JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 24, No. 3, MARCH 1978

Multilevel Stackelberg Strategies in Linear-Quadratic Systems 1

J. M E D A N I C 2 AND D. R A D O J E V I C 3

Communicated by G. Leitmann

Abstract. Open-loop multilevel Stackelberg strategies in deter- ministic, sequential decision-making problems for continuous linear systems and quadratic criteria are developed. Characterization of the Stackelberg controls via the solution of a higher-order square-matrix- Riccati differential equation is established; also, the basic structural properties of the coefficient matrices of this differential equation are established, and the basic structural properties of its solution are inferred.

Key Words. Sequential decision-making problems, Stackelberg stra- tegies, matrix-Riccati differential equation.

1. Introduction

The paper develops multilevel Stackelberg strategies in deterministic, sequential decision-making problems for continuous linear systems and quadratic criteria. Basic results for two-level Stackelberg problems have been presented in Refs. 1-3 and surveyed in Ref. 4; additional results have been presented in Refs. 5-7. These references treat open-loop, closed- loop, and feedback (e.g., signaling-free in Ref. 7) strategies for both the continuous case and the discrete case and in both the deterministic setting and the stochastic setting. The results established so far give the necessary conditions for Stackelberg strategies for the general case; they characterize these strategies more completely for the class of linear-quadratic problems.

1 This work was supported in part by the Energy Research and Development Administration, Contract No. E R D A E(49-18)-2088.

2 Visiting Research Associate Professor, Coordinated Science Laboratory, University of Illinois, Urbana, Illinois; on leave from the Mihailo Pupin Institute, Belgrade, Yugoslavia.

3 Research Associate, Mihailo Pupin Institute, Belgrade, Yugoslavia.

485

0022-3239/78/0300-0485505.00/0 @ 1978 Plenum Publishing Corporation

486 JOTA: VOL. 24, NO. 3, MARCH 1978

The general multilevel Stackelberg problem has been described in Ref. 8 and is now being studied. This paper shows that, for linear-quadra- tic problems, the results obtained for the two-level problem can be generalized directly to the multilevel case with an arbitrary number of sequential decision-makers. For the sake of clarity, results are presented in detail for the three-level problem. This suffices to show how the general- ized case with an arbitrary number of decision-makers is solved.

The basic results established in the paper are (i) the characterization of the Stackelberg controls through the solution of an (n. 2k-1)th order square-matrix-Riccati differential equation, where k is the number of decision-makers and n is the dimension of the dynamic system which they control, (ii) the establishment of the basic structural properties of the coefficient matrices of that matrix-Riccati differential equation, and (iii) the inference of the basic structural properties of the solution.

2. Problem Formulation and Development of Necessary Conditions

Consider an interconnected linear dynamic system in which three subsystems may be distinguished. Each subsystem is characterized by the ability to exert partial control over the system and by a performance criterion Ji(Ul, u2, u3, Xo) through which it evaluates system performance. It is assumed that a decision-making sequence is defined and that, once a decision-maker selects a strategy, all decision-makers following him in the decision-making sequence become aware of the selected strategy. For this class of control problems, the Stackelberg strategy is the accepted solution concept.

The decision-making sequence {N3, N2, N1} implies that subsystem N3 is the leader and selects its strategy first; N2 is the first follower and selects its strategy second; and NI is the second follower and selects its strategy last. Consequently, in making his decision, N1 knows the controls uz and u3; N2 must take into account the reaction of N1, but knows u3; and N1 must take into account the sequential reactions of both N2 and N1. The simplest problem is solved by N1 (an optimal control problem); a more complex problem is solved by N2 (a two-level Stackelberg problem); and the most complex problem is solved by N3 (a three-level Stackelberg problem). The complete solution of the problem is obtained by the solution of the leader's control problem, since the leader must solve problems faced by both N~ and N2 to determine their reactions to a given u3 in order to select that control which is best with respect to 33 taking these reactions of the followers into account. The leader's control problem is solved by deducing sequentially, in the order {N1, N2, N3} retrogressive with respect

JOTA: VOL. 24, NO. 3, MARCH 1978 487

to the decision-making sequence, the Stackel-berg controls for the sub- systems. The effect of this procedure is that, as each new control is charac~ terized, the dimensionality of the associated system, that represents a differential constraint in the determination of the optimal strategy for the next control in the sequence, is doubled.

Let the interconnected system be defined by the linear differential equation

2 = Ax +B!ul+B2u2+B3u3, (l)

performance criteria associated with each subsystem be and let the quadratic:

T

" t o - f = 1

where the usual positive-(semi)definiteness conditions are imposed on A, Bi, O~, F~, Rii, i, j = 1, 2, 3, as in the associated optimal control problem. Consider now the problem of characterizing the open-loop Stackelberg strategies for the subsystems.

Consider the problem solved by N!. Because N~ knows the open-loop strategies selected by N2 and N3, the problem is reduced to an optimal control problem defined by (1) and (2), for i = 1, given u2 and u3. The necessary conditions take the form

2 =Ax+BlUl+B2u2+B3u3, X(to)=Xo, (3-1)

1)1 = --Oax --A Tpl , pl(T)= FIx(T), (3-2)

0 = R1~ul +Brlpl; (3-3)

from (3-3), the control ul may be expressed as

. 1 = - n (4) and is an implicit function of the controls u2 and u3, since they influence the costate vector p.

Consider now the problem solved by N2. In deciding on the control u2, N~ must take into account the reaction of 1~5 to a given u2. The appropriate way of characterizing the reaction of N1 to u2 and u3 is to substitute ul as given by (4) into (3-1) and obtain the two-point boundary-value problem:

2 =Ax-Slpl-bB2u2+B3u3, $1 =B1RT~!B~, X(to)=Xo, (5-1)

Pl = - 0 1 x - A r p l , p l (T)=F~x(T) . (5-2)

It may be viewed that (5-2) characterizes the reaction of N~ on the strategies selected by N2 and ?43, while (5-1) characterizes the dynamics of

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the interconnected system taking this reaction into account. Also, substi- tuting (4) into (2) for i = 2, the performance criterion for N2 taking into account the reaction of N1 to given u2 and u3 takes the form

T

J2 = ½ It (xTO2x+p~S21pl+urR22u2+ufR23u3)dt+½x(T)TF2x(T), (6) o

where

$21 = B1R ~ R21R ~ B r.

The necessary conditions that characterize u2 minimizing (6) under the constraints (5) take the form

ic =Ax -Slpl+Bzu2+B3u3, X(to)=Xo, (7-1)

P l = - - Q I X - A rpl, pl(T)= Flx(T), (7-2)

#2=-Q2x-Arp2+Q1nl , p2(T)=F2x(T)-Flnl(T), (7-3)

h l = - -S 2 1 P l + $1p2 + A n 1 , n~(to) = 0, (7-4)

with

R - 1 B T U2 = - 22 2p2- (8 )

Finally, consider the problem solved by N3. After substitution of u2 given by (8) into (7-1), the necessary conditions are reduced to

ic = Ax - S l P l - $2P2 +B3u3, S2 = B2R2~B~, X(to) = Xo, (9-1)

#1 = - Q l x - A rpl, pl(T) = F~x(T), (9-2)

# 2 = - Q 2 x - A rp2+Qlnl, p2(T)=F2x(T)-F~nl(T), (9-3)

h1=-S21pa+Slpz+An1, nt(to)= 0. (9-4)

As in the previous case, (9-2) characterizes the reaction of N1 to the strategy selected by N2 and N3; (9-3) and (9-4) characterize the reaction of N2 to the strategy selected by N3; and (9-1) characterizes the dynamics of the interconnected system taking into account both the reactions of N2 and N1. The performance criterion J3(ua, u2, u3, Xo), with these reactions taken into account, takes the form

T ¢* y~ | ( x r T T r

= Q 3 x W p l S 3 a P l + P 2 S 3 2 P 2 + u 3 R 3 3 u a ) d t + ½ x ( T ) T F 3 x ( T ) , ( 1 0 ) at o

where --1 --1 T S3j =BiRjj R3iR# B i , j = 1, 2;

JOTA: VOL. 24, NO. 3, MARCH 1978 489

and the necessary conditions for the control u3 minimizing (10) under the constraints (9) take the form

2 = A x - Slpx - $2P2 - $3p3, x (to) = Xo, (11-1)

Pl = - -01X - - A T p l , P l ( T ) = F l x ( T ) , (11-2)

/)2 = - O 2 x - A r p 2 + O l n l , p z ( T ) = F e x ( T ) - F l n ( T ) , (11-3)

h l = - S 2 ~ p l + S l p 2 + A n l , nl(to)= O, (11-4)

~3 = - O 3 x - A rp3 + O~n2 + Ozn3,

p 3 ( T ) = F 3 x ( T ) - F l n 2 ( T ) - F z n 3 ( T ) , (11-5)

h 2 = - S 3 1 p l + S l p 3 + A n 2 + S z ~ W , n2(t0)= O, (11-6)

t i 3 = - S 3 2 p 2 + S 2 p 3 + m n 3 - S l w , n3(to)= O, (11-7)

rV = - O l n 3 - A r w , w ( T ) = F l n 3 ( T ) , (11-8)

with --1 T

U3 = - R a 3 B 3 P 3 . (12)

3. Characterization of Optimal Strategies

Because (11) is a linear system and the boundary conditions are linear, we look for linear relations between the variables w, Pi, ni, i = 1, 2, 3, and the state x of the form

Pi = Kix, i = 1, 2, 3, (13-1)

n~ = P~x, i = 1, 2, 3, (13-2)

w = Wx. ( t3-3)

It follows that, for (11) and (13) to be consistent, IV, Ki, P~ must satisfy the following relations:

I~1 + K I A + A TK1 -- K1S1K1 - K1S2K2 - K1S3K3 + Q1 = O,

K I ( T ) =/71; (14-1)

Ii22 + K z A + A rK2 - K2S1K1 - K2S2Kz - K2S3K3 + 0 2 - - 01P1 = O,

K2(T) = F~ - F I P I ( T ) ; (14-2)

490 JOTA: VOL. 24, NO. 3, MARCH 1978

I~3 + K 3 A + A rK3 - K3S1KI - K3SaK2 - K3S3K3 + 0 3 - 01P2 - 02P3 = 0,

K 3 ( T ) = F 3 - F 1 P 2 ( T ) - F 2 P 3 ( T ) ; (14-3)

P l + P 1 A - AP1 - P1S1K1 - P 1 S 2 K 2 - PIS3K3 + $21K1 - S t K a = O,

P~(to) = 0; (14-4)

P2 + P 2 A - A P 2 - P2S1K1 - P2S2K2 - P2$3K3 + $31K~ - $1K3 - $21 W = 0,

Pa(to) = 0; (14-5)

P3 + P 3 A - A P 3 - P 3 S 1 K 1 - P 3 S 2 K 2 - P 3 S 3 K 3 + S32K2 - $2K3 + $1 W = 0,

P3(to) = 0; (14-6)

Ii¢+ W A + A r W - W S t K 1 - W S 2 K 2 - W S a K 3 + Q1P3 = 0,

W ( T ) = P3(T) . (14-7)

The solution of (14) will provide W, Ki, Pi; and substitution of (13) into (4), (8), (12)determines the open-loop Stackelberg strategies in the form

ui = - R ~ B i K i ( t , t0)Xo, i = 1, 2, 3, (15)

where ~b(t, to) is the fundamental matrix of the system (14-1) with Pi, i = 1, 2, 3, expressed through (13):

Yc = ( A - S1K1 - 8 2 K 2 - S3K3)x, x (to) = Xo. (16)

The solution of (14) is difficult, because it reduces to the solution of a nonlinear two-point boundary-value problem, and because effective methods for the solution of general two-point boundary-value problems involve considerable numerical complexity. More importantly, there do not exist proven methods for the study of the associated infinite-time problem. Therefore, it is of interest to transform Eqs. (14) to a better- known form for which there exist reliable methods of solution. It will be shown that (14) can be transformed to the solution of a higher-order (4n ×4n)-dimensional matrix-Riccati, differential equation with a given terminal condition.

Note that, in (14), initial conditions are given for x ( t ) and ni(t), i = 1, 2, 3, and terminal conditions are given for w( t ) and pi(t) , i = 1, 2, 3; moreover, these terminal conditions are linear functions of x ( t ) and ni(t) . Therefore, reorder the system (14) by grouping x ( t ) and n~(t) into the augmented vector £(t), by grouping w ( t ) and pi(t) into the augmented

JOTA: VOL. 24, NO. 3, MARCH 1978 491

vector/~(t), and bring it to the form:

2 ffl

fi2 _ti_3

,02

w

a ' , 01 0 o 0 A [ 0 0

0 0 A 0 _ _ O - , _ - - ,0 0 A

_:q~_l o', o o 0 0 --02 01 .I

--O3 0 --O1 --O2 0 0 0 01

--8111 --S21

--$21 Sl I

-$3 0 0 0

--S31 0 51 821 0 --$32 $2 --Sl f i

-A~I, o l o o o - a r l o o

t

0 0 - A r 0

0 0 0 - A r

x l

n i l

n2 l

n31

Pl l

P21 ps I w l

(17)

Dashed lines have been used to delineate the equations that define the classical optimal control problem and the two-layer Stackelberg problem, respectively. The system can now be written in the compact form

= ~ - ~ , ~(to) = ~o, ~o = Ix o ~, o, o, ol ~, (18)

~= -O;-A~, ~(r )= P;(r) ,

where the augmented matrices A, S, (), P are given by

Li ° r _,2:_] ° 1 a

~ = t o : A ] = o 0

I -O1

L(~21 !O223

-F1

LF:~ I F:23

0 .

A

0 AA

____--sz _l_ _o o l-s~ -s~|' & 2 1 - & & _1

0 ' , 0 O ] - 0 1 ~ 0 0

o ~-0~ -0~ ' o l o 0~]

° l o,

- E l I 0 I.

0 I -F1 --F: ' J

0 l 0 F1

(19)

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Everywhere in the sequel, tilded variables refer to 4n x4n matrices, "hatted" variables refer to 2n x 2n matrices, and bare variables refer to the original n x n matrices, where the relations between the various variables are uniquely defined by (19). The system (18) is now solved in the usual way by defining the linear transformation

p(t) = K (t)x(t), (20)

whereby it follows that K( t ) must satisfy the Riccati equation

I~ + A TI~ + K A - I ( S I ( + ( ) =0, K ( T ) = F . (21)

If a solution of (21) exists (in particular, if there exists the solution of the associated algebraic Riccati equation for the infinite-time problem), this method of characterization is more expedient, despite the increased order of the Riccati matrix, since well-known algorithms for its solution have been developed. Furthermore, this characterization allows the powerful algebraic methods developed for the algebraic Riccati equation in optimal control (Refs. 6-7) to be brought into the Stackelberg solutions. However, the usual conditions of positive (semi)definiteness and sym- metry, assumed there, do not apply here, and the problem must be studied separately.

Finally, note that if the solution to (21) exists and is represented in the form

^ I .,, / l Kll [ gl__l/K12 q [ K21

R = L K 2 , K22 J = K3~

K4t

we have, from (15) and (24),

K12 K13 K14~ K= K23 K2_.4 l,

K32 K33 K34L K42 K43 g44J

K1 = K n + K12P1 + K13P2 + K14P3,

K2 = K21 + K22P1 + K23P2 + K24P3,

K3 = K31 + K32P1 + K33P2 + K34P3,

(22)

(23)

W = K41 + K42P1 +K43P2-t-K44P3.

Substituting Ki, i = 1, 2, 3, and W from (23) into (14-4) and (14-6) and solving for P~, i = 1, 2, 3, from t = to one obtains the optimal controls (15) with Ki given by (23).

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4. Structural Properties of the Associated Matrix-Riccati Differential Equation

The open-loop solution of the multilevel Stackelberg controls is intrinsically related to the matrix-Riccati differential equation (21). It is immediately evident in this case that the coefficient matrices {fi~, S, (~,/~} do not possess the symmetricity and semipositivg-definiteness properties present in the optimal control problem. For that reason, none of the properties of the solution K, present in that much studied case, can be assumed to hold, including the existence of the solution on a given time interval [to, T]. The question of a finite blow-up time is at present open, save for the conservative bounds that can be placed on it from the general norm considerations related to the growth of the solution of the Riccati differential equations. But, assuming that, in a particular case, the solution does exist on a time interval [to, T], it is possible to derive some of the structural properties of the solution. Recalling the defining relations (19) and (22) for the various matrices, call the quadruple {A, 5~, Q, F} as the coefficient matrices of the matrix-Riccati differential equation. Note the following properties of the various matrices:

(a) all basic matrices defined in (9) are symmetric, except the matrix A;

~i - S i i , i , j = 1, 2, i # j ; A T

(c) Oii =Oii , O_u = - Ojj T, i, j = 1, 2, i # j;

(d) P,~ =/~o', /'~, = - P ii T, i, j = 1, 2, i # f,

(24)

It is possible to state the following result.

Theorem 4.1. The solution /~(t) of the associated matrix-Riccati differential equation (8), for coefficient matrices {e{, 5~, Q, F} satisfying conditions (b)-(d), has the structure characterized by the properties

/~r =/£/i, (25-1) ii A T

K;i = Kii, (25-2)

for i,/" = 1, 2 and i # j. The proof of this theorem relies on the uniqueness theorem of

differential equations, if a solution to the equation considered on the finite interval [to, T] exists. Assuming that the solution exists, the proof proceeds

494 JOTA: VOL. 24, NO. 3, MARCH 1978

by showing that the solution, which then is unique, possesses properties (25). To that end, decompose .A, S, 0 , F and K into 2n × 2n matrices, i.e., utilize the "hatted" variables to write (21) as a system of four coupled matrix differential equations:

K11 + A rI~n + K11A - K11(SilK11 + S12K21) - K12(S21Kll - $11K21)

+ O n = o, g n ( r ) = F n ; (26-1) ,~ A A A ^ .* ,., ^ ,* a A A a T a

K12 + ~ TK12 + K12A - K n (& 1K12 + &2Kz2)- K12(SzlK12 - S t IK22) = 0,

RI2(T)=6; (26-2) x . Z a ^ A ^ A A A A A ^ a a A

K21 + A K21 + K21A -K21(S l IKn + $12K2t)-K22($21K11- S~IK21)

+021=6, R2~(T) = P21; (26-3)

= + A r R = + K = A - K21 (g~ iKt2 + &2K=)- K=(&~K12 - 'SnK=) ^ " ~ T - 0 r l = 0, /~22(T) = - F n . (26-4)

Clearly, if K is to enjoy properties (25), /~T /~T 1 ~2 and must satisfy the same differential equations as/£12 and/~21, respectively, while the negative transpose of the matrix differential equation for/~22 must reduce to the same equation satisfied by/£~1. Therefore, suppose that (25-1) holds, and consider whether it then satisfies (25-2). Transpose and premultiply by a negative sign the equation for/~2a, with the result

A T " . ~ A T ^ T * T A T T "~

- ~ f 2 + ( - K = ) A A (-K22)-[(-Kz2);~12+Klz~ll]K21

- [ ( - K = ) S n + K a 2 $ z ~ ] ( - K 2 2 ) + 0 1 ~ = O , ( T ) = F ~ ; (27)

or, after rearranging the terms,

A T * A ^ ^ ~ a T -K22 + A T (-KT2 )+ (--KT2)A - (--/(T2)(~ I ( - -RT2 )']- S~2K~ ) ^ T "* A T T ' * T A - -

- K l a ( S 2 ~ ( - K 2 2 ) - ~ l l K 2 1 ) + Q l l - O . (28)

Clearly, --KT2 will satisfy the same matrix differential equation as/~11, see (26), if

^ T A ^ T a

K12 = K12, K21 = K21.

Thus, symmetricity of/£12 and/(2~, which has been assumed, is sufficient to guarantee that

A T A

K22 = - K I >

JOTA: VOL. 24, NO. 3, MARCH 1978 495

On the other hand, we show that the antisymmetry of K l l and /~2z is sufficient to guarantee that the symmetry of/¢ij, i =/', holds. Transposing the

A equation for K12, one obtains

"~T A T ~ ~ T T~'T ) _ ~ ( ~ T ~ T "~" A T ~ A .... K12 + K 1 2 ( A - o11~"~ 11 +K22Sll)K12 --KT2s2~KT~

*'~ 22o12r~- 11 ~--- 0 , KT2 (T)= 0; ( 2 9 )

and, rearranging terms and utilizing the mutual antisymmetry of/~11 and K22, we obtain

/~1T2 "Jr- ( A - R l l S l l ) R I T 2 -{- K T 2 ( ~ + ~ T 1 R 2 2 ) - - R T 2 ~ 2 1 R T 1 - - R l l S 1 2 R 2 2 1 0 ,

R~2(r)= 0; (30) A T A

therefore, K12 satisfies the same matrix ~tifferential equation as K12, with the same terminal condition, which proves its symmetry. The symmetry of /~21 is shown similarly, which completes the proof.

Theorem 4.2. If a solution /¢ exists on the interval [to, T], the submatrices /eli, i, j = 1, 2, i #j , are positive semidefinite and monotone nonincreasing functions of time.

Proof. Consider first/~21. Assuming that Rll is the solution of (26-1) and taking into account that

A T - K 2 2 = Kll ,

the differential equation for/C21 may be written in the form A ~ ~ A T A A A A A A A A

K21 + ( A - S l l K l i ) K21 + K21(A - S 1 1 K 1 1 ) - K21S12K21 + 021 AT A A A A

+KllS21K11 = 0, Kal(T) =F21. Because

E l 2 ~ 0 , 0 2 1 --1- R T l g 2 1 R 1 1 ~" 0 , E l 2 ~" 0 ,

and because/~,2, 021, S21, $12 are symmetric, it is immediately clear that R21 enjoys all the well known properties of the optimal gain matrix of an associated linear regulator problem (in particular, positive semidefiniteness and monotone nonincreasing character). A similar conclusion is drawn for /~2 by observing that it satisfies the equation

/~12 + (~ -- ~ITIR 1T1 )TR12 -k R12(~Z ~ e T r o t * ^ - - O i l "t* 11 ) - - R 1 2 S 2 1 K 1 2

+ R . g l : R rl = o, K12(T) = 0;

again, the characteristic properties of ~2 and $21 are sufficient to establish the properties of R12 cited in the theorem.

496 JOTA: VOL. 24, NO. 3, MARCH 1978

Sufficient conditions for the existence of open-loop Stackelberg strategies were considered by Simaan and Cruz (Ref, 3) in the general Hilbert space setting and, as a particular case, have provided conditions that apply to problems defined by linear differential equations and quad- ratic criteria. By extending this method to multilevel problems, it is possible to state the following result.

Theorem 4.3. For the multilevel Stackelberg control problem defined by (1) and (2) with i = 1 , . . . , n, open-loop Stackelberg strategies exist over arbitrary time intervals if

Oi->0, Rii>-O, i# L Rii>-O.

The proof is a direct extension of the proof for the two-level case given in Ref. 3.

5. Conclusions

The paper develops explicit expressions for multilevel open-loop Stackelberg strategies for continuous linear-quadratic sequential decision- making problems. It is shown that the two-level solution (Ref. 3) can be generalized directly to multilevel problems. In particular, it is shown that the problem can always be reduced to the characterization by an associated Riccati solution. The order of the associated Riccati equation is n • 2 k-l, where n is the order of the system and k is the number of sequential decision-makers.

As a consequence of the properties deduced in the paper, the burden of computing the exact solution K(t) is precisely equal to the burden associated with solving the standard Riccati equation of order 4n × 4n with symmetrical coefficient matrices and involves the solution of a system of 8n2+ 2n differential equations. If it is assumed that the computation time increases with the cube of the number of equations to be solved simul- taneously, then the computation time for a direct solution is proportional to

(8n 2 + 2n) 3 + (3n 2)3;

here, the second term is due to the necessity of solving for Pi, i = 1, 2, 3; on the other hand, if K iterations are needed for satisfactory convergence to the solution of the two-point boundary-value problem (4)-(6), then the computation time is proportional to

K[(4n2) 3 + (3n2)3].

JOTA: VOL. 24, NO. 3, MARCH t978 497

Therefore, for a one-dimensional system (n = 1), the Riccati method would be superior if

K > 1 0 ;

and as the dimensionality of the system K increases, it would be superior in the limit for

K > 6 .

On the other hand, the importance of the Riccati equation with coefficient matrices Q, S, F that are nonsymmetric (in particular, nondefinite) in the class of sequential decision-making problems may spur more general study of the Riccati equation itself and the deduction of its intrinsic properties in cases other than those arising from the study of the optimal control problem and repeated problems. The study of the matrix-Riccati differ- ential equation in Stackelberg-type problems (in particular, the study of the related algebraic quadratic equation) will be pursued in following papers.

References

1. VON STACKELBERG, H., The Theory of the Market Economy, Oxford Uni- versity Press, Oxford, England, 1952.

2. CHEN, C. I., and CRUZ, J. B., JR., Stackelberg Solution for Two-Person Games with Biased Information Patterns, IEEE Transactions on Automatic Control, Vol. AC-17, No. 6, 1972.

3. SIMAAN, M., and CRUZ, J. B., JR., On the Stackelberg Strategy in Nonzero-Sum Games, Journal of Optimization Theory and Applications, Vol. 11, No. 5, 1973.

4. CRUZ, J. B., JR., Survey of Nash and Stacketberg Equilibrium Strategies in Dynamic Games, Annals of Economic and Social Measurement, Vol. 4, No. 2, 1975.

5. SIMAAN, M., and CRUZ, J. B., JR., Additional Aspects of the Stackelberg Strategy in Nonzero-Sum Games, Journal of Optimization Theory and Appli- cations, Vol. 11, No. 6, 1973.

6. BASAR, T., On the Relative Leadership Property of Stackelberg Strategies, Jour- nal of Optimization Theory and Applications, Vol. 11, No. 6, 1973.

7. CASTANON, D., and ATHANS, M., On the Stochastic Dynamic Stackelberg Strategies, Proceedings of the 6th World Congress of the International Federa- tion of Automatic Control, Boston, Massachusetts, 1975.

8. CRUZ, J. B., JR., Stackelberg Strategies for Multilevel Systems, Directions in Large-Scale Systems, Edited by Y. C. Ho and S. K° Mitter, Plenum Press, New York, New York, 1975.