CC BY
86
Izv. Inst. Mat. Inform. Udmurt. Gos. Univ.
2018. Volume 52
MSC2010: 91A23, 91A13, 91A10 © Yu. Averboukh
STACKELBERG SOLUTION OF FIRST-ORDER MEAN FIELD GAME WITH A MAJOR PLAYER
The paper is concerned with the study of the large system of identical players interacting with the environment. We model the environment as a major (exogenous) player. The main assumption of our model is that the minor players influence on each other and on the major (exogenous) player only via certain averaging characteristics. Such models are called mean field games with a major player. It is assumed that the game is considered in the continuous time and the dynamics of major and minor players is given by ordinary differential equations. We study the Stackelberg solution with the major player playing as a leader, i.e., it is assumed that the major player announces his/her control. The main result of the paper is the existence of the Stackelberg solution in the mean field game with the major player in the class of relaxed open-loop strategies.
Keywords: mean field game, Stackelberg solution, game with infinitely many players.
DOI: 10.20537/2226-3594-2018-52-01
Introduction
The mean field game theory studies the behavior of the large number of small players by examination the limit case when the number of players tends to infinity. The main assumptions of the mean field game approach are that: (i) the players are identical; (ii) the interaction between the players is performed by certain averaging characteristics. The study of mean field game theory starts with seminal papers by Lasry and Lions [23,24] and Huang, Caines and Malhame [17,18].
There are several approaches in the mean field game theory. First one reduces the original game with infinitely many players to the initially-boundary value problem for the coupled system of PDEs. Within the framework of this approach the existence of solution is proved [23-25,27]. Moreover, based on this approach one can construct an approximate equilibrium in the game with the finite number of players [27].
The second approach is called probabilistic [10-12,20]. It regards a solution of mean field game as a Nash equilibrium in a infinite player game. This approach is fruitful in the analysis of the limit of open-loop Nash equilibria in finite player games [15,21]. Furthermore, it was used to study the deterministic limit of stochastic mean field games [3].
The third approach is concerned with so called master equation which is a partial differential equation in the space of probabilities [6,9,27]. This approach was used to study the link between feedback Nash equilibria in the finite player game and mean field game [8,19,22]. Notice that the existence results for the master equation are obtained for the non-degenerate stochastic mean field games [8]. Additionally, the short time existence was delivered [11].
The mean field game methodology can be used to analyze the interaction of the large group of identical small players with the exogenous player. This problem is called mean field games with a major player [13,14,16,26,28,29]. In this case we study the system of infinitely many identical minor players and one major players. It is assumed that the minor players can influence on each other and on the major player only via certain mean-filed characteristics.
There are several game designs suitable for the mean field game with a major player. First one assumes that the major and minor players choose their strategies simultaneously. It leads to the Nash solution for the mean field game with a major player. Another game design appears when we assume that the major player chooses his/her strategy first and announce it. This solution concept refers to the Stackelberg game with the minor players as followers and the major player as leader. Previously, the Stackelberg solution of the mean field game with a major player was studied only for the linear-quadratic games (see [4,5,30]).
In the paper we study the Stackelberg solution of the first order mean field game with a major player, i.e., we assume that the dynamics of the minor and the major players are given by ODEs.
The main result of the paper is the existence of the Stackelberg solution. The paper is organized as follows. First, we introduce the general notations. In Section 2 we define the mean field game with a major player and formulate the main result. In the last section we examine the properties of the motions and prove the main result.
§ 1. Preliminaries
Given a Polish space (X, pX), denote by P(X) the set of Borel probabilities on X. We endow P(X) with the narrow convergence. Further, let P1 (X) be the set of probabilities m € P(X) satisfying, for some (and, thus, any) x0 € X,
£(m) = / pX(x,x0)m(dx) < to.
Jx
We introduce on P 1(X) the 1-Wasserstein metric by the rule [2]: if m1,m2 € P1 (X), then W1(m1 , m2) = inf / p(x1, x2)n(d(x1 , x2))
nen(mi,m2) JXxX
sup
^eLipi(x)
/ 0(x)m1(dx) — / 0(x)m2(dx)
/X JX
Here n(m1 ,m2) is the set of plans between m1 and m2 i.e.
n(m1 ,m2) = {n € P1 (X x X) :
n(r x X) = m1 (X),n(X x r) = m2(X) for any measurable r C X};
Lip1(X) stands for the set of 1-Lipschitz continuous functions from X to R. Note that the convergence in W1 implies the narrow convergence [2].
If (Q 1, S1), (Q2, ^2) are measurable space, m is the probability on £1, h : Q1 ^ Q2 is measurable, then denote by h#m the probability on £2 defined by the rule: for r € £2,
(h#m)(r) 4 m(h-1(r)).
If (X, pX), (Y, pY) are Polish spaces, n is a finite Borel measure on X, then denote by A(X, n, Y) the set of measures on X x Y with the marginal on X equal to n. As above, we endow A(X, n, Y) with the narrow convergence. Notice that A(X, n, Y) is itself a Polish space. If ^ € A(X, n, Y), then ^(•|x) stands for the disintegration of ^ along n, i.e., for each x, ^(-|x) is a probability on Y, for any 0 € Cb(X x Y), the function
x ^ J 0(x, y)^(dy|x)
is measurable and
/ 0(x,y)^(d(x,y)) = / 0(x,y)^(dy|x)n(dx).
jX xX JxJy
The existence of the disintegration is proved in [7, Theorems 7.10.6].
Further, if (Z, pz) is a Polish space, ^ € A(X, n, Y), v € A(X, n, Z), then denote by ^ * v the measure on X x Y x Z defined by the rule: for any 0 € Cb(X x Y x Z),
/ 0(x,y,z)(^ * v)(d(x, y, z)) = / / / 0(x, y, z)^(dy|x)v(dz|x)n(dx).
./XxYxZ JX JX JY
The existence of ^ * v follows from [7, Theorem 10.6.6.]. Notice that ^ * v € A(X, n, Y x Z).
Below we consider Rd as a phase space for the control problem of each player. Further, we regard Rd+1 to be an extended phase space. We often denote the elements of Rd+1 as pairs w = (x,z), where x € Rd, z € R. If w = (x,z), then set p(w) 4 x, q(w) 4 z. For a fixed T > 0, let C stand for C([0,T], Rd+1). If t € [0, T], then denote by et, et the operators from C onto Rd and Rd+1 respectively defined by the rules: for w(-) = (x(-),z(-)),
et(w(-)) 4 x(t), et(w(-)) 4 w(t).
§ 2. Mean field game with a major player
We assume that the system consists of one major player and infinitely many minor player. Let the dynamics of the major player be given by
d
-x(t) = f(t,x(t),m(t),u(t)), (2.1)
whereas the dynamics of each minor player be given by
d
-y{t) = g{t, x{t),y{t), m(t),u(t),v(t)). (2.2)
Here t € [0, T]; x(t) € Rd, y(t) € Rd are states of major and minor players respectively; u(t) € U, v(t) € V stand for the controls of major and minor players, m(t) € P:(Md) is a distribution of minors player at a time t. The initial distribution of the minor players m0 and the initial state of the major player xo are assumed to be given.
It is assumed that the objective function of the major player is
f T
I(x^),mt),u() = £(x(T),m(T)) + / f 0(t, x(t), m(t),u(t))dt;
o
whereas the representative minor player wishes to maximize the outcome equal to
f t
J (xt),ytf,mt),u(t),v(t)) ^ Z (x(T),y(T), m(T)) + / g0 (t, x(t),y(t),m(t),u(t),v(t))dt.
Jo
As it was mentioned above, we are seeking for the Stackelberg solution with major player as the leader. This means that the minor players solve their mean field game for the given control of the major player, whereas the major player maximizes his/her outcome subject to the solution of the mean field game of the minor players. We impose the following conditions.
(C1) The sets U and V are metric compacts.
(C2) The functions f, f0, g, g0, £ and Z are continuous.
(C3) The functions f, f0 are Lipschitz continuous w.r.t. x and m.
(C4) The functions g, g0 are Lipschitz continuous w.r.t. x, y and m.
(C5) The functions f, f0, g, g0 are bounded.
(C6) The initial distribution of minor players m0 lies in P1 (Rd).
Now, let us introduce the extended dynamics. For t € [0,T], x,y € Rd+1, m € P 1(Rd), u € U, v € V, we put
f(t, x, m, u) = (f (t, p(x), m, u), f0(t, p(x), m, u)), g(t,x,y,m,u,v) = (g(t, p(x), p(y),m,u,v),g0(t, p(x), p(y),m,u,v)). The extended dynamics for the major player is
d
—x(t) = f(t, x(t),m(t),u(t)), ®(0) = (xo, 0);
whereas the dynamics of the representative minor player is
d
-y(t) = g(t, x(t),y(t), m(t),u(t),v(t)).
Here, m(t) is a distribution of the minor players in the original phase space at the time t. Note that in this case the major player wishes to maximize the outcome
e(x(T), m(T)) 4 e(p(x(T)), m(T)) + q(x(T)).
Whereas the representative minor player wishes to maximize
C(x(T), y(T), m(T)) 4 ((p(x(T)), p(y(T)), m(T)) + q(y(T)).
Obviously, the function f is Lipschitz continuous w.r.t. x and m, whereas g is Lipschitz continuous w.r.t. x, y and m. Moreover, these functions are bounded. This means that, for some non-negative K and C0,
||f (t, x, m, u) — f (t, x', m', u)|| < K||x — x'|| + KW1(m, m'),
||g(t, x, y , m, u, v) — g (t, x', y', m', u, v)|| < K||x — x'|| + K||y — y'|| + KW1(m, m'),
||f (t, x, m, u)||, ||g(t, x,y,m,u, v)| < Co. (2.3)
As it was mentioned in Introduction, we consider the Stackelberg solution of the mean field game in the class of relaxed open-loop strategies. Put U 4 A([0, T], A, U), V 4 A([0, T], A, V). Notice that U and V are the sets of relaxed controls of the major and minor players. Moreover, it is convenient to consider the distributions of motions instead of the flow of probabilities on the phase space. Below let x € P(C), ^ € U. Denote by x(-,x, the solution of the equation
x(t) = (xo, 0) + / f (t, x(t ), er #x, u)^(d(r, u)).
J[0,t]xU
If, additionally, y0 € Rd, x € C, v € V, then denote by y(-, y0, x, x, v) the solution of the equation y(t) = (yo, 0) + / g(T,x(T),y(T),er#x,u,v)(u * v)(d(T,u, v)).
i[0,t]xU x V
Denote the mapping that assigns to y0,x,x, ^ and v the motion y(-) by
Further, set A 4 A(Rd,m0,U). Elements of A are distributions of the minor players' controls. A distribution of controls generates the distribution on C by the following rule: if x € P 1(C), ^ € U, Y € A, x is equal to x(-, x, then 2(x, y) 4 $(•, x, x, ■)#Y.
Further, we say that x is a distribution of minor players generated by the distribution of the minor players' controls y and the major player's control ^ if
2(x,^,Y) = x.
Below we denote by y) the distribution of the minor players generated by ^ and y.
Proposition 2.1. Given a relaxed control of the major player, and a distribution of the controls of minor players y, there exists a unique distribution of the minor players y)-
This proposition is proved in Section 3.
Definition 2.1. We say that y is an equilibrium distribution of the minor players' controls corresponding to ^ if, for x* = x = x(-,x*,^), Y*-a.e. (y0, v) and any v' € V,
e (x(T, x*, ^),y (T, y0, x, x*, v), eT #x)
> e(x(T,x*,^),y(T,y0,x,x*,v'), eT#x).
Denote by E(u) the set of equilibrium distributions of the minor players' controls corresponding
to
Definition 2.2. We say that € U and y * € A provide a Stackelberg solution of the mean field game with a major player if
(a) y* €E(^);
(b) |(X(T, tt(^*,y)^*),eT,y*)) > supjl(X(T, U),eT#^*,y)) : » € V,y €E(^)}. The main result of the paper is the following.
Theorem 2.1. There exists at least one Stackelberg solution in the mean filed game with major player.
This statement is proved in Section 3.
§ 3. Properties of the motions
This section is concerned with the continuous dependence of the motions introduced in the previous section on its parameters.
Proposition 3.1. The dependence (x, ^ X(-, x, is continuous.
Proof. Assume that ^ ^ and Wi(xn, x) ^ 0 as n ^ to. We shall prove that ||X(, xn, ^n) —
0.
Denote Xn = X(-,xnX = X(-,x,^). Recall that K is a Lipschitz constant for / and g. For t € [0,T], we have that
|Xn (t) — x(t)|
'[Q,t]xU
/ (t, Xn (t ), er #xn, u)^„(d(r, u))
/(t, X(t ), er #x, u)^(d(T, u))
J[Q,t]xU
< / ||/(t,Xn(T),er #xn,u) — / (t, X(t ), er #x,u)||^n (d(T,u))
J[Q,i]xU
+
'[Q,t]xU
/ (t, X(t ), er #x, u)(^n (d(T, u) — ^(d(T, u)))
< K ||Xn(t) — X(T)||dT + KtWi(xn,x) Q
/ (t, X(t ), er #x, u)(^n (d(T, u) — ^(d(T, u)))
+
'[Q,t]xU
Since ^ we have that, for any 0 € [0, T],
'[O,0]xU
/ (t, X(t ), er #x, u)(^n(d(T, u) — ^(d(T, u)))
(3.1)
converges to 0 as n —>• 00. Let L be a natural number, I £ 1,L, tL = IT/L. We have that
n
rn = sup
'[0,tL]xU
/ (t, X(t ), er # x, u)(un(d(T, u) — ^(d(T, u)))
converges to 0 as n ^ to for any natural L. Further, we have that
'[0,t]xU
/ (t, X(t ), er # x, u)(^n (d(T, u) — ^(d(T, u)))
< rn + 2CqT/L.
Here C0 satisfies (2.3). This and (3.1) imply
\\xn(t) - x(t) || < K f \\xn(r) - x(t)\\dr + KtWi(xn, x)+ rt + 2C0T/L. 0
Using the Gronwall's inequality, we get
\\Xn(t) - x(t)\\ < eKT(KTWi(xn,x)+ rt + 2C0T/L). Since W1(xn,x) — 0 and rt — 0 as n — for any natural L, we conclude that
lim \\xn - x\\ < 2eKT C0T/L.
n—<x
Further, letting L — to, we get that
lim \\xn - x\\ = 0.
n—^^o
□
Proposition 3.2. The dependence (y0,x, x, P, v) — y(^,y0,x, x, P, v) is continuous.
This proposition is proved in the same way as the previous one.
Let Y be the set of probabilities x on C satisfying the following properties:
1. e0#x is concentrated on Rd x {0};
2. e0#x = m0;
3. x is concentrated on the set of C0-Lipschitz continuous functions from [0,T] to Rd+1.
Since m0 € P 1(C), we have that Y is a compact set in P1 (C) [2]. Additionally, notice that, for any x € P 1(C), p €U, y € A,
^(x,P,l) €Y. (3.2)
Proposition 3.3. The dependence (Y, U, A) 3 (x, p, y) — ^(x, P,Y) € Y is continuous.
Proof. Let W1(xn, x*) — 0, pn — P*, Yn — Y* as n — to. Set xtn = x(,xn ,Pn), x* = x(,x*,P*). By Proposition 3.1,
\\xn - x*\\ — 0 as n — to. (3.3)
Further, let gn € P 1(Rd x U x V) be such that, for any 0 € Cb(Rd xUxV),
/ 0(x,p,v) Qn (d(x,p,v))= 0(x,pn ,v)Yn(d(x,v)).
Analogously, define q* € P 1(Rd x U x V) by the rule:
/ 0(x,p,v)Q* (d(x,p,v))= 0(x,p*,v)Y*(d(x,v)).
J Rd xUxV JRdxV
The existence of the probabilities Qn and q* follows from [7, Theorem 7.10.6]. Notice that
Qn — Q* as n — to. (3.4)
We have that E(xn,Pn,Yn) = ^(•,0Cn,xn, •, •)#Qn, E(x*,p*,Y*) = $(,x*,x*, •, •)#Q*. Thus,
W1(~(xn,Pn,Yn), E(x*,p*,Y*)) < W1($(;Xn,xn, •, •)#Qn, $(,x*,x*, •, •)#Qn) 5)
+ W1($(• ,x*,x*r, • )#Qn, • ,x*,x*r, • )#Q*)-
Choosing the plan between $(-,Xn, xn, ■)#£n and $(-,X*,x*, ■)#£n equal to ,Xn, ■), $(-,X*, x*, ■))#we get
<
!x«xV
(3.6)
To estimate ||$(yo,x„,Xn,J",v) - $(yo,X*,x*,^, v)|| recall that ||$(yo,Xn,Xn,v) -$(yo,X*,x*,^, v)|| is equal to supie[o;T] ||yn(t,yo,Xn,Xn,^,v) - y(t, yo, X*, x*, v)||. Further,
||yn(t, yo, Xn, Xn, v) - y(t, yo, X*, x*, v)||
< K / ||yn(r,yo,Xn,xn,^,v) - y(r,yo,X*,x*,^,v)|dr + Kt||Xn - X*|| + KtWi(xn,x*)-
o
The Gronwall's inequality yields that supte[oT] ||yn(t, yo,Xn,xn,^, v)-y(t, yo, X*, x*, v)|| converges to zero uniformly w.r.t. yo, and v. Thus, by (3.6) we have that
Wi($(-,x„,Xn, ■)#$(-,X*,x*, ■)#£n) ^ 0 as n y to.
(3.7)
To prove that W1($(-, X*, x*, ■, 0#£n, $(•, X*, x*, ■, ■)#£*) ^ 0 as n ^ to, recall that converges to narrowly. Consider the compact set CR = {y € C : ||y (0)|| < R}. We have that the restriction of $(-,X*,x*, ■, ■)#on CR converges narrowly to the restriction of $(-,X*,x*, ■, ■)#on CR. Further, for any 0 € Cb(C),
'c\cr
0(y)xn(dy)
/C\Cr
0(y)x(dy)
< ||0|mo(Rd \ br),
where stands for the ball of radius R centered at the origin. Therefore,
we conclude that $(-,X*,x*, ■, ■)#£n converges narrowly to $(-,X*,x*, ■, ■)#£*. Since $(-,X*, x*, ■, ■)#£n, $(-,X*, x*, ■, ■)#£* are concentrated on Y and Y is compact, we conclude that Wi($(-,X*, x*, ■, ■)#£n, $(-,X*, x*, ■, ■)#£*) ^ 0 as n y to.
Hence, taking into account (3.5) and (3.7), we get the consequence of the proposition. □
Proof of Proposition 2.1. The existence of the ^(^,7) follows from the compactness of Y, (3.2), the continuity of the mapping P(C) 9 x ^ ^(x, ^,7) € Y for all ^ and 7 and the famous Schauder fixed-point theorem [1, Corollary 17.56].
To prove the uniqueness, let us consider, for 0 € [0,T], the space C0 = C([0,0],Rd+1). Notice that C = CT. Further, denote by r0 the natural projection from C on C0. Now assume that there exist two probabilities x1 and x2 such that x» = 2(xi,^,7). Let X» = X(-,^,xi). Notice that there exists a constant c1 such that
11X1 (t) - X2(t)|| < CiWi(rt#xi, r4#x2). Using the Lipschitz continuity of the function g and (3.8), we have that
, yo, Xi, xi, v )—y (t, yo, X2, x2, v)
(3.8)
< K
< K
Î ||y(T,yo,Xi,xi,^,v) - y(r,yo,X2,x2,^,v)|dr ■Jo
+ K / ||Xi(T) - X2(T)||dT + K / Wi(rT#xi, rT#x2) Jo ./o
/ ||y(T,yo,Xi,xi,^,v) - y(r,yo,X2,x2,^,v)|dr ■Jo
+ K (ci + 1^ Wi(rT #xi, rT #x2)dT. o
(3.9)
The Gronwall's inequality gives the following estimate, for some constant c2:
Wv(t,V0,Xl ,Xi,p,v) - y(t,yo,X2 ,X2,p,v)\\< C2 i Wi(rT#Xl, rT#X2)dr.
J 0
Therefore, we have that, for any 9 € [0,T],
sup \\y(t,yo ,xi ,Xi,p,v) - V(t,Vo ,%2 ,X2,p,v)\\< C2 f Wi(rT#xi, rT#X2)dr. ie[o,0] Jo
Integrating this inequality w.r.t. the probability y, we get the estimate
i e
Wi(rd#XI, rd#X2) < C2 Wi(rT#X1, rT#X2)dT.
Jo
Using the Gronwall's inequality once more time, we conclude that, for any 9 € [0,T],
Wi(re#Xi, rd#X2) < 0.
Since rT = id, we obtain the uniqueness. □
Proposition 3.4. For any p €U, the set E(p) is nonempty. Moreover, the mapping p ^ E (p) is upper semicontinuous.
Proof. Given y € A, let us construct the multifunction r[p, y] taking values in A by the following rule. First, set X — ^(p,Y). Further, let B[p,Y] be the set of pairs (w,v*) € ld x V such that, for any v € V,
Z(x(T, X, p),y(T, w, X, p, v*),eT#X) > Z(x(T, X, p),V(T, w, X, P, v), ex#X). (3.10)
Put r[p, y] be equal to the set of all distributions y' € A such that supp(Y') C B[p,y\. Notice that y € E(p) iff
Y € r[p, y].
Let us show that r is upper semicontinuous. Indeed, assume that pn ^ p*, Yn ^ Y*, Yn € r[pn,Yn], Yn ^ Y*. We shall prove that supp(Y*) C B[p*,Y*]. Pick (w*,v*) € supp(Y*). By [2, Proposition 5.1.8] we have that there exists (wn, vn) € supp(Yn) such that wn ^ w*, vn ^ v*. Since supp(Yn) C B[pn,Yn], we have that, for Xn — ^(pn,Yn), and any v € V,
((X(T, X^ pn),У(T, wn, Xm p^ vn), eT#Xn)
> Z(x(T, Xn, pn),V(T, wn, Xn, pn, v),ex#Xn).
Using the continuity of all functions and passing to the limit, we get that, for any v € V,
((x(T, X*, P*),y(T, w*,X*,P*,v*), eT#X*)
> Z(x(T, X*,pn),y(T, wn, X*,p*,v), ex#X*)-
This means that y* € r[p*, y*].
Further, let us show that r[p, y] is nonempty. To this end, given w € ld, pick
v(w) € AYgmax{((x(T,X,p),y(T,w,X,p,v),eT#X) ■ v € V}. (3.11)
Here we denote X — ^(p,Y). Without loss of generality one can choose the function v to be measurable. Define y' by the rule: for any 0 € Cb(ld x V),
/ 0(w, v)y'(d(w, v)) — / 0(w,v(w))mo(dw). ild+1xV J ld
Obviously, supp(Y') = {(w,v(w)) ■ w € supp(mo)}. Therefore, y' € A. Furthermore, (3.11) implies that inequality (3.10) is fulfilled for every (w,v*) € supp(Y') and any v. Hence, y' € r[p, y]. Additionally, the set r[p, y] is convex by the construction. Thus, by the Fan-Gliksberg theorem [1, Corollary 17.55], for each p €U, there exists a fixed point of the multifunction y ^ r[p, y]. Since E(p) = {y € A ■ y € r[p, y]} and the multifunction r is upper semicontinuous, we have that the E is also upper semicontinuous. □
Proof of Theorem 2.1. By Definition 2.2, we have that 7*) is the Stackelberg solution iff
(^*,7*) € Argmax{|(X(T, ^,7),^),eT#^,7)) : T €U,7 €E(u)}.
Notice that, by Proposition 3.4, the set
gr(E) 4 {(^,7): ^ €U,7 €E}
is compact. Additionally, by construction, we have the set U is also compact. Thus, using the continuity of the functions X and we get the existence result for the Stackelberg solution in the first-order mean filed game with a major player. □
Funding. The research was supported by the Russian Foundation for Basic Research (project no. 17-01-00069).
REFERENCES
1. Aliprantis C., Border K. Infinite dimensional analysis: a Hitchhiker's guide, Berlin: Springer, 2006.
2. Ambrosio L., Gigli N., Savare G. Gradient flows: in metric spaces and in the space of probability measures, Lectures in Mathematics, ETH Zurich, Basel: Birkhauser, 2005.
3. Averboukh Yu.V. Deterministic limit of mean field games associated with nonlinear Markov processes, Applied Mathematics & Optimization, 2018. DOI: 10.1007/s00245-018-9486-9
4. Bensoussan A., Chau M.H.M., Lai Y., Yam S.C.P. Linear-quadratic mean field Stackelberg games with state and control delays, SIAM Journal on Control and Optimization, 2017, vol. 55, no. 4, pp. 2748-2781. DOI: 10.1137/15M1052937
5. Bensoussan A., Chau M.H.M., Yam S.C.P. Mean field Stackelberg games: Aggregation of delayed instructions, SIAM Journal on Control and Optimization, 2015, vol. 53, no. 4, pp. 2237-2266.
DOI: 10.1137/140993399
6. Bensoussan A., Frehse J., Yam P. Mean field games and mean field type control theory, Springer Briefs in Mathematics, New York: Springer, 2013.
7. Bogachev V.I. Measure theory, Volume 2, Berlin: Springer, 2007.
8. Cardaliaguet P., Delarue F., Lasry J.-M., Lions P.-L. The master equation and the convergence problem in mean field games, 2015, arXiv: 1509.02505 [math.AP]. http://arxiv.org/pdf/1509.02505.pdf
9. Carmona R., Delarue F. The master equation for large population equilibriums, Stochastic Analysis and Applications 2014, Berlin: Springer, 2014, pp. 77-128. DOI: 10.1007/978-3-319-11292-3_4
10. Carmona R., Delarue F. Probabilistic theory of mean field games with applications I, New York: Springer, 2018. DOI: 10.1007/978-3-319-58920-6
11. Carmona R., Delarue F. Probabilistic theory of mean field games with applications II, New York: Springer, 2018. DOI: 10.1007/978-3-319-56436-4
12. Carmona R., Delarue F., Lachapelle A. Control of McKean-Vlasov dynamics versus mean field games, Mathematics and Financial Economics, 2013, vol. 7, no. 2, pp. 131-166. DOI: 10.1007/s11579-012-0089-y
13. Carmona R., Wang P. An alternative approach to mean field game with major and minor players, and applications to herders impacts, Applied Mathematics & Optimization, 2017, vol. 76, no. 1, pp. 5-27. DOI: 10.1007/s00245-017-9430-4
14. Carmona R., Zhu X. A probabilistic approach to mean field games with major and minor players, The Annals of Applied Probability, 2016, vol. 26, no. 3, pp. 1535-1580. DOI: 10.1214/15-AAP1125
15. Fischer M. On the connection between symmetric n-player games and mean field games, The Annals of Applied Probability, 2017, vol. 27, no. 2, pp. 757-810. DOI: 10.1214/16-aap1215
16. Huang J., Wang S., Wu Z. Mean field linear-quadratic-gaussian (LQG) games: major and minor players, 2014, arXiv: 1403.3999 [math.OC]. http://arxiv.org/pdf/1403.3999.pdf
17. Huang M., Caines P.E., Malhame R.P. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized e-Nash equilibria, IEEE Transactions on Automatic Control, 2007, vol. 52, issue 9, pp. 1560-1571. DOI: 10.1109/TAC.2007.904450
18. Caines P.E., Huang M., Malhame R.P. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information and Systems, 2006, vol. 6, no. 3, pp. 221-251. DOI: 10.4310/CIS.2006.v6.n3.a5
19. Kolokoltsov V.N., Troeva M. On the mean field games with common noise and the McKean-Vlasov SPDEs, 2015, arXiv: 1506.04594 [math.PR]. http://arxiv.org/pdf/1506.04594.pdf
20. Lacker D. Mean field games via controlled martingale problems: existence of Markovian equilibria, Stochastic Processes and their Applications, 2015, vol. 125, no. 7, pp. 2856-2894.
DOI: 10.1016/j.spa.2015.02.006
21. Lacker D. A general characterization of the mean field limit for stochastic differential games, Probability Theory and Related Fields, 2016, vol. 165, pp. 581-648. DOI: 10.1007/s00440-015-0641-9
22. Lacker D. On the convergence of closed-loop Nash equilibria to the mean field game limit, 2018, arXiv: 1808.02745 [math.PR]. http://arxiv.org/pdf/1808.02745.pdf
23. Lasry J.-M., Lions P.-L. Jeux a champ moyen. I — Le cas stationnaire, Comptes Rendus Mathematique, 2006, vol. 343, pp. 619-625. DOI: 10.1016/j.crma.2006.09.019
24. Lasry J.-M., Lions P.-L. Jeux a champ moyen. II — Horizon fini et controle optimal, Comptes Rendus Mathematique, 2006, vol. 343, pp. 679-684. DOI: 10.1016/j.crma.2006.09.018
25. Lasry J.-M., Lions P.-L. Mean field games, Japanese Journal of Mathematics, 2007, vol. 2, pp. 229-260. DOI: 10.1007/s11537-007-0657-8
26. Lasry J.-M., Lions P.-L. Mean-field games with a major player, Comptes Rendus Mathematique, 2018, vol. 356, pp. 886-890. DOI: 10.1016/j.crma.2018.06.001
27. Lions P.-L. College de France course on mean-field games, College de France, 2007-2011.
28. Nguyen S.L., Huang M. Mean field LQG games with a major player: Continuum parameters for minor players, 2011 50th IEEE Conference on Decision and Control and European Control Conference, IEEE, 2011. DOI: 10.1109/cdc.2011.6160790
29. Nguyen S.L., Huang M. Mean field LQG games with mass behavior responsive to a major player, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), IEEE, 2012, pp. 5772-5797.
DOI: 10.1109/CDC.2012.6425984
30. Shi J., Wang G., Xiong J. Stochastic linear quadratic Stackelberg differential game with overlapping information, 2010, arXiv:1804.07466 [math.OC]. http://arxiv.org/pdf/1804.07466.pdf
Received 15.10.2018
Averboukh Yurii, PhD, Senior Researcher, Department of Control Systems, Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia;
Associate Professor, Department of Applied Mathematics and Mechanics, Ural Federal University, ul. Tur-geneva, 4, Yekaterinburg, 620000, Russia. E-mail: [email protected]
Ю.В. Авербух
Решение по Штакельбергу игры среднего поля с ведущим игроком
Цитата: Известия Института математики и информатики Удмуртского государственного университета. 2018. Т. 52. С. 3-12.
Ключевые слова: игры среднего поля, решение по Штакельбергу, игры бесконечного числа лиц.
УДК: 519.837.4, 517.977.8
DOI: 10.20537/2226-3594-2018-52-01
Статья посвящена исследованию системы большого числа однотипных игроков, взаимодействующих с внешним окружением. Мы моделируем это окружение как ведущего (внешнего) игрока. Основное предположение нашей модели состоит в том, что малые игроки могут влиять друг на друга и на ведущего игрока лишь через те или иные усредненные характеристики. Подобные модели носят название игр среднего поля с ведущим игроком. Мы предполагаем, что время непрерывно и динамика ведущего и малых игроков описывается обыкновенными дифференциальными уравнениями. Мы рассматриваем решение по Штакельбергу с ведущим игроком-лидером, то есть предполагается, что ведущий игрок объявляет заранее свое управление малым игрокам. Основной результат статьи состоит в доказательстве существования решения по Штакельбергу игры среднего поля с ведущим игроком в классе обобщенных программных управлений.
Финансирование. Работа выполнена при финансовой поддержке РФФИ (грант № 17-01-00069).
Поступила в редакцию 15.10.2018
Авербух Юрий Владимирович, к. ф.-м.н., старший научный сотрудник, отдел управляемых систем, Институт математики и механики им. Н.Н. Красовского УрО РАН, 620990, Россия, г. Екатеринбург, ул. С. Ковалевской, 16;
доцент, кафедра прикладной математики и механики, Уральский федеральный университет, 620000, Россия, г. Екатеринбург, ул. Тургенева, 4. E-mail: [email protected]
