the other players, and this is true for all players.55 Considering Definition 5.2, one may investigate the existence of equilibria in a Bayesian game as in a regular simultaneous game. In other words, the NE of the game 2−BG (Definition 5.1) is determined by the NE of the game 2−BG′ (Definition 5.2). Observe that Definition 5.1 can be generalized for a number q > 2 of players: DEFINITION 5.3. A q-player Bayesian game is defined as a five-tuple ⎛ q−BG = ⎝Q, q  i=1 Si , q  i=1 Ti ,  t1 ∈T1 ,...tq ∈Tq t ...tq h1 q i=1 Si , q  i=1 ⎞ i ⎠ , (39) where (i) (ii) (iii) (iv) Q = {1, . . . , q} is a finite set of players with cardinality q, Si = {Si,1 , . . . , Si,ni } is the strategy set of player i ∈ Q, with cardinality ni , Ti = {ti,1 , . . . , ti,mi } is the type set of player i ∈ Q, with cardinality mi , and i = {φi,1 , . . . , φi,mi } is probability distribution of the set type of the player i ∈ Q, with cardinality mi . Analogously to what it was shown for two-player Bayesian game, one can easily obtain an equivalent definition q−BG′ (analogous to Definition 5.2), so that the problem of finding NE of a q-player Bayesian game q−BG is the problem of finding NE considering q−BG′ . Thus, in the rest of the paper, we will consider just the case of two players, having in mind that all the results and notions can be easily extended for the case of q > 2 players. In the following subsections we introduce the fuzzy and interval fuzzy Bayesian games, by considering fuzzy and interval fuzzy probabilities, respectively. International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 15 5.1 The Two-Player Fuzzy Bayesian Games Assume that the fuzzy probabilities distributions  = {φ1 , . . . , φl }, of the l ′ ′ possible types for the player JAL , and ′ = {φ1 , . . . , φm }, of the m possible types for the player JBM , are known. Then we consider these probabilities in the respective payoffs contained in the matrix GLM AB (Equation 33), using the definition of fuzzy mean (Equation 20), to define the fuzzy matrix Z AB , which contains the probabilistic fuzzy payoffs of the game. Let A∗ = (A1 , . . . , Al ) and B ∗ = (B 1 , . . . , B m ) be the vectors containing com′ binations of strategies chosen by each player JAt and JBt , respectively. The set HAA∗ B ∗ =    l1 l 1  1 1 1m 1 m lm l m h11 A (A ,B ), . . ., hA (A ,B ) , . . ., hA (A ,B ), . . ., hA (A ,B ) (40) contains the vectors whose coordinates constitute the payoffs of a player JAt , for t ∈ L, for each combination of vectors A∗ and B ∗ when the player JB varies its type ′ t ′ in M, represented by the players JBt . Such payoffs are expressed by the matrices ′ tt RAB (Equation 34), according to each corresponding t and t ′ . Let   k 1 km k m A HAA∗ B ∗ |k = hk1 A (A , B ), . . . , hA (A , B ) ∈ HA∗ B ∗ (41) be the payoff vector of the player JAk , with specific type k ∈ L, when JB varies its type t ′ in M. Now, based on Equation 41, define the following set, for 0 ≤ α ≤ 1: ∗ ∗ (A B ) WA,α = H A∗  g HAA∗ B ∗ |1 ′  ,α (e), . . . , g HAA∗ B ∗ |l ′  ,α  (e) | e = (e1 , . . . , em ) ∈ Sα ′ , (42) ∗ |k where g ′A B : S α ′ → R is defined for k ∈ L by Equation 19 and S α ′ is given by  ,α   Equation 15. Observe that the arithmetic restriction is respected in the definition of (A∗ B ∗ ) . the set WA,α From Equation 42, for each k ∈ L, we compute a fuzzy mean related to k, whose α-cuts, for 0 ≤ α ≤ 1, are given as HAA∗ B ∗ |k μ ′  ∗ ∗ (A B [α] = P rojk (w) | w ∈ WA,α ∗ ) , ∗ (43) (A B ) where P rojk : WA,α → R is the projection function for the coordinate of k order from vector w, and the fuzzy mean is calculated as in Equation 20, where HAA∗ B ∗ |k is given by Equation 41. International Journal of Intelligent Systems DOI 10.1002/int 16 ASMUS, DIMURO, AND BEDREGAL Finally, the fuzzy payoffs of the player JAL , for a given combination of vectors of strategies (A∗ , B ∗ ), are obtained, for 0 ≤ α ≤ 1, through its α-cuts as L ∗ (A B w hA (A∗ , B ∗ )[α] = g,α (e) | e = (e1 , . . . , el ) ∈ Sα ∧ w ∈ WA,α  μw [α] =  ∗ ) (44) ∗ ∗) (A B w∈WA,α where  is the probability distributions of the l possible types for the player w [α] defines the α-cuts of the fuzzy mean as in Equation 20, g,α : Sα → R JAL , μw  ∗ ∗ (A B ) is defined by Equation 19, Sα is given by Equation 15, and WA,α is defined in Equation 42. The payoffs of the player JBM are computed in an analogous way. The fuzzy payoff matrix is denoted by R AB . Then, considering the discussion in the previous paragraphs, the definition of a two-player fuzzy Bayesian game, analogous to 2−BG′ , can be stated as follows: DEFINITION 5.4. A two-player fuzzy Bayesian game is defined as a six-tuple  L L 2−F BG′ = L, M, A, B, hA , hB . (45) To find a NE, we have to use a method for ordering two fuzzy numbers. In this work, we adopt the AD-order relation given in Definition 2.1. Observe that, by generalizing the concept of Bayesian game to fuzzy Bayesian game, the first becomes a particular case of the latter, if we consider only the cores of the fuzzy payoffs. 5.2. The Two-Player Interval Fuzzy Bayesian Games l }, of  = {φ 1 , . . . , φ Assume that the interval fuzzy probabilities distributions  ′ = {φ 1′ , . . . , φ m′ } of the m possible the l possible types for the player JAL , and  types for the player JBM , are known. Then, one can weight these probabilities by the respective payoffs contained in the matrix GLM AB , given in Equation 33, using AB the definition of interval fuzzy mean (Equation 29), to produce the final matrix Z containing the interval fuzzy probabilistic payoffs of the game. α1 ,α2 α1 ,α2 and S as given by Equation 23, In what follows, consider the sets S  ′ A and the set HA∗ B ∗ |k (Equation 41). Define the following set, for 0 ≤ α1 ≤ α2 ≤ 1: ∗ ∗ (A B ) VA,[α = 1 ,α2 ]   HAA∗ B ∗ |1 HAA∗ B ∗ |l α1 ,α2 , (46) q (e), . . . , q (e) | e = (e1 , · · · , em ) ∈ S ′ ′ ,[α ,α ] ′ ,[α ,α ]  1 2 H A∗ 1 2 ∗ |k α1 ,α2 A B where the function q → R is defined, for each k ∈ L, by Equation ′ ,[α1 ,α2 ] : S′ 28. Observe that the arithmetic constrain is respected in the definition of the set (A∗ B ∗ ) VA,[α . 1 ,α2 ] International Journal of Intelligent Systems DOI 10.1002/int 17 INTERVAL-VALUED FUZZY BAYESIAN GAMES For each k ∈ L, we calculate the interval fuzzy mean (Equation 29) related to k, whose [α1 , α2 ]-cuts for α1 , α2 ∈ [0, 1] and 0 ≤ α1 ≤ α2 , are given as H A∗ B ∗ |k A  μ ′ ∗ ∗ ∗ (A B ) , [α1 , α2 ] = P rojk (v) | v ∈ VA,[α 1 ,α2 ] (47) ∗ (A B ) where P rojk : VA,[α → R is the projection function for the coordinate of k order 1 ,α2 ] (A∗ B ∗ ) from the vector v and VA,[α1 ,α2 ] is defined by Equation 46. The interval fuzzy payoffs of the player JAL , for a given strategy vector combination (A∗ , B ∗ ), are obtained, for 0 ≤ α1 ≤ α2 ≤ 1, through its [α1 , α2 ]-cuts as follows: ∗ ∗ (A B ) α1 ,α2 v  ∧ v ∈ VA,[α hLA (A∗ , B ∗ )[α1 , α2 ]= q ,[α1 ,α2 ] (e) | e = (e1 , . . . , el ) ∈ S  1 ,α2 ]   μv (48) =  [α1 , α2 ] ∗ ∗ (A B ) v∈VA,[α ,α ] 1 2 where  μv  [α1 , α2 ] defines the [α1 , α2 ]-cuts of the interval fuzzy mean as in Equation (A∗ B ∗ ) α1 ,α2 v 29, q → R is defined by Equation 28, and VA,[α is defined in ,[α1 ,α2 : S  1 ,α2 ] Equation 46. The payoffs of the player JBM are computed analogously. Then, considering the discussion in the previous paragraphs, the definition of a two-player interval fuzzy Bayesian game, analogous to 2−BG′ , can be stated as follows: DEFINITION 5.5. A two-player interval fuzzy Bayesian game is defined as a six-tuple   hLA ,  hLB . 2−F BG′ = L, M, A, B,  (49) PROPOSITION 5.1. The interval fuzzy payoff  hLA (A∗ , B ∗ ) of an interval fuzzy Bayesian game can be obtained through its generator fuzzy payoffs, for 0 ≤ α1 ≤ α2 ≤ 1, by L L  hLA (A∗ , B ∗ )[α1 , α2 ] = hA (A∗ , B ∗ )l [α1 ] ∩ hA (A∗ , B ∗ )u [α2 ]. (50) L L Proof. It is immediate, since  hLA (A∗ , B ∗ ) = (hA (A∗ , B ∗ )l , hA (A∗ , B ∗ )u ), for 0 ≤ α1 ≤ α2 ≤ 1.  The result of Proposition 5.1 can be stated for the fuzzy payoff  hM B analogously. ′ Denote the interval fuzzy probabilities φ̂ and φ̂ by (φ̄l , φ̄u ) and (φ̄ ′ l , φ̄ ′ u ), respectively. From an interval fuzzy Bayesian game, we obtain two fuzzy Bayesian games: the first for the lower fuzzy probabilities φ̄l and φ̄ ′ l (α1 ∈ [0, 1]), and the second for the upper fuzzy probabilities φ̄u and φ̄ ′ u (α2 ∈ [0, 1]). The results of each International Journal of Intelligent Systems DOI 10.1002/int 18 ASMUS, DIMURO, AND BEDREGAL game, for each pair of vectors (A∗ , B ∗ ), constitute the generator fuzzy payoffs of the interval fuzzy payoffs of the original game, expressed as  L  L  hLA (A∗ , B ∗ ) = hA (A∗ , B ∗ )l , hA (A∗ , B ∗ )u   M ∗ ∗ M ∗ ∗ ∗ ∗  hM B (A , B ) = hB (A , B )l , hB (A , B )u . Thus, it is possible to state the equivalent definition: DEFINITION 5.6. A two-player interval fuzzy Bayesian game is defined as a six-tuple   L L   L L  2−F BG′′ = L, M, A, B,  hBl ,  hAu ,  hBu . hAl ,  (51) To investigate the existence of equilibria, we consider the interval AD-order (Definition 10). Observed that an interval fuzzy Bayesian game may be treated as a family of fuzzy Bayesian games, covering various fuzzy payoff modelings. 6. EXAMPLE 1: THE BAYESIAN HIRING GAME In this section, we analyze the proposed approach considering the Bayesian Hiring Game (BHG), a two-agent strategic interaction related to a hiring decision making, in which at least one of the agents is of unknown type. This kind of strategic interaction is very common in many areas, such as management and business. In particular, for MAS decision making, it can be used in situations involving agent negotiation, e.g., in resource or team allocation to accomplishment of a task or a goal.35 In this example, we show a case in which the standard BHG presents more than one Nash equilibria. However, the use of (interval) fuzzy probabilities ensures just one strict NE solution, whenever the (interval) AD-order relation is used. The hiring game can be stated as follows: A well-known and conservative company (the player denoted by JB ) is considering to hire the services of a new advertising agency that has just appeared in the market (the player denoted by JA ). The set of actions available to the player JB is B = {H, NH }, where H and NH denote the actions of “to hire” and “not to hire,” respectively. On the other hand, the player JA may choose a strategy in the action set A = {AC, MC}, where AC and MC denote the actions of “to adopt/propose an aggressive advertising campaign” and “to adopt/propose a moderate advertising campaign,” respectively. Although the conservatism of the company JB is well known, the type of the player JA is unknown in the market, but varying in the set L = {a (audacious), c (classic)}. The conservatism type of the player JB is denoted by b. The payoffs of the games for each particular type of JAL may be seen in Table I, assuming that they are established by experts considering an analysis of advantages, International Journal of Intelligent Systems DOI 10.1002/int 19 INTERVAL-VALUED FUZZY BAYESIAN GAMES Table I. Payoffs considering an advertising agency JAL , with types in L = {a (audacious), c (classic)}. (I) Type a (audacious) R JAa AC MC (II) Type c (classic) JB Aa B R H NH 2, −2 −1, 1 1, 0 0, −1 JB Ac B JAc AC MC H NH −2, 2 2, 2 −1, 0 0, −1 Table II. Probabilistic payoffs of the BHG. Lb RAB a JA , JAc JB AC, AC AC, MC MC, AC MC, MC H NH 4pa − 2, 2 − 4pa 2, 2 − 4pa pa − 2, 2 − pa 2 − 3pa , −pa + 2 2pa − 1, 0 pa , pa − 1 pa − 1, −pa 0, −1 risks, and losses. Observe that each game in Table I has a strict NE, namely, the strategy combinations (AC, NH ) for the game (I) and (MC, H ) for the game (II). 6.1. Defining the Bayesian Hiring Game To solve the hiring game, according to an expert analysis, the probability pa is assigned for the possibility of the agency being of the type a (audacious). Then, the probability of the agency being of the type c (classic) is pc = 1 − pa . The payoffs of the BHG are obtained by calculating the expected value of the payoffs for each combination of strategies, considering Equations 35 and 36. The probabilistic payoffs are shown in Table II, where the actions of the player JAL are represented by pairs composed by the actions of each type JA may assume in L. Now, consider the following situations: Case (BHG 1). Assuming that pa = 0.7, we obtain the payoffs presented in Table III (I), and the NE is the combination of actions ((AC, MC), NH ), meaning that with a chance of 70% of the advertising agency being audacious, the best rational choice for the company is “not to hire the services of this agency.” Case (BHG 2). On the other hand, whenever pa = 0.6, then the resulting payoffs are as shown in Table III (II), and, in this case, there exist two Nash Equilibra, namely, the strategy combinations ((AC, MC), H ) and ((AC, MC), NH ). Unless other circumstantial factors are analyzed (through the study of focal points or Schelling points56 ), in this situation the company would not know which is the best strategy to adopt. International Journal of Intelligent Systems DOI 10.1002/int 20 ASMUS, DIMURO, AND BEDREGAL Table III. Payoffs of the BHG. (I) pa = 0.7 Lb RAB JAa , (II) pa = 0.6 Lb RAB JB JAc AC, AC AC, MC MC, AC MC, MC H NH 0.8, −0.8 2, −0.8 −1.3, 1.3 −0.1, 1.3 0.4, 0 0.7, −0.3 −0.3, −0.7 0, −1 6.2. JAa , JB JAc AC, AC AC, MC MC, AC MC, MC H NH 0.4, −0.4 2, −0.4 −1.4, 1.4 0.2, 1.4 0.2, 0 0.4, −0.4 −0.4, −0.6 0, −1 The Fuzzy Bayesian Hiring Game Clearly, the assigned probability pa is essential to estimate the payoffs of the BHG. However, in many cases, these probabilities are difficult to be accurately estimated. In the case of this example, perhaps the specialists would not agree with a single value of pa . To avoid such impasse and to model the uncertainty in this modeling, one may adopt the fuzzy probabilities pa and pc , obtaining a FBHG. Considering the nonempty subsets Y = {a}, Z = {c} ⊆ L = {a, c}, the fuzzy probabilities are assigned in a way that pa = P (Y ), pc = P (Z) and P (L) = P (Y ∪ Z) = 1, respecting the arithmetic restriction. Then we obtain a probability distribution  = {pa , pc }. Observe that ′ = {1}, as JB does not vary its type. In the following, let A∗ be the vectors containing the combinations of actions chosen by JAa and JAc and B ∗ be the unitary vector containing the action chosen by JB . (A∗ B ∗ ) (A∗ B ∗ ) and WB,α Consider the sets Sα and S α ′ given by Equation 15, and the sets WA,α  c b (hab (Aa ,B b ),hcb A (A ,B )) A defined in Equation 57. Consider also the functions g,α a b (hab A (A ,B )) ′ : Sα → R c b (hcb A (A ,B )) ′ and g ,g : S α ′ → R defined by Equation 19.  ,α  ,α  We obtain the payoffs of the FBHG by calculating the expected value using the definition of fuzzy mean with the payoffs contained in Tables I (I) and (II). By Equation 40, the set of vectors of the payoffs of the player JAL is given by HAA∗ B ∗ =    cb c b  a b hab A (A , B ) , hA (A , B ) . (52) By Equations 41 and 42, we have, for 0 ≤ α ≤ 1, that ∗ ∗  ab a b  (h (A ,B )) (hcb (Ac ,B b )) g ′A (1), g ′A (1)  ,α  ,α  ab a b cb c b  = hA (A , B ), hA (A , B ) . (A B ) = WA,α International Journal of Intelligent Systems DOI 10.1002/int (53) 21 INTERVAL-VALUED FUZZY BAYESIAN GAMES From Equation 44, we obtain the fuzzy payoffs of the player JAL through its α-cuts for a given combination of strategy vectors (A∗ , B ∗ ), for 0 ≤ α ≤ 1, as L ∗ (A B w hA (A∗ , B ∗ )[α] = g,α (e) | e = (ea , ec ) ∈ Sα ∧ w ∈ WA,α c b (hab (Aa ,B b ),hcb A (A ,B )) A = g,α c b (hab (Aa ,B b ),hcb A (A ,B )) = μ A (hab (Aa ,B b ),hcb (Ac ,B b )) A where μ A 20. It follows that ∗ ) (54) (e) | e = (ea , ec ) ∈ Sα [α], [α] defines the α-cuts of the fuzzy mean as in Equation   L a b cb c b α hA (A∗ , B ∗ )[α] = hab A (A , B ) · ea + hA (A , B ) · ec | e = (ea , ec ) ∈ S . (55) Analogously, by Equation 40, the a set of the single payoff vector of the player JB is given by HAB∗ B ∗ =   a b cb c b hab B (A , B ), hB (A , B ) . (56) By Equations 41 and 42, for 0 ≤ α ≤ 1, we have that ∗ ∗  HB |  A∗ B ∗ b g,α (e) | e = (ea , ec ) ∈ Sα  ab a b cb c b  (hB (A ,B ),hB (A ,B )) = g,α (e) | e = (ea , ec ) ∈ Sα    a b cb c b α = hab B (A , B ) · ea + hB (A , B ) · ec | (ea , ec ) ∈ S , (A B ) WB,α = H B |b (57) ij : Sα → R is defined by Equation 19. in which g,α By Equation 44, the fuzzy payoffs of the player JB , for a given combination of strategy vectors (A∗ , B ∗ ), are obtained through its α-cuts, for 0 ≤ α ≤ 1, as ∗ ∗ (A B ) hB (A∗ , B ∗ )[α] = g w′ (1) | w ∈ WB,α  ,α  c b (hab (Aa ,B b )·ea +hcb B (A ,B )·ec ) = g ′B (1) | (ea , ec ) ∈ Sα  ,α =  (ea ,ec )∈Sα cb a b c b (hab B (A ,B )·ea +hB (A ,B )·ec ) μ ′  International Journal of Intelligent Systems [α], DOI 10.1002/int (58) 22 ASMUS, DIMURO, AND BEDREGAL Table IV. Fuzzy payoffs of the FBHG for pa = (0.6/0.7/0.8). Lb R̄AB JAa , JAc JB AC, AC AC, MC MC, AC MC, MC H NH (0.4/0.8/1.2), (−1.2/ − 0.8/ − 0.4) 2, (−1.2/ − 0.8/ − 0.4) (−1.4/ − 1.3/ − 1.2), (1.2/1.3/1.4) (−0.4/ − 0.1/0.2), (1.2/1.3/1.4) (0.2/0.4/0.6), 0 (0.6/0.7/0.8), (−0.4/ − 0.3/ − 0.2) (−0.4/ − 0.3/ − 0.2), (−0.8/ − 0.7/ − 0.6) 0, −1 cb a b c b (hab B (A ,B )·ea +hB (A ,B )·ec ) where μ ′  Equation 20 and g [α] define the α-cuts of the fuzzy mean as shown in cb a b c b (hab B (A ,B )·ea +hB (A ,B )·ec ) ′  ,α : S α ′ → R is defined by Equation 19,  It follows that   a b cb c b α hB (A∗ , B ∗ )[α] = hab B (A , B ) · ea + hB (A , B ) · ec | (ea , ec ) ∈ S . (59) Now consider the following situations: Case (FBHG 1). Suppose that the experts have assigned the triangular fuzzy number pa = (0.6/0.7/0.8) to model the case in which the fuzzy probability of the advertising agency be of the audacious type is “around” 70%, which represents a fuzzy modeling for the Case (BHG 1) discussed in Section 6.1. The corresponding α-cuts of pa are pa [α] = [0.6 + 0.1α, 0.8 − 0.1α] and one has that pc = (0.2/0.3/0.4) and pc [α] = [0.2 + 0.1α, 0.4 − 0.1α]. Considering the fuzzy means for pa and pc , we obtain the fuzzy payoffs for both players shown in Table IV. Observe that the arithmetic restriction is respected. To investigate the existence of a strict NE, we use the AD-order relation given in Definition 2.1, obtaining the strategy combination ((AC, MC), NH ). This result coincides with the crisp version of the game, when pa = 0.7. Case (FBHG 2). On the other hand, suppose that one determines the fuzzy probabilities as pa = (0.5/0.6/0.7) and pc = (0.3/0.4/0.5), International Journal of Intelligent Systems DOI 10.1002/int 23 INTERVAL-VALUED FUZZY BAYESIAN GAMES Table V. Fuzzy payoffs of the FBHG for pa = (0.5/0.6/0.7). Lb R̄AB JAa , JAc JB AC, AC AC, MC MC, AC MC, MC H NH (0/0.4/0.8), (−0.8/ − 0.4/0) 2, (−0.8/ − 0.4/0) (−1.5/ − 1.4/ − 1.3), (1.3/1.4/1.5) (−0.1/0.2/0.5), (1.3/1.4/1.5) (0/0.2/0.4), 0 (0.5/0.6/0.7), (−0.5/ − 0.4/ − 0.3) (−0.5/ − 0.4/ − 0.3), (−0.7/ − 0.6/ − 0.5) 0, −1 which represents a fuzzy modeling for the Case (BHG 2) discussed in Section 6.1. The resulting payoffs are shown in Table V. By using the AD-order, a NE is found in the strategy combination ((AC, MC), NH ), meaning that, even with a smaller chance of the advertising agency being of the audacious type (“around” 60%), it is still safer for the company not to hire its services. Observe that the use of fuzzy probabilities produced a strict NE, unlike the crisp case using pa = 0.6. 6.3. Interval-Valued Fuzzy Bayesian Hiring Game In this case, it is considered that the experts not only are unsure about which probabilities they must assign to each unknown type that the agency may assume, but they cannot reach a consensus on the form of the fuzzy numbers for the modeling the uncertainty in these probabilities. Then, we adopt interval fuzzy numbers to represent the probabilities of the agency being audacious or classical, obtaining an interval-valued fuzzy Bayesian hiring game (IFBHG). Considering the nonempty subsets Y = {a}, Z = {c} ⊆ L{a, c}, the interval (Y ), pc = P (Z) and P (L) = fuzzy probabilities are assigned in a way that pa = P  P (Y ∪ Z) = 1, respecting the arithmetic restriction for interval fuzzy probabilities. ′ = {  = {pa , pc }. Observe that  Then we obtain a probability distribution  1}, as JB does not vary its type. In the following, let A∗ be the vectors containing the combinations of actions chosen by JAa and JAc , and B ∗ be the unitary vector containing the action chosen by JB , (A∗ B ∗ ) α1 ,α2 α1 ,α2 and S given by Equation 23, and the sets VA,[α and and consider the sets S  ′ 1 ,α2 ] ∗ c b (hab (Aa ,B b ),hcb A (A ,B )) ∗ (A B ) A defined by Equation 60. Consider also the functions q VB,[α ,[α ,α ] 1 ,α2 ] α1 ,α2 S  a b c b (hab (hcb A (A ,B )) A (A ,B )) q ′ ,[α1 ,α2 ] , q ′ ,[α1 ,α2 ] 1 2 : α1 ,α2 S ′ → R and : → R, defined by Equation 28. We determine probabilistic interval fuzzy payoffs of the IFBHG for each combination of payoff vectors A∗ and B ∗ , using the definition of interval fuzzy mean, and cb a b c b the already defined HAA∗ B ∗ = {(hab A (A , B )), (hA (A , B ))} (Equation 52). Then, by Equation 46, we obtain the following expression, with 0 ≤ α1 ≤ α2 ≤ 1: ∗ ∗  ab a b  c b (hA (A ,B )) (hcb A (A ,B )) q (1), q (1) ′ ′  ,[α1 ,α2 ]  ,[α1 ,α2 ]   ab a b cb c b  = (hA (A , B ), hA (A , B )) . (A B ) = VA,[α 1 ,α2 ] International Journal of Intelligent Systems DOI 10.1002/int (60) 24 ASMUS, DIMURO, AND BEDREGAL Based on Equation 48, the interval fuzzy payoffs of the player JAL , for each combination of strategy vectors (A∗ , B ∗ ), are obtained through its [α1 , α2 ]-cuts, for 0 ≤ α1 ≤ α2 ≤ 1, as ∗ ∗ (A B ) α1 ,α2 w  ∧ v ∈ VA,[α } hLA (A∗ , B ∗ )[α1 , α2 ] = {q ,[α1 ,α2 ] (e) | e = (ea , ec ) ∈ S  1 ,α2 ] c b (hab (Aa ,B b ),hcb A (A ,B )) A = q ,[α ,α ] 1 2 c b (hab (Aa ,B b ),hcb A (A ,B )) A = μ  α1 ,α2 (e) | e = (ea , ec ) ∈ S  [α1 , α2 ], (61) (hab (Aa ,B b ),hcb (Ac ,B b )) A A where  μ [α1 , α2 ] defines the [α1 , α2 ]-cuts of the interval fuzzy  mean as in Equation (29). Then one has that  a b cb c b  hLA (A∗ , B ∗ )[α1 , α2 ] = hab A (A , B ) · ea + hA (A , B ) · ec | e α1 ,α2 . = (ea , ec ) ∈ S  (62) Analogously, we obtain the payoffs of the player JB . Considering Equation 56, then one has that   a b cb c b HAB∗ B ∗ = hab B (A , B ), hB (A , B ) , and we obtain the following set for 0 ≤ α1 ≤ α2 ≤ 1: ∗ ∗  HB |  α1 ,α2 A∗ B ∗ b q (e) | e = (ea , ec ) ∈ S  ,[α1 ,α2 ]  ab a b cb c b  (hB (A ,B ),hB (A ,B )) α1 ,α2 = q,[α (e) | e = (ea , ec ) ∈ S  ,α ] (A B ) VB,[α = 1 ,α2 ] 1 H B |b (63) 2   α1 ,α2 a b cb c b , = hab B (A , B ) · ea + hB (A , B ) · ec | (ea , ec ) ∈ S  α1 ,α2 ij where q → R is defined by Equation 28.  ,[α1 ,α2 : S By Equation 48, the (probabilistic) interval fuzzy payoffs of the player JB , for a given combination of strategy vectors (A∗ , B ∗ ), are obtained through its [α1 , α2 -cuts, for 0 ≤ α1 ≤ α2 ≤ 1, as ∗ ∗ (A B ) v  hB (A∗ , B ∗ )[α1 , α2 ] = q ′ ,[α1 ,α2 (1) | v ∈ VB,[α1 ,α2 ] c b (hab (Aa ,B b )·ea +hcb B (A ,B )·ec ) B = q ′ ,[α ,α ] 1 =  2 α1 ,α2 (ea ,ec )∈S  α1 ,α2 (1) | (ea , ec ) ∈ S  c b (hab (Aa ,B b )·ea +hcb B (A ,B )·ec ) B  μ ′ International Journal of Intelligent Systems (64) [α1 , α2 ] DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 25 Table VI. Payoffs of the lower FBHG for pa l = (0.65/0.7/0.75). Lb R AB JB JAa , JAc AC, AC AC, MC MC, AC MC, MC H NH (0.6/0.8/1), (−1/−0.8/−0.6) 2, (−1/−0.8/−0.6) (−1.35/−1.3/−1.25), (1.25/1.3/1.35) (−0.25/−0.1/0.05), (1.25/1.3/1.35) (0.3/0.4/0.5), 0 (0.65/0.7/0.75), (−0.35/−0.3/−0.25) (−0.35/−0.3/−0.25), (−0.75/−0.7/−0.65) 0, −1 c b (hab (Aa ,B b )·ea +hcb B (A ,B )·ec ) B where  μ ′ [α1 , α2 ] defines the [α1 , α2 ]-cuts of the interval fuzzy c b a (A ,B b )·ea +hcb (hab α1 ,α2 B (A ,B )·ec ) B mean as in Equation (29), q : S → R is defined in Equa′ ′ ,[α1 ,α2 ] tion 28. Finally one obtains: α1 ,α2 a b cb c b  . hB (A∗ , B ∗ )[α1 , α2 ] = hab B (A , B ) · ea + hB (A , B ) · ec | (ea , ec ) ∈ S  (65) Now consider the following situations: Case (IFBHG 1). Suppose that one assigns interval fuzzy numbers for pa and pc to analyze the resulting payoffs when modeling the case in which the fuzzy probability of the advertising agency being of the audacious type is “around” 70%, but the experts did not come into consensus on the modeling of this fuzzy number, which represents an interval fuzzy modeling for the Case (FBHG 1) discussed in Section 6.2. Then, consider the interval fuzzy number pa = (pa l , pa u ) = ((0.65/0.7/0.75), (0.55/0.7/0.85)), which encompasses several fuzzy modelings of the expression “around” 70%. As stated in the definition of interval fuzzy probability, one has that pc = (pc l , pc u ) = ((0.25/0.3/0.35), (0.15/0.3/0.45)). According to Proposition 5.1, we can build two FBHGs, one for each generator fuzzy number of pa to obtain the payoffs of the analyzed IFBHG. Thus, by assigning pa l = (0.65/0.7/0.75), a lower FBHG is constructed, and calculating the fuzzy means, we find the payoffs shown in Table VI. Analougouly, we assign pa u = (0.55/0.7/0.85) and obtain, through the α-cuts, the payoffs of an upper FBHG shown in Table VII. Based on Equation 50, for each combination of payoff vectors (A∗ B ∗ ), for the player JAL we have that  L  L  hLA (A∗ , B ∗ )[α1 , α2 ] = hA (A∗ , B ∗ )l [α1 ] ∩ hA (A∗ , B ∗ )u [α2 ] , International Journal of Intelligent Systems DOI 10.1002/int (66) 26 ASMUS, DIMURO, AND BEDREGAL Table VII. Payoffs of the upper FBHG for pa u = (0.55/0.7/0.85). L− R AB JB JAa , JAc AC, AC AC, MC MC, AC MC, MC H NH (0.2/0.8/1.4), (−1.4/−0.8/−0.2) 2, (−1.4/−0.8/−0.2) (−1.45/−1.3/−1.15), (1.15/1.3/1.45) (−0.55/−0.1/0.35), (1.15/1.3/1.45) (0.1/0.4/0.7), 0 (0.55/0.7/0.85), (−0.45/−0.3/−0.15) (−0.45/−0.3/−0.15), (−0.85/−0.7/−0.55) 0, −1 Table VIII. Payoffs of the player JAL for pa = ((0.65/0.7/0.75), (0.55/0.7/0.85)). JB JAa , JAc AC, AC AC, MC MC, AC MC, MC H NH ((0.6/0.8/1), (0.2/0.8/1.4)) 2 ((−1.35/−1.3/−1.25), (−1.45/−1.3/−1.15)) ((−0.25/−0.1/0.05), (−0.55/−0.1/0.35)) ((0.3/0.4/0.5), (0.1/0.4/0.7)) ((0.65/0.7/0.75), (0.55/0.7/0.85)) ((−0.35/−0.3/−0.25), (−0.45/−0.3/−0.15)) 0, −1 Table IX. Payoffs of the player JB for pa = ((0.65/0.7/0.75), (0.55/0.7/0.85)). JB JAa , JAc AC, AC AC, MC MC, AC MC, MC H NH ((−1/−0.8/−0.6), (−1.4/−0.8/−0.2)) ((−1/−0.8/−0.6), (−1.4/−0.8/−0.2)) ((1.25/1.3/1.35), (1.15/1.3/1.45)) ((1.25/1.3/1.35), (1.15/1.3/1.45)) 0 ((−0.35/−0.3/−0.25), (−0.45/−0.3/−0.15)) ((−0.75/−0.7/−0.65), (−0.85/−0.7/−0.55)) 0, −1 for 0 ≤ α1 ≤ α2 ≤ 1. Furthermore, it holds that  L  L  hLA (A∗ , B ∗ ) = hA (A∗ , B ∗ )l , hA (A∗ , B ∗ )u . (67) Thus, by composing each interval fuzzy payoff through their generator fuzzy numbers, we obtain the final payoff matrix for the player JAL , shown in Table VIII. Following the same procedure for the player JB , one obtains the payoffs of the player JB , as shown in Table IX. Finally, using the interval AD-order, we find that there exists a strict NE in the strategy combination ((AC, MC), NH ), which coincides with the solution found for both BHG and FBHG, in the cases (BHG 1) (Section 6.1) and (FBHG 1) (Section 6.2), respectively. Case (IFBHG 2). By assigning pa = ((0.55/0.6/0.65), (0.45/0.6/0.75)), we obtain an IFBHG representing an interval fuzzy modeling for the Case (FBHG 2) discussed in Section 6.2. Similarly to developed for the Case (IFBHG 1) above, we obtain the interval fuzzy payoffs for the players JAL and JB , as shown in Tables X International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 27 Table X. Payoffs of the player JAL for pa = ((0.55/0.6/0.65), (0.45/0.6/0.75)). JAL JAa , JAc AC, AC AC, MC MC, AC MC, MC H NH ((0.2/0.4/0.6), (−0.2/0.4/1)) 2 ((−1.45/−1.4/−1.35), (−1.55/−1.4/−1.25)) ((0.05/0.2/0.35), (−0.25/0.2/0.65)) ((0.1/0.2/0.3), (−0.1/0.2/0.5)) ((0.55/0.6/0.65), (0.45/0.6/0.75)) ((−0.45/−0.4/−0.35), (−0.55/−0.4/−0.25)) 0 Table XI. Payoffs of the player JB for pa = ((0.55/0.6/0.65), (0.45/0.6/0.75)). JB JAa , JAc AC, AC AC, MC MC, AC MC, MC H NH ((−0.6/−0.4/−0.2), (−1/−0.4/0.2)) ((−0.6/−0.4/−0.2), (−1/−0.4/0.2)) ((1.35/1.4/1.45), (1.25/1.4/1.55)) ((1.35/1.4/1.45), (1.25/1.4/1.55)) 0 ((−0.45/−0.4/−0.35), (−0.55/−0.4/−0.25)) ((−0.65/−0.6/−0.55), (−0.75/−0.6/−0.45)) −1 and XI, respectively. We obtain a NE in the strategy combination ((AC, MC), NH ), which coincides with the solution found for the FBHG, in the case (FBHG 2) (Section 6.2). Therefore, even with a smaller chance of the advertising agency being of the audacious type (“around” 60%), considering the uncertainty of the experts in the fuzzy modeling, it is still safer for the company not to hire its services. Observe that the use of interval fuzzy probabilities produced a strict NE, like in the fuzzy case, but unlike the crisp case using simply pa = 0.6. 7. EXAMPLE 2: THE BAYESIAN PRISONER’S DILEMMA The Prisoner’s Dilemma is a paradox in decision analysis in which two individuals acting for their own best interests adopt a course of action that does not result in the ideal outcome. The standard Prisoner’s Dilemma is stated in such a way that both parties choose to protect themselves at the expense of the other partner. As a result of following a purely rational reasoning process to help oneself, both players conduct themselves to a worse solution than if they had cooperated with each other in the decision-making process. The iterated and evolutionary approaches of Prisoner’s Dilemma are commonly used to analyze/promote the emergence of cooperation/coalition among individual agents, e.g., when the agents may evolve their strategies by reinforcement learning considering social relationships.36 Zhang et al.38 have pointed out that psychological experiment studies revealed that human interaction behaviors are often not the same as what game theory predicts, due to several reasons, e.g., relevant constraints (in general, established by the social/organizational/physical context) are not considered when the players choose their best strategies. International Journal of Intelligent Systems DOI 10.1002/int 28 ASMUS, DIMURO, AND BEDREGAL Differently from the proposal of Zhang et al.,38 which considered fuzzy constraints in the Prisoner’s Dilemma, in an attempt to reflect human behaviors (e.g., fairness, altruism, self-interest), we consider that the agents can assume different types defined by moral standards,37 establishing a Bayesian Prisoner’s Dilemma with Moral Standards, so that the model can predict the outcomes more adequately. The Prisoner’s Dilemma can be illustrated by the following situation: Two thieves are arrested by the police, but there is no evidence to incriminate them irrefutably. Then each of the suspects is isolated and interviewed separately by the police. Both thieves receive the same proposal: If a suspect does not confess and the other does, the one who confessed is released for helping the police and the other takes maximum time in jail. If both of them confess, then the complaint of crime partner loses value and each one takes a medium penalty. However, it is of common knowledge that if no one confesses the crime, then each one takes a minimum penalty for the lack of evidence. The last situation of both thieves not confessing represents cooperation in the decision-making process. It is well known that the Prisoner’s Dilemma has a NE when both agents choose not to cooperate, confessing the crime.1 Although this combination of strategies provides more safety for the players, collectively it produces less gain than the case where the two decide to cooperate (not confessing). The problem is precisely the possibility of the player who decided not to confess being “betrayed” by the other, who may confess. In this case, the first would receive maximum penalty, whereas the latter not. 7.1. Introducing the Prisoner’s Dilemma with Moral Standards Consider the case in which there is the possibility of having different types of players, determined by moral standards.37 The strategic interaction consists in a simultaneous game, since there is no communication between the players neither a history of past interactions. The difference is that there exists a common belief about the type of each player, and how the interactions would occur between these types. The interaction takes place between players JA and JB , each of which have to choose between the strategies C (“to cooperate”) or NC (“not to cooperate”). The first moral standard to be defined is the egoism, when the agent is more concerned with itself than others. This is the type that describes the behavior of players in the classic Prisoner’s Dilemma. By considering the combinations of possible strategies, the egoistic player has the following preferences: (NC, C) > (C, C) > (NC, NC) > (C, NC). (68) On the other hand, the altruism standard (when the agent is regardful of others), leads to the following preferences of strategies: (C, C) > (NC, C) > (NC, NC) > (C, NC). (69) The altruistic player prefers mutual cooperation over other strategies. Finally, we define the selflessness standard (when the agent acts motivated by no concern International Journal of Intelligent Systems DOI 10.1002/int 29 INTERVAL-VALUED FUZZY BAYESIAN GAMES Table XII. Payoffs of the Prisoner’s Dilemma with Moral Standards. (I) JAa × JBa (II) JAe × JBa JB RAa Ba (III) JAs × JBa JB R A e Ba JB R A s Ba JA C NC JA C NC JA C NC C NC 3, 3 2, 0 0, 2 1, 1 C NC 2, 3 3, 0 0, 2 1, 1 C NC 2, 3 0, 0 3, 2 1, 1 for itself but only for the others), with the following preferences: (C, NC) > (C, C) > (NC, NC) > (NC, C). (70) Then, assume that the player JA varies its type in L = {e, a, s}, in which e, a, and s denote the egoistic, altruistic, and selfless types, respectively, and the players are denoted by JAL . The player JB is known to be of the altruistic type, denoted by JBa . The payoffs of the interaction between the players JBa and JAL are shown in Table XII. In this table, the maximum, medium, and minimum penalties are represented by 0, 1, and 2, respectively. No penalty is indicated by 3. In the case of Table XII (I), there are two NE, namely, the strategy combinations (C, C) and (NC, NC), that is, there is no strict NE. Considering the situation (II), the NE is found in the strategy combination (NC, NC), as in the standard Prisoner Dilemma. On the other hand, in the case (III), there is a NE in the strategy combination (C, C). 7.2. The Bayesian Prisoner’s Dilemma with Moral Standards We follow the same procedures developed for the BHG in Section 6.1. First we assign probabilities for each possible type of the player JAL and thereby construct a single probabilistic game. Denoted by pa , pe , and ps , the probabilities the player JAL to be of the altruistic, egoistic, and selfless types, respectively, with pa + pe + ps = 1. The matrix with probabilistic payoffs, obtained through the weighted averages, may be seen in Table XIII. Now, we assign values for the probabilities pa , pe , and ps to investigate the solutions of the game. We analyze the following situations: Case (BPD 1). Consider the case of total uncertainty about the probabilities to be assigned for each type, that is, pa = pe = ps = 13 . Then we obtain the payoffs shown in Table XIV (I), which presents two Nash equilibria in the strategy combinations ((C, NC, C), C) and ((NC, NC, C), NC). In this case, the player JBa would not have a strictly more advantageous action to choose, having to make use of other circumstantial information (focal points) to decide what action to take. Case (BPD 2). Now, assign pa = 0.2, pe = 0.2, and ps = 0.6, that is, there is a good chance (60%) that the player JAL is of the selfless type. The matrix containing International Journal of Intelligent Systems DOI 10.1002/int 30 ASMUS, DIMURO, AND BEDREGAL Table XIII. Probabilistic payoffs of the Bayesian Prisoner’s Dilemma for JAL and JBa . R A L Ba JB JAa , JAe , JAd C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C NC 3pa + 2pe + 2ps , 3pa + 3pe + 3ps 3pa + 2pe , 3pa + 3pe 3pa + 3pe + 2ps , 3pa + 3ps 3pa + 3pe , 3pa 2pa + 2pe + 2ps , 3pe + 3ps 2pa + 2pe , 3pe 2pa + 3pe + 2ps , 3ps 2pa + 3pe , 0 3ps , 2pa + 2pe + 2ps ps , 2pa + 2pe + ps pe + 3ps , 2pa + pe + 2ps pe + ps , 2pa + pe + ps pa + 3ps , pa + 2pe + 2ps pa + ps , pa + 2pe + ps pa + pe + 3ps , pa + pe + 2ps pa + pe + ps , pa + pe + ps Table XIV. Payoffs of the Bayesian Prisoner’s Dilemma with Moral Standards. (I) pa = pe = ps = R A L Ba JAa , JAe , JAs C, C, C C, C, N C C, N C, C C 7 3, 5 3, 8 3, R A L Ba JB JAa , JAe , JAs NC C NC 3 1, 2 C, C, C 2.2, 3 2 5 3 5 3 4 3 5 3 4 3 4 3 C, C, N C 1, 1 1 3, 4 3, 2 3, 4 3, 2 3, 5 3, 0 1, 1 N C, N C, N C 2 C, N C, N C 2, 1 2, 2 N C, C, N C 4 3, 5 3, 5 3, 1 N C, N C, N C (II) pa = 0.2, pe = 0.2, ps = 0.6 JB N C, C, C N C, N C, C 1 3 1.2 C, N C, C 2.4, 2.4 C, N C, N C 1.2, 0.6 N C, C, C 2, 2.4 1.8, 2 0.6, 1.4 2, 1.8 0.8, 1.2 2, 1.8 N C, C, N C 0.8, 0.6 0.8, 1.2 N C, N C, C 2.2, 1.8 2.2, 1.6 1, 0 1, 1 the payoffs of this interaction is shown in Table XIV (II). There exists a strict NE in the strategy combination ((C, NC, C), C), meaning that it is more advantageous for JBa to cooperate in this interaction. Case (BPD 3). Assume that whenever we assign a high probability to pe , the NE is some combination of strategies in which JBa chooses NC (“not to cooperate”). In fact, this always happens for pe > 0.5. To exemplify this case, we assign pa = 0.299, pe = 0.501, and ps = 0.2. The payoffs for these probabilities can be seen in Table XV (I). As expected, there exists a NE in the strategy combination ((NC, NC, C), NC), meaning that is safer for JBa not to cooperate. Case (BPD 4). However, by assigning pe = 0.5, a difference of 0.01 from the Case (BPD 3), we get the same equilibrium solutions found for the case where there is complete uncertainty about the types that the player JAL may assume (i.e., pa = pe = ps = 13 ). As an example, the payoffs for pa = 0.3, pe = 0.5 ,and ps = 0.2 may be seen in Table XV (II). International Journal of Intelligent Systems DOI 10.1002/int 31 INTERVAL-VALUED FUZZY BAYESIAN GAMES Table XV. Payoffs of the Bayesian Prisoner’s Dilemma with Moral Standards for pe around 0.5 (I) pa = 0.299, pe = 0.501, ps = 0.2 R A L Ba JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C JB (II) pa = 0.3, pe = 0.5, ps = 0.2 R A L Ba C NC 2.299, 3 1.899, 2.4 2.8, 1.497 2.4, 0.897 2, 2.103 1.6, 1.503 2.501, 0.6 2.101, 0 0.6, 2 0.2, 1.8 1.101, 1.499 0.701, 1.299 0.899, 1.701 0.4, 1.501 1.4, 1.2 1, 1 JB JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C 2.3, 3 1.9, 2.4 2.8, 1.5 2.4, 0.9 2, 2.1 1.6, 1.5 2.5, 0.6 2.1, 0 NC 0.6, 0.2, 1.1, 0.7, 0.9, 0.4, 1.4, 1, 2 1.8 1.5 1.3 1.7 1.5 1.2 1 Observe that the situation of Case (BPD 4) is very difficult to model. How can one accurately estimate the chance of the another player be of the egoistic type? As the player JBa is altruistic, it is important to estimate whether or not it should rely on the another player, or if at least there is some the possibility of mutual cooperation. 7.3. The FBPD with Moral Standards To deal with the problem discussed in the Case (BPD 4) of Section 7.2, one may use fuzzy probabilities, defining a FBPD. For the set L = {a, e, s}, with nonempty subsets X = {a}, Y = {e}, and Z = {s}, we have that pa = P (X), pe = P (Y ), ps = P (Z), and P (L) = P (X ∪ Y ∪ Z) = 1, respecting the arithmetic restriction. Therefore, the set  = {pa , pe , ps } denotes the fuzzy probability distribution. Analogously to how it was developed in the Fuzzy Hiring Game, we obtain the matrix of the fuzzy probabilistic payoffs of the FBPD, by weighting the payoffs and probabilities according to the definition of fuzzy mean expressed by Equation 20. We consider the following situations: Case (FBPD 1). We may obtain a fuzzy version of the Case (BPD 4) discussed in Section 7.2, assigning a probability “around” 50% of the player JA being of the egoistic type, modeled as pe = (0.4/0.5/0.6), a probability “around” 20% of that player being of the selfless type, given by ps = (0.1/0.2/0.3), and a probability “around” 30% of the player being of the altruistic type, given by pa = 0.3. As the calculations of the fuzzy means must respect the arithmetic restriction, one should always get a crisp value from each operated fuzzy number whose sum results in 1. That is, for xa ∈ pa [α], xe ∈ pe [α] and xs ∈ pd [α] we have that xa + xe + xs = 1. However, as pa [α] = [0.3, 0.3], then it follows that xa = 0.3. One has that 0.3 + xe + xs = 1 xs = 0.7 − xe . International Journal of Intelligent Systems (71) DOI 10.1002/int 32 ASMUS, DIMURO, AND BEDREGAL Table XVI. Expressions for the probabilistic payoffs of the FBPD. R A L Ba JB JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C NC 2.3, 3 0.9 + 2xe , 0.9 + 3xe 2.3 + xe , 3 − 3xe 0.9 + 3xe , 0.9 2, 2.1 0.6 + 2xe , 3xe 1 + xe , 2.1 − 3xe 0.6 + 3xe , 0 2.1 − 3xe , 2 0.7 − xe , 1.3 + xe 2.1 − 2xe , 2 − xe 0.7, 1.3 2.4 − 3xe , 1.7 1 − x e , 1 + xe 2.4 − 2xe , 1.7 − xe 1, 1 Table XVII. Fuzzy payoffs of the FBPD for pa = 0.3, pe = (0.4/0.5/0.6), ps = (0.1/0.2/ 0.3). R A L Ba JB JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C NC 2.3, 3 (1.7/1.9/2.1), (2.1/2.4/2.7) (2.7/2.8/2.9), (1.2/1.5/1.8) (2.1/2.4/2.7), 0.9 2, 2.1 (1.4/1.6/1.8), (1.2/1.5/1.8) (2.4/2.5/2.6), (0.3/0.6/0.9) (1.8/2.1/2.4), 0 (0.3/0.6/0.9), 2 (0.1/0.2/0.3), (1.7/1.8/1.9) (0.9/1.1/1.3), (1.4/1.5/1.6) 0.7, 1.3 (0.6/0.9/1.2), 1.7 (0.4/0.5/0.6), (1.4/1.5/1.6) (1.2/1.4/1.6), (1.1/1.2/1.3) (0.7/1/1.3), 1 The expressions of the probabilistic payoffs of the FBPD are presented in Table XVI. By assigning all possible values of xe ∈ pe [α] = [0.1α + 0.4, −0.1α + 0.6], with 0 ≤ α ≤ 1, we obtain the final matrix with fuzzy payoffs of FBPD, shown in Table XVII. Comparing the payoffs using the AD-order, we obtain a strict NE in the strategy combination ((NC, NC, C), NC). That happens by the fact that when the player JAL chooses the combination (C, NC, C), the player JBa must choose between the following combinations of strategies: ((C, NC, C), C) or ((C, NC, C), NC). In the crisp case (Table XV), the payoffs for such strategy combinations of strategies are equal, that is, haB ((C, NC, C), C) = haB ((C, NC, C), NC) = 1.5, which makes the strategy combination ((C, NC, C), C) be a nonstrict NE. However, in the fuzzy case, the strategy combinations ((C, NC, C), C) and ((C, NC, C), NC) produce the payoffs a hB ((C, NC, C), C) = (1.2/1.5/1.8) a hB ((C, NC, C), NC) = (1.4/1.5/1.6), International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES a 33 a Figure 3. Fuzzy payoffs hB ((C, N C, C), C) and hB ((C, N C, C), N C). respectively (see Figure 3). Using the AD-order (Definition 2.1), we have that (1.2/1.5/1.8) < (1.4/1.5/1.6), since it holds that 1.2 < 1.4 ∧ 1.6 < 1.8 ∧ 1.5 ≤ 1.5 ⇒ (1.2/1.5/1.8) < (1.4/1.5/1.6). It follows that a a hB ((C, NC, C), C) < hB ((C, NC, C), NC), which means that the strategy combination ((C, NC, C), C) does not constitute a NE. Thus, by considering a probability “around” 50% of the player JAL being of the egoistic type, it is still safer for the player JBa not to cooperate in the interaction. Case (FBPD 2). Now, we consider a slightly lower probability of the player JAL being of the egoistic type (49%) to analyze the solution of the game. Considering pe = (0.39/0.49/0.59), pa = 0.3, and ps = (0.11/0.21/0.31), one has the fuzzy payoffs shown in Table XVIII. We now compare the following payoffs of the player JBa (see Figure 4): a hB ((C, NC, C), C) = (1.23/1.53/1.83) and a hB ((C, NC, C), NC) = (1.41/1.51/1.61), using the AD-order. In this case, we consider the degree of imprecision ρ = 0.8. This degree determines how relevant is the accuracy of the information given by the fuzzy number in this context. International Journal of Intelligent Systems DOI 10.1002/int 34 ASMUS, DIMURO, AND BEDREGAL Table XVIII. Fuzzy payoffs of the FBPD for pa = 0.3, pe = (0.39/0.49/0.59), ps = (0.11/ 0.21/0.31). R A L Ba JB JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C NC 2.3, 3 (1.68/1.88/2.08), (2.07/2.37/2.67) (2.69/2.79/2.89), (1.23/1.53/1.83) (2.07/2.37/2.67), 0.9 2, 2.1 (1.38/1.58/1.78), (1.17/1.47/1.77) (2.39/2.49/2.59), (0.33/0.63/0.93) (1.77/2.07/2.37), 0 (0.33/0.63/0.93), 2 (0.11/0.21/0.31), (1.69/1.79/1.89) (0.92/1.12/1.32), (1.41/1.51/1.61) 0.7, 1.3 (0.63/0.93/1.23), 1.7 (0.41/0.51/0.61), (1.39/1.49/1.59) (1.22/1.42/1.62), (1.11/1.21/1.31) (0.7/1/1.3), 1 a a Figure 4. Fuzzy payoffs hB ((C, N C, C), C) and hB ((C, N C, C), N C). Consider a F1 = (a1 /u1 /b1 ) = hB ((C, NC, C), C) = (1.23/1.53/1.83) and a F2 = (a2 /u2 /b2 ) = hB ((C, NC, C), NC) = (1.41/1.51/1.61). One has that a1 < a2 ∧ u1 > u2 ∧ b1 > b2 . (72) ∀α : (0 ≤ α ≤ ρ = 0.8) ⇒ (0.3α + 1.23 ≤ 0.1α + 1.4). (73) and International Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 35 Then, according to Definition 2.1, from Equations 72 and 73, we have that F1 < F2 , meaning that a a hB ((C, NC, C), C) < hB ((C, NC, C), NC). a In this case, even with a higher core, the payoff hB ((C, NC, C), C) is ranked a lower than hB ((C, NC, C), NC), since the first carries less information accuracy as compared to the latter. Thus, the combination ((C, NC, C), C) does not constitute a NE, which means that there is a strict NE in the combination ((N, NC, C), NC). A decrease from “around” 50% to “around” 49% of the chance of the player JAL being of the egoistic type did not change the equilibrium solution, and it remains safer for the player JBa not to cooperate with JAL . Observe that if we assign ρ > 0.85, the ranking is reversed. In fact, let δ = 0.85 be the y-coordinate of the intersection point between the α-cuts 0.3α + 1.23 and 0.1α + 1.4. Whenever ρ > δ, the higher the fuzzy number’s core, the greater the number.34 As an example, considering ρ = 0.9, it holds that ∃α : (0 ≤ α ≤ ρ = 0.9) ∧ (0.1α + 1.4 < 0.3α + 1.23). (74) It follows that a a hB ((C, NC, C), NC) < hB ((C, NC, C), C), resulting in two nonstrict NE: one in the strategy combination ((C, NC, C), C) and another in the strategy combination ((NC, NC, C), NC). Observe that, besides the value of ρ, the ways in which the probabilities pe and ps are modeled are important to define the solutions of the game. Different modeling of these probabilities would produce different values of such δ, which are essential to compare with ρ and define the payoffs’ ordering. This is an example of a situation in which we may face the uncertainty in the shape of the fuzzy probabilities. 7.4. Interval FBPD with Moral Standards To deal with the problem discussed in the Case (FBPD 2) of Section 7.3, we adopt an interval FBPD, by using interval fuzzy probabilities to address the uncertainty in the modeling of fuzzy probabilities. The extension to the interval fuzzy game is made by replacing pa , pe , and (X), pe = P (Y ), ps = P (Z), and ps by pa , pe , and ps , respectively, so pa = P   P (L) = P (X ∪ Y ∪ Z) = 1, respecting the arithmetic restriction for interval fuzzy  = {pa , pe , ps }. probabilities. Thus, we obtain a probability distribution  International Journal of Intelligent Systems DOI 10.1002/int 36 ASMUS, DIMURO, AND BEDREGAL Table XIX. Interval fuzzy payoffs of the player JAL . R A L Ba JB JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C NC 2.3 ((1.78/1.88/1.98), (1.58/1.88/2.18)) ((2.74/2.79/2.84), (2.64/2.79/2.94)) ((2.22/2.37/2.52), (1.92/2.37/2.82)), 2 ((1.48/1.58/1.68), (1.28/1.58/1.88)) ((2.44/2.49/2.54), (2.34/2.49/2.64)), ((1.92/2.07/2.22), (1.62/2.07/2.52)) ((0.48/0.63/0.78), (0.18/0.63/1.08)) ((0.16/0.21/0.26), (0.06/0.21/0.36)) ((1.02/1.12/1.22), (0.92/1.12/1.42))) 0.7 ((0.78/0.93/1.08), (0.48/0.93/1.38)) ((0.46/0.51/0.56), (0.36/0.51/0.66)) ((1.32/1.42/1.52), (1.12/1.42/1.72)) ((0.85/1/1.15), (0.35/1/1.45)) Table XX. Interval fuzzy payoffs of the player JBa . R A L Ba JB JAa , JAe , JAs C, C, C C, C, N C C, N C, C C, N C, N C N C, C, C N C, C, N C N C, N C, C N C, N C, N C C NC 3 ((2.12/2.37/2.52), (1.92/2.37/2.82)) ((1.38/1.53/1.68), (1.08/1.53/1.98)) 0.9 2.1 ((1.32/1.47/1.62), (1.02/1.47/1.92)) ((0.48/0.63/0.78), (0.18/0.63/1.08)) 0 2 ((1.74/1.79/1.84), (1.64/1.79/1.94)) ((1.46/1.51/1.56), (1.36/1.51/1.66)) 1.3 1.7 ((1.44/1.49/1.54), (1.34/1.49/1.64)) ((1.16/1.21/1.26), (1.06/1.21/1.36)) 1 With respect to the Case (FBPD 2) of Section 7.3, we then consider the interval fuzzy probabilities pe = ((0.44/0.49/0.54), (0.34/0.49/0.64)) ps = ((0.16/0.21/0.26), (0.06/0.21/0.36)) and pa = 0.3. The interval fuzzy payoffs are calculated analogously to the IFBHG in Section 6.3. We find the payoffs of the lower FBPD using only the lower generator fuzzy probabilities of pa , pe , and ps , and similarly we determine the payoff of the upper FBPD. The interval fuzzy payoffs of the player JAL and JBa are shown in Tables XIX and XX, respectively. The ranking between the interval fuzzy payoffs  haB ((C, NC, C), C) = ((1.38/1.53/1.68), (1.08/1.53/1.98))  haB ((C, NC, C), NC) = ((1.46/1.51/1.56), (1.36/1.51/1.66)) depends on the assigned value for the degree of imprecision ρ. As in the Case (FBPD 2) of Section 7.3, we first assign ρ = 0.8. According to the interval ADInternational Journal of Intelligent Systems DOI 10.1002/int INTERVAL-VALUED FUZZY BAYESIAN GAMES 37 order (Definition 10), for this degree of imprecision, we have   haB ((C, NC, C), NC). haB ((C, NC, C), C) < Then, applying interval fuzzy probabilities for ρ = 0.8, we obtain a NE in the strategy combination ((NC, NC, C), NC), as in the FBPD for the same degree of imprecision. However, by assigning ρ = 0.9 and applying the interval AD-order, we again obtain a NE in the strategy combination ((NC, NC, C), NC), indicating that the player JBa should not cooperate. Therefore, for ρ = 0.9, the interval fuzzy game produces a result different from the result obtained by the fuzzy version, which have presented two nonstrict NEs. Observe that when comparing (1.08/1.53/1.98) with (1.36/1.51/1.66), one has that δ = 0.9333, which means that δ > ρ. Thus, whenever the interval AD-order is adopted, we always consider the fuzzy modeling with higher uncertainty (upper generator fuzzy number) to rank approximate interval fuzzy numbers with different cores. Observe that the value of δ for a particular ordering can vary as we extend the game from a fuzzy version to an interval fuzzy version, for the same ρ. This means that the interval fuzzy game modeling may produce different results than the fuzzy version that has originated it, since it considers scenarios with more fuzzyness. 8. CONCLUSION Game theory has been largely applied for decision making and support in strategic interactions in several fields, as well as for risk and influence assessments, knowledge discovery and flow, social simulation, and MAS. Some of these applications may consider the uncertainty concerning the parts involved in the interactions. In this context, when probabilities are assigned for the different types of the interacting parts, one may apply Bayesian games. However, when the various experts have to deal both with vagueness and uncertainty, that is, when those probabilities cannot be determined exactly and it is difficult to come to a consensus about the fuzzy modeling, one may adopt intervalvalued probabilities. In this paper, we introduced the concept of interval-valued fuzzy Bayesian games, using interval-valued fuzzy probabilities for modeling the types of agents involved in the interaction, represented by symmetric triangular interval fuzzy numbers. To investigate the NE solutions of fuzzy and interval-valued fuzzy Bayesian games, we applied the (interval) AD-order relation, which had guaranteed the existence of equilibrium solutions. We presented two different examples, comparing the results obtained with the crisp, fuzzy and interval fuzzy approaches. In particular, we studied the (Interval) FBHG and the (Interval) FBPD with Moral Standards. The first can be a useful tool for the analysis of any kind of hiring and/or exchange of services in several contexts. On the other hand, the latter basically provides a framework for understanding International Journal of Intelligent Systems DOI 10.1002/int 38 ASMUS, DIMURO, AND BEDREGAL how to strike a balance between cooperation and competition and is a very useful tool for strategic decision making, finding application in diverse areas ranging from business, finance, economics, social networks, data and political sciences to philosophy, psychology, biology, and sociology. 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