The solution of a class of non-antagonistic games

D. A. Molodtsov

67

REFERENCES 1.

GERMRER, Yu. B., Inmduction in operations research theory (Vvedenie v teoriyu issledovaniya operatsii), Nauka, Moscow, 1971.

2.

ROZEN, V. V., Games with ordered outcomes, Third Allwdon Conference on Game Theory, Abstracts of papers, Izd-vo Odesskogo un-ta, 203-204, Odessa, 1974.

3.

A~~,

4.

AUMANN, R. J., Utility theory without the completeness axiom: a wrrection,Econometrica, 32, No. l-2, 210-212,1964.

5.

YANOVSKAYA, E. B., Equilibrium situations in games with non-Archimedean utilities, in: Mathematical methods in the social sciences (Matem. metody v sotsial’nykh naukakh), No. 4, Izd-vo In-ta fii. i matem., Akad. Nauk LitSSR, 98-118, Viius, 1974.

6.

YANOVSKAYA, E. B., Equilibrium situations in general coalition-free games, Third All-Union Conference on Game Theory, Abstracts of papers, Izd-vo Odesskogo un-ta, 22-26, Odessa, 1974.

7.

VOROB’EV, N. N., Modem state of game theory, Ckp.mat.Nuuk, 25, No. 2 (152), 81--140,197O.

8.

OWEN, G., Game theory, Saunders, 1968.

R. J., Utility theory without the completeness axiom, ~~norne~jca, 30, No. 3,445-462,1962.

THE SOLUTION OF A CLASS OF NON-~A~NISTIC

GAMES*

D. A. MOLODTSOV Moscow (Received 2 1 Februav

1975)

THE SOLUTION of a non-antagonistic two-person game with information exchange is reduced to a maximization problem. If the first player has imperfect information about the second’s pay-off function, the game reduces to a game without undetermined factors.

1. Approximate reduction of the solution of a game I’1to to a maximization problem Consider the two-person game ri (F, G, X, Y) , where F, G are the pay-off functions, and X, Y are the strategy sets of the first and second players respectively. Initially the first player chooses SEX, and tells the second, then the second maximizes G. By solution of the game we mean fmding the first player’s maximum guaranteed result uo, where sup fa (z) ) xe:x

aao,

inf F(s,y), VoNaWIP) N,[G3(r)=(y~YIG(s,y)~maxG(s,z)-a}, Ua =

Jz(z:)=

ZEY

and finding his strategies realizing uu, possibly up to E. This game statement was originally due to Germeier [l] . In [2-41 some general aspects of the numerical solution of a game f, are considered, and games of a particular type are solved. In [3] the solution of I’1 is reduced a proximately to a maximization problem of the type max u-C” [F (5, y) SC@ (2, y) -U, J W.=)Pg9 Y -IF(x,y)+C@(x, ?B.v~chisl Mat. mat. Rz., 16,6,1451-1456,1976.

y)-UI

68

D.A. Molodtsov

where @ky)=f

[G(s,&-G(

z,z)-IG(5,y)-G(~,z)~]‘~u(z).

Y It can be seen that this problem is very complicated, even when F and G are reasonably simple functions. Extra difficulties arise because the game rl is ill-posed [2]. In the present paper we outline a simpler method for the approximate reduction of the solution of a game r1 to a maximization problem. For fl> 0 we consider the function GB(5, y) =G (~7 Y) --P’(T Let the constants m and M be such that, given any SEX

Y). and g=Y

we havem
E/)GM-l-m.

If F and G are continuous on the product of compacta of the metric spaces XX Y, then, and (Y> 0, we have given any 7~30, j3>0, x=X, y=N,[GB] (x)

f~+iw (2)Q Proof: Let y=N,[Gp]

a+7

(x,Y)G -

B

+f&).

(z), then

G(2,Y>-BF (5,Y)b max[ G (5, z) -/3F (x, z) ] -7 IEY

max

3

[G(x,z)-pF(x,z)]-7

-NCLCGIP) h

max [ maxG(x, =NOL I’JIW VeT

V)-a-pF(x,z)]--7

= max G (5, z) -a-,y-pfaL (x) BG (x, y) -a-y-/3fa

(x) .

ZGY

Hence F (x, y) <‘f=(x) f (a-i-7)/p. Again let y@V,[ Gp] (x), then

G (x:,Y>-PmaG (5,Y>-BFb, Y) 2 max [G (r, z) -BF (x, z) l-7

IEY

2 max G (x, z) -pm-pill-y, IEY

i.e., G(x, y) 2 max G (I, z) -pill-y. IEY Hence y&VV+eM[G] (z)

and fr+Vf (4 =

The

lemma is proved.

min

F(x,y)GF(x,y).

The solution of a classof non-antagonisticgames

69

Consider the maximization problem

h(P)= From the lemma, given b>O, y>O, a>0

max F(x, Y). 1~x3u=N1fGplW

(1)

we have

For a! = 0 we have

It was shown in [2] that lim uor= uo. cb-ro+ In short, we have proved: Theorem 1

If F and G are continuous on the product of compacta of metric spaces XX Y, then

Note. The relation 7 / fl+ 0 is essential. Consider an elementary example: F(Z, y) =y, G(z, J) =O, X=Y= [0, I]. It is easily seen that uO=O, and v,.(P) =min {r/P, 1). It now has to follow, from uy(13)+uo, that 7/ /3+ 0. Notice that a stronger assertion than Theorem 1 can in fact be derived from the lemma, namely, instead of the solution of problem (l), we can maximize a function F only with respect to x=X, while y is an arbitrary member of the set N, [ Gal (x) . An assertion similar to Theorem 1 holds for the quantity thus obtained. The resulting approximate problem is simpler than problem (l), inasmuch as a complete knowledge of the set ‘N,[ Gk] (x) , is unnecessary, and a knowledge of any point of it is sufficient. This fact reveals that fairly simple algorithms can be devised for the numerical solution of the game r 1. We can write problem (1) as ur (p) = max u %I,U under the constraints x=X, uGp(x, z)-1 for all ZEY. Using the device described in [ 11, we contract this group of contraints into a single constraint: CD(5, y, p, y) ~0, the function @ being non-negative throughout its domain of definition. Weput RF(u)=&, yEXXY]F(X, Y)>Ul. Then, if F and G are continuous, and X and Y are compact, we can write problem (1) in the form q(p) = max u under the constraints UE[~,

n+M]

and

70

D.A. Molodtsov

In other words, since Q,is non-negative, u,(/3) may be equated to the maximum root of the equation OrB(u) =O. This result can also be regarded as a generalization of the method of “discrepancies” for seeking a maximum [5] . As Cpwe can take e.g., the function @ = max { max CGa(CC,z) 4s

(2, Y) 1-T; O>

ZEY

or the integral convolution CD=

s [max{G~(~,a)-G~(~,y)-7,0}lPdo(~),

Y

where, given a suitable choice of the parameter p, we can achieve the required smoothness of Cp, provided that F and G are sufficiently smooth. We shall mention individually the case when Y= (1, 2, . . . , n}, we put

, i.e., is finite. For KEY

Then, we can write problem (1) as v,@)=max ieY

max F(qi). XfPT

(2)

(it.)

It was shown in [4] that, when Y consists of n elements, solution of the game rl demands the solution of 2” maxmin problems. It follows from (2) that we are now able to reduce the solution of rl to n fairly simple maximization problems: to find

max F (2, i) xlzx under the constraints GB(2, i) >G, (5, j) -y, j=Y.

2. The game rl in the presence of undetermined factors One of the main assumptions that makes uo the first player’s guaranteed pay-off is the assumption that the first player knows exactly the pay-off function of the second player. A more general assumption, and one that more accurately reflects an actual situation, is to the effect that the first player has inaccurate knowledge, not only of the second player’s pay-off but also of his own. We shah consider the statement of the game rl in this case in more detail. The first player DEB, where knows two familiesof functions F(z, y, a). and G(z, y, b), ZEX, y=Y, F(x, y, 0) and G(x, y, p) are the first and second player’s pay-off functions (from the point of view of the first player). In reality, the second player’s pay-off function G(x, v) is known exactly by the second player, the first knows this, and he knows that G(x, y) belongs to the family G(XY,

0).

The solution of a classof non-antagonisticgames

71

We shall assume that X, Y, and B are compacta of complete metric spaces, and that F and G are continuous in XX Y X B. Put

No[Gl(s,~)={y~YIG(s,y,~)=maxG(~,z,B)}. LEY

We shall assume that the first player knows that the second can, for any x, exactly maximize his and informs the second (we own pay-off function. Hence, if the first player chooses r=X assume that this is possible), then he can guarantee himself a pay-off equal to j(5) = min xsB The

F(s, y, B).

min

vd'a[Cl(r,B)

first player’s maximum guaranteed result is obviously 1

e

uo = sup f(z) = X-=X

sup min xrX BEB

min uSNo[G](X,~)

In the game l?,(P, G, X, Y) =I’,(F, G, X, Y) (see [6]), where y is the set of all mappings of Y into X, such an introduction of an indeterminate factor considerably complicates the solution of the game (see e.g., [7,8] ). In our case, the matter is much simpler. We put

z=b, B)?

Z=YXB,

F (5, z) =F(t,

Y9B) 9

P).

Q,(2, z> =G b, Y, B>- max G b, u, UEY

Theorem 2 The game rl above described, with an indeterminate factor, is equivalent to the game I’,(F, 0, X, 2) without an indeterminate factor. (The equivalence of games is understood in the sense of the best guaranteed results and the first player’s optimal strategies being the same.) Proof: The first player’s best guaranteed result in the game

uo = sup x=X

where

min

rl (F, a, X, 2)

F(z,z),

r~No[O](r)

No[~,l(t)={z~ZI~((z,z)=max~(s,~)}, vez ={y,

pYXBIG(z, y,PI= maxG(r, u, a>>. UEY

This follows from the fact that max 0 (5,~) =O. ZEZ Hence we obtain F(5)=

min

F(z, z),

r~No[‘Dl(x)

whence the theorem follows.

is equal to

D. A. Molodtsov

72

Notice individually the case when the first player’s pay-off function is independent of the indeterminate factor, i.e., F(x, y, /3)=F(s, y). Then, putting y((Z,y)=max[:G(z,y,B)-maxG(s,v,B)l, VEY BEB the game PI with an indeterminate factor is equivalent to the game follows from the fact that

Fl (F, Y, X,

Y).

This

max Y (5, y) =O. Y-=Y The indeterminate factors in the game PI can be of a different kind; for instance, there may be errors in communicating the first player’s strategy to the second, or the first may have inaccurate knowledge of the set of second player’s strategies. Whatever the case, if the first player knows in some way that, if he chooses XEX, then the second will choose his strategy from the set N(x) C Y, then the game is equivalent to the game I’,(F, H, X, Y), where I?(IL’, y) =-d(y, N(s)), d and d is the Hausdorff metric. To summarize, we have shown that a wide class of games rl with an indeterminate factor can be reduced to some equivalent games VI without an indeterminate factor. Translated by

D. E. Brown.

REFERENCES 1.

GERMEIER, YU. B., Introduction to operations research theory (Vvedenie v teoriyu issledovaniya operatsii), Nauka, Moscow, 1971.

2.

MOLODTSOV, D. A., and FEDOROV, V. V., Appro imationof two-person games with information exchange, Zh. @chhisLMat. mat. Fiz., 13,6, 14 k g-1484,1973.

3.

GORELIK, V. A., Approximate determination of a maxmin with constraints connecting the variables, Zh. vFhisL Mat. mat. Fiz., 12,2,510-517, 1972.

4.

MOLODTSOV, D. A., A class of games with unopposed interests, Zh. vehisl. Mat. mat. Fiz., 15, No. 3, 789-795,197s.

5.

GEMEIER, YU. B., and KRYLOV, I. A., Search for maxmins by the method of “discrepancies,” Zh, vj%hisLMat. mat. Fiz., 12,4,871-881, 1972.

6.

GEMEIER, YU. B., On two-person games with a fiied sequence of moves, DokL Akad. Nauk SSSR, 198, No. S, lOOl-1004,197l.

7.

VATEL’, I. A., and KUKUSHKIN, N. S., Optimal behaviour of the player having the right to the first move, when he has inaccurate knowledge of the second player’s interests, Zh. vj%hisLMat. mat. Fiz., 13, No. 2,303-310,1973.

8.

KONONENKO, A. F., The role of information about the opponent’s target function in a two-person game with a fixed sequence of moves, Z/r. vj?hisL Mat. mat. Fiz., 13,2,311-317,1973.