CC BY
19UDC: 519.833.2 MSC2010: 91A06
BERGE-VAISMAN EQUILIBRIUM FOR ONE LINEAR-QUADRATIC
DIFFERENTIAL GAME
© V. I. Zhukovskiy
Moscow State University named after Lomonosov Faculty of Computational Mathematics and Cybernetics Department of Optimal Control Leninskiye Gory, GSP-1, Moscow, 119991, Russian Federation e-mail: zhkvlad@yandex.ru
© L. V. Smirnova
Moscow State University of Technologies and Management named after Rasumovskiy Zemlyanoy Val, 73, Moscow, 109004, Russian Federation e-mail: smirnovalidiya@rambler.ru
Berge-Vaisman equilibrium for one linear-quadratic differential
game.
Zhukovskiy V. I., Smirnova L. V.
Abstract. We obtained coefficient criteria for the existence of the Berge-Vaisman equilibrium in a non-cooperative positional linear-quadratic game of two persons with small parameter.
Keywords: non-cooperative positional linear-quadratic differential game, dynamic programming, Berge-Vaisman equilibrium, Nash equilibrium, continuous dependence and analyticity of the solution by parameter
Introduction
The notion of "Berge equilibrium" appeared in Russia in 1994 during the critical discussion of the published book [1] by Claude Berge in Paris. Berge equilibrium (BE) removes "selfish" nature promoted in [1] Nash equilibrium due to the altruistic approach, dictated by the concept of BE. In 1995 Constantin S. Vaisman (then a graduate student of Zhukovskiy) defended his Ph.D. thesis on Berge equilibrium, at the Leningrad University in 1995. Unfortunately, Vaisman died three years after the Ph.D. defense of the thesis, before the age of 36 years. During these three years he published 19 works, a list of which we present at the end of the article. We observe also that Vaisman together with the first author of this article wrote individual chapters in two books [9, 19].
Vaisman merit lies in the fact that he presented the example that the property of individual rationality for BE, generally speaking, does not take place. Therefore Vaisman added this requirement in the definition of Berge equilibrium, after which BE was naturally called as the equilibrium by Berge-Vaisman.
BE has not got the bright destiny. Because of Vaisman's death, who was the greatest enthusiast of Berge equilibrium, they suspended the investigation of it in Russia. Furthermore, the publication of the book [1] aroused the acute review of Martin Shubic. However, the Algerian trainees of Zhukovskiy Radjef Mohamed Said and Larbani Moussa managed to publish the works [21, 22], which caused widespread interest in the West to BE. As shown by the review [23], right now the research of BE stuck at an early stage. Namely, they are the initial accumulation of facts, the formalization of modification BE, a comparison with Nash equilibrium. Futhermore, basically, all the studies are limited to only finite non-cooperative games. We believe it is time to proceed to the second heuristic stage, that is to answer the following two questions:
1) Is there Berge equilibrium and how to build it?
2) How should one take into account the dynamics of the conflict?
The recently published book [24] was dedicated to the answer to the first question, it is true only within static version of non-cooperative games. We expect to dedicate a separate book to Dynamic version of the problem (within the mathematical formalization of the positional differential game proposed by Russian academician Krasovsky). Though this article is devoted to a partial answer to the second question.
1. Formulation of the problem
Let us consider a non-cooperative positional linear-quadratic differential game of two persons, where one of the players has small influence on the rate of the change of the phase vector
({1, 2}, E, {Ai}^, { Ji(U, to, Xo)}i=i,2>. (1)
Here {1, 2} is a set of the serial numbers of the players. Variation of the control system E with respect to the time t is described by the linear equation
x = A(t)x + u1 + eu2, x(t0) = x0,
where the n x n-dimensional matrix A(-) E Cnxn[t0, $]; t0 > 0 is the moment of beginning and $ > t0 is the fixed moment of finishing of the game; x = (x1,... ,xn) E is a phase n-column-vector. The pair (t,x) E [t0,$] x Rn forms a position of the game, (t0,x0) is the initial position of the game, (t,x(t)) is the current position of the game at the moment of the time t E [t0,$]; u E is a control action of the i-th player (i = 1, 2),
the constant e > 0 is the small scalar parameter. A set of strategies of the i-th player is Ai = [Ui + Uj(i,x) = Qj(i)x}, i. e., a strategy U of the i-th player identified with a vector-valued function ui(t,x) (denoted by Ui ^ ui(t,x)), that is linear with respect to x G and continuous with respect to t. Thus, the choice of strategy Ui for the ith player means the choice of an n x n-dimensional matrix Q(^) G Cnxn[0,$) with the continuous elements on interval [0,$). A system U = (Ui, U2) G A = A1 x A2 is called a situation of the game (1).
The game (1) proceeds in the following way. Each i-th player chooses its strategy Ui ^ ui(t,x) = Qi(t)x, Ui G Ai (i = 1, 2). Substituting ui = ui(t,x) in (1), we have an inhomogeneous linear system of differential equations containing continuous coefficient with respect to t:
x = [A(t) + Qi(t) + eQ2(t)]x, x(to) = xo.
For any fixed e = const > 0 this system has the unique continuous solution x(t), which can be extended for interval [to,$]. Using x(-) G Cn[to,$], we construct realizations ui[t] = Qi(t)x(t) of the chosen by the players strategies Ui G Ai. On continuous triples (x(t),u1 [t],u2[t] | t G [to,$]) we define the payoff functions of the players, which are given by quadratic functionals:
i9
Ji(U,to,xo) = x'($)C/x($) + J(u/1[t]DiiMi[t] - u'2[t]Di2U2[t] + x'(t)Gix(t))dt,
to i9
J2(U,to,xo) = x'($)C2x($) + J( u/[t]D2iui[t] + u'2[t]D22U2[t] + x'(t)G2x(t))dt,
to
where the prime on the top indicates the operation of transposition, n x n-dimensional matrices Ci, Gi, Dij- (i,j = 1, 2) are symmetric and constant. Further, the fact that the quadratic form u'Mu is definitely negative (positive, nonnegative and nonpositive) we denote by M < 0 (>, >, <).
Definition. A situation UB = (U-f, Uf) G A is called a Berge-Vaisman equilibrium for the game (1) if the following conditions for any choice of the initial position (to, xo) G [0,$) x Rn hold:
Ji(U/B,U2,to,xo) < Ji(UB,to,xo) VU2 G A2,
J2(Ui,U2b,to,xo) < J2(UB,to,xo) VUi G Ai, (2)
maxmin Ji(Ui,U2,to,xo) < Ji(UB,to,xo),
Ui U2
maxmin J2(Ui,U2,to,xo) < J2(UB,to,xo).
2. Existence of the Berge-Vaisman equilibrium
So, we will consider a differential positional linear-quadratic game of two persons (1). Recall that dynamics of the game is described by the ordinary linear differential equation
x = A(t)x + u1 + eu2, x(to) = xo,
where x,u G A(-) G Cnxn[to,$], e = const is a small parameter, constant $ > to > 0. A set of strategies of the i-th player is
A = {U - ui(t,x) = Qi(t)x | Vt G [0,$),x G Rn,Q(-) G Craxra[0,$)} (i = 1, 2), payoff functions of the players are:
Ji(Ui,U2,to,xo) = x'($)Cix($) + J(u1 [t]DiiUi[t] - u'2[t]Di2U2[t] + x'(t)Gix(t))dt,
to 9
j2(Ui,U2,to,xo) = x' ($)C2x($) + J (-ui[t]D2iUi[t] + u'2 [t]D22U [t] + x'(t)G2 x(t))dt,
to
where symmetric constant matrices Ci, Gi, Dij are such that C2 < 0, G2 < 0, Dj > 0 (i,j = 1, 2). Berge-Vaisman equilibrium is formalized by definition 1.
Statement. If the constant e > 0 is sufficiently small and the matrices
C2 < 0,G2 < 0,Dij > 0 (i, j = 1, 2), (4)
then a situation of Nash equilibrium doesn't exist, but a situation of Berge-Vaisman equilibrium,
(UB,U2B) - (uB(t,x),uB(t,x)), UB G Ai (i = 1, 2) exists in the game (1) for any choice of the initial position (to,xo) G [0,$) x Rn. Moreover, uB(t,x) can be represented in the form uB(t,x) = QB(t,e)x (i = 1, 2), where the matrices QB(t,e) G Cnxn[to,$] for mentioned constant e > 0.
Proof. First of all we note two facts. Firstly, according to Dn > 0 and (or) D22 > 0 the situation of Nash equilibrium doesn't exist in the game (1) [25, p. 115]. Secondly, maximins in the left parts of the inequalities (3) don't exist too (by D12 > 0, D21 > 0 and [26, p. 272273]). Therefore, to prove the existence of Berge equilibrium it is sufficient to establish the correctness of the implication [D12 > 0, D21 > 0] ^ [3 V(t0, x0) € [0, x Rn]. We can establish this fact using the method of the dynamic programming and Poincare's theory about small parameter [27]. According to the method of the dynamic programming we construct two scalar functions
Wi (t,X,Ui,U2,Vi, V2)
ÔVÎ "dt
+
dVi
dx
[A(t)x + ui + £«2]+
+ u/iDiiui — m'2Di2m2 + x'Gix,
W2(t,x,Ui,U2,Vi, V2)
dV2 dt
+
dx
[A(t)x + ui + £u2] —
In the usual way (such as in [25, p. 62-67]) the correctness of the next proposition can be established, where V = (V1, V2) G R2, 0n is the n-dimensional zero vector; 0nxn is the n x n-dimensional zero matrix; /dem{u ^ u*} implies that in the expression which is situated in the braces u is replaced by u*:
Assume that for the game (1) we find two continuously differentiable scalar functions V(t,x) (i = 1, 2) such that the following conditions hold:
V($, x) = x'Gix Vx € Rn; (6)
2. there are two n-dimensional vector-valued functions «¿(t, x, V) (i =1, 2) such that following expressions
max{Wi(t, x, ui(t, x, V), u2, V)} = /dem(u2 ^ u2(t, x, V)},
U2 (7)
max{W2(t, x, ui, u2(t, x, V), V)} = /dem{ui ^ ui(t, x, V)}
ui
are valid for any t G [0, $), x G Rn, V G R2;
3. there are the continuously differentiable solutions Vj(t,x) (i = 1, 2) of the system of two partial differential equation
Wj[t,x, V] = Wi(t,x,ui(t,x, V),u2(t, x, V),V(t,x)) = 0, (i = 1, 2) (8)
with boundary conditions (6);
4. the inclusions Uf ^ uf(t, x) = «(t, x, Vi(t, x), V2(t,x)) G Aj (i = 1, 2) hold.
Then:
a) situation Berge-Vaisman equilibrium has a form: UB = (Uf, Uf );
b) the payoff of the players are Ji(UB, to,xo) = Vi(to, xo) (i = 1, 2) for the any initial position (to,xo) G [M) x
Now, we use this proposition. Firstly, the requirements (7) are valid, if
dWi(t,x,ui(t,x, V),u2, V)
d«2
dW2(t,X,Ui,U2(t,X, V), V)
dV1
= ^"dx1 - 2Di2U2(t, x, V) = °n,
ÔUi
From (9) we get
ui=ui(t,x,V )
dx
- 2D2iUi(t,x,V) = °n
d2Wi(t, x, ui(t, x, V),u2, V)
du2
d2W2(i,x,Mi,«2(t,x, V), V) 5«!
-2Di2 < 0,
-2D2i < 0.
U2(t,x,V) = |D12i ddxi =
«i(t,x,V ) =1 Diiii • Taking into account (10) and (8) from (5) we have
:iq)
dV
Wi[t,x,V ] = +
dVi
dx
A(t)x + -
1 / dVi
D
_i dV2
2 V dx y 2i dx _i dV2
+ £2f' D_idVi + U^ ' D_iD D
+ T V"dx) D 12 nx + H"dx) D_ DllD21
+ x'G ix = 0, V($, x) = x'Cx,
dx
+
dV
W2[t,x,V ] = ^ +
<9t
dx
4 \ dx
dx
e2
+ e>
dx
' D_ i dVi , e2 f ÔVi yn_i
2
dx
+ —
n D _2 D22D i2 nx
_ idVi
11)
+ x'G2x = 0, V2(#, x) = x'G2x.
We search the Lyapunov-Bellman functions Vi(t, x) (i = 1, 2) in the kind of the quadratic form
+
Vi(t, x) = x'ei(i)x, 8i(i) = 6i(t) (i =1, 2). (12)
Substituting (12) in (11) and taking into account = 2e^ (i = 1, 2), we obtain
the quadratic forms with respect to the components of the vector x. Equating to zero the coefficients of these quadratic forms, we get the system of two matrix equations
ei + ei(A + D2-11e2)+(A/ + e2D2-11 )ei + Gi+ +e2D2-i1DnD2-i1e2 + e2^i(eb e2) = 0raxra, ei($) = G1,
(13)
e 2 + e2A + A'e2 + G2 + e2 D-1e2+
+e2^(eb e2)=0„xn, e2($) = G2,
where by symbol ^i(e1, e2) the addends are denoted. They are quadratic with respect to the elements of the matrices e1 and e2.
Let's prove that the system (13) has the extendable to interval [0,$] solution (e1(t), e2(t)) for sufficiently small e.
Indeed, for e = 0 from (13) we get
^i + ei(A + D-11e2)+(A/ + e2D-11)ei + Gi+
(14)
+e2D-i1DiiD-i1e2 = 0nxn, ei($) = Gi,
e2 + e2a + A'e2 + G2 + e2D-i1e2 = 0„xn, e2($) = G2. (15)
Now we note that since G2 < 0, D21 > 0 and G2 < 0 then the matrix equation (15) has the unique extendable to interval [0,$] continuous solution e2(t) [28, p. 207]. Substituting given e2 = e2(t) in (14), we obtain the matrix system of the linear inhomogeneous equations which are relative to e1 with the continuous (with respect to t) coefficients. This equation also has the unique extendable to interval [0,$] continuous solution e1(t).Thus, for e = 0 the system (13) and hence the system (11) has the extendable to interval [0,$] solution (e1(t), e2(t)). According to the theorem about the continuous dependence of the solution on parameter [27, p. 8], it follows that for sufficiently small e the solution of the system (13) is defined on the same interval. Moreover, by Poincare's theorem about analyticity of the solution by parameter [27, p. 8] this solution can be represented in the form of the uniformly convergent on interval [0,$] series
e*(t,e) = e*(t) + £eke(k)(t) (i = 1,2). k=1
This fact gives the possibility to search the solution (13) in the form of a series in terms of powers of e. In conclusion, we notice that by (10) we obtain
uf (t, x, e) = uf (t, x, V2(t, x, e) = x'02(t, e)x) =
= D-162(t,e)x, Uf ^ uf (t,x,e);
uf (t, x, e) = uf (t, x, Vi(t, x, e) = x'O 1 (t, e)x) =
= eD-216i(t,e)x, U2f ^ uf (t,x,e).
With that the payoffs of the players for any initial positions (t0,x0) G [0,$] x Rn are Ji(Uf ,t0,x0) = x0©j(t,e)xo (i = 1, 2). This completes the proof.
□
Conclusion
We obtained coefficient criteria for the existence of the Berge-Vaisman equilibrium in a non-cooperative positional linear-quadratic game of two persons with small parameter.
References
1. BERGE, C. (1957) Théorie Générale des Jeux a n Personnes Games. Paris: Gauthier-Villar. BERGE, C. (1957) General game theory of several persons. Paris: Gauthier-Villar.
2. Вайсман, К. С. Равновесие по Бержу в одной дифференциальной игре / К. С. Вайсман // Сложные динамические системы: сборник научных трудов. — Псков: Псковский педагогический институт, 1994. — C. 58-63.
VAISMAN, K. (1994) Berge equilibrium in one differential game. Complex Dynamical Systems: collection of scientific articles . . p. 58-63.
3. VAISMAN, K. (1994) The Berge equilibrium for linear-quadratic differential game. Multiple criteria problems under uncertainty: Abstracts of the 3-d International Workshop. . p. 96.
4. Вайсман, К. С., Жуковский, В. И. Свойства равновесия по Бержу / К. С. Вайсман, В. И. Жуковский // Математические проблемы экологии: тез. докл. — Чита, 1994. — C. 27-28. VAISMAN, K. and ZHUKOVSKIY, V. (1994) Properties of Berge equilibrium . Mathematical problems of ecology: Abstr. of Intern. Workshop. . p. 27-28.
5. Вайсман, К. С. Структура равновесных по Бержу решений / К. С. Вайсман, В. И. Жуковский // Понтрягинские чтения-V: тез. докл. — Воронеж, 1994. — C. 29.
VAISMAN, K. and ZHUKOVSKIY, V. (1994) The structure of Berge equilibrium solution . Pontryagin reading-V: Abstr. of Intern. Workshop. . p. 29.
6. VAISMAN, K. and ZHUKOVSKIY, V. (1994) The Berge equilibrium under uncertainty. Multiple criteria problems under uncertainty: Abstracts of the 3-d International Workshop. . p. 97-98.
7. ZHUKOVSKIY, V., SALUKVADZE, M. and VAISMAN, K. (1994) The Berge equilibrium: preprint. Tbilisi: Institute of Control Systems.
8. Житомирский, Г. И. О равновесии по Бержу / Г. И. Житомирский, К. С. Вайсман // Сложные динамические системы: сборник научных трудов. — Псков: Псковский педагогический институт, 1994. — C. 52-57.
ZHITOMIRSKIY, G. and VAISMAN, K. (1994) About Berge equilibrium. Complex Dynamical Systems: Collection of scientific articles. . p. 52-57.
9. Вайсман, К. С. Равновесие по Бержу / К. С. Вайсман // Линейно-квадратичные дифференциальные игры / В. И. Жуковский, А. А. Чикрий. — Киев: Наукова думка, 1994. — C. 119-143.
VAISMAN, K. (1994) The Berge equilibrium. In the book Zhukovskiy, V. and Chikrii, A. Linear-quadratic differential games. . p. 119-143.
10. Вайсман, К. С. Существование гарантированного равновесия по Бержу в одной дифференциальной игре / К. С. Вайсман // Понтрягинские чтения-VI: тез. докл. — Воронеж, 1995. — C. 19.
VAISMAN, K. (1995) Existence of Berge equilibrium for one differential game . Pontryagin reading-VI: Abstr. of Intern. Workshop. . p. 19.
11. Вайсман, К. С. Об одном решении строго выпуклой бескоалиционной игры / К. С. Вайсман // Сложные управляемые системы: Межвуз. сб. науч. тр. — М.: РосЗИТЛП, 1996. — C. 13-16.
VAISMAN, K. (1996) About one solution for strictly convex differential game . Complex Dynamical Systems: Collection of scientific articles. . p. 13-16.
12. Вайсман, К. С. Равновесие по Бержу в дифференциально-разностной игре / К. С. Вайсман, Н. Ж. Аймуханов // Сложные управляемые системы: Межвуз. сб. науч. тр. — М.: РосЗИТЛП, 1996. — C. 90-93.
VAISMAN, K. and AYMUKHANOV, N. (1996) Berge equilibrium for the differential game . Complex Dynamical Systems: Collection of scientific articles . . p. 90-93.
13. VAISMAN, K. (1996) About differential game under uncertainty. Nonsmooth and Discontinuous Problems of Control and Optimization: Abstr. of Third Intern. Workshop. . p. 45-48.
14. Вайсман, К. С. Равновесие по Бержу: автореферат дис. ... канд. физ. мат. наук / К. С. Вайсман. — СПбГУ, 1995. — 15 c.
VAISMAN, K. (1995) The Berge equilibrium: abstract of diss. ... cand. of phys. math. sciences. SPbSU.
15. ZHUKOVSKIY, V. and VAISMAN, K. (1996) About one solution in non-cooperative game. Game Theory and Economics: Abstr. of N.N. Vorob'ev Memorial Conference. . p. 77.
16. ZHUKOVSKIY, V., MOLOSTVOV, V. and VAISMAN, K. (1997) Non-cooperative games under uncertainty. Game Theory and Application. 3. p. 189-222.
17. VAISMAN, K. (1997) Nash equilibriums routing and Networks. Game Theory and Application. 3. p. 147-160.
18. Zhukovskiy V. I., Vaisman К. S. To a problem about Berge equilibrium // Вестник Псковского Вольного ун-та. Математика и информатика. — Псков, 1997. — Вып. 1. — C. 49-70.
ZHUKOVSKIY, V. and VAISMAN, K. (1997) To a problem about Berge equilibrium. Mathematics and Informatics. Bulleten of Pscov Free University. 1. p. 49-70.
19. Вайсман, К. С. Арбитражная схема Нэша при неопределенности / К. С. Вайсман // Кооперативные игры при неопределенности и их приложения. — М.: Эдиториал URSS, 2010. — C. 231-249.
VAISMAN, K. (2010) Arbitration scheme Nash under uncertainty. In the book Zhukovskiy, V. Cooperative games under uncertainty and their applications. . p. 231-249.
20. SHUBIC, M. (1961) Review of C. Berge «General theory of n-person games». Econometrica. V. 29. (4). p. 1-821.
21. RADJEF, M. (1998) Sur l'existenced'un equilibre de Berge pour un jeu différentiel a n porsonnes..
Cahiers Mathématiques de l'Université d'Oran. 1. p. 89-93.
22. LARBANI, M. and LEBBAH, H. (1998) A concept of equilibrium for a game under uncertainty. Europ. J. Oper. Res.. 117. p. 145-156.
23. COLMAN, A., KORNER, T., MUSY, O. and TAZDAIT, T. (2011) Mutual support in games: some properties of Berge equilibria. Journal of Mathematical Psychology, Article in Press. . p. 1-10.
24. Гусейнов, А. А. Математические основы Золотого правила: альтруистский способ разрешения конфликтов в противоположность эгоистическому равновесию по Нэшу / А. А. Гусейнов, В. И. Жуковский, К. Н. Кудрявцев. — М.: USRR, 2016. — 256 c.
GUSEYNOV, A., ZHUKOVSKIY, V. and KUDRYAVTSEV, K. (2016) Mathematical foundations of the Golden Rule: the altruistic way of the conflict solution as opposed to the selfish Nash equilibrium. Moscow: USSR.
25. Жуковский, В. И. Линейно-квадратичные дифференциальные игры / В. И. Жуковский,
A. А. Чикрий. — Киев: Наукова думка, 1994. — 320 c.
ZHUKOVSKIY, V. and CHIKRII, A. (1994) Linear-quadratic differential games. Kiev: Naukova Dumka.
26. Жуковский, В. И. Кооперативные игры при неопределенности и их приложения /
B. И. Жуковский. — М.: Эдиториал USSR, 2010. — 336 c.
ZHUKOVSKIY, V. (2010) Cooperative games under uncertainty and their applications. Moscow: Editorial URSS.
27. Мищенко, Е. Ф., Розов, Н. Х. Дифференциальные уравнения с малым параметром и релаксационные колебания / Е. Ф. Мищенко, Н. Х. Розов. — М.: Наука, 1975. — 247 c. MISCHENKO, E. and ROZOV, N. (1975) Differential equation with small parameter and relaxation fluctuations. Moscow: Nauka.
28. Ли, Э. Б. Основы теории оптимального управления / Э. Б. Ли, Л. Маркус. — М.: Наука, 1972. — 547 c.
LEE, E. and MARCUS, L. (1972) Foundations of Optimal Control Theory. Moscow: Nauka.
