Open access peer-reviewed chapter - ONLINE FIRST

Existence of Open Loop Equilibria for Disturbed Stackelberg Games

By T.-P. Azevedo Perdicoúlis, G. Jank and P. Lopes dos Santos

Submitted: October 15th 2019Reviewed: March 20th 2020Published: May 11th 2020

DOI: 10.5772/intechopen.92202

Downloaded: 124

Abstract

In this work, we derive necessary and sufficient conditions for the existence of an hierarchic equilibrium of a disturbed two player linear quadratic game with open loop information structure. A convexity condition guarantees the existence of a unique Stackelberg equilibria; this solution is first obtained in terms of a pair of symmetric Riccati equations and also in terms of a coupled of system of Riccati equations. In this latter case, the obtained equilibrium controls are of feedback type.

Keywords

  • differential games
  • linear quadratic
  • Riccati differential equations
  • Stackelberg equilibrium
  • worst-case disturbance

1. Introduction

The study of linear quadratic (LQ) games has been addressed by many authors [1, 2, 3, 4]. This type of games is often used as a benchmark to assess the game equilibrium strategies and its respective outcomes. In a disturbed differential game, each player calculates its strategy taking into account a worst-case unknown disturbance. In non-cooperative game theory, the concept of hierarchical or Stackelberg games is very important, since different applications in economics and engineering exist [1, 5]. This is also the case of gas networks where a hierarchy may be assigned to its controllable elements—compressors, sources, reductors, etc… Also, for this application, the modelling as a disturbed game makes a lot of sense, since the unknown offtakes of the network can be modelled as unknown disturbances. Further research on Stackelberg games can be found for instance in AbouKandil and Bertrand [6]; Medanic [7]; Yong [8]; Tolwinski [9].

No assumptions/constraints are made of the disturbance. To be easier to understand the hierarchical concept, we consider only two players. Therefore, we study a LQ game of two players with Open Loop (OL) information structure where the players choose its strategy according to a modified Stackelberg equilibrium. Player-1 is the follower and chooses its strategy after the nomination of the strategy of the leader. Player-2, the leader, chooses its strategy assuming rationality of the follower. Both players find their strategies assuming a worst-case disturbance.

In this work, we consider a finite time horizon, where for applications this is chosen according to the periodicity of the operation of the problem being studied.

The disturbed case of the representation of optimal equilibria for noncooperative games has been studied [10, 11] considering a Nash equilibrium. It is the aim of this paper to generalise the work of Jank and Kun to Stackelberg games and extend the results presented in Freiling and Jank [12]; Freiling et al. [13] to the disturbed case. To calculate the controls, we use a value function approach, appropriately guessed. Thence, we obtain sufficient conditions of existence of these controls and its representation in terms of the solution of certain Riccati equations. Furthermore, a feedback form of the worst-case Stackelberg equilibrium is obtained.

In a future paper, we expect to present analogous conditions using an operator approach.

In Section 2, we define the disturbed LQ game and define Stackelberg worst-case equilibrium. In Section 3, we derive sufficient conditions for the existence of a worst-case Stackelberg equilibrium under OL information structure and investigate how are these solutions related to certain Riccati differential equations. Section 4 concludes the paper and outlines some directions for future work.

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2. Fundamental notions

We start with the concept of best reply:

Definition 2.1.(Best reply) Let ΓNbe a N-player differential game. For i1N,

γiγ1γi1γi+1γNjiUj.

We say that γ˜iis the best reply against γiif

Jiγ1γi1γ˜iγi+1γNJiγ1γN

holds for any strategy γiUi. We denote the set of all best replies by Riγi.

We study games of quadratic criteria, defined in a finite time horizon t0tfRand subject to a linear dynamics, controlled players and also an unknown disturbance. Hereby also consider uj=γjtηj,where ηjis the information structure of Player-j. In this case, ηj,j=1,,N,is of OL type.

Definition 2.2.(Linear Quadratic (LQ) differential game) Let ΓNbe an Nplayer differential game finite time horizon T=t0tf.Suppose further that:

  1. the dynamics of the game are assumed to obey a linear differential equation

ẋt=Atxt+j=1NBjtujt+Ctwt,xt0=x0.E1

In this equation, tT,where the initial t0and the final tfare finite and fixed, the state xtis an ndimension vector of continuous functions defined in Tand with xtf=xf. The controls ui,i=1,,N,are square (Lebesgue) integrable and the midimension vector of continuous functions is also defined in T. Also, the disturbance wtLmT.The different matrices are of adequate dimension and with elements continuous in T.

  1. the performance criteria are of the form

Jiuiuiw=Kxtf+t0tfΨuiuiwdt.E2

where

Kxtf=xTtfKifxtfE3
Ψuiuiw=xTtQitxt+wTtPitwt+j=1NujTtRijtujt,E4

with symmetric matrices KifRn×nand symmetric, piecewise continuous and bounded matrix valued functions QitRn×n,RijtRmi×mjand PitRm×m,i=1,2,,N.

We observe that no cost functional is assigned to the disturbance term because no constraints can be applied on an “unpredictable” parameter. In what follows, we consider N=2.To extend the theory to N>2,since this is an hierarchical solution, we need to define the structure of the leaders and followers in the game. We can even have more than two hierachy levels. We assume that Player-2 is the leader and Player-1 is the follower.

The leader seeks a strategy u2tin OL information structure and announces it before the game starts. This strategy is found knowing how the follower reacts to his choices. The follower calculates its strategy as a best reply to the strategy announced by the leader.

Problem 2.1.Find the control uiUi,i=1,2,in Tfor which Jiuiuiw,i=1,2,is minimal when subject to constraints uit=γitηit,i=1,2,and (1) and considering a worst-case disturbance.

Consider Ui,i=1,2,the sets of functions such that (1) is solvable and Jiexists, with ui,i=1,2,in these conditions Ui,i=1,2,Ware said the sets of admissible controls and disturbance, respectively.

Definition 2.3.(Stackelberg equilibrium) Let Γ2be a 2-person differential game, we define the Stackelberg/worst-case equilibrium in two stages.

  1. A function ŵiuWis called the worst-case disturbance, from the point of view of the ith player belonging to the set of admissible controls, if

    JiuiuiŵiJiuiuiw,i=1,2,E5

holds for each wW. There exists exactly one worst-case disturbance from the point of view of the ith player according to every set of controls.

  • We say that the controls u1u2form a worst-case Stackelberg equilibrium if

    1. The leader chooses u2such that

      maxγ1R1u2J2γ1u2ŵ2maxγ1R1u2J2γ1u2ŵ2

    for all u2U2.

  • The follower then chooses u1such that

    R1u2=u1J1u1u2ŵ1J1γ1u2ŵ1.

  • To guarantee the uniqueness of OL Stackelberg solutions, matrices are assumed to satisfy Kif0,Qi0,Rij>0,ijand Rii0,i,j=1,,Nin TSimaan and Cruz [14].

    In what follows, we drop the dependence of the parameters in tto reduce the length of the formulas.

    3. Sufficient conditions for the existence of OL Stackelberg equilibria

    In this section, we withdraw sufficient conditions for the existence of the worst-case Stackelberg equilibrium, using a value function approach.

    A disturbed differential LQ game as defined in Definition 2.2 is said playableif there exists a unique Stackelberg worst-case equilibrium.

    Theorem 3.1.Let the solution of the Riccati differential equation

    Ė1=E1AATE1Q1+E1S1+T1E1,E1tf=K1f,E6

    with S1=B1R111B1Tand T1=CP11CTexist on T.

    For any given admissible OL control of the leader, u2,define e1Rn,d1Rby

    ė1=E1S1+T1e12E1B2u2ATe1T,e1tf=0E7
    ḋ1=u2TR12+e1TB2u2+14e1TS1+T1e1,d1tf=0.E8

    Then, the following identity holds:

    2J1u1u2=x0TE1t0x0+x0Te1t0+d1t0+t0tfz1tR112dt+t0tfztP12dt,E9

    where z1R112=z1R11z1with

    z1=u1+R111B1TE1x+12e1

    and zP12=zP1zwith

    z=w+P11CTE1x+12e1

    and xa solution of (1).

    Proof:The proof is similar to the analogous result for the non-disturbed case Freiling et al. [13].

    Theorem 3.2.Let the solution E1of (6) exist on T. Then the unique response of the follower to the leader’s OL strategy u2tis given by:

    u1=R111B1TE1x+12e1,E10

    where the maximum disturbance,

    w1=P11CTE1x+12e1,E11

    was considered. E1and e1are the solutions of (6)(7) and xis then the solution of

    ẋ=AS1+T1E1x12S1+T1e1+B2u2,E12
    xt0=x0.E13

    The corresponding minimal costs then are

    J10=2J1u1u2=x0TE1t0x0+x0Te1t0+d1t0.E14

    Proof:We have that the unique OL response of the follower to the leader’s announced strategy u2(10) under worst-case disturbance (11), that we substitute in the trajectory (1) to obtain:

    ẋ=AS1+T1E1x12S1+T1e1+B2u2.

    The cost functional minimal value is obtained when we substitute in (9) the minimal control and themaximal disturbance.

    Notice that J10u2is not depending on u1.This, as a matter of fact, is only true if we consider OL information structure, since otherwise u2would depend on the trajectory xand hence, via (1), also on u1.In OL Stackelberg games, the leader tries next to find an optimal OL control u2that minimises J2u1u2u2while u1u2is defined by (10).

    Theorem 3.3.Let the solution of the Riccati differential Eq. (6) and the solution of

    Ė2=E2HHTE2Q+E2S+TE2,E2tf=K2f000,E15

    with S21B1R111R21R111B1T,S2B2R221B2Tand T2CP21CT.Also HAS1Q1E1T1AT,QQ200S21,SS2000and TT2T2E1E1T2E1T2E1exist in T, where E2R2n×2n.Also BB20m1×n.

    For any given control u2of the leader, define functions e2R3n,v1,vw,xRnand d2Rin Tby the following initial and terminal value problems:

    ė2=HT+E2S+Te2,e2tf=0E16
    ḋ2=14e2TS+Te2,d2tf=0E17
    v̇1=Q1x+E1T1ATv1+E1Cw,v1t0=v10E18
    ẋ=AxS1v1+B2u2+Cw,xt0=x0,E19

    with v1E1+12e1.

    Then, we obtain

    u1=R111B1Tv1,w1=P11CTv1,

    and the following identity

    2J2u1u2w2=x0Tv10E2t0x0v10+x0Tv10e2t0+d2t0+t0tfz2R222dt+t0tfzP22dt,

    where y=xv1,z2R222=z2R22z2and

    z2=u2+R221B2T0m1×nE2y+12e2

    and 0mi×n,i=1,2the mi×ndimensional zero matrix and zP22=zP2zand

    z=w2+P21C1TE2y+12e2.

    Proof:Consider (10): u1=R111B1TE1x+12e1v1.Then, differentiate v1and substitute the derivatives into the obtained expression using (6), (7) and (8). Also, the optimal control u1and disturbance w1in (11). Hence:

    v̇1=Q1xATv1,ẋ=AxS1v1+B2u2+Cw.

    Hence defining HAS1Q1E1T1AT,BB2On×m2and C1IE1C.We define yxv1to write these two equations as: (??) as:

    ẏ=Hy+Bu2+C1wE20

    Next, we consider the following value function

    V˜2t=V2tyt=yTE2y+e2Ty+d2E21

    for some mappings E2:TR2n×2n,e2:TR2n, and d2:TR2,where E2is symmetric for each tT.

    We consider (21), where we substitute (20):

    dV˜2dt=ddtyTE2y+e2Ty+d2=ẏTE2y+yTĖ2y+yTE2ẏ+ė2Ty+e2Tẏ+ḋ2+xTQ2x+u1TR21u1+u2TR22u2+wTP2wψ2=yTHT+u2TBTwTC1TE2y+yTĖ2y+yTE2Hy+Bu2+C1w+ė2Ty+e2THy+Bu2+C1w+ḋ2+yTQ200S21Qy+u2TR22u2+wTP2wψ2

    Now we associate certain terms

    =yTHTE2+Ė2+E2H+Qy+u2y2TR22u2y2+y2TR22u2+u2TR22y2y2TR22y2+wαTP2wα+wTP2α+αTP2wαTP2α+u2TBTE2y+BTe2+yTE2B+12e2TBu2+wTC1TE2y+C1Te2+yTE2C1+12e2TC1wė2T+e2THy+ḋ2ψ2

    and furthermore

    =yTHTE2+Ė2+E2H+Qyψ2+u2y2TR22u2y2y2TR22y2+wαTP2wααTP2α+u2TBTE2y+BT12e2+R22y2+yTE2B+12e2TBy2TR22u2+wTC1TE2y+C1T12e2+P2α+yTE2C1+12e2TC1αTP2wė2T+e2THy+ḋ2ψ2

    Consider

    R22y2+B2TOm2×nE2y+12e2=0

    and also

    C1TE2y+12e2+P2α=0

    If R22>0then y2=R221B2TOm2×nE2y+12e2.If P2>0then α=P21C1E2y+12e2.

    Define SS2000and TT2T2E1E1T2E1T2E1. Substitute this y2and αin the calculations:

    =yTHTE2+Ė2+E2H+QE2S+TE2y+u2y2TR22u2y2+wαTP2wα+ė2T+e2THE2S+Ty+ḋ214e2TS+Te2ψ2

    Considering:

    HTE2+Ė2+E2H+QE2S+TE2=0ė2T+e2THe2TS+TE2=0ḋ214e2TS+Te2=0

    that is

    Ė2=HTE2E2HQ+E2S+TE2ė2=HT+E2S+Te2ḋ2=14e2TS+Te2

    We end up with

    dV˜2tdt=u2y2TR22u2y2+wαTP2wαψ2E22

    Integrating yields:

    V˜2tfV˜t=ttfu2y2TR22u2y2+wαTP2wαsttfψ2.

    Further, we assume the mappings E2,e2,d2to be chosen in such a way that the following terminal values hold:

    E2tf=K2fe2tf=0d2tf=0

    Then, we obtain V˜2tf=yTtfKyfytfand substituting:

    V˜2t=yTtfKyfytfttfu2y2TR22u2y2+wαTP2wα+ttfψ2E23

    Observe that the rhs of (23) does not depend of ut0tand the rls of (23) does not depend of u2ttf.Then considering now the infimal value, we recall that:

    V2ty=infu2ttfttfψ2τŷτuτ+yTtfK2fytf

    Now, we substitute this into (23) and consider the infimal values over all possible control functions in ttf:

    V˜2t=infyTtfK2fytf+ttfψ2V2tyinfttfu2y2TR22u2y2+wαTP2wα

    then we have:

    V2ty=V˜t+infut,tfttfu2y2TR22u2y2+wαTP2wα

    V2tyequals V˜2tif u2y20tTand wα=0.As the leader chooses his strategy assuming rationality of the follower and worst-case disturbance, the follower should take also the worst-case disturbance into account.

    To conclude, consider t=t0and hence:

    V2t0y=V˜2t0+infut0,tft0tfu2y2TR22u2y2+wαTP2wα

    Then from (21):

    V2t0y=y0TE2t0y0+e2Tt0y0+d2t0+infut0,tft0tfu2y2TR22u2y2+wαTP2wα

    Defining z2u2y2=u2+R221B2T0E2y+12e2and zwα=w+P21C1E2y+12e2,we have:

    V2t0y=y0TE2t0y0+e2Tt0y0+d2t0+t0tfz2R222+zP22dt

    Now, we substitute y0=x0v10.

    The leader may choose its best answer either by accounting directly for its worst-case disturbance or by considering that the follower knows that there is a worst-case disturbance. In this work, the leader takes the worst-case disturbance directly into account.

    Notice that in the term

    J20=x0Tv10E2t0x0v10,E24

    x0,E2t0,do not depend on the choice of u1,u2.Since we shall study the situation for Player-2 when Player-1 applies his optimal response control defined in (10), we have to set v1=E1x+12e1.From (7), we can see that v1t0=v10depends on e1t0and hence also on u2.

    In order to derive from Theorems (3.1) and (3.3) sufficient conditions for the existence of a unique worst-case Stackelberg equilibrium, we must get rid of the u2-dependence on v10.Therefore, we propose to restrict the set of admissible controls to functions representable in linear feedback form. This is what we do next.

    Theorem 3.4.Let the solutions E1tRn×n,E2R2n×2nof (6) and (15) exist in T, respectively. Let further the coupled system of equations

    K̇1=Q1K1AATK1+K1S1+T1K1+K1S2K2,E25
    K̇2=Q2K2AATK2+Q1p+K2S1K1+K2S2+T2K2+K2T2E1p,E26
    ṗ=pAS21K1+S1K2+AT1E1p+pS1K1+pS2+T2K2+pT2E1p,E27

    admits a solution in T.

    Then, there exists a unique open loop disturbed Stackelberg equilibrium in feedback synthesis which is given by

    u1t=R111tB1TtK1txt,E28
    u2t=R221tB2TtK2txt,E29

    considering worst-case disturbamces wiand where xtis a solution of the closed loop equation

    ẋ=AS1K1S2+T2K2T2E1px,xt0=x0.E30

    The minimal cost for the follower, J10u2,is as in (14), and for the leader is

    J20u1u2=12x0TInK1Tt0E2t0t0InK1t0x0+e2Tt0InK1t0x0+d2t0

    where e2t0,d2t0are determined by (16) and (17), respectively.

    Proof:The proof is similar to the analogous result for the non-disturbed case [13].

    From the convexity assumptions, it follows that S1,S,Q1,Qand E1tf,E2tfare all semidefinite. Therefore, as far as the convexity conditions hold, the standard Riccati matrix Eqs. (6) and (15) are globally solvable in tf[15].

    It still remains the following questions to be answered (i) direct criteria for solvability of these equations if the convexity assumption is guaranteed as well as (ii) solvability of the coupled system of Eqs. (25)(27).

    Actually, this system of equations can also be written as a single, nonsymmetric Riccati matrix differential equation. Hence:

    K̇1K̇2ṗ=Q1Q20K1K2pA+AT000ATQ1S21S1AT1E1K1K2p+K1K2pS1+T1,S2,0K1K2pK1K2ptf=K1fK2f0.E31

    As it can be easily observed, all these Riccati equations are of nonsymmetric type:

    Ẇ=B21WB11+B22W+WB12W,Wtf=Wf,E32

    where Wis a matrix of order k×nwhose coefficients are of adequate size. See AbouKandil et al. [16] for results on the existence of solution of Riccati equations.

    4. Discussion and conclusions

    High dimension problems appeal to the use of hierarchic and decentralised models as differential games. One example of these problems is large networks, as for instance the management and control of high pressure gas networks. Since this is a large dimension and geographically dispersed problem, a decentralised formulation captures the non-cooperative nature, and sometimes even antagonistic, of the different stake-holders in the network.

    The network controllable elements can be seen as players that seek their best settings and then interact among themselves to check for network feasibility. The equilibrium sought by the players depends on the way the players are organised among themselves. It makes some sense to have some autonomous elements that run the network and others follow, as is the case of a main inlet point of a country, as it happens with the inlet of Sines in the portuguese network. The ultimate goal of the network is to meet customers’ demand at the lowest cost. As the main variation of the problem is due to the off-takes, these may be seen as perturbations to nominal consumption levels of a deterministic model.

    Therefore, it makes some sense to view the gas transportation and distribution system as a disturbed Stackelberg game where the players play against a worst-case disturbance, that means a sudden change in weather conditions from one period of operation to the other. Neverthless, the theory is not ready, and also having in mind the development of algorithms, direct solution methods, and explicit solution representations need to be further investigated. In this work, we have obtained sufficient conditions for the existence of the solution of a 2-player game. However, direct criteria for solvability of this problem needs more work. Also, the solvability of the coupled system of Eqs. (25)(27) has to be further investigated. Also, we would like to solve the same problem using an operator approach.

    Similarly to what we have done in the past for Nash games, we would like to study this problem considering the underlying dynamics as a repetitive process, that seems to be adequate to capture the behaviour seemingly periodic of the network. Also, the boundary control of the network depends on the type of strategy sought by the players. The structure of these versions of the problems need to be examined.

    The obtained results, in every stage of the work, should be applied to a single pipe and ideally using some operational data. Furthermore, we expect to apply the work to a simple network, which is not exactly a straightforward extension.

    Acknowledgments

    I would like to thank the reviewer for his valuable suggestions.

    This work has been financed by National Funds through the Portuguese funding agency, FCT – Fundao para a Cincia e a Tecnologia under project: (i) UID/EEA/00048/2019 for the first author and (ii) UID/EEA/50014/2019 for the third author.

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    T.-P. Azevedo Perdicoúlis, G. Jank and P. Lopes dos Santos (May 11th 2020). Existence of Open Loop Equilibria for Disturbed Stackelberg Games [Online First], IntechOpen, DOI: 10.5772/intechopen.92202. Available from:

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