SINCE von Neumann and Morgenstern [1] established modern

Transcription

1 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 4, AUGUST Linear Quadratic Uncertain Differential Game With Application to Resource Extraction Problem Xiangfeng Yang and Jinwu Gao Abstract Uncertain differential game investigates interactive decision making of players over time, and the system dynamics is described by an uncertain differential equation. This paper goes further to study the two-player zero-sum uncertain differential game. In order to guarantee the saddle-point Nash equilibrium, a Max Min theorem is provided. Furthermore, when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic, the existence of saddle-point Nash equilibrium is obtained via the solvability of a corresponding Riccati equation. Finally, a resource extraction problem is analyzed by using the theory proposed in this paper. Index Terms Riccati equation, resource extraction, saddlepoint Nash equilibrium, uncertain differential game. I. INTRODUCTION SINCE von Neumann and Morgenstern 1 established modern game theory in 1944, game theory has been proven useful in many fields such as economics, management science and sociology, etc. Motivated by the missile versus enemy aircraft pursuit schemes, Isaacs 2 introduced differential game which studies the conflict in the context of a dynamical system. Due to its important applications in military area, lots of work sprang up in the field of differential game thereafter. In 1964, Berkovitz 3 developed a variational approach to differential game. In 1966, Pontryagin 4 solved differential game in open-loop solution in terms of the maximum principle. In 1967, Leitmann and Mon 5 investigated the geometry of differential game. In 1971, Friedman 6 introduced discrete approximation sequence methods to establish the values of differential game and existence of saddle point. Ever since, a solid mathematical foundation had been laid for differential game theory. A special case of differential game is the linear-quadratic differential game, in which the system dynamics is described by a linear differential equation, and the performance index function is quadratic. Linear quadratic differential game gained popularity not only among dynamic game theorists but also among economists in the area of policy coordination, resource extraction, and capital accumulation. In 1965, Ho et al. 7 discussed a class of linear quadratic differential game: pursuit-evasion game. In 1969, Starr and Ho 8 investigated a sufficient condition for the existence of closed-loop strategies that was obtained Manuscript received January 29, 215; revised June 25, 215 and July 31, 215; accepted August 24, 215. Date of publication October 5, 215; date of current version August 2, 216. This work was supported in part by the National Natural Science Foundation of China under Grant , Grant , and Grant X. Yang is with the Department of Mathematical Sciences, Tsinghua University, Beijing 184, China yangxf14@mails.tsinghua.edu.cn. J. Gao is with the School of Information, Renmin University of China, Beijing 1872, China. jgao@ruc.edu.cn. Digital Object Identifier 1.119/TFUZZ based on the existence of the solution of a Riccati equation. In 197, Schmitendorf 9 studied both open-loop and closed-loop strategies. Later then, many researchers did important works on linear quadratic differential game e.g., However, in real-world situations, the state evolution is often affected by the interference of noise. The noise may be added to the players observations of the system state or to the state equation itself. When noise is modeled by the Wiener process and system evolution can be described by a stochastic differential equation, the differential game will evolve into stochastic differential game. Results obtained by Fleming 13 in stochastic control made it possible to analyze differential game situations with stochastic state dynamics. Later on, Azevedo et al. 14 analyzed a stochastic optimal control problem for a Markov-switching jump-diffusion stochastic differential equation. Temocin and Weber 15 developed a numerical approximation method for solving the optimal control problem for controlled autonomous stochastic hybrid systems with jumps. For stochastic linear quadratic differential game, Mou and Yong 16 studied open-loop strategies by means of Hilbert space method in 26. Sun and Yong 17 investigated open-loop and closed-loop saddle-point equilibriums in 214. Sometimes, no samples are available to estimate the probability distribution. For such situation, we have no choice but to invite some domain experts to evaluate the belief degree that each event will occur. Then, fuzzy set theory offers an appropriate alternative. In the literature, fuzzy strategic game has attracted much attention in the past three decades. For example, Garagic and Cruz 18 developed a solution concept of Nash equilibrium for N-person static fuzzy noncooperative games; Chakeri et al analyzed fuzzy payoff games using different fuzzy preference methods; Bector and Chandra 24 and Li 25 presented the fuzzy Nash equilibriums as well as their solution methods for fuzzy matrix games. Gao et al proposed a spectrum of credibilistic strategic game that studied the credibilistic Nash equilibriums as well as their properties. Fuzzy cooperative game has also been discussed by some researchers. For instance, Butnariu 34, Aubin 35, and Butnariu and Kroupa 36 studied n-person games with fuzzy coalitions; Shen and Gao 37 and Gao et al. 29 proposed the solutions of credibilistic core and credibilistic Shapley value for fuzzy coalitional game, respectively; Tan et al. 38 presented a Banzhaf function for fuzzy coalitional game. Recently, researchers have been paying attention to fuzzy dynamic game due to its important applications. In 22, Chen et al. 39 introduced fuzzy differential games for nonlinear stochastic systems. In 28, Sharma and Gopal 4 proposed a Markov game hybrid fuzzy controller. In 213, Gao and Yu 41 proposed the credibilistic subgame perfect equilibrium for fuzzy extensive game. In 215, IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See standards/publications/rights/index.html for more information.

2 82 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 4, AUGUST 216 Zhang et al. 42 introduced a leader-based optimal coordination control for the consensus problem via fuzzy differential game. Uncertainty theory was proposed by Liu 43 in 27 and refined by Liu 44 in 21 to handle belief degree. Nowadays, uncertainty theory has become a new branch of mathematics for modeling indeterminate phenomena based on normality, duality, subadditivity, and product axioms. In order to deal with dynamical uncertain phenomena, uncertain process was proposed by Liu 45 as a sequence of uncertain variables indexed by time. Liu 44 designed a Lipschitz continuous uncertain process with stationary and independent increments, which is now called canonical Liu process. Uncertain differential equation was proposed by Liu 46 as a type of differential equations driven by a canonical Liu process. Based on these concepts, Chen and Liu 47 proved the existence and uniqueness theorem for the solution of an uncertain differential equation under linear growth condition and Lipschitz continuous condition. Yao and Chen 48 proved that the solution of an uncertain differential equation can be represented by a spectrum of ordinary differential equations. Meanwhile, Yao and Li 49 introduced uncertain alternating renewal process and its application, and Yao 5 studied uncertain differential equation with jumps. Nowadays, uncertain differential equation has been widely applied in many fields. In 21, Zhu 51 introduced uncertain optimal control and gave equation of optimality by means of uncertain differential equations. These results made it possible to analyze differential game situations with uncertain state dynamics described by an uncertain differential equation. In 213, Yang and Gao 52 proposed uncertain differential game in which the noise is modeled by the canonical Liu process and gave a sufficient condition to guarantee the existence of feedback Nash equilibrium. In this paper, we consider the zero-sum uncertain differential game and discuss a special case: linear quadratic uncertain differential game. The rest of this paper is organized as follows. Section II reviews some basic results of uncertainty theory. Section III presents a Max Min theorem of a two-player zerosum uncertain differential game. Meanwhile, Section IV proves a sufficiency condition and a necessity condition of the linear quadratic uncertain differential game that are equivalent to the solvability of the Riccati equation. Finally, a resource extraction problem is discussed in Section V. II. PRELIMINARIES Uncertain measure M is a real-valued set function on a σ- algebra L over a nonempty set Γ, which satisfies normality, duality, subadditivity, and product axioms. The triplet Γ, L, M is called an uncertainty space. Definition 2.1 see 43: Let L be a σ-algebra on a nonempty set Γ. AsetfunctionM : L, 1 is called an uncertain measure if it satisfies the following axioms: Axiom 1 Normality Axiom: M{Γ =1for the universal set Γ. Axiom 2 Duality Axiom: M{Λ + M{Λ c =1 for any event Λ. Axiom 3 Subadditivity Axiom: For every countable sequence of events Λ 1, Λ 2,...,we have { M Λ i M{Λ i. Besides, in order to provide the operational law, Liu 45 defined the product uncertain measure on the product σ-algebra L as follows. Axiom 4 Product Axiom: Let Γ k, L k, M k be uncertainty spaces for k =1, 2,... The product uncertain measure M is an uncertain measure satisfying { M Λ k = M k {Λ k k=1 k=1 where Λ k are arbitrarily chosen events from L k for k =1, 2,..., respectively. Definition 2.2 see 43: An uncertain variable is a function ξ from an uncertainty space Γ, L, M to the set of real numbers such that {ξ B is an event for any Borel set B. In order to describe uncertain variable in practice, uncertainty distribution Φ:R, 1 of an uncertain variable ξ is defined as Φx =M{ξ x. An uncertain variable ξ is called normal if it has a normal uncertainty distribution Φx = 1 + exp πe x 3σ 1 denoted by N e, σ, where e and σ are real numbers with σ>. Definition 2.3 see 45: The uncertain variables ξ 1,ξ 2,...,ξ m are said to be independent if { m m M {ξ i B i = M{ξ i B i for any Borel sets B 1,B 2,...,B m of real numbers. The expected value and variance of uncertain variable ξ are defined by + Eξ = M{ξ rdr M{ξ rdr provided that at least one of the two integrals is finite V ξ =Eξ Eξ 2 respectively. Definition 2.4 see 46: Let Γ, L, M be an uncertainty space and let T be a totally ordered set e.g., time. An uncertain process is a function X t γ from T Γ, L, M to the set of real numbers such that {X t B is an event for any Borel set B at each time t. An uncertain process X t is said to have independent increments if X t,x t1 X t,x t2 X t1,..., X tk X tk 1 are independent uncertain variables, where t is the initial time and t 1,t 2,...,t k are any times with t <t 1 < <t k.an uncertain process X t is said to have stationary increments if, for any given t>, the increments X s+t X s are identically distributed uncertain variables for all s>.

3 YANG AND GAO: LINEAR QUADRATIC UNCERTAIN DIFFERENTIAL GAME WITH APPLICATION TO RESOURCE EXTRACTION PROBLEM 821 Definition 2.5 see 45: An uncertain process C t is said to be a canonical Liu process if 1 C =and almost all sample paths are Lipschitz continuous; 2 C t has stationary and independent increments; 2 every increment C s+t C s is a normal uncertain variable, whose uncertainty distribution is Φx = 1 πx 1 + exp,x R. 3t Definition 2.6 see 46: Suppose C t is a canonical Liu process, and f and g are some given functions. Then dx t = ft, X t dt + gt, X t dc t 1 is called an uncertain differential equation. A solution is a Liu process X t that satisfies 1 identically in t. The existence and uniqueness theorem of solution of uncertain differential equation was first proved by Chen and Liu 47 under linear growth condition and Lipschitz continuous condition. Later on, Zhang and Chen 53 proposed multidimensional canonical Liu process, Yao 54 proposed uncertain calculus with multidimensional canonical Liu process, and Ji and Zhou 55 proved a sufficient condition for a multidimensional uncertain differential equation s having a unique solution. For more detailed exposition of uncertainty theory with applications, the readers may consult Liu s recent book 56. In 21, Zhu 51 proposed an uncertain expected value optimal control problem. Assume that R is the return function and W is the function of terminal reward. If we want to maximize the expected return on,t by using an optimal control, then we have the following optimal control model: J,x =supe Rt, xt,utdt+w T,xT u subject to: dxt =ft, xt,utdt + gt, xt,utdc t xt =x. 2 In order to find the optimal control, we write Jt, x=supe Rs, xs,usds+w T,xT 3 u t where t,t and xt =x. Theorem 2.1 see 51: Let Jt, x be twice differentiable on,t R; then, we have J t t, x =sup{rt, x, u+j x t, xft, x, u 4 u where J t t, x and J x t, x are the partial derivatives of Jt, x with respect to t and x, respectively, and the boundary condition is JT,x=W T,x. 5 III. ZERO-SUM UNCERTAIN DIFFERENTIAL GAME An uncertain formulation for quantitative differential game of prescribed duration involves a vector-valued uncertain differential equation dxt =ft, xt, u 1 t, u 2 t,..., u n tdt + gt, xt, u 1 t, u 2 t,..., u n tdc t x = x 6 that describes the evolution of the state, and n objective functions sup E u i R i t, xt, u 1 t, u 2 t,..., u n tdt + W i xt,i N = {1, 2,...,n. 7 In 6, xt R m denotes the state variables of game, u i R k i is the control of player i, C t is a k-dimensional canonical Liu process, x is the given initial state, and f :,T R m R k 1... R k n R m and g :,T R m R k 1... R k n R m R k are functions that have continuous partial derivatives and satisfy linear growth condition and Lipschitz condition. In 7, T>is the terminal time of the game, and E denotes the expectation operator performed at time zero. Each player has perfect observations of the state vector xt at every moment t,t and constructs his strategy in the game 6 7 as an admissible feedback control of the following type: where u i t =u i t, xt u i, :,T R m R k i. In 213, Yang and Gao 52 defined the feedback Nash equilibrium for uncertain differential game and presented a sufficient condition to guarantee feedback Nash equilibrium. Definition 3.1 see 52: A set of strategies {u 1s, x, u 2s, x,..., u ns, x is called a feedback Nash equilibrium for the n-player uncertain differential game 6, 7, and {x s,t s T is the corresponding state trajectory, if there exist real-valued functions V i t, x :,T R m R, satisfying the following relations for each i N: V i t, x =E R i s, x s, u i s, x, u is, x ds t + W i x T V i T,x =W i xt E R i s, x i s, u i s, x i, u is, x i ds t + W i x i T u i,,t R m, x R m

4 822 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 4, AUGUST 216 where in the time interval t, T : dx s =fs, x s, u i s, x, u is, x ds + gs, x s, u i s, x, u is, x dc s x t =x; dx i s=fs,x i s,u i s, x i,u is, x i ds x i t =x. + gs,x i s,u i s, x i,u is, x i dc s Theorem 3.1 see 52: An n-tuple of strategies {u i t, x; i N provides a feedback Nash equilibrium to the n-player uncertain differential game 6, 7 if there exist real-valued functions V i t, x :,T R m R,i N, satisfying the partial differential equations: Vt i { t, x = sup Ri t, xt, u i t, x, u it, x u i + x V i t, xft, x, u i t, x, u it, x = R i t, x, u i t, hx, u it, x V i T,x =W i xt. + x V i t, xft, x, u i t, x, u it, x Now, we consider the two-player zero-sum uncertain differential game that described by 6 and 7. For instance, two competing firms resource extractors fight a battle for same resource, just like cutting a cake. If one takes a larger quantity of resource, then it reduces the amount of resource available for the other. Then, two objective functions reduce to a performance index function that player 1 wants to maximize the index function, while player 2 wants to minimize the index function. A saddle-point Nash equilibrium solution can be characterized as follows. Theorem 3.2 Max Min Theorem: A pair of strategies {u i t, x Rk i ; i =1, 2 provides a feedback Nash equilibrium solution called saddle-point Nash equilibrium to the twoplayer zero-sum of the game 6 7 if there exists a real-valued function V t, x :,T R m R, satisfying the partial differential equations: V t t, x = max min{rt, x, u 1, u 2 u 1 u 2 + x V t, xft, x, u 1, u 2 = min max{rt, x, u 1, u 2 u 2 u 1 + x V t, xft, x, u 1, u 2 = Rt, x, u 1, u 2+ x V t, xft, x, u 1, u 2 V T,x =W xt. Proof: This result follows as a special case of Theorem 3.1 by taking n =2,R 1 = R 2 =. R, and W 1 = W 2 =. W, in which case V 1 = V 2. = V and existence of a saddle point is equivalent to interchangeability of the min max operations. The theorem is proved. IV. LINEAR QUADRATIC UNCERTAIN DIFFERENTIAL GAME Let us consider the two-player zero-sum uncertain differential game, where the evolution of the dynamic system is described by a linear uncertain differential equation t is suppressed dx =Ax + B 1 u 1 + B 2 u 2 dt + gt, x, u 1, u 2 dc t 8 with initial condition x = x. The performance index function is Ju 1, u 2 =E x T Mx +2u T 1 S 1 + u T 2 S 2 x + u T 1 R 1 u 1 + u T 2 R 2 u 2 dt+xt T GxT. Here, C t is a k-dimensional canonical Liu process on an uncertainty space, x R m is the state variable, u i R k i i =1, 2 are control variable taken by player i; A,B i, and S are bounded measureable matrix function on,t with dimensions m m, m k i, and k i m, respectively; M and R i are bounded measureable symmetric matrix function on,t with dimensions m m, k i k i, respectively; G is an m m constant symmetric matrix; and R 1,R 2 are positive-definite matrices. In linear quadratic uncertain differential game 8, 9, player 1 wants to maximize the index function while player 2 wants to minimize the index function. A. Sufficiency Condition Theorem 4.1: The two-player zero-sum linear quadratic uncertain differential game 8, 9 has a saddle-point Nash equilibrium solution if the following Riccatic equation with the time argument t suppressed has a solution P t: P + A T P + PA+ M 2 Bi T P + S i T R 1 Bi T P + S i= 1 P T =G where P t is bounded measureable symmetric matrix function on,t with dimensions m m. Moreover, saddle-point and the optimum value of performance index function are {u i = Ri 1 Bi T P + S ix; i =1, 2 and Ju 1, u 2=x T P x, respectively. Proof: Let P t be the solution of Riccatic equation 1 and let xt be the solution of uncertain differential equation 8 corresponding to control u 1 t, u 2 t. Using the fundamental theorem of uncertain calculus see 54, we have t is suppressed dx T P x = x T P + A T P + PAx +2 i u T i Bi T P x dt +...dc t. 9

5 YANG AND GAO: LINEAR QUADRATIC UNCERTAIN DIFFERENTIAL GAME WITH APPLICATION TO RESOURCE EXTRACTION PROBLEM 823 Taking integrations on,t and expectations, we obtain E x T T P T xt = x T P x +E x T P + A T P + PAxdt + E 2u T i Bi T P xdt. 11 Substituting 11 into the 9, Ju 1, u 2 can be reduced to Ju 1, u 2 =E x T Mx +2u T 1 S 1 + u T 2 S 2 x +u T 1 R 1 u 1 + u T 2 R 2 u 2 dt + x T P x +E x T P + A T P + PAxdt + E 2u T i Bi T P xdt = x T P x +E x T P + A T P + PA+ Mxdt + = x T P x + E 2u T i Bi T P + S i x + u T i R i u i dt E + R i u i + Ri 1 Bi T P + S i x dt. The theorem is proved. T u i + Ri 1 Bi T P + S i x B. Necessity Condition If a linear feedback control is optimal for the linear quadratic uncertain differential game 8, 9, then it must be optimal also in the class of linear feedback controls of the following form: u i = K i x where K i is a k i m constant symmetric matrix. The corresponding system is dx = A + 2 B i K i x + gt, x,k 1,K 2 dc t 12 x = x. Noting Xt =Extxt T, then using the fundamental theorem of uncertain calculus, X satisfies the following differential matrix equation: Xt = A + 2 B i K i X + X A + 2 T B i K i X = X Ex x T 13 with the associated performance index function J, expressed equivalently as JK 1,K 2 = T + Tr M +2 Ki T S i Ki T R i K i X dt + Tr GXT. Theorem 4.2: If linear feedback controls {u i = K ix; i = 1, 2 are optimal for the two-player zero-sum linear quadratic uncertain differential game 8 9, then Riccatic equation 1 Bi T P + S i. Proof: Since the given feedback controls {u i = K ix; i = 1, 2 are optimal over the set of all admissible controls, it must in particular be optimal over the class of all linear feedback must have a solution P. Moreover, K i = R 1 i controls. Therefore, as shown earlier, K i must solve the following deterministic optimal control problem: s.t. max K 1 min K 2 + T Tr M +2 Ki T S i Ki T R i K i X dt + Tr GXT Ẋt = A + X = X. B i K i X +X A + T B i K i 14 This is a matrix optimal control problem. According to matrix minimum principle see 57 and 58, we obtain that the Hamiltonian function is Tr M +2 + A + Ki T S i + Ki T R i K i X B i K i X + X A + T B i K i P T

6 824 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 4, AUGUST 216 and the adjoint equation is P = M 2 2 Ki T S i 2 Ki T R ik i P T =G A + 2 T B i K i P P A + 2 B i K i =2S i X + R i K i X + R i K i X T +Bi T PXT + Bi T P T X. 15 Note that in the above calculation, we have used the following formula: X TrAX =AT, X TrAXT =A and X TrAXBXT =A T XB T + AXB. Since R i,x, and P are symmetric and R i is nonsingular, the third equation of 15 is reduced to K i = Ri 1 Bi T P + S i. Substituting K i into the first equation of 15, we can see by a simple calculation that P satisfies P + A T P + PA+ M Bi T P + S i T Ri 1 Bi T P + S i =. Then, the theorem is proved. Remark 4.1: Whatever diffuse coefficient gt, x, u 1, u 2 of the dynamic system is linear or not, Theorems 4.1 and 4.2 hold, too. V. RESOURCE EXTRACTION GAME The resource extraction problem is a classical problem in the field of economics, in which economic agents firms or countries exploit natural resources. Under the assumption that the resource dynamics follows a differential equation, Clark 59 studied a common-property fishery resources problem by a differential game approach. Taking into account the noise added to a standard procedure in differential game, Jørgensen and Yeung 6 investigated stochastic differential game model of a common property fishery. If we assume that the resource dynamics follows an uncertain differential equation driven by a canonical Liu process, then an uncertain differential game model of resource extraction can be studied. Suppose that there are two firms resource extractors. They exploit a renewable resource fisheries. The lease for resource extraction begins at time and ends at time T, T>.Letu i t denote the quantity of resource extraction of firm i at time t, i =1, 2, where each extractor controls its quantity of extraction. Let xt denote the size of the resource stock at time t, xt > and the equation of resource dynamics is dxt =axt u 1 t u 2 t dt + σxtdc t, x = x where a > denotes the growth rate of resource, the initial state x is given, the uncertain process C t is a 1-D canonical Liu process that defined on an uncertain space Γ, L, M. The performance index function is Ju 1,u 2 =E xt 2 u 1 t 2 + u 2 t 2 dt + xt 2. Extractor 1 seeks to maximize the performance index function, while extractor 2 seeks to minimize the performance index function. Invoking Theorem 4.1, we have the following Riccati equation: { P +2aP +1= P T =1. The only solution of the above equation is / P t = 2a +1e 2aT t 1 2a. The saddle-point Nash equilibrium are u 1t = 1 2a +1e 2aT t 1 xt 2a u 2t = 1 2a +1e 2aT t 1 xt. 2a The optimum value of performance index function is Ju 1,u 2= 1 2a +1e 2aT 1 x 2 2a. VI. CONCLUSION In this paper, we have proved a Max Min theorem as a sufficient condition to guarantee the saddle-point Nash equilibrium in a two-player zero-sum uncertain differential game. Furthermore, for the linear quadratic case, we derived a sufficient condition and a necessary condition for the existence of a saddle-point Nash equilibrium. Finally, we analyzed the resource extraction problem by use of the uncertain differential game theory. This study, along with our former work 52, constructed a theoretical framework of uncertain differential game. Therefore, there are lots of problems for further research. On the one hand, one may consider uncertain differential game with jumps in which the system dynamics can be described by an uncertain differential equation with jumps. On the other hand, one may study cooperative uncertain differential game, in which groups of players coalitions may enforce cooperative behavior. In addition, one may study the applications of uncertain differential game such as common property fishery problem and military confrontation. REFERENCES 1 J. von Neumann and D. Morgenstern, The Theory of Games in Economic Bahavior. New York, NY, USA: Wiley, 1944.

7 YANG AND GAO: LINEAR QUADRATIC UNCERTAIN DIFFERENTIAL GAME WITH APPLICATION TO RESOURCE EXTRACTION PROBLEM R. Isaacs, Differential Games. New York, NY, UA: Wiley, L. D. Berkovitz, A variational approach to differential games, Adv. Game Theory, vol. 52, pp , L. S. Pontryagin, On the theory of differential games, Russ. Math. Surv., vol. 21, no. 4, pp , G. Leitmann and G. Mon, Some geometric aspects of differential games, J. Astronaut. Sci., vol. 14, no. 2, pp , A. Friedman, Differential Games. New York, NY, USA: Wiley, Y. C. Ho, A. E. Bryson, and S. Baron, Differential games and optimal pursuit-evasion strategies, IEEE Trans. Autom. Control, vol. AC-1, no. 4, pp , Oct A. W. Starr and Y. C. Ho, Nonzero-sum differential games, J. Optim. Theory Appl., vol. 3, no. 3, pp , W. E. Schmitendorf, Existence of optimal open-loop strategies for a class of differential games, J. Optim. Theory Appl., vol. 5, no. 5, pp , P. Bernhard, Linear-quadratic, two-person, zero-sum differential games: Necessary and sufficient conditions, J. Optim. Theory Appl., vol. 27, no. 1, pp , P. Zhang, Some results on two-person zero-sum linear quadratic differential games, SIAM J. Control Optim., vol. 43, no. 6, pp , M. C. Delfour, Linear quadratic differential games: saddle point and Riccati differential equations, SIAM J. Control Optim., vol. 46, no. 2, pp , W. H. Fleming, Optimal continuous-parameter stochastic control, SIAM Rev., vol. 11, no. 4, pp , N. Azevedo, D. Pinheiro, G. W. and Weber, Dynamic programming for a Markov-switching jump-diffusion, J. Comput. Appl. Math., vol. 267, pp. 1 19, B. Z. Temocin and G. W. Weber, Optimal control of stochastic hybrid system with jumps: a numerical approximation, J. Comput. Appl. Math., vol. 259, pp , L. Mou and J. Yong, Two-person zero-sum linear quadratic stochastic differential games by a Hilbert space method, J. Ind. Manag. Optim., vol. 2, no. 1, pp , J. Sun and J. Yong, Linear quadratic stochastic differential games: openloop and closed-loop saddle point, SIAM J. Control Optim., vol. 52, no. 6, pp , D. Garagic and J. B. Cruz, An approach to fuzzy noncooperative Nash games, J. Optim. Theory Appl., vol. 118, no. 3, pp , A. Chakeri, A. N. Dariani, and C. Lucas, How can fuzzy logic determine game equilibriums better? in Proc. 4th Int. IEEE Conf. IEEE Intell. Syst., 28, vol. 1, pp A. Chakeri, J. Habibi, and Y. Heshmat, Fuzzy type-2 Nash equilibrium, in Proc. Int. Conf. Comput. Intell. Modell. Control Autom., 28, pp A. Chakeri, N. Sadati, and S. Sharifian, Fuzzy Nash equilibrium in fuzzy games using ranking fuzzy numbers, in Proc. IEEE Int. Conf. IEEE Fuzzy Syst., 21, pp A. Chakeri and F. Sheikholeslam, Fuzzy Nash equilibriums in crisp and fuzzy games, IEEE Trans. Fuzzy Syst., vol. 21, no. 1, pp , Feb S. Sharifian, A. Chakeri, and F. Sheikholeslam, Linguisitc representation of Nash equilibriums in fuzzy games, in Proc. Annu. Meeting North Amer. IEEE Fuzzy Inf. Process. Soc., 21, pp A. Bector and S. Chandra, Fuzzy Mathematical Programming and Fuzzy Matrix Games. New York, NY, USA: Springer-Verlag, D. F. Li, Mathematical-programming approach to matrix games with payoffs represented by Atanassov s interval-valued intuitionistic fuzzy sets, IEEE Trans. Fuzzy Syst., vol. 18, no. 6, pp , Dec J. Gao and B. Liu, Fuzzy multilevel programming with a hybrid intelligent algorithm, Comput. Math. Appl., vol. 49, no. 9, pp , J. Gao, Credibilistic game with fuzzy information, J. Uncertain Syst., vol. 1, no. 1, pp. 74 8, J. Gao, Z. Q. Liu, and P. Shen, On characterization of credibilistic equilibria of fuzzy-payoff two-player zero-sum game, Soft Comput., vol. 13, no. 2, pp , J. Gao, Q. Zhang, and P. Shen, Coalitional game with fuzzy payoffs and credibilistic Shapley value, Iran. J. Fuzzy Syst., vol.8,no.4,pp , J. Gao and X. Yang, Credibilistic bimatrix game with asymmetric information: Bayesian optimistic equilibrium strategy, Int. J. Uncertain. Fuzz., vol. 21, no. supp1, pp. 89 1, J. Gao, Uncertain bimatrix game with applications, Fuzzy Optim. Decision Making, vol. 12, no. 1, pp , R. Liang, Y. Yu, J. Gao, and Z. Q. Liu, N -person credibilistic strategic game, Front. Comput. Sci-Chi., vol. 4, no. 2, pp , X. Yang and J. Gao, Uncertain core for coalitional game with uncertain payoffs, J. Uncertain Syst., vol. 8, no. 1, pp , D. Butnariu, Stability and Shapley value for an n-persons fuzzy game, Fuzzy Set. Syst., vol. 4, no. 1, pp , J. P. Aubin, Cooperative fuzzy games, Math. Oper. Res., vol. 6, no. 1, pp. 1 13, D. Butnariu and T. Kroupa, Shapley mappings and the cumulative value for n-person games with fuzzy coalitions, Eur. J. Oper. Res., vol. 186, no. 1, pp , P. Shen and J. Gao, Coalitional game with fuzzy information and credibilistic core, Soft Comput., vol. 15, no. 4, pp , C. Tan, Z. Jiang, X. Chen, and W. H. Ip, A Banzhaf function for a fuzzy game, IEEE Trans. Fuzzy Syst., vol. 22, no. 6, pp , Dec B. S. Chen, C. S. Tseng, and H. J. Uang, Fuzzy differential games for nonlinear stochastic systems: suboptimal approach, IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp , Apr R. Sharma and M. Gopal, Hybrid game strategy in fuzzy Markov-gamebased control, IEEE Trans. Fuzzy Syst., vol. 16, no. 5, pp , Oct J. Gao and Y. Yu, Credibilistic extensive game with fuzzy payoffs, Soft Comput., vol. 17, no. 4, pp , H. Zhang, J. Zhang, and G. Yang, Leader-based optimal coordination control for the consensus problem of multiagent differential games via fuzzy adaptive dynamic programming, IEEE Trans. Fuzzy Syst., vol. 23, no. 1, pp , Feb B. Liu, Uncertainty Theory, 2nd ed. Berlin, Germany: Springer-Verlag, B. Liu B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty. Berlin, Germany: Springer-Verlag, B. Liu, Some research problems in uncertainty theory, J. Uncertain Syst., vol. 3, no. 1, pp. 3 1, B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., vol. 2, no. 1, pp. 3 16, X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optim. Decision Making, vol. 9, no. 1, pp , K. Yao and X. Chen, A numerical method for solving uncertain differential equations, J. Intell. Fuzzy Syst., vol. 25, no. 3, pp , K. Yao and X. Li, Uncertain alternating renewal process and its application, IEEE Trans. Fuzzy Syst., vol. 2, no. 6, pp , Dec K. Yao, Uncertain differential equation with jumps, Soft Comput., vol. 19, no. 7, pp , Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybern. Syst., vol. 41, no. 7, pp , X. Yang and J. Gao, Uncertain differential games with application to capitalism, J. Uncertainty Anal. Appl., vol. 1, p. 17, T. Q. Zhang and B. Chen, Multi-dimensional canonical process, Inf.- Tokyo, vol. 16, no. 2A, pp , K. Yao, Multi-dimensional uncertain calculus with Liu process, J. Uncertain Syst., vol. 8, no. 4, pp , X. Ji and J. Zhou, Multi-dimensional uncertain differential equation: Existence and uniqueness of solution, Fuzzy Optim. Decision Making, vol. 14, no. 4, pp , B. Liu, Uncertainty Theory, 4th ed. Berlin, Germany: Springer-Verlag, M. A. Rami, J. B. Moore, and X. Y. Zhou, Indefinite stochastic linear quadratic control and generalized differential Riccati equation, SIAM J. Control Optim., vol. 4, no. 4, pp , M. Athans, The matrix minimum principle, Inf. Control, vol. 11, pp , C. W. Clark, Restricted access to common-property fishery resources: A game-theoretic analysis, in Dynamic Optimization and Mathematical Economics. New York, NY, USA: Springer, 198, pp S. Jørgensen and D. W. Yeung, Stochastic differential game model of a common property fishery, J. Optim. Theory Appl., vol. 99, pp , 1996.

8 826 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 4, AUGUST 216 Xiangfeng Yang received the B.S. degree from the South-Central University for Nationalities, Wuhan, China, in 211, and the M.S. degree from the Renmin University of China, Beijing, China, in 214. He is currently working toward the Ph.D. degree with the Department of Mathematical Sciences, Tsinghua University, Beijing. He has authored or coauthored eight articles on several journals including International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Journal of Intelligent & Fuzzy Systems, Journal of Intelligent Manufacturing, Journal of Uncertainty Analysis and Applications, and Journal of Uncertain Systems. His current research interests include uncertain systems, uncertain differential equations, and their applications. Jinwu Gao received the B.S. degree in mathematics from Shaanxi Normal University, Xi an, China, in 1996, and the M.S. and Ph.D. degrees in mathematics from Tsinghua University, Beijing, China, in 25. He is currently an Associate Professor with the School of Information, Renmin University of China, Beijing, China. His current research interests include fuzzy systems, uncertain systems, and their application in optimization, game theory, and finance. He has authored or coauthored more than 3 papers that have appeared in IEEE TRANSACTIONS ON FUZZY SYSTEMS, Fuzzy Optimization and Decision Making, Journal of Intelligent Manufacturing, Soft Computing, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Computer & Mathematics with Applications, and other publications. Dr. Gao has been the Co-Editor-in-Chief of the Journal of Uncertain Systems since 211 and Executive-Editor-in-Chief of the Journal of Uncertainty Analysis and Applications since 213. He served as the Vice President and President of Intelligent Computing Chapter of Operations Society of China from 27 and 215, respectively, the President of International Consortium for Electronic Business from 212 to 213, the Vice President of International Consortium for Uncertainty Theory since 213, and the Vice President of International Association for Information and Management Science from 21 to 213.