A class of non-antagonistic games

239 Theorem 2

SOhtiOn

The solutions v of the difference problem converge in any interior subdomain 52’to the Of the differential problem u at the rate 0 (WZ) in the net norm Of f+%,h (n’2h).

Theorems 1 and 2 are also true in the case where s2 is a rectangular parallelepiped; then in the proof of Theorem 1 the function t must be constructed in the form t=roth I#, and in the proof of Theorem 2 both sides of the equation must be multiplied scalarly by the function rot, (5” rot/, Y). Translated by J. Berry REFERENCES 1.

LADYZHENSKAYA, 0. A., Mathematicalproblems of the dynamics of a viscous incompressible fluid (Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti), Nauka, Moscow, 1970.

2.

LEBEDEV, V. I., Difference analogues of orthogonal decompositions, basic differential operators and some boundary value problems of mathematical physics. I. Zh. vj%hisl.Mat. mat. Fiz., 4,3,449-465, 1964.

3.

NIKOL’SKII, S. M., Approximation of functions of many variablesand imbedding theorems (Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya), Nauka, Moscow, 1969.

A CLASS OF NON-ANTAGONISTIC GAMES* D. A. MOLODTSOV

Moscow (Received 25 September 1973)

A GAME with a finite set of strategies of the second player is considered. The determination of the best guaranteed result for the fast player is reduced to a finite number of problems at the maximum. A bimatrix game is solved. We consider a game I’(Pi(s), Gi(s), X, Z),where Fi(z), G,(s) are the pay-off functions, and X and I={I, 2,. . . , n} are the sets of strategies of the first and second players. We will assume that Fi and Gi are continuous on the compact X of a metrical space. The best guaranteed result for the first player, making his move first and communicating it to the second player, is u, =

SUP

min Fi (3))

XEXiEI(X)

(11

where I(s) = {iEZIGi(x) = max Gj(r)}. FI Let w be a subset of the set I, R(o)=(x~Xll(s)=o}. Obviously, (1) can be written as follows: max

u =

0CI.

*Zh. vjkhisl. Mat. mat. Fiz., l&3,789-795,

1975.

R(o)#0

sup minFj(z). rsR(o)

j,a

(2)

24C

D. A. Molodtsov

If w is fmed and R(o) #0, then the calculation of programming problem

sup min Fj(s) reduces to the non-linear xrR(o)

jso

sup u *,u

(3)

with the constraints Fj(X)-U30,

j’0,

Gj(x)-Gh(x)>O,

k=Z\o,

j”~,

(4)

i, i”0.h

Gj(x)-Gi(x)=O,

If problem (3), (4) is solved, then (2) can be represented as the discrete programming problem

maxf(o), OEO

where a is the set of all the subsets I, and f (@I =

SUP minFj(s),

R(o)#fl,

if

x~R(o) ien if R(o)=0 -m,

if 0~0.

In principle problem (5) can be solved by a complete scan. However for sufficiently large n this method becomes practically unrealizable, since the number of elements of the set a may be very large. We show how this scan can be shortened. Let o E Q we denote by I w I the number of elements in theseto.Fork=l,2

,... ,nwe

write 1 X”=(x=XI

U(x) l
Let Xh be everywhere dense in X. Then

max

u= IolGk,

R(o)#0

SUP raR(o)

min

Fj(Z).

j,o

Let U = min

Fj(X*),

js w*

x’=Rm;

if z*=R(cI)*), thenx* is the best guaranteeing strategy, if x’zR(o’), lying sufficiently close to x* is an e-optimal guaranteeing strategy. hoof: Let xcX\xk

then any point x=R(o’)and

and e>O, 6=

max Gi(x) - max Gj(x) > 0. ier j=r\r(x)

Since Xk is everywhere dense in X, there exists a YEX~ such that IPi

Then

ICY

and

-Fi(y)

I
Vi=Z(x),

min Fj(y)> min F<(x)-e. if=(V) iEr(r)

IGj(x)-Gj(y)

IG6/3

v,i~:I.

241 Since 11(y) I
max IolSk.

R(o)ZB

ys~(o)

> min Fi (I) - E, is*(x)

j=m

consequently, sup min

max IolGk,

Since E>O is arbitrary,

R(m)#0 UER(W)jea

Fj(y)

2

U--E.

the first part of the theorem is proved. The proof of the second part of the

theorem is obvious. Definition 1. The functions fix) and g(x) are called locally coincident neighbourhood

of the point x0 exists in which these functions

at the point x0, if a

are identically

Definition 2. The functions f(x) and g(x) are called locally non-coincident if they are not locally coincident

equal. at the point x0,

at this point.

Theorem 2 If x==

then Xk is everywhere dense in X if and only if for every x=int X\XA there exist

i, j E I (z) such that Gj and Gi are locally non-coincident

at the point x.

ProoJ We first show that the condition of the theorem is sufficient for xk to be everywhere dense in X. It is obvious that for this it is sufficient to show that for any x=int X\Xk and any e>O there exists ayEP,where mGlZ(x) I-l such that p(z, y) -CE. I_et xEint X\Xk ande>O. If Iz(z) I =n, non-coincident I I(Y) I
then there exist i, j=Zsuch

that Gi and Gi are locally

at the point x, that is, there exists a y=X, p&z, y)
I. But if I I (5) I a,

such that min Gi(y)<

k,(T)

max Gj(y)

EI\I(X)

for any y such that p (x, y) (6. Let i, j =I (x) be such that Gi and Gi are locally non-coincident the point x. Accordingly, so chosen thatl(y)cZ(s).

at

there exists a Y, p (5, Y)
We now prove necessity. Let Xk be everywhere dense in X and x=int X\XA. We assume that Gi and Gj are locally coincident at the point x for any i, FEZ, then there exists a neighbourhood of the point x in which all the functions Gi, i= I (x) , are identically equal, and consequently, for any pointy of this neighbourhood dense in X.

II(y) Ia IZ(r) I ~=k+l. This contradicts

the fact that xk is everywhere

We notice that Theorem 1 makes it possible to shorten the scanning in the solution of the discrete problem to ~!!~c”’ (Cni is the number of combinations of n things i at a time). Finding (1) becomes especially skple in the case where X1 is everywhere dense in X. Then lL=

max

i
R(z)+0

sup Fi(Z), x=R(i)

D. A. Molodtsov

242

that is, it is necessary to solve not more than n mathematical programming problems of the form sup Fi (4 XEX with

the

constraints Gi(x)>Gj(x), j=Z, j+i.

We present some conditions sufficient to satisfy the condition of local non-coincidence of two functions at some point. Statement 1

Let f(x) and g(x) be differentiable in the direction I at the point x0, let x belong to the m-dimensional Euclidean space E,,, . Then if Of(x0) -#-,

e PO)

a1

a1

then f(x) and g(x) are locally non-coincident at the point x0. Statement 2

Let Ax) and g(x) be differentiable at the point x0, XC%,. Then if grad f (XO)+grad g (x0), then f(x) and g(x) are locally non-coincident at the point x0. Remark. If the first-order differentials of the functions f and g are equal at the point x0, we can formulate a similar condition with differentials of the second order, and so on. Statement 3

Letf(x)=Scz(t)dx, and g(s)=SB(t)dx,wherexisaprobabilitymeasuredefinedonthe u-algebra of subsets of the set T, a compact subset of E, , and or(t) and p(t) are continuous on T, Then if a(t) *p(t), then f andg are locally non-coincident at the point x for any x. Statement 4

The analytic functions Ax) and g(x),z~E,,,, are locally coincident at some point if and only if

f(s)=g(x).

The proof of these statements is trivial. We will apply these results to a bimatrix game. As shown in [l] , the first player may consider that the second player uses only pure strategies, from this the best guaranteed result of the first player, making the first move, is unchanged. Therefore, we consider the game J?(Fi (p), Gi (p), s”, I), where m m Fi(p)=

c

%Pj,

G(p)

=

j=i

btjpj,

c j=i

m

sm= p = (Pi,.

. . , Pm)

pj IC

[={I,

2,. . . ) n},

=

1, PjSO,

,==I

bi= (bii, . . ., bi,).

j=l,Z,.

. .,

m

,

243

Short communications

We renumber the strategies of the second player so that Z={l,

where IZjI>l, j=t+l,. for any k, i=Zj.

. . , s,

2,. . . , 4 It+,, . . . , I,},

andif bk=bi, then there exists a j, t+l
such that k, iEZj; bh=bi

In other words the subscripts for which the vectors bi are the same, are combined into the separate sets Ii. Since the strategies of the first player, belonging to any of the sets Ii, are equivalent, then he in fact makes a choice of the sets I’= {1,2,. . . , s}.

Then if we introduce the new functions: for i=l, 2, . . ., t, for k=t+l, . . . , s, for i=l, 2,. . . , t, for k=t+l,. . . , s,

Gi’(P) =Gi (p) G,*(p)=Gl(p),wherel=Zk,Fi*(P)=F*(P)

Fk* (p) = min Fj(p)

j= Ik

then the game l?(Fi*(p), Gi’(p), S”, Z’)d be equivalent to the game r(Fi(p), Gi(p), 5’“‘.I) in the sense of equality of the best guaranteed result and the optimal strategy of the first player. Let m G,*(p)=

z j=i

bij’pj,

bi’ = (bii.9 . . . ) him*),

t=Z’.

We GUI apply the results obtained to the game r(Fi*(p), Gi’(p),Sm, Z*)SimXgradGi*=b;*and all the bi* are different, then u = sup min F<(p) = pssm

max isI*,

iel(p)

where R(i)={p~S~lGi*(~)>Gj*(p),j~I’,

R(i)#0

max Fi* (p) , jER(i)

j+i}. Let

ai = max PEP

min k#i.

[Gi’ (p) - Gk*(p) ]

hsl’

and pi be a point realizing the maximum. Statement 5 The set R (i) ~0

Z+oof: Let R(i)+@

if and only if a+O. If R(i)+@,

then pi=R(i).

andpER(i),that is,

Gi’(p)-Gj’(p)>O,

j’I*,

ifi,

min [Gi*(p) - Gj’(p) I> 0; jfi, j=f

consequently, CC~~O. NOWlet ai>O, min [Gi’(p*)- Gk*(pi)]>

k#i,

0,

hEI*

that is,G,*(pi)>G,*(p’), k+i, k=I’or pair. We give another formulation of the criterion of non-emptiness of R(i):

244

D. A. Molodtsov

=

max

p.=sm

min

r

qEs8-f

LI

r

=

min

(bij*-bkjt)Pjqk=

U

j=l

kfi

k=t,

(bij*- bkj*) qk =

max

qESs-1

min

iCj4m k=l,

k-i,

k#i

k+i

If aiSO, then there exists a SKY*-’ such that

2

,zjyrn ( bij* -

k-i,

)

bki*4% s 0,

k#i

or II

bij’ -

c

k=i.

bkjt < 0,

j=l, 2t*..rm,

kfi

and this means that the vector bj* dominates the convex combination of the remaining vectors bk’, k+i, k=l’.

If

CZi>O,

then

,~I~,(bij* -

2 k=i,

bk;p,)

> 0

for any

qESS-‘,

kpi

and this means that the vector bi* does not dominate any combination of the remaining vectors. Therefore, we have proved the following: Statement 6

The set R (i) # thevectors bk*, k*i, Statement

if and only if the vector bi * does not dominate any convex combination of

0

k=l’.

7

If R(i)*@, then R(i) =(p=SllrIG*(p) Z+oof:

The

S,‘(p),

jd’}.

inclusion -

follows from the continuity of the functions Gj*, j=I’.

Giq(p’)

>Gj*(p’),

jEl*,

i+i.

245

Short communications

We take @=zq(l-q)Gj’(p)+qGj*(P’),

j~l*,

i#i.

Because of the linearity of the functions j=I’, jfi,

Gi’([l-qlp+qpi)‘Gj’([l-_qlp+qpi), this means that the point (I-q)p+qp+R(i)

and

(I-q)p+qp’-+p

as

q+O,that is, p~q$.

Therefore, we have obtained that u =

max

max{

iCiCt,

where for l
max

u1,

1+1CkCs,

ai>0

Qii, ak>O

ai>O m

max

Ui =

p=Sm

(6)

aijpj

IS

j=!

subject to the constraints m

y.

and for t+lGkGs,

(b<,‘-bkj*)pj

k=!.2,

3 0,

. . . ,s,

(7)

a,,>0 m vk =

max min WJj pESm lG,k c j=1

subject to the constraints m

E

(bk,'-b,j')pj

2 G,

r=l.?

,

I.

3=1

(8) subject to the constraints m

m Jl

u

(bkj’-brj*)pj

> 0,

(9)

T-1, 2,. . . , S, 1u j=i

j=i

The discovery of the c+ also reduces to a linear programming problem

subject to the constraints m

c j:=l

(b,j’-bkj*)pj

3 ~2,

l
k+i.

(11)

V. I. Prokopenko

246

Therefore, we have found that to find the best guaranteed result in a bimatrix game for a player not possessing information and making the first move, it is necessary to solve n problems of the form (IO), (11) and not more than n problems of the form (6), (7) or (8), (9). We will now show how to find an e-optimal strategy. Suppose that u =

max Fi’ (p) = Fi’ (p”),

then for sufficiently small positive values of 6 the strategies P= (I-G)p”+Gp*d?(i)

are e-optimal.

Translated by J. Berry REFERENCES 1.

GORELIK, V. A., The principle of the guaranteed result in non-antagonistic two-person games with exchange of information, in: Operationalresearch (baled. operatsii), No. 2,102-118, VTs Akad. Nauk SSSR, Moscow, 1971.

A GAME WITH A “SLUGGISH” PARTNER* V. 1. PROKOPENKO

Moscow (Received 26 November 1973)

A GAME of two “sides” with non-antagonistic interests is considered, where one of the sides, having accurate information about the strategies of the second, possesses the right to the first move, and the behaviour of the second side is characterized by a certain “sluggishness”. Let fl(X1, x2), fz(zl, x2) be real-valued functions, defmd and continuous on the compact XI XX,. We consider the following game of two sides.

The interests of the first side (A), choosing zi=Xi, are described by the pay-off function fj ($1, $2); the interests of the second side (B), choosing sz~Xz, are described by the pair [jz(zi, XZ), A], where the number A>0 measures the insensitiveness of the side B, so that this side necessarily prefers the situation (~~1,x2') to the situation (sf2, ~2) only if j~(x,‘, x2) -fz (51~~xz2)>A.But if O
*Zh. @chid. Mat. mat. Fiz., 15, 3,795-799, 1975.