Existence of Berge equilibrium in mixed strategies Текст научной статьи по специальности «Математика»

Ученые записки Таврического национального университета им. В. И. Вернадского

Серия «Физико-математические науки» Том 27 (66) № 1 (2014), с. 261-279.

УДК 519.833.2 MSC2000: 91A06

V. I. Zhukovskiy, S. N. Sachkov, L. V. Smirnova

EXISTENCE OF BERGE EQUILIBRIUM IN MIXED

STRATEGIES

We formalize a guaranteed solution notion for a non-cooperative game of n persons under uncertainty. This notion is based on the appropriate modification of maximin and the Berge-Vaisman equilibrium. We obtain existence conditions for the guaranteed solution in the class of mixed strategies (probability measures).1

Keywords: probability measure, mixed strategy, weak compactness in itself, guarantee, Berge-Vaisman equilibrium, Nash equilibrium, maximin.

E-mail: zhkvlad@yandex.ru

Introduction

The Berge equilibrium concept was introduced intuitively by French mathematician Claude Berge [1]. A brief review of Berge's book by Shubik [2] scared economists and contributed to its subsequent neglect in the English-speaking world. Particularly it was marked: "The arguments have been presented in a rather abstract manner and no attention has been paid to applications to economics. The book will be of a little direct interest to economists". The Berge's book was translated into Russian in 1961, and V.Zhukovskiy in 1994-1995 formalized the Berge equilibrium for linear-quadratic differential games under uncertainties [3], [4].

Note that Nash equilibrium is a common optimality concept for non-cooperative games. The key difference is that in case of Nash equilibrium an individual player's deviation from the equilibrium cannot increase the player's own payoff whereas; at the same time in case of Berge equilibrium a deviation by one or more players can reduce the payoff of a player, who does not deviate from an equilibrium situation. The Berge

1Работа выполнена при поддержке РФФИ, проект №14-01-90408 Укр_а.

equilibrium concept formalizes mutual support among players motivated by the altruistic social value orientation in such games.

Now turn to formal definitions. Consider a non-cooperative game of three persons

r = ({1, 2, 3}, (Xi}i=i;2,3, {/i(x)}i=l,2,3> , where Xi c Rli is a set of strategies xi of the i-th player, /i(x) is his payoff function, a

3

situation x = (x1,x2,x3) € X = n Xi.

i=1

K. Vaisman called a couple (xv, /v) € X x R3 a Berge equilibrium solution for the game r3 [5]-[7] if the following conditions

(10) a situation xv = (xV,xV,xV) satisfies a Berge equilibrium condition, i.e. /i(xV,x2,x3) < /i(xv) V x3 € X3 (j = 2, 3), /2(xi, xv, x3) < /2(xv) V xk € Xk (k = 1, 3), /3(xi,x2,xv) < /3(xv) V xr € Xr (r = 1, 2);

(20) a property of individual rationality holds for all players, i.e.

/1(xv) > max min /1(x),

xi X2 ,X3

/2(xv) > max min /2(x),

X2 xi ,X3

/3(xv) > max min /3(x)

X3 Xi ,X2

hold.

The "game"sense of the condition (20) is as follows. If the property of individual rationality (20) holds then every player provides himself a payoff /i(xv) (i = 1, 2, 3) which is at least not less than the i-th player's maximin. The condition (20) first was proposed by Zhukovskiy's doctoral student Konstantin Vaisman in 1994 [5]-[7]. He constructed some examples such that a situation satisfying the Berge equilibrium conditions (10) provides some players payoffs which are less than their maximins. In order to overcome this negative property of the Berge equilibrium Vaisman proposed to use additionally condition (20). Moreover, the following results were obtained by Vaisman:

— in some cases the Berge equilibrium exists, when there is no Nash equilibrium;

— in some games (The Prisoners' Dilemma, The Environmental Protection [8, p. 193]) if players simultaneously choose Berge equilibrium strategies, then everyone receives a larger payoff than if they chose Nash equilibrium strategies.

Konstantin Vaisman died suddenly at the age of 35 in 1998. He owned a remarkable trait: he had been helping everyone and forgetting himself. The authors of this paper think that Vaisman's researches of Berge equilibrium provide a basis to call the above mentioned solution (xv, /v) of the non-cooperative game r3 the Berge-Vaisman equilibrium.

Sufficient existence conditions of the Berge equilibrium were obtained by Zhukovskiy [9] in the form of existence conditions for a saddle point (x°, zv) € X xX of the Germayer convolution max(fi(x\\zi) — fi[z]), zi € Xi, z = (z1,..., zn) € X = n Xi. The ideas of

i i&N

this approach have been used in the current research. The aim of this paper is

— to formalize the Berge guaranteed solution for the non-cooperative game of n persons under uncertainty, when we have only the limits of variations of these uncertainties;

— to prove existence of the Berge guaranteed solution in the class of mixed strategies (probability measures).

1. Auxiliary data

1.1. Existence conditions for continuous selector. First introduce some facts from mathematical programming [10], [11].

Suppose that

(1) a set X c Rl (Rl is the Euclidean ¿-dimensional space) is a compact one;

(2) a set Y c Rm is a convex compact one;

(3) a scalar function F(x, y) is determined and continuous on X x Y, x € X and

y € Y;

(4) for any x € X the function F(x, y) is strictly convex in y € Y, i.e.

F(x, Xy(1 + (1 — X)y(2^>) < XF(x, y(1)) + (1 — X)F(x, y(2))

for all y(j) € Y (j = 1, 2) and any X = const € (0,1). Then there exists a continuous m-vector function y(x) : X ^ Y such that

min F(x,y) = F(x,y(x)) Vx € X. y&Y

1.2. Maximin in terms of hierarchical game. In game theory a maximin strategy xg and a maximin Fg are defined by the chain of equalities

maxmin F (x,y) = min F (xg ,y) = Fg. (1)

xGX y£Y V&Y

For the process of accepting a guaranteed solution (xg ,Fg) € X x R we can suggest the following interpretation in terms of bilevel hierarchical game [10]. Two players are participating in the game: Center and a player at a lower level on the hierarchy. Assume that Center forms his own strategy x € X and the player at the lower level constructs an uncertainty y(x) : X ^ Y, y(-) € C(X, Y) (see Fig. 1). The game proceeds as follows.

The first move is made by Center. He informs the lower level player of his possible strategies x € X.

Pic. 1

The following (second) move is transferred to the lower level player, who forms an uncertainty y(x) : X — Y such that for every x € X

min F (x,y) = F (x,y(x)) = F [x], (2)

y&Y

and informs Center about a specific type of the uncertainty y(x).

Finally (third move): Center forms a pair (xg, Fg) which is defined by the condition

max F (x,y(x)) = F (xg ,y(xg)) = Fg. (3)

x&X

Thus, Center can use the strategy xg. In this case Center provides himself the guarantee Fg whatever uncertainty y(x) € Y has been realized because Fg < F(xg,y) Vy € Y. Since Fg > F[x] Vx € X, the guarantee Fg is the largest of all guaranties F[x] . The given above "hierarchical"approach will be applied in Section 2.

1.3. Mathematical model of conflict. Assume that a mathematical model of conflict is represented by a non-cooperative game of N persons under uncertainty

rN = N, {Xi}ieN, YX, {fi(x, y)heN).

Here N = {1,... ,n} is a set of the agents (players) numbers. A strategy of the f-th player xi € Xi <z Rli (i € N). The players choose their strategies independently of each other in the game rN. Each i-th player formes and uses his own strategy xi € Xi (i € N). As a result we get a situation x = (x\,..., xn) € X = n Xi c R (l = li). A set of

i&N i&N

uncertainties y(x) : X — Y c Rm is denoted as YX. In the terminology of the theory of zero-sum games y(x) is a countersituation. We define a payoff function fi(x,y) of the i-th player on the sets (x,y(x)). The i-th player obtains the payoff fi(x,y(x)) which is

equal to the value of his payoff function in the concrete couple (x, y(x)). The aim of the i-th player is to choose a strategy Xi € Xi such that his payoff is rational according to his point of view. By choosing their strategies the players need to focus on possibility of realization of any uncertainty y(x) € YX.

Let us now turn to the notion of a guaranteed solution of the game rN.

2. Guaranteed solution of game rN

2.1. Definition. To formalize a solution of the game Tn we shall use the approach from Subsection 1.2. The only difference is that we replace formation of the interior minimum from (2) by formation of n minimums (for every i-th player)

fi[x] = min fi(x, y) = fi(x, y(i)(x)) Vi € N, x € X. (4)

yëY

Moreover we replace formation of the outer maximum from (3) by two following operations:

a) find a set Xv of all situations xv in the "game of guaranties"

r = N, {Xi}i&N, {fi[x] = fi((x,y(i)(x))}i&N ) such that the Berge equilibrium condition is satisfied, i.e.

max fi[x\\xV] = fi[xv] (i € N), (5)

xN\{i}^XN\{i}

where

[x \\xv ] - [x1 , ■ ■ ■ , xi-1, xv , xi+1, • • • , xn], xN\i - [x1] ■■■ 7 xi-1, xi+1, ■ ■ ■ , xn],

XN\{i} = H Xj;

j&N,j=i

b) find a Slater maximal situation xv € Xv in the n-criteria problem

(XV, {fi[x]}i&N) (6)

such that the system of strict inequalities

fi[x] >fi[xv] = fS (i € N) (7)

is inconsistent for any x € Xv.

Then the couple (xv, fs) € X x Rn is called a Berge strong guaranteed solution (equilibrium) in the game rN. Here the n-vector f = (f1t ■ ■ ■, fn) € Rn. We present a construction process of the Berge strong guaranteed solution in the game rN in Fig. 2.

Now introduce a formal definition.

Pic. 2

Definition 1. A couple (xv, fs) € X x Rn in the problem rN is called a Berge strong guaranteed solution (BVSGS) if there exist n continuous m-vector functions y(i\x) (i €

€ N) such that

min fi(x,y(x)) = fi(x,y(i(x)) = fi[x] Vx € X (i € N),

y(-)& X

and

10) there exists a situation xv € X in the non-cooperative "game of guarantees"

(N, {Xi}i&N, {fi[x]}i&N) (8)

such that the equality (5) is satisfied. The set of all situations xv is designated by Xv ;

20) the situation xv is Slater maximal for the n-criteria problem

(Xv, {fi[x]}iGN), i.e. for any x € Xv there exists an index j(x) = j € N such that

fj (x) < fj (xv) = ff.

This condition is equivalent to inconsistency of the system fi[x] > fi[xv] = ff (i € N) for any x € Xv. 30) the n-vector fS = (f1[xv],..., fn[xv]) = f,fS).

2.2. Sufficient conditions for the Berge equilibrium. We assign the Germayer convolution [12]

p(x, z) = max(fi[x\\zi] - f[z]) (9)

i&N

to the "game of guarantees"(8). Here

[x\\zi] = [xi,... ,x—i, Zi,xi+1,... ,x,n] € X = JJ Xi, Zi € Xi (i € N),

ieN

z = [zi,... ,zi,... ,zn] € X. A saddle point (x0,zv) of the scalar function p(x,z) is determined by the chain of inequalities

p(x, zv) < y(x0, zv) < y(x0, z) Vx, z € X. (10)

Taking into account (9) from the left inequality in (10) for zv = x0 we get

V(x°,x°) = miax(fi[x°\x0] - fi[x0]) = 0.

i&N

<p(x,zv) = max(fi[x\\zj] - fi[zv]) < 0 Vx € X.

i&N

Then (10) yields Hence for every i € N

fi[x\\z]] - fi[zv]) < 0 Vx € X.

Thus, we get for all x € X

fi[x\\zj] < fi[zv]) (i € N). (11)

Fulfillment of the conditions (11) for all x € X (i € N) means that the second component zv = xv € X of the saddle point (x°,zv) satisfies the Berge equilibrium condition (5).

Remark. Construction of the Berge equilibrium situation zv = xv € X (which is determined by (5)) is reduced to construction of the saddle point (x°,zv) € X2 of the scalar function (9). The second component zv € X of the saddle point (x°,zv) satisfies the Berge equilibrium condition (5).

2.3. Continuity of function p(x,z).

Proposition. Suppose that in the "game of guarantees"(8) the following conditions

1. the sets Xi are compact ones (i.e. closed and bounded) in Rli;

2. the payoff functions fi[x] are continuous on X (i € N)

take place. Then the scalar function ^(x, z) from (9) is continuous on X x X.

This proposition follows immediately from the well-known property [11]: suppose that the function ^(u, w) is continuous on U x W and the set W is compact; then the function

n(u) = max^(u,w) is continuous on U.

wew

3. Mixed strategies

3.1. Borel a-algebra. We consider the segment Y * = [y\,y2] C R. A collection 9 of subsets of Y = {y € R|yi < y < y2} is called a-algebra if it satisfies the following three properties:

1. [y1 ,y2] is an element of 9;

2. if T C [y1,y2] is an element of 9, then its complement [y1,y2] \ T is an element of 9 as well;

oo

3. if Tk C [y1,y2] (k = 1,2,...) are an elements of 9 then their union |J Tk is an

k=1

element of 9 too.

If every element of a-algebra 9(1) is an element of a-algebra 9(2 then one can say that a-algebra 9(2 contains a-algebra 9(1).

First, we consider any a-algebra 9 which contains all segments [a, /3] C [y1,y2]. One can prove that there exists a smallest a-algebra B(Y*) such that

1. it is an element of any other a-algebra;

2. all closed segments from [y1,y2] are the elements of B(Y*).

This a-algebra B(Y*) is called the Borel a-algebra. Elements of the Borel a-algebra B(Y*) are called Borel measurable sets. Thus for the segment [y1,y2] the Borel a-algebra is the smallest a-algebra over [y1,y2] containing all closed subsets of [y1,y2].

Second, for the set Y * = {y = (y 1,..., ym) | yi € [yt\yf2] (i = 1,...,m)} a-algebra is a collection of subsets of Y* such that

1. Y * is an element of

2. 9 is closed with respect to the complementation operation Y*\Yk for all Yk €9 (k = 1, 2,...);

oo

3. 9 is closed with respect to the operation of countable unions (J Yk.

k=l

The Borel a-algebra B(Y *) is the smallest a-algebra over Y * containing all closed subsets of Y*.

Third, we consider a set Y € Rm . Let Y be compact and hence bounded. Then there exist numbers y(l), y(2) (i = 1,... ,m) such that

Y C Y * = {y = (yi,...,ym) | y(l) < y < y(2) (i = 1,... ,m)}.

Let us construct B(Y*). Then

B(Y) = B(Y*) p| Y = {Ykf) Y I Yk € B(Y*)}.

The Borel a-algebra B(Xi), where the set Xi (i € N) of pure strategies xi of the i-th player is a compact set in Rli, is constructed in the same way.

3.2. Mixed strategies and situations in mixed strategies. Assume that in the class of pure strategies xi € Xi (i € N) there does not exist a situation xv satisfying the Berge equilibrium condition (5). Then one can follow the approach proposed by Borel, von Neumann and Nash. The approach is that the set Xi of pure strategies xi should be extended to the set of mixed strategies; then for the game (8)

N, {Xi}i&N, {fi[x]}iGN)

existence of situation satisfying the Berge equilibrium condition can be established on a class of mixed strategies.

For this construct the Borel a-algebra B(Xi) for every set Xi (i € N) and construct

the Borel a-algebra B(X) for the set of situations X = n Xi. Assume that B(X)

ieN

contains all Cartesian products of elements of the Borel a-algebras B(Xi) (i € N).

In game theory a mixed strategy vi(-) of the i-th player is a probability measure on the compact set Xi. Consider the definition from [8]. Assume that B(Xi) is a Borel a-algebra over a compact set Xi C Rli .A probability measure is a nonnegative scalar function vi(-) which is defined on B(Xi) and satisfies the following two conditions:

10) for every sequence {Q^^OO^ of mutually disjoint elements from B(Xj) the relation

vi(U Q^) = U Vi(Qk) kk

holds. We call this the property of countable additivity of the function vi(-); 20) the equality

Vi(Xi) = 1

takes place. We call this the property of normability.

Hence Vi(Q(i)) < 1 VQ(i) e B(Xi).

Denote the set of mixed strategies vi(-) of the i-th player by {vi} (i e N).

Let ¿(-) be Dirac function. Then a measure of the form 6(xi — x*)(dx) is also a mixed strategy from the set {vi} (i e N). Note that the measure-products v(dx) = = vl(dxl) ■... ■ vn(dxn) determined in [13], [14] are the probability measures on the set X of situations (in pure strategies). Denote the set of probability measures v(dx) by {v}. The measure v(dx) is called a situation in mixed strategies.

Note that for constructing the measure-product v(dx) we use the smallest a-algebra B(X) over X1 x ... x Xn = X such that B(X) contains all Cartesian products Q^x x ... x Q(n), where Q(i) e B(Xi) (i e N).

By [15], [16] the sets of all possible probability measures vi(dxi) (i e N) and v(dx) are weakly closed and weakly compact in itself sets. This means (for {v}) that from every infinite sequence {v(k"1} (k = 1,2,...) we can choose a subsequence {v(kj)} (j = 1,2,...) such that {v(kj)} weakly converges to a measure v^t) e {v}. In other words, for any scalar function ^(x) which is continuous on X, we have

Aki)(d.x) = I m(x)v°(

lim / p(x)v(kj\dx) = p(x)v°(dx) J J

X X

and ve {v}.

Since ^(x) is continuous, the integrals (mathematical expectations)/ p(x)v(dx) exist.

X

By Fubini's theorem we have

J p(x)v (dx) = j ...J p(x)vn(dxn)... vi(dxi),

X Xi X„

where the order of integration can be changed.

3.3. Mixed extension of the game (8). We put into correspondence to the "game of guarantees"in pure strategies (8) its mixed extension

{N, {vi}i&N, {fi[v] = j fi[x]v(dx)}i&N}. (12)

X

Here (as in (8)) N is a set of players' numbers, {vi} is a set of mixed strategies vi(■) of the i-th player (i e N). In the game (12) each i-th player chooses his own strategy vi(■) e {vi}. As a result the situation v(■) e {v} in mixed strategies is composed. Further we introduce the payoff function (mathematical expectation) fi[v] = / fi,[x]v(dx) of i-th

X

player on the set {v}.

For the game (12) the situation in mixed strategies vv() € {v} satisfies the Berge equilibrium condition if

max fi[v\\vj] = fi[vv] (i € N), (13)

vN\{i}t)£{vN\{i}}

where

VN\{i}(dxN\{i} = vi(dxi)... vi-i(dxi-i)vi+i(dxi+i)... vn(dxn), [v\\vj] = [vi(dxi)... v—i(dxi-i)vj(dxi)vi+i(dxi+i)... vn((lx,n)\, vv(dx) = v\(dx\)... vV(dxn).

For the game (12) the condition (13) determines the analogue of Berge equilibrium situation xv satisfying (5). Denote the set of situations in mixed strategies vv(-) € {v} satisfying (13) by {vv}.

3.4. Properties of situations in mixed strategies satisfying the Berge equilibrium condition.

3.4.1. Weak compactness in itself of the set {vv}. We establish the weak compactness in itself of the subset {vv} c {v}.

Assume that x] is an arbitrary continuous on X scalar function. Suppose that the elements v (k')() (k = 1,2,...) of the infinite sequence {v ^X-)}^x belong to the set {vv}. Then, since {vv} c {v}, it follows that {v(k)}']!°=l c {v}. As {v} is weakly compact in itself (see Subsection 3.2) there exists a subsequence {v(kj\-)}':j=l and a measure v°() € {v} such that

lim / i^[x]v(kj)(dx) = p[x]v°(dx).

J J

X X

Now we prove the validity of the inclusion v°(-) € {vv}. Assume the contrary. Then for a rather large j there exists a number i € N and a situation v(-) € {v} such that fiMv?] > fi[vkj]. This inequality contradicts the inclusion {v(kj)(-)}°=l c {vv}. Thus for the game (12) we obtained weak compactness in itself of the set of situations

{vv }.

Compactness of the set f[{vv}] = (J f[v] (n-vector f = (fl,...,fn)) in the criteria space Rn can be established in the same way.

3.4.2. Auxiliary property 1. Consider scalar functions (9) yi(x, z) = fi[x \\ z,] — fi[z] and p(x,z) = maxyi(x,z). According to Subsection 2.3 it follows that if f,[x] (i € N) are

i£N

continuous and the set X c Rl of situations x is compact, then y(x, z) is determined and continuous on X X.

We have pi(x,z) < p(x,z) = maxpi(x,z). Integrating the both parts of this

i£N

inequality by an arbitrary measure-product ¡i(dx)v(dz), where fi(-) € {v} and v(■) € {v}, we get

/ pi(x, z)^(dx)v(dz) < / maxpi(x, z)^(dx)v(dz) J J i&N

XxX XxX

for all i € N. Therefore

max I pi(x, z)^(dx)v(dz) < / maxpi(x,z)y(dx)v(dz). i€N J J ieN

XxX XxX

Taking into account the form of pi(x, z) from (9) we have

max / (fi[x || zi] - fi[z\)^(dx)v(dz) < / max(fi[x || zi] - fi[z\)^(dx)v(dz). (14) i&N J J i&N

Xx X Xx X

Remark. The inequality (14) is a generalization of the well-known property: maximum of sum do not exceed sum of maxima.

3.4.3. Auxiliary property 2. Now consider an auxiliary two-person zero-sum game

r2 = ({1, 2}, {Xi = X, X2 = X}, p(x, z)).

In the game r2 the set of strategies x of the first player is X1 = X, the set of strategies z of the second player is X2 = X. The payoff function p(x, z) of the first player is of the form (9). The aim of the first player is to choose his strategy x € X such that to get the largest possible value of the payoff function p(x,z). The aim of the second player is to choose a strategy z € X such that the function p(x, z) takes the least possible value. The solution of the game r2 is a saddle point (x°, z°) € X x X. It satisfies the relation

p(x,zv) < p(x0,zv) < p(x0,z)

for yx € X and ^z € X.

Now assign a mixed extension

r2 = ({1, 2}, {v}, {ri,p(v,^))

for the game r2.

Here {v} is the set of mixed strategies v(■) of the first player,= {v} is the set of mixed strategies !!,(■) of the second player, the payoff function (mathematical expectation) of the first player is p(v,^) = f pi(x,y)y(dx)v(dy).

Xx X

The solution of the game T2 is a saddle point (v0where (vis defined by the inequalities

p(v,Vv) < p(v0,vv) < p(vV) (15)

for all v(■) € {v} and ¡i(^) € {v}. This solution is called a saddle point in mixed strategies for the game r2.

Glicksberg proved in 1952 the theorem of existence of Nash equilibrium in mixed strategies for a non-cooperative game with n > 2 players [17]. For the special case of a non-cooperative game of n > 2 players, namely for a two-person zero-sum game r2, this theorem implies the following proposition.

Proposition. Suppose that the set X c Rl is compact and the payoff function of the first player y(x, z) is continuous on X x X in the game r2. Then there exists a solution (v°,jv) satisfying (15) in the game r2, i.e. there exists a saddle point in mixed strategies for the game r2.

Taking into account (9) we can present the inequalities (15) as

/ max(fi[x \\ z,] — fi[z])^(dx)vv(dz) < / max(fi[x \\ z,] — fi[z])^°(dx)vv(dz) < J i&N J i&N

X xX XxX

< f max(fi[x \\ z,] — fi[z])v°(dx)v(dz) (16)

Xx X

for all j() € {v} and v(■) € {v}. Setting v(dz) = (dx) the equality

v(j°,v) = max(fi[x \\ z,] — f,[z])j0(dx)v(dz)

J i^N

Xx X

implies

y(j°,v ) = 0. Hence, taking into account (16) we obtain

/ max(fi[x \\ z,] — fi[z])j(dx)vv(dz) < 0. J

XX

By (14) we get

max / (f,[x \\ z,] — f,[z])j(dx)v (dz) < 0, ■i€N J XX

max

' f,[x \\ Zi]J(dx)vv(dz) —J f^^dx^O < 0.

Xx X Xx X

Then

J fi[x \\ zl]j(dx)vv(dz) < J f,[z]j(dx)vv(dz) Vj(^) € {v}.

Xx X Xx X

Taking into account normalization of probability measure j(^) (see Subsection 3.2) we have f j(dx) = 1. Then the previous inequality implies

X

J f,[x \\ z,]j(dx)vv(dz) < J f,[z]j(dx)vv(dz) Vi € N.

X X X X

Using notations from (12), taking into account fi[i || vi] = f fi[x || zi]^(dx)vv(dz),

X xX

we get

fi[I || vV] < fi[uv] (i € N),

i.e. condition (13) holds.

Thus, if the sets Xi c Rli (i € N) are compact and the payoff functions fi[x] of every i-th player are continuous on X in the game (8), then there exists a situation in mixed strategies vv(-) € {v} satisfying Berge equilibrium condition (5) in the game (8).

4. Existence

4.1. The notion of strong guaranteed Berge equilibrium in mixed strategies.

In this section we present the main result of present paper. We establish existence of a strong guaranteed Berge equilibrium in mixed strategies for the game

rN = (N, {Xi}i&N, YX, {fi(x, y)}i&N).

For this we assign a quasimixed extension

rN = (N, {Vi}i&N, YX, {fi(v, y)}i&N)

to the game rN.

Recall that in rN the sets Xi c Rli (i € N) are compact ones. N = {1,... ,n} is the set of players' numbers in the game rN (as in rN). In rN every i-th player (i € N) can use both pure strategies xi € Xi c Rli (i € N) (as in rN) and mixed strategies (probability measures) v(■) determined on the Borel a-algebra B(Xi) over the compact set Xi c Rli (see subsection 3.2). YX is the set of uncertainties y(x) : X Y, Y c Rm. The payoff function of the i-th player is of the form

fi(v,y) = J fi(x,y)v(dx). (17)

X

Similarly to Subsection 2.1, we introduce the notion of a strong guaranteed equilibrium (vV, fs) € {v} x Rra (f = (fi,..., fn)) in mixed strategies for the game rN. For this we use three stages:

Stage 1. Taking into account the relation

min fi(x,y) = fi(x,y(i)(x)) = fi[x] Vx € X (i € N)

V&Y

we construct n vector-functions y(i\x) € YX.

Stage 2. For the non-cooperative "game of guarantees"of n persons

(N, {Vi}i&N, {fi[v }}i&N ),

(18)

where the payoff function of the i-th player is defined by the equality f,[v] = f f,[x]v(dx)

X

(i € N), we find a set {vv} c {v} of situations in mixed strategies vv() such that

max f,[v\\vj] = f,[vv] (i € N). (19)

Here the measure-product vN\{,)(dxN^^ = vi(dxl)... v^i^x^^v^i^x^i)... vn(dxn), i.e. vv(■) satisfies the Berge equilibrium condition in the game (18). Stage 3. For the n-criteria problem

{{vV}, {Mv]},&N)

construct a Slater maximal solution vv(■) € {v} such that for any vv(■) € {vv} the system of strict inequalities

Mv] >Mvv]= Vf (i € N)

is inconsistent.

Then the couple (vv, fs = (ff,..., f n)) is called a Berge strong guaranteed equilibrium in mixed .strategies for the game rN. The situation in mixed strategies Vv(■) is called a guaranteeing situation; n-vector fs is called a vector guarantee.

4.2. Proof of existence. Assume that the following conditions hold for the game rN:

(10) the sets X, c Rli (i € N) and Y c Mm are compact and Y is convex as well; (20) the payoff functions f,(x,y) (i € N) are continuous on X x Y (X = n X,);

(30) for any x € X the payoff functions f,(x,y) (i € N) are strictly convex in y € Y, i.e. for all A = const € (0,1) and y€ Y (j = 1,2) the following strict inequalities hold

fi(x, + (1 — A,)y(2)) < fx, yw) + (1 — A,)Mx, yW) (i € N).

We prove that fulfillment of conditions 10 — 30 provides existence of strong guaranteed equilibrium of Berge in mixed strategies in the game rN .In other words, we are going to prove that conditions 10 — 30 yield existence of a couple (v(■), fs) satisfying the requirements of Stages 1-3 from Subsection 4.1.

Stage 1. By Subsection 1.1 fulfillment of conditions 10 — 30 yields existence of a continuous m-vector function

y(,)(x) : min f,(x,y) = f,(x,y,(x)) = f,[x] Vx € X (i € N). (20)

V&Y

Note that the functions f,[x] = f,(x, y('l') (x)) (i € N) are continuous on X (as a superposition of continuous functions f,(x,y) and y^(x)). By (20) for every x € X we have

f,[x] < f,(x,y) Vy € Y(i € N). (21)

Integrating the both parts of the inequality (21) by an arbitrary measure v(-) € {v} we get

fi[v] = J fi[x]v(dx) < j fi(x,y)v(dx) = fi(v, y) Vy € Y (i € N). (22)

Due to the inequality (21) we can assign the "game of guarantees"

r = (n, {Xi}ieN, {fi[x]}ieN)

to the game rN.

Since for all y € Y we have

fi[x] < fi(x,y) (i € N),

then the vector guarantee f [x] = (f\[x],..., fn[x]) corresponds to each situation x € X in rg. In the same way, due to the inequality (22) in the "game of guarantees"in mixed strategies

r = (N, {Vi}i&N, {fi[v] = j fi[x]v(dx)}i&N)

X

the vector guarantee f [v] = (f\[v},...,fn[v}) corresponds to each situation in mixed strategies v(■) € {v}.

Since inequalities (21) and (22) hold for all i € N then it follows that the guarantees f[x] and f[v] are the "smallest". This is a main reason to use the term "strong guaranteed".

Stage 2. Since the sets Xi (i € N) are compact and the function fi[x] is continuous on

X = Xi (see conditions 10 — 20 and Stage 1), taking into account Subsection 3.4.3,

ieN

there exists a situation in mixed strategies vv(■) satisfying the requirement of Berge equilibrium (19) in the game rg. Therefore the set {vv} = 0. By Subsection 3.4.1 the set {vv} is weakly compact in itself, then the set of values of payoff functions

* = f [{vv}]= U f [vv] (heref [v] = fv],..., fn[v])) (23)

is a compact set in Rn (see the proof in Subsection 3.4.1).

Stage 3. Let ai = const > 0 and ^ ai > 0. Consider a linear convolution ^ aifi

i&N i&N

determined on $ (see (23)). Since ^ aifi is continuous on $ and taking into account

ieN

Weierstrass theorem we get that there exists a constant n-vector f s = (f (,..., fn)) € $

such that ma^ aifi = aif f. Due to Karlin's Lemma [19] the alternative fs is

fi€N ieN

maximal by Slater in the n-criteria problem

($, {fi}i&N),

i.e. for any f € $ the system of inequalities

fi > fis (i € N)

is inconsistent. Taking into account the construction way of the set $ (see (23)) one can state that there exists a situation Vs() € {v} such that fs = f[vs\. This situation in mixed strategies vs(■) is Slater maximal in n-criteria problem ({vv}, {f,[v]},,&N). According to the definition from Subsection 4.1 the couple ( v, f s) is a Berge strong guaranteed equilibrium in mixed strategies for the game rN. Thus, if for the game rN the following conditions

— the sets X, c Rli (i € N) and Y c Mm are compact;

— the set Y is convex;

— the scalar payoff functions f,(x,y) (i € N) are continuous on X x Y;

— for any x € X the payoff functions f,(x,y) (i € N) are strictly convex in y € Y

hold, then there exists a Berge strong guaranteed solution (BVSGS) in mixed strategies in the game rN.

Remark. The "game sense"of the notion of BVSGS is as follows. Every situation has a corresponding vector guarantee. Among these guarantees we have to select the ones which correspond to the Berge equilibrium situations. Then from such guarantees a Slater maximal guarantee (with respect to the selected guarantees) has to be chosen. The couple (the equilibrium situation and the corresponding vector guarantee) is offered as a "good"solution (BVSGS) for the game rN. In fact, whatever uncertainty is realized in the game rN, the players (using the situation from BVSGS) provide themselves "the largest"guaranteed payoffs. For each player this guaranteed payoff coincides with the corresponding component of the vector guarantee.

5. Conclusions

In this paper two new basic results of game theory have been established. These results concern Berge equilibrium (see the review [18]). First, for the non-cooperation game

r = N, {Xi}ieN, {fi[x]}ieN ),

where N = {1,... ,n} is the set of players' numbers, the set of strategies Xi of the i-th

player is Xi C Rli (i € N ), the situations x = (xi, ...,Xn) € X = n Xi C Rl, the

ieN

i-th player payoff function fi[x] is determined on X, the following proposition have been obtained.

Proposition 1. If Xi are compact and fi[x] are continuous on Xi (i € N) then in the game there exist the situations in mixed strategies VV(-) € {v} satisfying the Berge equilibrium condition (19) such that vv(-) is Slater maximal with respect to all other situations satisfying the Berge equilibrium condition.

Second, we considered a non-cooperation game of n persons under uncertainty

rN = (N, {Xi}ieN, YX, fx, y)h&N),

where N, xi, Xi, x, X are the same as in , YX is the set of uncertainties y(x) : X — — Y c Rm, on the set X xY the payoff function fi(x, y) of any i-th player is determined.

In subsection 4.1 for the game rN we have introduced the notion of Berge strong guaranteed equilibrium in mixed strategies.

Proposition 2. Let in the game rN the following conditions

— the sets Xi c Rli (i € N) and Y c Rm are compact;

— the set Y is convex;

— the scalar payoff functions fi(x,y) (i € N ) are continuous on X x Y ;

— for any x € X the payoff functions fi(x,y) (i € N) are strictly convex in y € Y hold, then there exists a Berge strong guaranteed solution in mixed strategies in the game rN.

References

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[3] Zhukovskiy V.I. and Chikrii A.A. Linear-Quadratic Differential Games // Naukova Dumka, Kiev, 1994 (in Russian).

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[7] Vaisman K.S. The Berge equilibrium for linear-quadratic differential game // Multiple criteria problems under uncertainty: Abstracts of the 2-d International Workshop, Orekhovo-Zuevo, Russia, 1994, pp. 27-28.

[8] Vorob'ev N.N. Game Theory for Economists-Cyberneticists // Nauka, Moscow, 1985 (in Russian).

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[10] Morozov V.V. Foundations of Game Theory // Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, 2002 (in Russian).

[11] Morozov V.V., Sukharev A.G. and Fiodorov V.V. Operations Research: Problems and Exercises // Vys'shaya shkola, Moscow, 1986 (in Russian).

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Loeve M. Probability Theory // Princeton, D. Van Nostrand Company Inc., London, New York, Toronto, 1955.

Dunford N. and Schwartz T. Linear Operators // Part 1, Interscience Publishers, New York and London, 1958.

Lusternic L.A. and Sobolev V.I. Elements of Functional Analysis // Nauka, Moscow, 1969 (in Russian).

Glicksberg I.I. A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points // Proc. Amer. Math. Soc., 3(1) (1952), 170-174. Colman A.M., Korner T.W., Musy O. and Tazdai't T. Mutual support in games: Some properties of Berge equilibria // Journal of Mathematical Psycology 55(2) (2011), 1-10. Karlin S. Mathematical Methods and Theory in Games, Programming and Economics // Pergamon Press, London-Paris, 1959.

Zhukovskiy V.I. Introduction to Differential Games under Uncertainty: Berge-Vaisman Equilibrium // PRASAND, Moscow 2010 (in Russian).

Существование равновесия по Бержу в смешанных стратегиях

Для бескоалиционной игры n лиц при неопределенности формализуется понятие гарантированного решения, основанного на подходящей модификации максимина и равновесия по Бержу. Получены условия существования гарантированного решения в классе смешанных стратегий (вероятностных мер).

Ключевые слова: вероятностные меры, смешанная стратегия, слабая компактность, гарантия, равновесие по Бержу-Вайсману, равновесие по Нэшу, максимин.

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