Stackelberg Solution for Two-Person Games with
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1 IEEE TRANSdCTIOKS ON AUTObL4TIC CONTROL, VOL. AC-17, KO. 6, DECEMBER [12] N. Dunford and J. T. Schwartz, Linear Opemtors, part 1. Kerr York: Interecience, [13] R. 4. Baker and A. R. Bergen, Lyapunov stability and Lyapunov funct.ions of infinite dimensional system, IEEE Trans. Automat. Cdr., vol. AG14, pp , June [14] I. W. Sandberg, Some stability results related t.o those of V. M. Popov, Bell. Sysf. Tech. J., pp , Nov [15] F. Riesz and B. Sz-Yagy, Functional Analysis. New York: Kngar J. <I. Holtzman, ~Yalinear System Theory. Englewood Cliffs, N.J.: PrenticeHall, F. N. Bailey, The app1i;ation of Liapunov s second met,hod to interconnected systems, J. SIAM Confr., ser. A, vol. 3, pp , V. A. Yacubovich, Frequency conditions for t.he absolute stability of control systems wit.h several nonlinear or linear nonstationary blocks, Automat. Rmot~ Catr. (USSR), pp , June Richard Estrada (S 67-M 69) was born in Philadelphia, Pa., on September 18, He received the B.S. degree in electrical engineering from the Universit,y of Pennsylvania, Philadelphia, and the MS. and Ph.D. degrees in electrical engineering from the University of California, Berkeley, in 1965, 1967, and 1969, respectively. While at the University of California, Berkeley, he held the National Science Foundation Traineeship. Since 1969 he has been a member of the technical stae in t.he I)epart.ment of Ocean Systems Research, Bell Telephone LaboraPories, Inc., Whippan;, K.J., where he has worked in applications of detection a.nd communication theory. He is t.he author of several papers on feedback stability. Dr. Estrada k a member of the S0ciet.y for Industrial and Applied Mathematics. Stackelberg Solution for Two-Person Games with Biased Information Patterns C. I. CHEK AND JOSE B. CRUZ, JR. Abstrucf-A strategy suggested by Stackelberg for static economic competition is considered and extended to the case of dynamic games with biased information pattern. This strategy is reasonable when one of the players knows only his own cost function but the other player knows both cost functions. As with Nash strategies for nonzero-sum dynamic games, open-loop and feedback Stackelberg strategies for dynamic games could lead to dieerent solutions, a phenomenon which does not occur in optimum control problems. Necessary conditions for open-loop Stackelberg strategies are presented. Dynamic programming is used to define feedback Stackelberg strategies for discrete-time games. A simple resource allocation example illustrates the solution concept. I I. IXTRODCCTIOK P; A TWO-PERSON zero-sum game [I], [a], the sum of the cost. functions of t.he t.wo players is equal to zero. Thus the amount that one pla.yer gains is equal to the amount that t,he ot,her loses. Since the objectives of the two players a.re exactly opposit.e, t,here can be neither cooperat>ion nor compromise. On the other hand, in a two-person game wit,h identical goa.ls, because the cost Manuscript received January 14, 1972; revised July 6, Paper recommended by I. B. Rhodes, Chairman of the IEEE S-CS Large Systems, Different.ia1 Games Committee. This work was supported in part by t.he U.S. Air Force under Grant. AFOSR D, in part by the Joint Services Electronics Program under Contract. DAAB C-0199 with the Kniversity of Ill., and in part by the National Research Council of Canada under Grant, NRC-A-4160 with Lava1 University, Quebec, P.Q., Canada. C. I. Chen was with the Coordinated Science Laboraton., University of Illinois, Urbana, Ill. He is noa with t.he Depart.ment of Electrical Engineering, Lava1 Vniversitp, Quebec, P.Q., Canada. J. B. Cruz, Jr., is with the Coordinat,ed Science Laboratory and the Depart,ment of Electrical Engineering, University of Illinois, Urbana, Ill functions for the players are identical, both players tend to co0perat.e with ea.ch other. The problem can then be solved as an optimization problem. In two-person nonzero-sum ga.mes [a], the objectives of the players are neither exa.ct.ly opposite nor do they coincide with each ot.her. There are several mays of defining a solut,ion under t.hese conditions. The optima.1 strategy depends on the rationalit,y assumed by ea.ch player. Some of the Strategies that, have been investigated are nlinmax [i],[4], Nash [5], and noninferior strategies [3],[SI, each of which has desirable charact.erktics. In this paper, a strategy suggest.ed by Stackelberg (discussed in [7] and [SI) for stat.ic economic competition will be considered a.nd extended to the case of dynamic competition with biased information pat.t.erns. Dejnition; Given a two-person game, where Player 1 wants to minimize a cost function Jl(ul,u2) and Player 2 want,s t,o minimize a cost funct.ion J2(u1,~) by choosing ui,u2 from admissible stra.tegy sets L71 and U2, respectively, then the strategy set, (u1*,u2*) is called a Stackelberg strategy zcith Player 2 as leader and Player 1 as follower if for any u2 belonging to U2 and u1 belonging t.o.ti where and u1* = UlO(U2*).
2 792 IEEE TRANSACTIONS ON AUTQN.4TIC CONTROL, DECEMBER 1972 Thus, a Stackelberg strategy xith Player 2 as leader is the optimal strategy for Player 2 if Player 2. announces his move first and if the goal of Player 1 is to minimize J1, while that of Player 2 is to minimize Jz. If Player 2 chooses any other strategy uz, then Player 1 will choose a strategy u1 that minimizes J1, but the resulting cost. for Player 2 will be greater t,han or equal to that Then the Stackelberg strat-egy with Player 2 as the leader is used. The Stackelberg strategy vith Player 2 as leader is an attractive strategy when the informat,ion pattern is biased in the sense that Player 1 does not know the cost function of Player 2, but. Player 2 knows both of the cost functions. By announcing his St.ackelberg strat,egy uz* first., Player 2 forces Player 1 t.o follow and use the St.ackelberg strategy u1*. In t.his paper, Stackelberg solutions for tmo-person nonzero-sum dynamic games are invest.igated. It is assumed that. the dynamic model, i.e., the stat.e equat,ions, and t.he state are know to both players, but only the leader knows b0t.h cost functions. The follower knows his own cost, funct.ion. As -&h Kash solutions for nonzerosum games, feedback and open-loop Stackelberg strategies could yield different, solut,ions. Kecessary conditions for open-loop Stackelberg solutions are derived using varia.- t,ional methods. For discrete-time games, feedback Stackelberg strat.egies are defined using dynamic programming. A simplified resource allocation example is presented to illustrate tmhe solut,ion concept. 11. STACKELBERG SOLCTION FOR TWO-PERSON STATIC GAXES Before considering nonzero-sum dynamic games, a few simple bimatrix games will be considered to illustrate some features of Stackelberg solutions. In all games considered in this section, each player wishes to minimize his own cost and is indifferent to the cost. borne by t.he other player. In the mat.rix game of Fig. 1, Player 1 chooses his st,rategy from the set (21,52,53), while Player 2 chooses his strat,egy from t,he set. (yl,y2,y3). The corresponding entries give t.he costs J1 and J2 for t.he two players, respect<ively. Clearly, the game is nonzero sum: If Player 2 announces that he would choose yl, then Player 1 would choose 21 so as to minimize J1. Therefore, the corresponding cost to Player 1 -would be 3 a.nd the corresponding cost to Player 2 would be 3. If Player 2 announces that he would choose y2, Player 1 would then choose 23, and t.he corresponding costs would be 1 and 5, respectively. Similarly, if Pla.yer 2 announces that, he would choose pa, then Player 1 would choose x2 and the corresponding costs would be 3 and 1. Since t.he goal of Player 2 is t.0 minimize J2, his best, choice would be y3 if he were t,he one t.0 commit himself first, assuming that the goal of Player 1 is solely to make J1 as small as possible. The set. (x2,y3) is the Stackelberg strategy n-it.h Pla.yer 2 as leader. The Kash strategy set. for the matrix game in Fig. 1 is (x1,y1). Furt.hermore, (xs,y2) and (x3,y2) constit.ute the noninferior strategy set. In general, Stackelberg strategies and Kash strat.egies need not be members of the non- TlAYEll 1 Fig. 1. A simple bimatrix game. inferior strategy set.. It. is also clear that y3 is the minmax strategy for Player 2. As anot,her exa.mple, u1 a.nd u2 are scalars, U1 and UZ > are R1, and the cost functions Jl(ul,u2) and J2(ullu2) are convex a.nd twice differentia.ble wit.h respect to both arguments. Player 1 wants to minimize J1, Rrhile Player 2 wants to minimize Js. Equicost contours in the space i U1 X LjF2 are plot.tred for J1 and J2 in Fig. 2. Suppose Player 2 announces that he sill choose uz = ~ 2 Player ~ ) 1 d l t.hen choose uh such that J1(uk,%J 5 J1(ul,uzz) for all u1 E UI. This is achieved by choosing u1 = uh such that a.t. u1 = ulz, the line ug = uzz is tangent to an equicost cont,our of Player 1. The locus of such p0int.s for all u2 E U2 are plotted as P&?l. P1@1 is called the rational readon curve or reaction curve for Player 1. Similarly, the reaction curve for Player 2 is obtained as PzQ2. Xon- consider a Stackelberg strategy xith Player 2 as leader. For any choice of ~2, Player 1 will elect to choose uio such Ohat (ul0,%) lies on his readon curve P1Q1. Thus, in order to minimize his om payoff, Player 2, while playing as leader, d l choose u2* so that for any (ulo,u2) E P1Q1,J2(ul*,u2*) 5 J2(u10,uz). In other words, Player 2 will choose u2 such that Jz is minimized with respect to u2 E Us, while (u1,u2) is constrained to be on the rea,ction curve PIQ1. This is achieved at point R, where t.he equicost contour of Player 2 is tangent to t,he reaction curve PIQl. A Stackelberg stra.t,egy nit.h Player 1 as leader can be found simila.rly t.o be point T, where the equicost, cont,our of Player 1 is tangent to t,he reaction curve P2Q2 of Pla.yer 2. The point ht, which is the intersect.ion of t,he two rea.ction curves PIQl and P2Q2, gives the Nash st.rategy for this game. In ot.her words, the strategy pair (uw,wn) ha,s the property J~(ULV,%~) 5 J~(u~,uw) and Js(Ulx,%N) 5 Jg(uw,uJ for any u1 E UI a.nd any 21.r E UT. The cost for the leader of the St.ackelberg strategy is the best he ca.n have for any pair of strategies on the rea.ction curve of the follower. So the cost, for t,he leader is certainly at. least. as good as his cost when Nash strategy is used, provided that Player 1 plays along his reaction curve. In a zero-sum game, a player who commit,s himself first cannot. get a payoff which is better than t.hat when he makes his choice second or when both players choose simultaneously. However, it is clear that,, for a nonzerosum game, a player who commits himself first can choose a Sta.ckelberg strategy with himself as leader t.o force a solution t.hat, is favorable to him. In Fig. 2 both Stackelberg strat.egies are better t,han the Wash strat,egy for both players. Situations also arise in which the follower d l be
3 CEIEN AND CRVZ: STACKELBERQ SOLUTION FOR QAb7ES '2N 2T ' "Z? Fig. 2. Equicost contours for twc-player nonzero-sum gee. Player 1 chooses u1 and Player 2 chooses up. 117 is Nash solution, R is Stackelberg solution with Player 2 as leader, and Tis Stackelberg solution with Player 1 as leader. UZ L/ I D worse off than when the Nash strategy pair is played. Fig. 3. A zero-sum game without, saddle However, if the leader actually chooses a strategy corresponding to Stackelberg strategy, the follower d l do worse by not following a Stackelberg strategy himself. Necessary conditions for a strategy to be the St,ackelberg strategy for static two-person nonzero-sum game where UX = 8' and Uz = Rr2 is given by the following proposition. Proposition: For sta.tic two-person nonzero-sum games where the admissible strategies for Player 1 are in R" and the admissible strategies for Player 2 are in R" and the cost functions for Player 1 and Player 2, J1(ul,uz) and Jz(ul,uz), are twice Merentiable with respect to both arguments u1 and ~2, then a Stackelberg strategy with Player 2 as leader (ul*,uz*), if it exists, must satisfy the following conditions: UT U1 point. G(~I,u~) = ajl(u1,uz) = (4) aul I and u; D Ul a - [JZ(UI,UZ) + XG(ul,U2)] = 0 (5) du, Fig. 4. A zerc-sun1 game with saddle point. for i = 1,2, where h is a.n rl-dimensional row vector multiwith either player as leader are all the same as the saddlepoint stra,t,egy by definition. exa,mple of such a, case & plier. Similar conditions for a St.ackelberg strategy with Shomm in ~ i 4 ~ where. (U1~,U2*) is t,he sadde-pobt Player 1 as 1ea.der are obtained by intercha.nging sub- strategy. scripts 1 a.nd 2 in (4) and (5), a.nd X is r2 dimensional. For zero-sum games where JZ = - J1, the Stackelberg 111. OPEN-LOOP AND FEEDBACK STACKELBERG SOLUTIONS strategy with Player 2 as leader is the minmax st.rategy FOR TWO-PERSON DYNAMIC GAMES for Player 2. In other words, if both players play the Stackelberg strategy with Player 2 as leader, Player 2 In a two-person dynamic game, Player 1 wishes to will get his minmax payoff (security payoff). In Fig. 3, choose prior t.0 the start of the game his control ul(l) (ul*,~*) is a Stackelberg strategy with Player 2 as leader and the security payoff for player 2 is ko. Note that a saddle-point st.rategy for this example does not exist since min,, ma,, Jz = ko while max,, min,, Jz does not exist a.t all, a.nd hence the Sta,ckelberg strategy with Player 1 as leader does not exist,. For a stmatic zero-sum game where saddle-point a strategy exists, minmax strategies, maxrnin strategies, and Stackelberg strategies for all t in the interval [to, if] to minimize
4 794 IEEE TRA~NSACTIONS ON AUTO~L~TIC CONTROL, DECEMBER x=x2 t=l x x=x3 t=l x=x4 t=l (c) 0 1 x=x- t=l t=o t=l t=2 Fig. 5. A discrete fhite-state multistage nonzero-sum game. both subject to the constraint 3i- = f(x,t,u1,uz), %(to) = x0 (8) where x is the st.ate vector. In a nonzero-sum differential game, it has been found that open-loop Na.sh solutions and feedback Nash solutions a.re different in genera.1 [9]. Thus, one cannot obtain the open-loop solutions from the feedback solutions of vice versa, as in optima.1 control. It d l be shown by exa.mple that feedback Stackelberg strat.egy and openloop Stackelberg strategy for dynamic games can be different. The example is a simple discret,e fde-state muhistage nonzero-sum two-person ga.me sinlilar to t.he one considered by Starr and Ho [9]. It is shown in Fig. 5 using the notation in 191. The feedback Stackelberg solution is defined by dynamic programming. Figs. 6(a)-(d) show the corresponding bimat.rix ga.mes at stage t = 1 for the four states. The Stackelberg solution m-ith Player 2 as leader a.t t = 1 is (0,O) if LC = x2, (1,O) if x = 23, (0,l) if x = 24, and (1,l) if x = 25. Assuming tha.t the players play their feedback Stackelberg strategies at t = 1, Fig. 6(e) shows the bimatrix game at stage t = 0. The feedback Stackelberg strat.egy at this stge is (1,l). Thus, the costs for Players 1 and 2 for the entire ga.me are 0 and -3, with an as- socia.ted t.raject.ory of ~(1) = x5 and 42) = ~ 2 ~. Nom- consider the open-loop sbrategies as shown in the bimat.rix game in Fig. 7. The Stackelberg st,rategy is the sequence pair (01, 10). The cost,s corresponding t.0 t,hese cont,rol sequences are 1 and -2 and the corresponding trajectory is z(1) = z3, ~(2) = xu. Note that the feedback ,D 1,-2 x=xl t=tl 1-2,l 0,-3 (e) Fig. 6. Sequence of bimatris games for determining the feedback Stackelberg st,rategy for the multist.age game of Fig. 5. (a)-(d) are the associated games at, t = 1. (e) is t.he bimatrix game at t = 0, assuming that the players use feedback Stackelberg strategies at. t = G PL4YER ,i Fig. 7. Bimatrix game for det,ermining the open-loop Stackelberg strategy for the multistage game of Fig. 5. Stackelberg strategy m<tlth Player 2 as leader (11, 11) is not a Stackelberg strategy in t,he open-loop table. It should be not,ed that the Stackelberg strat,egy for zero-sum discrete games corresponds to t.he minmax strategy considered by Propoi [lo]-[e!]. Thus, for the leader of a zero-sum discrete game, a feedback Stackelberg strategy always yields a cost that is better than or equal to the cost using open-loop St.ackelberg strategy. One reason for this difference between t,he open-loop and feedback St.ackelberg solutions with t.he same player as leader is that, for the feedback Stackelberg solution,
5 CHEN U D CRVZ: STACKELBERG SOLDTIOW FOR GAMES 795 t several control sequences are eliminat,ed from consideration at t = 0 by t,he a.ssumption that the players always attempt to opt.imize (in the sense of Stackelberg) the transit,ion t.0 t,he next, stage based on t.he current state, assuming t,hat, for the remaining stages, feedback StJackelberg strat.egies will be used. IV. KECESSARYCOICDITIONS FOR A STRATEGY TO BE -4N OPES-LOOP STACKELBERG STRATEGY In this section, necessary conditions for a strategy pair to be an open-loop Stackelberg strategy pair are derived based on fist-order variations. The difterentia.1 game under consideration is given by (6)-(8). The st.ate vector x is assumed to be of dimension n a.nd u1 and ~2 are of dimension TI and f2, respectively.?rto inequality constra.ints on the controls or stat,e variables are considered here. The terminal time ty may be fixed or variable; we shall generally consider it, fixed. Dynamics of the system and initial state are assumed to be known to both players, but no measurement of the state vect,ors is available to any player throughout the course of t,he game. Necessary conditions t,hat a strategy pair (ul*(t),%*(t)) must satisfy for it t.o be an open-loop St,ackelberg st,ra,t,egy pair wit,h Player 2 as leader are sought. For any given ~ (t), necessary conditions for minimizat,ion of J1 by Player 1 yields and ah1 - = o aul ah1 Plf + - = 0, PI&) = 0 ax where H1 is given by HI = L(x,~,uI,u~) + z)i'~(z,~,ui,u~) (11) and pl is an n-dimensional vector. By adjoining these equat,ions as well as the system equation to the minimization problem faced by Player 2, the following additiona,l necessary condit,ions are obtained : The main difference between these necessary conditions and those for the open-loop Kash stxategy [6] hinges on the fact.s t,ha.t, in forming t.he variations for Player 2, ul(t) is assumed to be fixed in the case of open-loop Nash strat.egy, u-hile u,(t) is assunled to be an implicit function of %(t) via relations (9) and (10) in the case of open-loop Stackelberg strategy. The necessary conditions for an open-loop St.ackelberg strat,egy with Player 1 as leader can be found similarly, Ivith the roles of t,he t.wo players interchanged. Similar approaches can be used to obtain the necessary conditions for open-loop Stackelberg strategies of discretetime dynamic games. V. DISCRETE-TIME SYSTEM AND DYNAMIC PROGR-4MhiIICG APPROA4CH The dynamic programming approach for solving a discrete-time optimizat,ion problem [13] can be used to define t,he feedback Stackelberg solutions for discretetime games. Let t,he transit.ion of st,at,es at st.age 7% be described by x(n + 1) = fn(x(fl),~,,vn) (17) where un and v, a.re decision variables of Player 1 and Player 2 at stage n. The objective funct.ions for Players 1 and 2 a.re.v Jx(2) = Lt(*)(x,u*,vJ. (19) i=o If n is considered as t.he initial stage at which decisions a.re to be made, and defining V,+l(')(x) as the value of the objective function of Phyer 1 for the last N-((n + 1) stra.ges when feedback St.ackelberg strat.egy with Player 2 a.s leader is used and when t,he state of the stream entering the stage (n + 1) is x, TTn(')(x) and = { T/Tn+l(')V,(x,~,*,~,*)] + Ln(')(z,un*,vn*) (20) TTn(2)(x) = min { TTn+l(21 [fn.(x,~.,o,~n)] E, + L,(2yx,u,O,~vn)} (21) where a.nd u,* = u,yu,*) (22.) where Hz = Lz(T,~,uI,N) + z)?'~(x,~,ui,wj (16) and p2 is an n-dimensiona.1 vect.or, an n-dimensiona.1 vect.or, and E an rl-dimensional vector. = min { V,+1(1)[fn(x,un,vn)] + Ln(l)(z,un,vn) f. ('2.3) Vn+l(') [fn(x,un0,2',) 1 + ~?L(')(x,un0,~7d W' This means t.ha.t t,he optima.1 decision in the sense of Stackelberg at. stage n for Pla.>-er 2 is t.he one that makes the sum of Vn+l(2) [f(z,~.~~,v,) ] and L.(2)(x,u,o,vn) minimum
6 796 knowing that, for any v,, Player 1 will choose uno such that Vn+l(l)[fn(x,~n,vn)] + L,(l)(zjun,un) is minimized. For the case of a zero-sum game, it can be shown that v,(~)(x) = max min { vn+1(l) [~,(x,u,,v,) 1 I'n un + -Lz(l)(X,un,vn) 1, (24) which is the ma,xmin condition for zero-sum games [11 I. VI. RESOURCE ALLOCATION EXkMPLE In this sect,ion a simplified model for a. dynamic research and development resource allocation problem under rivalry [14] is considered. It is assumed that two firms, labeled as firm A and firm B, are compet,ing with each other for a share of.the market for a specific consumer good. The size of the market is assumed to be fixed for t.he int,erval of consideration and it is assumed that it. will not. change with time nor will it be affect.ed by the research a.nd development effort of either firm. However, each firm's share of the market does depend on the quality of their product and in turn depends on the research and development effort of each firm. It is further assumed that the product is of such a nature that all ot.her costs, such as the cost of modifying plants, are negligible compared with t,he cost of research and development. For highly technical products, this assumption may be fairly reasonable. Let VA(~) and VB(t) be the amounts of money invested in resea.rch a.nd development by fims A a.nd B at time t. Let x be a cerhin kind of measurement of technical gap between these two firms. The evolution of this gap is modeled by 3i. = v*1/2 - (I1vp. (25) Thus, it is assumed that the effect of these investments on the improvement of technical level is t,he Cobb- Douglas type [E] so that the ra.te of the gap equals the difference of two such functions. The multiplying factor (112 1 accounts for the fact tha.t it is easier for a developing iirm, firm B, to catch up than for firm A, which is technica.lly advanced, to innovate. It is assumed that the shares of the market are equal when t.here is no difference in technical levels. When there is a gap betxeen t.echnica.1 levels of these two firms, a portion of the market. of t.he firm lagging t,echnically is temporarily taken over by its rival. This portion is assumed to be proportional to the square of the difference in technical levels. It is assumed that there is no perma.nent or lasting effect of this take-over of t.he market. In other words, the share of the market at each instant is determined by the technical levels at. that instant only. Thus, the revenues of these two firms for finite horizon are IEEE TRANSACTIONS ON AEMMATIC CONTROL, DECEMBER 1972 where x0 is some constant such that, when x reaches xo, the market is completely t.aken over by firm A, P is the quasi-rent that is assumed to be const,ant and y is the discount rate. The goa.is of both firms are to maximize their own revenues. The controls to achieve these goals are the amount of the money invested in research and development, V, and VA. From (25) it is seen that an increase in VA t,ends to increase the gap x and an increase in T7B tends to reduce the gap. However, the respective tendencies for improvement,s in JA and JB in (26) and (=), due to a change in x, are partly canceled by t.he increases in VA and VB. In t.his paper, only solut.ions that satisfy z(t) 5 ro will be considered. The problem can be converted to a. standard linear quadmtic differential game by t,he following change of variables [le]: Y = exp (- (Yt/2)) x (28) u1 = VA~/~ exp (- (y/2)t) (29).& = VB1/2 exp (-- (~/2)t) (30) T7 (35) Note t.hat the last terms in (34) and (35) are constant. The open-loop Stackelberg strategy with Player 2 (the technically lagging firm) as leader is given below. Applying the necesmry condit,ions for open-loop St.ackelberg st.rategy with Player 2 as leader, SI - + SI2 + Q + (U2S1Sz = 0, S SIS~ $- a2sz2 - Q - QP = 0, u1* = SlZ (36) uz* = -as22 (37) Sl(t,) = 0 (38) S2(tf) = 0 (39)
7 CHEN AND CRUZ: STACKELBERG SOLUTION FOR GMvlES 797 ment program, shown in Fig. S(c), but it continues to invest up to a period of 2 time units. 01 I I I I Tlme. t 0.5 h (a) o VII. CONCLUSION In this paper, a strategy suggested by Stackelberg for sta,tic economic competition has been extended to the case of dynamic ga.mes with biased informa.tion patt.ern. This strat,egy is reasonable when one of the players knows only his own cost function but the ot,her player knows both cost functions. When cooperation or negotiation are permitted, this may also be a rea.sonable strategy for hint,ing for initiating cooperation or for threa.tening. Also, this strategy may phy an important role in dynamic ga.mes where t.ime delays in the information ava.ilable t,o the players are different,. Open-loop and feedback Sta.ckelberg st,rat.egies are different for both zero-sum and nonzero-sum dynamic games. Furthermore, Stackelberg strat,egies with different players as leader are generally different. In this paper, necessary conditions for open-loop Stackelberg st,rategies for differential games have been presented. A dynamic progmmming algorit,hnl for feedback Stackelberg st,rat.egies for discrete-time games has also been present.ed. A simple resource allocation example illustrates the solution concept. ACKNOWLEDGMENT Helpful discussions with M. Simaan are gratefully acknowledged. Tlrne, t (C) Fig. 8. (1) Plot of z versus t. (b) Plot. of VA versus t. (c) Plot. of VB versus t. The paramet.er values for all plots are 7 = 0.1, a = 3, T, = 2.5, Q = 0.1. and z(t) sat.isfies Fig. S shows plots of a(t), VA(t), and VB(t). It should be noted that u.~ and u2 are only candidates for open-loop Stackelberg strategies which satisfy the necessary conditions. Using these st.rategies, Fig. S(a) shows that, over a time period of 2.5 units, the technical gap is reduced by a factor of about 7. The t,echnically advanced firm represent,ed by Player 1 uses a decreasing resea.rch and development investment program witlth practically no investment, after 1.5 t,ime units, as shown in Fig. S(b). The technically lagging firm also uses a decreasing research and develop- REFERENCES J. von Neumann and 0. Morgenstern, Th.e TheoTry of Games and Ewmnic Behavior. Princeton, N.J.: Princeton Univ. PT~SS, 1 Q47. R.-Luce and H. Raiffa., Games a.nd Den sions. Nerr York: AT7Tiley, V. Pareto, Xanuel d ewnomique Politique, 2nd ed. Paris: Giard, J. M. Danskin, The Theory of ~lfm-xlin. New York: Springer, F: Nash, Jr., Equilibrium points in X-person games, Proc. A at. Acud. Sei. US., vol. 36, pp. 4849, A. W. Starr and Y. C. Ho, Nonzero-sum differential games, J. Optinziz. Theory and Appl., vol. 3, Mar H. von Stackelberg, Xarkfform an.d Gleich.gxeicht. Vienna, Berlin: Springer, M. D. Intrilligator, Na.thenaa.ticu1 Optimization and Econ.mnic Theory. Englewood Cliffs, N.J.: Prentice-Hall, FTT. Starr and Y. C. Ho, Furt,her properties of nonzero-sum differential games, J. Optimiz. Theory and Appl., vol. 3, Apr A. I. Propoi, Minimax control problem with a priori information, Avtomat. Telemekh., pp , July , I Minimax problems of control under successively acquired informat.ion, Automat. Telemekh.., pp , Jan , On the theory of dynamic games, in Colloquium on. Meihods of Optimization, X. N. Moiseev, Ed. Berlin: Springer, R. Bellman, Dyrmnaic Programming. Princeton, K.J.: Princeton Univ. Press, F. M. Scherer, Research and development. resource allocation under rivalry, Quart. J. Econ., vol. 81, pp , Aug H. Brems. Quantitative Economic Theary. New York: Wiley, B. D. 0. Anderson and J. B. Moore, Linear system optimization with prescribed degree of st.ability, Proc. Inst. Elec. Eng., vol. 116, Dec
8 798 IEEE TR.4NSACTIONS ON -4TJTOMATIC CONTROL, DECEMBER 1972 Cheng-I Chen (S ) was born in Taiclkg, Taiwan, on J~ly 19, He received the B.S. degree from the National Taiwan University, Taipei, Taiwan, in 1964, the pi1.s. degree from Oklahoma State Universit.y, Stillwater, in 1967, and the Ph.D. degree from the University of Illinois, Urbana, in 1971, all in electrical engineering. He was a Teaching and Research Assistant at. Oklahoma State Universit.? from 1966 to 1967, a Teaching Assistant. in the Department of Electrical Engineering, University of Illinois, in 1967, and a Research Assistant in the Coordinated Science Laboratory, University of Illinois, from 1967 to Since September 1971, he has been an KRCC Postdoctoral Fellow at Lava1 University, Quebec, P.Q., Canada. His current research interests are in t.he areas of generalized opt.imization theory and application of system control t.heory in traffic engineering and power systenls. Jose E. Cruz, Jr. (S 56-M 57Sk1 61-F 68) was born in Bacolod City, Philippines, on September 17, He received the B.S.E.E. degree (summa cum laude) from t.he University of the Philippines, Diliman, in 1953, the S.M. degree from the Massachusetts 1nst.itute of Technology, Cambridge, in 1956, and the Ph.D. degree from the University of Illinois, Urbana, in 1959, all in electrical engineering. From 1953 to 1954 he taught at. the Universit,y of the Philippines. He was a Research Assistant. in the M.I.T. Research Laboratory of Electronics, Cambridge, from 1954 t,o Since 1956 he has been with the Department of Electrical Engineering, University of Illinois, where he was an Instructor until 1959, an Assistant. Professor from 1959 to 1961, an Associate Professor from 1961 to 1965, and Professor since Also, he is currently a Research Professor at t,he Coordinated Science Laboratory, University of Illinois, where he is group leader of the Control Systems Group. In 1964 he was a visit.ing Associate Professor at the University of California, Berkeley, and in 1967 he xas an Associate of the Center for Advanced Studies, University of Illinois. He is coauthor (with XI. E. Van Valkenburg) of Introductory Signals and Circuits (Boston: Ginn, 1967), coauthor (with W. R. Perkins) of Engineerin.9 of Dynamic Systems (New York: Wiley, 1969), and Editor of Feedback System (Kea York: McGraw- Hill, 1972). His research interests include sensit,ivity, optimal control of systems m-ith uncertain parameters, nmltivariable Systems, and differential games. Dr. Cruz is a member of the American Association of Universit,y Professors, Eta Kappa Nu, Phi Kappa Phi, and Sigma Xi. He has been a member of the Theory Committee of the American Automatic Control Council since He is listed in American Men of Science and Who s Who in America and is the recipient of t,he Curtis SV. McGraa Research Award of the American Society for Engineering Educat,ion. He was an Associate Editor of the IEEE TRAXSACTIONS os CIRCUIT THEORY from 1962 to 1964, Chairman of the Linear Systems Comndtee of t,he IEEE G-AC from 1966 to 1969, a member of the A4dministrat.ive Commit.tee of the IEEE G-.IC (now the IEEE SCS) since 1966, a member of the IEEE Fellow Committee since 1970, and Editor of the IEEE TRASSACTIOX~ os ACTOMATIC CONTROL since January Short Papers Lyapunov Functions for Quadratic DifEerential Equations with Applications to Adaptive Control GERD LUDERS AND KUMPATI S. NARENDRA Absfracf-Conditions are derived for the existence of quadratic Lyapunov functions for vector differential equations with quadratic terms. These conditions are used to establish a set of nonlinear equations for which the null solution is asymptotically stable in the whole. Many of the recent results in the stability of model reference adaptive systems are particular cases of these general conditions. I. IKTRODUCTION Since 1963 several authors have considered the design of model reference adaptive systems using Lyapunov s direct method. Their objective has been t,o obtain on-line schemes for adjusting the parameters of a plant so that, its out,puts mat.ch those of a reference model for a general class of inputs. The plant (prior t.0 adding t.he adapt,ive control inputs) and the reference model chosen are considered to be linear in all the cases. Adaptive control is accomplished Manuscript received December 30, 1971: revised May 31, Paper recommended by G. Saridis, Chairman of the IEEE SC8 Adaptive end Learning Systems, Pattern Recognition Committee. The authors are with the Department. of Engineering and Applied Science. Yale Uniremity, Kea. Haren, Conn by adjusting some of the plant parameters by using the correlat.ion of two signals that me state variables or inputs of the overall system. For example, if aij is a plant parameter that can be adjusted, then the control law is generally of the form u..(t) 11 = -Bzizi or &(t) = -BXCU~ (1) where zi and x, are state variables of the system, and u, is an input. This in turn makes the plant parameters a;jt) additional stat,e variables. The overall adapt,ive system can therefore be represented by a nonlinear direrent.ial equation containing only linear and bilinear terms in the state variables and inputs. Because of this reason, in all the schemes suggested, the crucial point is the determination of conditions under which t.he ent.ire system is asymptotr ically st,able in the whole. Hence, in most of the model reference adaptive syst,ems considered in the lit.erat.ure, a quadrat.ic Lyapunov function candidate V(z) is proposed and the equat,ion for the adaptive controller is derived so that V(z) is a Lyapunov function for t,he system, thus ensuring stabilit.y. Since the analysis of t.he class of adaptive systems discussed so far uhinlately reduces to a study of a special class of nonlinear d8erent.ial equations, the aim of this paper is to determine a general class of such differential equations having a quadratic Lyapunov function. The result. is a quite general set of quadratic differential equat,ions that. are asympt.otically st.able in the whole. It. is then illustrat.ed through two examples that the class of adaptive systems suggested in t.he past corresponds t.0 a very part,icular form of the general equations derived.
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