Mixed strategies in two-person positional games with information exchange

Mixed strategies in two-person positional games

11. ROZENMULLER, I., Cooperative games and markets (Russian

45

translation),Mir, Moscow, 1974.

12. PELEG, B., An inductive method for constructing minimal balanced collections of finite sets, Naval Res Logist. Quart., 12, No. 2, 155-162, 1965. 13. OWEN, G., Game theory, Saunders, New York, 1968. 14. SHAPLEY, L. S., Cores of convex games, Internat. J. Game Theory, 1, No. 1, 1 l-26,

197 1.

MIXED STRATEGIES IN TWO-PERSON POSITIONAL GAMES WITH INFORMATION EXCHANGE*

T.N.DANIL'CHENKO Moscow (Received 20February 1975) THE USE of mixed strategies is considered in a class of two-person positional games in which the first player immediately tells the second of his strategy function at all n moves. The first player’s optimal guaranteeing strategy and optimal guaranteed results are found. Kuhn’s definition [l] of a positional N-person game embraces a wide class of games. Very general assumptions can be made about the mutual amounts of information of the players at the instant of deciding any intermediate move; and accordingly, the concept of a player’s strategy is wide, including randomization of pure strategies on the whole or successive randomization of moves (a probability distribution in alternatives for each information set of the players). However, a solution of the game is understood in quite a narrow sense, namely, as a search for equilibrium strategies. At the present time, however, especially in two-person games (N= 2) the principle of the maximum guaranteed result is being widely used. Study is evolving in two directions from the point of view of strategies: the use of mixed strategies in two-person games, where the first player expects information to reach him just once about the second player’s strategies vector [2] , and the discussion of dynamic two-person games in pure strategies with a sequential step-by-step inflow of information about the opponent’s strategies vector [3,4] . In the present paper an attempt is made to combine these two trends. Dynamic two-person games with perfect information (the players know the prehistory) are considered, under different assumptions about the knowledge of concrete realizations of the players’ choices. The following method of randomization is used: at the j-th move each player chooses a probability distribution on alternatives of the j-th move, j = 1,2, . . . , n. The first player’s strategy may be communicated to the second in different ways: a priori communication of the strategies immediately at all n steps of the game; step-by-step communication of the strategies only at each j-th move in turn; and various combinations of the a priori and step-by-step methods of strategy communication at the n interaction steps. We shah confine ourselves to finding the optimal guaranteed result and the fust player’s optimal strategies, in the case when the first player tells the second of his strategy-functions immediately at all steps of the game, and the second player is cautious, and confines himself to the principle of the maximum guaranteed result.

*Zh. vychisl. Mat. mat. Fiz., 16,5,

1136-1145,

1976.

T. N. Danil’chenko

46

1. Formulation of the problem Let the sets of player’s choices at the j-th step be specified by the metric compacta X, Yj, j=l, 2,. . . , n; ff (x, y) , i=l, 2, are the continuous pay-off functions, specified on XX Y. Here,

x=(t,, . . . 5*)=X,

Y=(Yi,

)

n

X=

YlEY,9

* * *, YJEY,

V-5,

=II ,=i II

Xj, IIj=i

Y

Yj.

Let Pi, & denote the sets of all probability measures in Xi, Yi respectively (or what amounts to the same thing, the first and second player’s sets of mixed strategies at the j-th move). The respective mixed strategies at the j-th move are PI, EP, and Q”,EQj . For simplicity it will be assumed throughout that j$ =p), Qqj =Qj. We introduce x,= (51, . . . ,x3, J’j’ ($I?, . - - , Yj>, IGjGn. Here, pi, qj are the distribution laws for the Pj’(Pi, ***, Pj)9 Qj=(Qi,..*tQj)9 quantities xi, vj respectively. We assume that each player agrees to take as his criterion the mathematical expectation of his pay-off. We consider the strategy in which each player’s choice is made successively in n steps, at each of which pj=Pi, qj=Qj, j=l, 2,. . . , n. Both players try to obtain the maximum guaranteed result, and the fust player knows that this is the second player’s principle of behaviour. The result of the game depends in an essential way on whether or not the players have at their disposal information about the concrete realizations of their opponent’s choices. We consider the case when the first player knows at the j-th move both the mixed strategy qj?Qj, and the concrete realization of the second player’s choice YjEYj, and also the over-all prehistory pj-1, Yj-1, l
Q*),

P”j=Pj(Pj-r, Yj, qj),

2GjGn.

(Here, p”j=pj‘(pj-1, Yj, qj) is the distribution law of the quantity xi.) We shall consider the problem when the first player tells the second immediately at all n moves his strategy functions

P(Y,S>={Fi=Pl

(Yi, qi), p”3=Pj(Pj-1, Yjl CIJ7 2sj+

(here and below; p= (pi,. . . ,P”), q= (q+, . . . , qn> ), after which the second player maximizes his own pay-off, which now depends solely on his choice, i.e., on the choice q= (q,, . . . , q,J . If the first player decided to use the functions p=p(Y,

q)={~l=Pi(Y,,

41); p”j’Pj(Pj-it

Yj, QJ, 2Gj+,

and the second chose the mixed strategy q= (Q~,. . . , q,,), where qjEQ1, lGj
then the

47

Mixed strategiesin two-personpositionalgames

(The assumed type of strategies is such that the order of integration can be changed.)

2. Lemma about the second player’s guaranteed result Let us find the result Lo which can be independently achieved by the second player by using mixed strategies, while confining himself to the principle of the guaranteed result. Let Fi be the set of all strategies of the type ~j=p~(pj-~, Jrj, ~j), and 5’ the set of all strategies of the type p”,‘=pj’(p+1, yj), IfjGn, where j&, p”~’are any functions of the arguments indicated, with values in Pi. Notice that the set @ is only pad of the set P,, l
PO,

qo,

YO

symbolize the absence of an argument.

1

If both players agree to use mixed strategies and during the game, the second has no information about the concrete realizations of the choices xi of the first, while the first, in addition to information about the mixed strategies of the second Q,EQ, counts on having information about the concrete realizations yjezYj, then Lo = max min.. YlEYl

PIEPI

. mas U,,EY

min 12P”EPn

I x,

...

J

fz(x,y)dp,..

.dp,.

Y,

Proof: Under the conditions described, the second player can guarantee himself the following result, whatever the actions of the first player: Lo’ = rn: max F, (t, q). P

Let us show that Lo’=Lo.

In

fact,

On the other hand, since the set of second player’s pure strategies is part of his set of mixec strategies, we have

T. N. Danil’chenko

Lo* =

min $jE

max I... 1 [s... \fP(x,

l
Pj,

Y,

q

Y,

min

>

Xl

max y i..;s

?;jzPj[Pj-l, YjgB(Y~)I, i
min

= Pj’E

max y I..;!

l
Pj’,

Mx,Y)d~,..&

*-?a

fz (x, Y) &’

x

r)d”p,...d~~]dq,...dq,

x,

. . . &I’.

72

Here and below, IQ) denotes the choice of y with probability 1. Hence,

= max min . . . max min s . . . s fz(x, y)dp,‘...d& YGYl 5,’ %IEY?I Pn’ Xl x, = max min...max

I/EYl PGPI U&Yn

min PnEPn

1...1 f2 (x, y) dp,

&

. . . dp, = Lo.

x,

Notice that Lo can be realized by the second player on pure strategies. The lemma is proved.

3. Auxiliary constructions We introduce the quantities

L(P,

q) =Fz(P, q),

Lj (Pi, qj) =

min max qjtlEQj+l PjtlE Pjtl

Lj+l

(Pi,

Pjtls qjtl)s

O\< j\
(3.1)

Let L,’ (p, y) =j Xl ~5’ (ph Yj) =

. . . s fi (x, y) dpn . . . dpi, Xl& max uj+i EYjti

min Li+i b+i, PHleP1ti

Yhi 1 ,

OGjGn-4.

(3.2)

By the same method as was used for proving Lemma 1, we can show that Lj(pj, Q)=J . . .J Lj’(pj, yj)dqj.. . dqs, Y’1 YJ

lGj
Lo=Lo’.

(3 *3)

Further, let Pj’(Pj-l, Yj) = {piEPjl

min Lj’(Pj7 Yj)‘L,‘(Pj-i7 PjEPj

Pi7 Yj)}9

IGjGn,

(3 -4)

49

Mixed strategies in two-person positional games

(3.5)

Ei’(Pj-1~Yj-1)={Zj~EYjl~~‘(pj-l,P~,Yf~,~j)=~j-*(Pj-*~Yj-l)}t

l
while on the other hand, from (3.3), (3.4) we have

i.e.,

minLj (Pj_1, pj~~

~jl

Sj)> Lj

(Pj_1,

Pj’7qj)7

l
j

The strategies pl* (p,-,, y,), j+l
L, (PI, qj) * We define the sets Ej(P+*, Yj-r)={qFB(YJ

IYFEjl(Pj-l,

Yj-l)}, lGj
(3 -6)

(the Ei’ are given by (3.5)). Assume that the first player has used a strategyp (Y, q) = {pj =Pj (Pj-1, Yj, qj), lGj
(Otherwise, the apparatus of e-strategies has to be introduced.) Given the first player’s strategy p(y, q), the second chooses any strategy qd the choice of strategies p (y, q) and qd?(p) we have

L,‘a

(p) . Given

max L(pi, sd,

(3.7)

0<,<7t--i

since it follows from (3.1)-(3.3) and (3.6) that,

only

by choosing Q&‘j(Pj-i, qi); O
can the second player guarantee himself the pay-off L”>Li(pi,

Yj-i), i+l
50

T. N. Danil’chenko

We put

(3.8)

Let M,/(P,Y)=

Mj'

J... JMGYML..~P,, xi XII

(Pj, Yj) =

inf , “j+lcZEj+l

(Pj* yj)

We can show, by the same method as was used in the proof of Lemma 1, that

Mj (pjpa) =

J .. JM~’ (pj,yj) dqj a

r1

s.

.

IG~G~,

dq,,

M~=M,‘.

(3.10)

rI

Let Py (Pj-*, YJ) realize

max Jf,’ (P,, YJ =M,‘(P,-i,

iqez,

PP, Yj),

(3.11)

P,EPj

where

maxMj (Pi_1'Pi’ qj) =

pj~~j

inf QjEEj(Pj-1,

Yj-1)

Mj(Pj_19

py,

qj), py -from(3.11); inf

MI (Pj-I9 Pj”, Qj) =

4jEgj’tPj-1s

J!fj(Pj_lr Pj”CSj), Yj-1)

i
We introduce the sets and quantities

D, =

(PA Qr)

SUP

Kj=

(3.12)

PiEPi, q,EQ<, l
M,(P,, qd,

(p, q>)=‘J) -m7

if

Dj+@, (3.13)

if

DI=O,

IGjGn.

We have: Lemma 2.1

If D,+ 0, for some j, then MoGKj. 2. If. for some i and j such that i < j, we have D,P0

and DjZ0,

then KSKj.

51

Mixed strategiesin two-personpositionalgames

Proo$ Let us prove assertion 2 (the proof of assertion 1 is similar). We have the inclusions t, (p,_-i,

(pm_,, y,+)

gm’

>L&_l(p,_,,

Yma?I’(Pm-~~

={pd=P,,

Jr,_,)

Y-1))

{pm=Pm, Y?J+Y~I~~ (pmym)

=

ym--i)}Et,“(Pm-~,Ym-l)=

Since D,+0, DjZ@, we obtain from (3.9), (3.10), (3.12), and (3.13), in the light of these inclusions, the chain of inequalities

inf

... “JeEj’

“P

(Pj-1, Yj-1) PjGPj

Mi’ (Pj, Yj)l ‘qi ...

aup aup (Pi*$)Eni &+l, qi+$)Qq+l(pi,

=G

.

sup

Pi)

' ' (Pj.QjE"j (Pj-1,Sj_1) I11 y ' *. 1

* ’ ’ dql

Mj’(pj, Yj) dqj

s

. . . de]

yi

aup

<

LPj,qj)EEj'i(Pjs Crj)lLj(Pj, Pi)>/ max L,

Mi(Pj7 Sj)

(P,. 9,))

O
=

(Pj, qj) = Kj*

cP,SyTEDMj

3’1

j

If, for arbitrary j, 1

4. Optimal strategy

=p;, ifqi I

We define the strategies Pi

.* = Pi=P:, P

The

E

i
El (pi_,,

Y&

I\< i < n,

Pia,

if if

q1 e El, qi’=% (pi_,, y&,

Pi’, r -I- 1
if

Q,,, e E,, (p,, y,).

point (P,r”, qi?) =(pljo, . . . , p,:, qi?, . . . , qd), Lj (pjj’, Q”)>

max

lGj
I
is such that

Lt (piio, Psfo) and Mj(pti”, qjjo) >Kj-8

oct
(here, pIjo, qij” are the distributions of the quantities xi,yi respectively);


(4.1)

52

T. N. Danit’chenko

i

pa

=

i

if

,“:;

if

a ’ P,=Pa*71
p,=

if

p& ( Pa’, m+f\(a
if if if

(4.2)

if if

(Ptl

We have: Theorem

Under the game conditions stated, the first player’s ~-optimal gu~anteed result is: max i, the strategy pt.8 of (4.2), (4.3) guarantees, possibly with r,
1) y-K, where j= e-accuracy, the result Kj; 2) y=Mo, if, for any MO is obtained.

i,

1 Qi


we

have D,=%; the strategy p** of (4.1) guar~tees that

we haveD,=0. Then, given any strategy Proo$ 1. Assume that, for any i, I-.M,. Assertion 2 of the theorem is proved. 2. Let j=rCi,ra; ~a ,i.e, D,+0, given any strategy’&y’, q), we have

and let all the sets D,===0 for all i, j

t

1G

i Q

n.

Then,

(4.4)

Mixed strategiesin two-personpositionalgames

53

If j = n, then L,‘3

mas

L(P,, qj).

O
If the strategy p(y, q) is such that L, * = LO, then nothing prevents the second player from choosing any strategy crlEB, (p-f, yi-,), 1-Gi
(4.5)

It will next be assumed that the strategy p(y, q) is such that L,* > Lo. If, given the strategy P_(Y,_91=(pI,=Pj(P,-I, P?, qtv.--t

- ] dl), 41

yj. qj>, lGiGn)~ then, by (3.13),

%Cp>= {it, 4~~~fi)E;:{q(I>,E.R(p)

ye-cl.

[ [pi, 1.. 3

(4.6)

If

then, by (3.7) and (4.41, for i < n, Lnf=Ln--1(p,,-,. (I,‘_-1)= . . . =L, (Pjs SJ =

max

L, (Pi, $1.

Z
(4.8)

Ifi = n, then L,’ = max L, h ,
qd.

Ln this case, any strategy &,I of (4.7) is equivalent from the point of view of the second player and realizes, in the light of (4.81, L“==&JPty,

&I,>,

. * . = Lm, (pm,, qm~-f.

4d=L,r i~‘ncm,c

(Pm, %) ===.L~(Prnll%nJ =. -1 . . .
1.

(4.9)

Assume that the second player has chosen any strategy (r,,, (W from the class of strategies such that the least number in (4.9) is m. If qi,m,’==(qi’n,. . . , (I,,*=, Q g+l. . . . , Y,~“‘), then, by (4.91, L,,‘=L,,Ip(y.q:~~),Y:~‘l=L!,,(p,,”.cf,n~a~>L.(pl~,qi”),

l~iaw-4,

(4. IO)

which implies, in view of (3.17), (p,,,“‘. q,,!“‘) dl,,.

(4.11)

The strategy q’={y,“‘, $GKGD?; ~Fk,fp,-,~ yt-,jt nz-!-i<‘tGn] equivalent to the strategy qtp)(‘n). For, from the form of q’, the definition of Ez(pI-~, y.._*) and (3.3j. we have L,“ZL,[p(y, q’), q’f3L,(p,“‘. (I,,:“‘) =L,.‘,i.e.. L,,‘== L,[p(y, q’), q’]~l~,~, (p,:,“!, qS,,“‘), the assumption that L,, ip(y, q’), q’] =L,(p,. (I, ! for r
54

T. N. Danil%henko

In short, we can carry out the estimation of 7 for q’. From (3.8)-(3.10) it follows that the use of the strategy q’ by the second player does not promise the fust more than M,[P(y, q') , $1 =&I (P, m, qmm), in which case, by (4.11) and Lemma 2,

sup

1G

J&n (p?n,qnr)=Km~&*

(P&x,)ED,

Thus,

given any strategy p(y, 4) and optimal second player’s behaviour, the estimate (4.6)

holds for 7. On the other hand, we can show, using a similar method to that in (31, that the strategy ~01 of (4.2), (4.3) guarantees the first player the result Kj - E. The theorem is proved.

5. Examples and notes Let USgive some examples to illustrate our theorem (n = 2, discrete case). ExampIe I. fi (xi,

x2,

pi,

~2.) =--xi

2+~2z-zA2+iQ,

fZ(YJ,

112)=5drr2--2Y22,

zt,

Y*=Q,

3

(i=l,

2).

In this example, D,= 0, i&=0, +&=O.

1

fi=1,

Example 2. ff (G .Q, Yi, 2). Here, D2=%, D,f0 Example

x,,

3. fi ($1,

yl=O, 1 (i=t,

2).

32,

yi,

~zf=-Xi2-X22f~i2+y~2,

(M,=ft~,-~/~),

Y2) =;5Si2-Xz2+51/iz+Yz2,

Here,D2#%, D,*0

f&r,

Yi, yzf =-Xi2-~~~2+~~2,

Xi,

~0,

y=K.j=6/3.

fz(Zi,

22,

yi,

~2)

=5~~~-~~~-10yi~-Yz~;

(.~,=~cK,=~~I~~K,=~~/~),

p=K~=f7/2,

Notes. 1. In the case ji (x, y) +f2(x, y) =O (which implies that the player’s interests are opposed), we find, on following formally the procedure described above, that the optimal guaranteed result for the .first player is y= min max.., !/I= l-1

PIEPI

min max vnE.

Yn

which is in general greater than the corre~ond~

PnPPn

Jx,. . .X” J

fj(x,y)dp,...dpj,

pay-off in pure strategies.

2. In the case when the first player knows only the concrete realizations of the second’s choices, the first player can, as in [S] , disregard the possibility of the second using mixed strategies while he fmds the guaranteed and optimal guaranteed results and the corresponding strategy. 3. If the first player knows only the second’s mixed strategies, the problem is obtained [3] with the pay-off functions i=s112; wp,q)=J J . ..JJfl(x.~)dp,d~,...dpidpl, Y, x,

Y,K”

p=(Pl, *a*, Pnf, q==(qJ, -**, q,,): pp==P, respectively at the j-th move, 1
q,=Q,

are the first and second player’s strategies

Problems of the transport type

55

The author thanks Yu. B. Germeier, K. K. Mosevich, and A. F. Kononenko for valuable discussions and comments. Translated by

D. E. Brown

REFERENCES 1.

KUHN, G. U., Positional games and the information problem, in: Positionalgames (Russian translations), Nauka, 13-40, Moscow, 1967.

2.

KONONENICO, A. F., and KUKUSHKIN, N. S., Mixed strategies in games with a fixed sequence of moves, Dokl. Akad. Nauk SSSR, 209, No. 6,1274-1277.1973.

3.

DANIL’CHENKO, T. N., and MOSEVICH, K. K., Multistep two-person games with a fixed sequence of moves, Zh. uychisl.Mat. mat. Fiz., 14, No. 4,1047-1052, 1974.

4.

DANIL’CHENKO, T. N., and MOSEVICH, K. K., Multistep two-person games with unopposed interests, Dokl. Akad. Nauk SSSR, 211, No. 6,1273-12751973.

5.

GERMEIER, Yu. B., Games with unopposed interests (Igry s neprotivopolozhnymi Mosk. un-ta, Moscow, 1972.

interesami), Izd-vo

ALGORITHMS OF FEASIBLE DIRECTIONS FOR SOLVING CERTAIN PROBLEMS OF THE TRANSPORT TYPE* N. D. ASTAKHOV Moscow (Received

3 1 March 1975)

ALGORITHMS of feasible directions, suitable for solving certain linear programming problems of the transport type, are described. By taking the example of the allocation problem, we show that the algorithms are applicable to a wide range of basically similar transport problems.

1. The transport problem with bounded sets of production points (closed model)

We consider the problem: to find

(1) under the conditions

c

x,j= b,,

j=J,

1EI

Here and throughout,

c

Xtj=ai,

i=I,

X,,>c),

id,

j=J.

JEJ

I= {I, 2, . . . , m}, J= {I, . . . , n}.

We shall assume that the condition for solvability [l] of this problem is satisfied, i.e.,

*Zh. v_?chisl.Mat. mat. Fir., 16,5, 1146-1154,

1976.