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5UDC: 519.977
MSC2010: 91A06, 91B50 DOI: https://doi.org/10.37279/1729-3901-2020-19-3-15-18
THE STABILITY OF COALITIONAL STRUCTURE IN DIFFERENTIAL LINEAR-QUADRATIC GAME OF FOUR PERSONS
© V. I. Zhukovskiy, S. P. Samsonov, V. E. Romanova
LoMONosov Moscow State University Faculty of Computational Mathematics and Cybernetics Department of Optimal Control Leninskiye Gory, GSP-1, Moscow, 119991, Russia e-mail: [email protected], [email protected], [email protected]
The stability of ooalitional structure in differential linear-quadratic game of four persons.
Zhukovskiy V. I., Samsonov S. P., RomanovaV. E.
Abstract. In article coefficient criteria of the stability of coalitional structure in differential linear-quadratic positional game of 4 persons are established. Following the approach adopted in the article, it is possible to obtain coefficient criteria of the stability of coalitional structures both in games with a large number of players and for other coalitional structures.
Keywords : coalitional games, threats and counterthreats, Pareto maximum,, stability of coalitional structure.
Introduction
In article coefficient criteria of the stability of coalitional structure in differential linear-quadratic positional game of 4 persons are established. Consider the coalitional differential four-player game:
, E, {Ui}i=1,2,3,4 , {Ji(U, t*, x*)}j=i,2,3,4^ • (!)
Here the controlled system E is described by the linear vector equation
4
X = A(t)x + ui, x(t*) = x*,
i=1
where x, ui G Rn, the moment $ = const 0 the game ends; starting position (t*,x*) G [0, $) x Rn, n x n-matrix A(t) elements are continuous on [0, define A(-) G Cnxn [0, $].
The game has coalitional structure P K2} formed by two coalitions K = {1,2} and K2 = {3,4}. Set of strategies of i player
U = {Ui - ui(i,x)|ui(i,x) = Qi (t)x, Qi(-) e Cnxn [0, tf]}
that is, the player's choice of his strategy Ui e U is actually comes down to the choice of a specific continuous on [0, $] of the matrix Qi(t), then in this case for the game
4
(1) is situation U = (U1,...,U4) e U = Ui, the strategies of the coalition K1
i=1
are UKl = (U1,U2) e UKl = U x U2, and strategies of the coalition K2 will be UK2 = (U3, U4) e UK2 = U3 xU4 (coalition is a subset of players united by the opportunity to have a joint choice of their strategy). The function of the payoff of i-player in the game is defined by the quadratic functional
" 4
Ji(U, t„ x) = x'(tf)Cix(tf) + ^ uj [t] DijUj [t] dt (i = 1, 2,3,4),
1 j=1
where D13 = £1D13, D24 = £2D24, D31 = £3D31, D42 = £4D42, (i = 1, 2, 3,4), n x n matrices Ci, Dij are supposed to be symmetric constants, £i > 0 is small parameter, the dash above means the transpositional operation.
The situation UP e U is Pareto-maximal in the game (1) if for any choice of the initial position (t*,x*) e [0, $) x Rn and all U e U the system of inequalities is inconsistent
Ji(U,t„x,) > Ji(UP,t„x,), (i = 1, 2, 3,4)
one of which is at least strict.
Next, M > 0(< 0) means that quadratic form z Mz is definitely positive (negative).
Lemma. Let there be constant ai > 0 (i = 1,..., 4), such that
4 4
C(a) = ^ ajC, < 0, A (a) = ^ ajDjt < 0 (j = 1, 2, 3,4). (2)
j=i j=i
It follows that Pareto-maximal situation exists for the game (1) and when £i > 0 (i = 1,..., 4) is sufficiently small it has the form
UP ^ -D-1(a)r(i)x, (i = 1, 2, 3,4), (3)
where
0*(t) = Y' C"1(a)+/[Y (tf, t )]D(a)Y ' (tf, r)dr\ Y (tf,t). (4)
The stability of coalitional structure in differential linear-quadratic game of 4 persons 17
4
We have D(a) = £ D"V), Y(tf,t) = X(tf)X"1(t) and X(t) is fundamental matrix of j=i
the system x = A(t)x.
The proof is similar [1, p.244-226].
Let us say that the coalition K threatens the coalition K2, tending to include the player 3 in Ki, if there exists the strategy U^ G and it is such that
Ja(U£, UP,Uf ,t„x„) > Ja(UP), jj(UT1 ,U3P,U4P,t„x„) > Jj(UP,t„x„) (j = 1, 2).
With such a threat the coalition Ki improves payoff of the player 3 through the use of its strategy U^ instead of U^ = (Up,Up) at the same time the coalition Ki doesn't «spoil» the gain of the players Ki compared with starting Jr(UP, ) (r = 1, 2, 3). Thus, the threat of Ki is as follows: when choosing U^ , the coalition Ki «lures away» the player 3 promising him to improve his payoff without spoiling its initial payoffs.
In response to such a threat the K2 has counterthreat, if there is such a strategy U4C G U4, that
Ji(UT1, Up,U4C,t„x„) > J(UT1 ,UaP,U4P(l = 3,4).
The counterthreat of the coalition K2 is as follows: the player 4 (who is not «lured away» by the coalition Ki) due to the choice of his strategy Uf7 G U4 will increase the payoff of the player 3 in comparison with the payoff that was formed under the threat. Such an «act» is beneficial to the player 4. Since in this case, his payoff will also increase.
The threat of the coalition Ki is defined similarly. The coalition Ki «breaks» the coalition K2 by luring the player 4 to itself. And in response to this threat there is a counterthreat of K2, that is carried out by the player 3 (due to choosing the strategy U3C G U3). In the same way, appropriate threats of K2 and counterthreats of Ki are identified.
Obviously, that if a coalition has counterthreat in response to any threat of any other coalition, so it makes no sense for players to change their strategies from the case UP.
Definition. The coalitional structure P = {Ki, K2} of the game (1) is called stable, if in response to any threat of any coalition, in which it tends to include any player of another coalition, this another coalition has a counterthreat.
«Таврический вестник информатики и математики», № 3 (48)' 2020
Statement. Let us suppose that
Dn > 0, D12 < 0, Di3£i > 0, D21 < 0, D22^2 > 0, D23 < 0, D3153 > 0, D32 < 0, D33 > 0, D41 < 0, D32e4 > 0, D43 < 0,
a)
D14 < 0 D24 > 0 D34 < 0 D44 > 0
b) there are such constants ai > 0 (i = 1, 2, 3,4) that the conditions (2) are fulfilled.
It follows that coalitional structure P of the game (1) is stable when £i is sufficiently small, and the corresponding situation UP is determined by the equalities (3), (4). The proof is based on the statements 4.3 and 4.4 from [2, p.46].
Remark 1. The proposed concept of stability of a coalitional structure is taken from general game theory [3], but for differential games it is used for the first time.
Remark 2. Following the approach adopted in the article, it is possible to obtain coefficient criteria of the stability of coalitional structures both in games with a large number of players and for other coalitional structures.
References
1. Жуковский, В. И, Чикрий, А. А. Линейно-квадратичные дифференциальные игры. — Киев: Наукова Думка, 1994. — 326 с.
ZHUKOVSKIY, V. I & CHIKRIY, A. A. (1994) Linear-quadratic differential games. Kiev: Naukova Dumka.
2. ZHUKOVSKIY, V. I & SALUKVADZE, M. E (1994) . The Vector-Valued Maximum. N. Y.: Academic Press. p. 404.
3. Оуэн, Г. Теория игр. — М.: Мир, 1971. — 230 с. OWEN, G. (1971) Game theory. Moscow: Mir.
