On a way of forming coalitions

ON A WAY OF FORCING COALITIONS* V. V.MOROZOV Moscow (Received

19701

5 October

THE desirability of combining players into coalitions, concept of obtainable vector, is discussed. Ill are considered.

formed by using the

The cooperative games defined in

1. Let r = {l , . . . , nl be the set of all players in a cooperative game r, xi the pure strategy of the 6th player, belonging to a compact subset Xi of finitedimensional Euclidean space, and Wi (xt, . . . , xnf the pay-off function of the i-th player, defined and continuous in the set

fiX&Y,X

*.. X X,.

i=l

The i-th player in the coalition K designates, in agreement with the other players of K, a number pi > 0 in the criterion (with this criterion, the functions Wi(xi, . . . , 5n) may be incommensurable, as distinct from the case e.g. of Von Neumann’s criterion 2 Wi (x1,. . ., ;z,), where it is assumed that Wi(xt, . . . , x,,) is the iEK

amount of infinitely divisible (see

czl, p.

commodity, freely transmitted between the players

34).

,,wf~,,...,%J-& i6K where

pi0 = max

*

Pi

min Wi(sl, X$X.’ 3c. 1=.x.2

, . . ,5,)

is the guaranteed pay-off of the

&-I

i-th player in pure strategies,

when he plays alone.

If the players of K have arranged about the numbers pi in the criterion selects the

fl.11, they will thereby have formed the coalition K. The coalition *Zh. uTchis2. Mat. mat. Fiz., 11, 3, 611422, 1971.

85

V. V. Morozov

86

n

st.ratw t K*E

in such a way as to maximize the criterion (1.1) as Xi; IEK far as possible, and since the criterion (1.1) includes the undetermined factors (see 131for the terminology)

it selects

x~* E XK* (Bi, I E K),

where

5I; 1max min min Wi (zK’ 5r\K) - pi” XK.xr\K SK Pi

=

= min min XI\K SK The players of K may arrange things incompletely: they may agree that the vector (pi, i E K) E E, where E is a compact set of k-dimensional Euclidean space. In this case the players of K either try to enter other coalitions, or they form a coalition K by agreeing on a criterion (1.1) in which (pi, i E K) E E are undetermined factors. The coalition K then selects a strategy x*~, maximizing the function

min (pi.IEK)EE

min min q\K

SK

Wi(~r,

* * * ,~n)-~fji” Pi

This concludes the definition of the cooperative

.

game I.

The significance for the players of the quantities pi and of the subsequent maximization of criterion (1.1) is then revealed by the following definition. Definition. The vector of numbers (d,, i E K) will be termed obtainable in the game I if a vector (pi, i-k K) exists, such that, whatever the strategy X1(’ E

X:;*

(fJir

i E K)

I3y negotiating about the Pi and maximizing the criterion (l.l), the players of K can guarantee a pay-off di to the i-th player. It seems natural to examine what vectors (di, i E K) are obtainable in the game I’. Theorem 1

The vector Proof.

(pZO,i E K)

is obtainable in the game I.

Let -113 be the max-min pure strategy of the i-th player when he plays

On a way of forming coalitions

alone.

We then have, for the strategy

xx0 =

Wi(5E’) - @i”2 0, -

(zio, i E K)

87

of the coalition K:

i E R;

hence, for all Bi > 0,

This means that, for all cSi > 0 and all xx* E Xl* (pi, i E K) w,(zR*) 3 fi?, i E K, which is what we wanted to prove. Let the vector (d,, i E K) be obtainable in the game I?, and let d,, < &,” for some i,. There is then no point in the i,-th player entering the volition K, since he can settee a payoff dil by playing alone. Throughout, therefore (except for Theorem 81, obtainable vectors id,, i E K) which satisfy the condition di > g,“, i E K. will be considered. For these vectors,

A vector (d,, i E K) is obtainable in the game r if and only if a strategy TK E ITXi exists, such that iEK

Put p: = di - pi”. Then

W&f+)

The proof of sufficiency

is obvious.

> 4, i

E

K.

Let W be the set of vectors guaranteeing pay-offs for the players of the coalition K when the coalition uses pure strategies. Then, by Theorem 2, the

88

V. V. Morozov

vectors of W are obtainable in r. Let ui be the maximum guaranteed pay-off of the i-th player in mixed strategies, when he plays alone. If the vector (ui, i :=: K) is obtainable, this is an argument in favour of forming the coalition K. For, the i-th player can obtain a pay-off vi by playing alone and using a mixed strategy. This entails a certain risk, since the player replaces the original function Wi (x1, . . . , xn) by its mathematical expectation. In the coalition K, the player can obtain a pay-off ui without any risk, if the vector (ui, i E K) is obtainable.

Further, let G: be the pay-off of the i-th player in some equilibrium situation in mixed strategies in an n-person non-cooperative game r, in which the pay-off function and the set of players’ strategies are the same as in the game I?. If K = I, and for all equilibrium situations in mixed strategies in the game T the corresponding vector (ii, i E 1) is obtainable in r, then, in view of the advantage of pure over mixed strategies

indicated

alternative:

K, to the alternative:

form a coalition

play in the non-cooperative Some sufficient f ui, i EC IO and (ii, n-person

2.

conditions

Define

should

prefer the

no one combines,

will next be found,

i E If are obtainable an element

and all

these

under which the vectors

in cooperative represent

games.

Cooperative

a generalization

of the

of repetition.

Let s &l be a positive

functions

the players

game p.

games will be considered;

game l7 and contain

above,

integer.

r” as the game r in which the strategies Wi (x1, . . . , nn) are replaced

respectively

xi E

Xi and the pay-off

by

I q(s)=

(.ri’,

* ..,XiS)&yiS

=

n

&f,

j=l

. . , ~~j) are the functions TVi(,r,, . . . , are replaced by nj, . . ., x j. YL, . . . , x n n

Here, Xii = Xi, j = 1, . . . , s and G) t in which the variables Notice

WZ(q3,.

that I”= I-‘.

The maximum guaranteed

pay-off

of the i-th player

in mixed strategies

in

On

the game I“

is ai, if he plays

of /-orming coalitions

a wa.v

RI)

alone.

The pay-off of the i-th player

in the game rs

q(s)

=

i

is

Q s-1,

j=l

situation where Ir,?, . . . , V,,j are the pay-offs of the players in an equilibrium in miscd strategies in the game r for all j. In r”, the vector (Ki (a), i E 1) is obtainable,

if all the vectors

Notice

that,

if a vector

(Vi’, i E I), j = is obtainable

1,. . . , s

are obtainable.

in I“, then it is obtainable

in Cs for

all s ‘b 1. By Theorem

2, the vector (d;, i E K) is obtainable

only if a coalition

strategy

ItK(s)

=

(,X1(s), i E K)

rnin )I;, (.cli (s), x,_ -iVi (.rIi (s)) = ‘r\h (5) Sufficient

conditions

that a given vector Let’; functions Let

will first be obtained

{(

i E K) 1(UT,, i E.K)

w,,

(s)) > d,,

if and

such that

iEK.

(2.1)

s, 3 1 to exist,

such

in the game r”o.

hull of the set W, defined

W, (x,, . . ., sn) are continuous,

R =

exists,

for an integer

(di, i E, K) is obtainable

be the convex

in the game rs

above.

Since the pay-off

the set W, and hence W, is compact. E r

and let there be no vector

(WI’, i E K) E W such that wI’ > wI, i E K}. In the case set R is the “north-east” part of the boundary of W.

K : iI, 2I, the

Theorem 3 Let the vector (di, i E K) E W\,R. such that the vector (d,, i E K) is obtainable Proof.

Obviously,

‘? is the set of vectors

exist.

Then

an integer

s, >,l esists,

in the game ITso. (u.~, i E K) such that

wi

where P (x,, 1 E K) is the distribution By hypothesis,

there

function

is a distribution

in

function

n_X,. 1E.X P* (x1, 1 E A) such that

v. v.

90

Introduce the random quantities (x/,

P’($,

Wi(LCt, I E K), S

EE K), having

and by P (A, s, the probability

lim P(Ai‘)=l, S400 For the events AT, and A:

‘2

P(Ai,“) > t

of the event

iEK.

(2.2)

(Ails

U Ai,“)

=

P

(Ail”

n AI,*).

P(A<,“) > t,

and

2t Applying

function

of the random

we have

P (Ai,‘) + P (Ai,“) - P If

the distribution

Denote by 3, the dispersion

2 E K), j = 1, . . . , s.

quantities

Morozov

1 < P (Ai,” fl A,,‘) s

(2.3)

(2.3) k - 1 times,

kt - (k - I> < P UI_ A:) Hence,

if t ‘> (k - 1)/k then

P(fl

A:)

> 0.

From (2.9,

an so exists

such

SK

that

rc- 1

P(Ai”)>t>T,

iEK;

We thus find from (2.4) that a strategy wi(x~(So)) and the proof follows Notes.

1.

Using

p ( JIK 4”?

xK (s,) can be found such that >

di,

i E K,

from (2.1). Chebyshev’s

inequality,

it can be seen that, for all

s 21, P (A;) > lHence,

> 0.

if s, is such that

Di

s (pi - di)” ’

i E K.

t2.4)

On a way of forming coalitions

(2.4) is satisfied

91

and the vector (di, i E K) is obtainable in the game I’%

2. If the vectors (ui7 i E K) and (ci, i*E I) satisfy the condition stated in Theorem 3, they are obtainable in rSo for some s,. But a stronger result also follows from Theorem 3: up to an arbitrary E > 0, given a sufficiently large s,, any vector is obtainable in the game r 80 which results from the players of the coalition using mixed strategies. ~x~~~~~ 1. Let max-min mixed strategies players, such that

(Notice that, if K = f = 11, 2, I, a non-~tagonisti~ condition (2.5) would be meaningless.) Then, the vector ( ui, i E Zi) E W \ R vector (ui, i E K) is obtainable in TSo.

F*,, 1 E K, be available to the

game is in question; otherwise,

and so 3 1 exists such that the

3. Sufficient conditions will now be found under which a vector is obtainable in a game l7”, where the number s is fixed. Some of the results will be given for two-person games (K = I = II, ZP, because their extension to n-person games is either la~rious or ~~0~ to the author. For simplicity, the examples will also refer to two-person games. It will be assumed that, for all i, the set Xi is the interval ‘0, 11 on the real axis. First take the game I” (s = If. For the two-person game Pi, let M = max W, (x1, z,), a%=¶

m=minW,(2,,32). xrs”

Consider the following function in tm, MI:

92

V. V. Morozov

If the function are obtainable Proof. second

in [m,Ml, the vectors

c$(w,) is concave

in the two-person

Let F* and G* be the max-min

players

(Vi’? v,‘) =

respectively.

Then,

i :,~(v&IQ~

(ul, UJ and (Cl, c-2)

game I’. mixed strategies

of the first and

the vector

dG’(s,),

i;i W, (xi, 4

dF’ (51) dG’ (z,) Jo%

ij and

vi’ 2

vi, vz’ 2

only needs Let

vz. Furthermore,

to be shown that any vector (wl, wz) E W.

obviously, of?

Thennumbers

(Vi, CZ) E w.

is obtainable hj>O,

Hence

it

in I’.

i=i,...Z,

xhj=i j=l

and vectors

(u$,

E W,

wi)

2

j =

1, . . . , I,

exist,

such that

i=f,2-

hjWi=Wi,

j=1 Moreover, w, =

ihjw,l< j=l

(w,, cp(w,))

The vector

from (3.1), the vector

(3.4)

~~~~j~('D,'~S~(~~iw,')=~6,). j=l

E

W

by definition

(wl, w,) is obtainable

of the function

r$(w,).

Hence,

in I’.

Corollary 1

If Wz(G, 52) = f(Wi(G, 41, concave

in Tm,M!,then the vectors

the two-person Proof.

(u,, u,) and

(Vi, V,)

are obtainable

game I’.

c$(w,) = f(w,) for all zu, E

If the set W is convex, in the two-person game I’. It is easily

f(w,)is defined

where the function

the vectors

shown that the function

'm.Ml.

(ul, u,) and

(ci, F~)

q5(10,) is concave

are obtainable

in [m,Ml.

and in

On a way of forming coalitions

Let the function IV =

{(w,,

are obtainable

in the two-person ~(uI,)

the conditions

are obtainable

If

in which the players of Corollary

-W,

(u,, u,) is obtairtable

Proof.

Let the vector

If TIV,(J1, J?) =

and let &)

in !m, Ml.

have identical

(Wi(z,,

interests

q)

2, so that the vectors

interests

= -Wz(xi,

(W, (xl, x2))

(u,, u,) and (E,, V.,)

I”.

3, the vectors

Wi(cci, x2) =

shown to be concave

and opposed

in such games

and by Corollary person game P.

vector

to its variables, (u,, u,) and ($,

game I’.

is easily

Games

2.

&) = Wz(&, zz) ) satisfy

@(w,, to’,) be convex with respect Then the vectors w2) =O}.

wz) fQ(w,,

The function Example

93

( Ui, II?)

(uz, u,) and

(x2, x1)

the

game I”.

W,(Z,,

(.F,, 2%) be such that

and

in the two-

Wz(xi, x2) = Wz(x2, xi),

and

in the two-person

-117,(X.a, .T,)

are obtainable

5%) = UZ.

TVz(xl, 2) = /(I[

(x, Q’-j-

function, defined in the interval to, 1 %I, (rr, -q)“]), where f is a continuous while the vector ([, $ is such that either ‘is < E < 1, I/:: SZ rl < 1, or 0 4 ; G I/?, 0 < q < ‘It, then the vector (z:,, u,) is obtamat>le in the twoperson

game I’“.

The proof will be given for the case similar

in the case

0 <

g <

‘12, 0 <

‘/z <

E<

n & r/z .

‘i, ‘/? <

‘q <

Take a vector

1.

It is

(0, T’,) such

v. v.

94

W,(O, &)

that

2

uz.

Morozov

Then, obviously,

‘h G P((E, rl),

(0,22)) <

P((& rl), (0, O)),

for arbitrary where o ((r,‘, G’), (z,, a) ) = V[ (51’ - 21)’ -I- (~2’ - XZ)“] Hence the circle, centre vectors (x’~, XI,) and (x1, zz) E [0,1] x [0, I]. (4, 11)and radius p ((& q), (0, &)), in which W2(x1, x,) is con$.a$, cuts the diagonal [(O, O), (1, 111 of the square 10, 11 x LO, 13 at a point (G, ZZ). Hence W,(&, &) = W,(O, 6)

> uz,

w, (ii, $2) = 0 =

Ul,

and the proof follows from (2.1). The next propositions Theorem

refer to n-person games.

6’

If, the function Wifzl, . . . , ziy . . . , 5,) i = 1, . . ., n - 1, is concave with respect to xi, the vectors (ui, i E I) and (ui, i E K), where K s I\ in), are obtainable in the game r’. Proof.

A point < E CO,11 exists,

such that

Wj(zi, . . . z+.~, 55, ZA+~, with i = 1, . .., n - 1 (see Xl, . . . , zic2, xi+,, . . . , t, . . . ) xn) 3 vi for all e.g. [41, p. 485). Hence the vector (vi, i E K), where K E I\Inl, is obtainable. Also, max W, (f,, . . ., ?;i, . . ., p12_-1,5,) 2 27,. % Take ?, such that TV,(fl,. . . , ii!,) 3 u,. Then TYi(fiy . . . ,Z,) 3 ui, i=l , . . . , n, and from (2.1), the vector (vi, i cs 1) is obtainable in the game r’. Theorem

7

If an equilibrium situation exists in pure strategies vector (ui, i E I) is obtainable in r’.

in the game I”, the

The proof is obvious and may be omitted. Now consider games r”, where s >, 2. Theorem

8

Let the players’ pay-off functions be specified as follows:

of forming coalitions

On a way

W,h ,, . . . , .yn) is an arbitrary

continuous

W?(&, . . . , 5,) = Ct’Wi(xi,

xi+i,

95

function, - . -

9

5717 Xi,

e

-

m f

Xi-I) + Cz(,

(3.2)

i = 2, . . . , n, where

czi, i = 2, . . . ,I?,

.cli,

Under these

conditions,

are constants

of players

The proof is given

11 X [0, 11

x2) =

Wi(z2,

oj,

x2) >

-l/4

2, . . . , n.

in the game Pern

{(G,

G) 1 (z,,

u, :, 0 and IJ, b 0.

x2) E [0,

< X, - x, < 11, while W, (x1,

IO, 11 x lo, 11. Define

VJ’~(X~~CZ-.?)=0 (W,(.r,,

vector (x1, x,) such that W’(z,, mean that (ul, u,) is unobtainable

S =

< x, - x, < 0, or l/2

of the square Obviously,

then

i =

in the Appendix.

and either

x1).

0,

i, for whom cIi = 0.

Example 5. Let W,(x,, x,) > 0 in the set x,) = 0 at other points

cli >

(ui, i E I) is obtainable

the vector

where m is the number

and

the function

If W,(z,,

s2) =

zz) > Hence

0).

W2(zi,

O.(Wz(sl, there is no

x2) 2 uI > 0, w (51, 52) 2 uz > 0. This in r’; but by Theorem 8, it is obtainable in r2.

Theorem 9 Let the functions i E

I) be

i

W,(X~, . . . , 5,))

such that integers

i=

1, . . . , n,

and the vector

pi > 0, i = 1, . . . , n, and s exist,

such that

pi = ST

i=l [pi

max LYi(X1, . . . . X,)+(S-_PJ XI, , xn

min

.x,,..., xn

IVi(Xl,

. . . . Zn)]s-‘~;di,

i = 1, . . . . n. The vector

Cd,, i E I) is then obtainable

Proof.

Let the vectors

in the game r”.

Cx:, . . . , xni) be such that

IV, (q., . . . . x,,) = TVi(s,‘,

max

. . . . x,~).

%,...,X* Then,

with i = 1, . . . , n,

n 2

PjWi (xl’, . . . . Z,,‘) S-l > [pi max .I,. ...(xn

j=l +(s-pi)

and the proof follows

min Wi(q, (xn

I,,

from (2.1).

. . ..x.)]s-~>c&

TV, (51, . . . . 5,)

+

(d,,

Exnmplr~

6.

Let the Sun&ions

f,(t),

i

1, 2, be defined,

continuous

and non-

1 for t E I-1, 11, while negative for 1 E I-L, 11, and periodic with period T min fi(t)=O, i= 1, 2. ,4lso, let integers p, ’ 0, pz ~ 0 and s ’ 0 exist, p1 t p2 z s and

i$?that

1

i = 1, 2.

, fi(t)dt~[P~~~~fi(t)ls-l, c Pllt

Wi(Xjy

Zz)

are obtainable

“= fi(S,

-

i 1, 1, 2, then the vectors f,(t), i = 1, 2, are periodic,

Za),

For, since

in rs.

(u,, u,) and ((I,, GJ

1 vi =

;i =

i=l,2,

fi(q.&, c

iJ and it only remains Theorem

to apply Theorem

9.

10

If there is an equilibrium situation in pure strategies in a non-cooperative n-person game rl, with first playe& strategies x, E X, = !O, 11, i-th players i = 2, . . . , n

strategies,

Zi

(S) = (til,

*a*

9

fl

Icj,‘) E

j=l,.

Xi',

. .,s,

j=l

where

Xij=

[O,l],j=l,

and i-th player’s

. . . . s,

pay-off function

S

the vector

(u

i E I) is obtainable

i ’

in the game r”.

Proof. Let (55, C(s), . . . , 35,(s) ) strategies in the game rl. Then, f: Wl(f,, j=1

fi,

. . . , Z”,,‘) s-l== max [i I

be an equilibrium

IV, (x1, ?,j,

situation

. . . . :P,$+]

jz1

A

z

i=2,

i \Vi (Z1. . . . . Z{-1, zij, 7{++,, . . . . FT,,j)S-I] > vi max x‘i’ . ..‘.YiSi j=1 . . . . n, and the proof follows

from (2.1).

in pure

> vl,

On a way

Let the i-th player,

in LO, 11, having

coal itions

i = 1, . . . , n - 1, have the max-min

where p,’ > 0 are integers function

of forming

such that

2

and JX is a distribution

pi’ _ Si

a jump of un$at

mixed strategy

the point X. Then the vector

(U i’

n-1 i E

I)

is obtainable

in the game rs,

where

s = fl

si,

i=l

Proof. Let the point x,” be such that cv, (XI, *.., G-I,

s

z,,‘) dF,’

(~1) . . . dF:_,

(x,_,)

> ZY,.

n-1 rI xi i=l

Then, j ,,.,,, jn_l,lqzBm,,

1=1, ,,,,

~,_lPj~Pj~...Pj~~-~‘~i(~~‘i~

v..t

dZ1~X,o)S-l

=

_+ zzz

w&q,

s

...

“-1 n .~i

, xn-I, x,,‘) clF,’ . . . dF:,_,

> ui

i=l

i=l

, -*a, n, and the proof follows

f(t) be defined and continuous

ExampZe 7. Let the function

tE[--l,

11,

i

from (2.1).

Plf(t

+ ;j

= 0,

for

LE[[-1,0],

l=O

and let p, > 0 be integers (XI -

X21

is obtainable

5 pl = s. I=0

such that

and W2(x,, x,) is an arbitrary

continuous

Then,

function,

in the game TS.

For, the first player

has the max-min F’ =

f

i

and it only remains

to apply Theorem

mixed strategy

pl J,,,,-, ,

1=0

11.

if wi(&,

22) =f

the vector

(ul, u,)

v. v.

!tx

‘l’hc ;uuhor thanks YII. IL &mcicr

‘Plor~)rcilJ

for guidance.

APPENDIX Proof of Theorem 8. It will be assumed initially that the c,~ ” 0, i n. Using (3.2), we get ui =: C&i + czi,

2, . . . ,

i = 2, . . . , n.

Now take an arbitrary vector (up,...,

&)) )) E w n = (Wi (4, ‘ . ., ZJ, _f ., w, fz1, . . .) 51L

and define the vectors

(w(:),. . ., I@)~

I = 2, . . . , n,

d”) ‘ = up (q, “E+l’f . *, St&,a, ‘ .I r&J,

as follows: i = 1,. . ., n_

If we put c,’ = 1 and c,’ = 0, it easily follows from (3.4) that Cl%& W!‘) t 2-=. i cl

(wf$_1 - ,;I-1 +-rni,

f=l,.*.,n,

1-i,..

‘,

where &+2-l= p-1-n P P

9

?.&‘I_,= w&_*,

P = 1, 2,

(3.6)

if i f 1 - 1 > n. We use the vectors

(u:?), , . ., ~2)) to form the vector (&, . . ., &,):

Using (3.5) and (3.61, it can be shown that & = c& -f- Ck,

i

=I

2, . . . ,

n.

(3.5)

In fact,

It can be shown by similar working that the components of the vectors (&1x1 , *a*, n, satisfy the equation

ID~S*J),

On a way of forming coalitions

99

WiJCli= 0.

i i=l

When the vector

(w/r ), i E I) runs over the set W, then, from (3.7), the vector

(li:,, i E I) runs over a set D on the straight w, =

line given hy the equations n.

i = 2, . . . ,

Cl’Wl + CT_‘,

This set D is a closed interval, since the set W is closed and connected. Further, r;.,o, i E 1) be the end of D which has the greater coordinates. by E, and E, the half-spaces defined by the hyperplane. i

u+li

= i

i=l

where E, contains

all the vectors

with large negative it isalso

contradicts

the definition

Hence,

p c

coordinates.

contained

in E,.

For, if the

of EO, it would follow from the above that

(Li, i E I)

vector

w.o/c;, 1

i=l

If the vector (w\‘), i E I) E W, vector were to belong to the interior the corresponding

Let denote

would belong

to the interior

of E,, whhch

(G,O, i E I).

of the vector

E, and from (3.3),

iii0 = 2 @n-1,

Vi<

i =I.

1=1

Hence,

if

cii

>

0, i = 2, ‘.. . , n, the proof is complete.

It can now be assumed c2i = 0,

Consider

without

loss of generality

i = 2, . . . , 7th+ 1,

czi > 0,

the game with n - m players,

numbered

that

i = m + 2, . . . , n. i =

1, m + 2, . . . , n, having

pay-

off functions Wi’(51,

where i=2

1,+2’

. . .,

5%) =

~0 ,. . , IO,,,+, , . . ..m+l.

wi

(51, DO, . . .,

are arbitrary

By what has been proved, the (n-&-person Hence,

x0 n+l,zm+2,

. . .,

fixed strategies

the vector

(vi, i E I\

5J7 i = I, mfL.4,

of the players

yd2’ {i$)

game rnmrn.

from (2.1), there exist

.x,O, zom+2, . . . , xylo such that

numbered

is obtainable

in

100

Wi’ (210,XL+%’. . ., x1,0)>

Since 5,‘)

Wi(Xl, >

Vi,

. . . , 5,)

=

Czi =

Vi,

i =

i = 1, m + 2, . . ., n.

Vi’

2, . . . , m + I,

we have w~(x~~,.

. . ,

1, * ee, n, and from (2.1), the vector (vi, i E I) is obtainable in

i =

the n-person game rnem. This completes the proof. Translated

by D. E. Brown

REFERENCES 1.

GERMEIER,

YU. B.

Player

concentrations

2.

VON NEUMANN, J. and MORGENSTERN, Behauiour, Princeton UP, 1953.

3.

GERMEIER, operatsii KARLIN,

Tekh.

kiber-

0.

Theory of Games and Economic

YU. B.

and the Theory

4.

in the study of systems,

2, 25-33, 1970.

netika,

i teorii

Methodic and Mathematical Foundations of Operations Research of Games (Metodicheskie i matematicheskie osnovy issledovaniya igr), Rotaprint VTS MGU, 1, 68-72, 1967.

S. Mathematical Methods in the Theory Addison-Wesley, 1959.

Economics,

of Games, Programming

and