Approximation of some zero-sum non-continuous game by a matrix game

Series I: COMMENTATIONES MATHEMATICAE XXX (1991) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1991)

An d r z e j Ce g ie l s k i (Zielona Gôra)

Approximation of some zero-sum non-continuous game by a matrix game

Abstract. We give a method of approximative solution of some zero-sum two-person game with non-continuous pay-off. The general game of timing is a particular case of our game. The method consists in solution of some matrix game which is the base for defining e-optimal strategies for both players.

1. Introduction. Let Г = (X, Y, К) be a two-person zero-sum game, where X = Y = [0, 1] and the pay-off function К: X x Y -*R is defined as follows:

where the functions L and M are defined on the closed triangles Ax = {(x, y)e e X x Y : O ^ x ^ y ^ l } and A2 = {(x, y )e X x Y : respective­

ly , and Ф is defined on [0, 1]. The general game of timing is a particular case, with L and M continuous on At and A2, respectively, and satisfying some monotonicity condition (see Karlin [1]). In Sections 2 and 3 we assume that L and M are continuous on A t and A2, respectively, and satisfy some monotonicity conditions weaker than those given by Karlin [1]. The game considered in Section 2 is a particular case of the game solved in Section 3. In Section 4 we weaken the continuity assumption on L and M.

2. A particular case. Let L and M be continuous on A1 and A2, respectively. Let

A = {^e[0, 1]: Vx>4 L(x, x) > M(x, x)}, L(x, y) for x < y, Ф(х) for x = y, M{x, у) for X > y,

We assume that L, M and Ф satisfy the following conditions:

(i) Vy>XoL ( ’, y) is strictly increasing in [0, x0], 00 Vx>XoM(x, •) is strictly decreasing in [0, x0], (iii) Vx>XoM(x, x) ^ Ф(х) ^ L(x, x).

262 A. C e g i e l s k i

Let Г = (Px, PY, К ) be the mixed extension of the game Г, where Px and PY are the spaces of probability measures on X and Y, respectively, and K: Px xPy ^ R is defined as follows:

K{ii, v) = j K (x, y)d(n x v)(x, y).

X * Y

Theorem 1. The game Г has a value v and for every e > 0 each of the players has an s-optimal strategy.

P ro o f. (We base on the proof of this theorem for x0 = 0 which can be found in [2].) L and M are continuous on the compact sets Ax and A2, and therefore uniformly continuous. Hence for each s' > 0 there exists <5 > 0 such that

(¹) and (2)

\Ц х i, у 1) - Ц х 2, y2)I < s’

if \xx- x 2\ < Ô, |у ! - у 21 < <5> (*i, Уд £ d j , i = 1, 2,

\M(xx, y l) — M (x2, y2)I < s'

if \xx- x 2\ < ô, \y1- y 2\ < ô, (x t, у{)е А 2, i = 1, 2.

Let e > 0 be arbitrary. Let Ô > 0 be such that (1) and (2) hold for s' = e. Divide the interval [0, 1] by points xx, ..., xr such that 0 = i x < ... < xr = 1 ,т/+1 — тг

< <5, i = 0, ..., r — 1 and xk = x 0 for some k , k = 1, . . . , r. Let Te be a matrix game with the pay-off matrix A = [ay] where atj = Х(тг, т,-), i, j = 1, ..., r.

Let xE = (xx, ..., xr) and yE = (y i, ..., yr) be optimal strategies and vE the value of the game ГЕ, that is,

Г

E ^{X i a iJ}

i = i

for j = 1 ,. . . , r and '

Г

E у f i n < v>

j = 1

for i = 1 ,. . . , r. Observe that from (i) and (ii) and from Theorem 2.2.4 of [1] we have

x 1 = x 2 = ... = xk_! = 0, y x = y2 = ... = = 0.

Indeed, let A x = [ay], i j = k, ..., r, A 2 = [fly], i = k , . . . , r , ; = 1, ..., /с 1, A 2 > I 1, к 1, j к , ..., r, A^ ? If j

= 1, ..., к — 1. Then

A = A 4 A 3 _A 2 A x_

By (ii), each column of A 2 strictly dominates the first column of A x, and by (i), each row of A 3 is strictly dominated by the first row of A x. Hence from Theorem 2.2.4 of [1] it follows that the submatrices A 2, A 3 and A4 can be dropped or, in other words, for arbitrary optimal strategies (xl 5 ..., xr) and (^ i, • • •, У,) of the game ГЕ the first к — 1 coordinates are zero.

Now we show that for every rje[0, 1]

(3) K{xE, t j ) ^ v E- e .

a) Let rj ^ x0. Then from (ii) it follows that

r r r r

K(xe> n) = Z xiK (Ti’ n ) ^ Y xiK (Ti’ xo) = Z xiaik = Z xiaik > ve-

i = k i = к i = к i = 1

b) Let rj ⁼ tj for some j, j = k + 1, ^{. . . ,} r. Then

r r

(4) K(xE, rj) = Y XiK(Tit tj) = Y Xidtj > ve.

i — 1 i= 1

c) Let Tj < rj < xj+1 for some j, j = k, ..., r — 1. Then K(xE, rj)= Y x tK(xt, rj) = X XiL(xit rj)+ Z xiM (Ti> n)-

i = k i —к i = j +1

From the definition of К it follows that

K(xE, Xj) = £ x£K(т„ xj) =

Y xMb>

^) + ^ Ф (^ )+ Z xiM{Xi, xj).

i = 1 i = l i = j + 1

Hence using (1), (iii) and (2) we have

j ~ i

K{xE, t]) — K(xE, Xj) = Y г]) — Ь(х{, Xjft + XjlLiXj, г])-Ф{т,)]

i" k

r

^{j - 1}

r

+ Z XiCM (Ti> Xi, Xj)] > - 8 Y xi~ z Z *.■ = - ( !-Xj)£ > -8 .

i = j^{+ 1} i —к i = j + 1

Hence and from (4) it follows that K{xE, rj) ^ K(xE, t,-) —e ^ vE—e.

The inequality (3) is thus shown. Hence it follows easily that K(xE, v) ^

^ vE — e for every veP, that is,

sup inf К{ц, v) ^ vE — e.

u e P x v e P y

In a similar way it can be proved that

inf sup K(fi, v) ^ vE + e.

vePY fiePx

Hence

vE — e ^ sup inf К(ц, v) ^ inf sup К(ц, v) < vE + e.

264 A. C e g i e l s k i

Since vE is bounded there exists a subsequence of {vB} which converges to a real number v as e -> 0. Then v is of course the value of Г and x£ and y£ are e-optimal strategies of Г.

3. General case. Theorem 1 can be used for approximative solution of some games of timing. The general model of the game of timing, the existence and the forms of optimal strategies for players are given in [1]. But for a concrete game of timing an approximative solution of the game obtained by using Theorem 1 would be often simpler. The problem is, however, that not all games of timing satisfy condition (iii) (e.g. the silent-noisy duel in Example 3,

§5.1 of [1]). Therefore in this section we weaken condition (iii).

Let L and M satisfy conditions (i) and (ii) given in Section 2. We do not assume that Ф satisfies (iii), but only that

(iv) Ф(0) lies between L(0, 0) and M(0, 0), and Ф(1) lies between L(l, 1) and M( 1, 1).

Theorem 2. I f conditions (i), (ii), and (iv) are satisfied then the game Г has a value and for every s > 0 there exist s-optimal strategies for both players.

P ro o f. Define Ф': [0, 1]->R as follows:

{

^Ф(х) if M(x, x) ^ Ф(х) ^ L(x, x) or x < x0, L(x, x) if Ф(х) > L(x, x) and x ^ x0,

M(x, x) if Ф(х) < M(x, x) and x ^ x0, and define К [0, 1] x [0, 1] ->R as follows:

f L(x, y) if x < y, K'(x, у) = К Ф'(х) if x = у, LM(x, у) if x > у.

Let Г' = ([0, 1], [0, 1], К'). Observe that Г' satisfies the assumptions of Theorem 1 since L and M satisfy (i) and (ii) and Ф' satisfies (iii). Let e > 0 be arbitrary. Let Ô' > 0 be such that (1) and (2) hold for s' = ^s and <5 = <5'. Divide the interval [0 ,1 ] by points т15. . . , т г such that 0 = < ... < тг = 1, Ti+ ! — tt < S ',i = 1 ,.. . , r — 1 and xk = x0 for some к, к = 1, ..., r. Let Ге' be a matrix game with pay-off matrix A' = [a'7] where а'ц = K'(xt, x7), i, j = 1 ,..., r. Let x' = (x'i, ..., x') and y'E = (y l, ..., y£) be optimal strategies and v'E the value of ГЕ. By Theorem 1, these strategies are ^s-optimal for Г', that is,

^j > 6 [0 , l ] K'(xe>У) ^{^ Ve~ 3 £ ,} ^ x e [ 0 , l ] Ув) ^ ^É + 3 £ -

As in the proof of Theorem 1 it can be shown that x t = ... = xk- k = 0 and y 1 = ... = yfc_i = 0. Let

B 1 = {ie N: к ^ i ^ r , Ф(х1) < М(тг, xj}, B2 = { jeN : k ^ j ^ г, Ф(т;.) > Ц хр Xj)},

ô = min{(ii + 1 — тг): i = k, ..., r - 1 } .

Let nij be the uniform distribution on the interval Uj = (тp т7 + <5) and let

<5f be the distribution concentrated at xt. Let

В

= X

x'imi+

Z

x’i3i’

v = Z

y'jmj+

Z

ieB i i*B i j e B 2 M B 2

Observe that by (iv) the supports of the measures ц and v are included in [0, 1].

We show that

Vxe[o(i] K{x, v) ^ v’E + e.

a) Let x = г, for i = k , r , and гфВ2. Since Ф(т,) ^ Ф'(т,) we have K(x, v) = K(xt, v)

= Z yW X (T«’ y)<H-(y)+ Z

y'jK (Ti’

tj

)

j e B 2 Uj j $ B 2

^ Z

yj S

x (Ti’ у)<*т /у )+ Z r7.)

je B 2 Uj j $ B 2

< Z У} f [ ^ ( Ti’ ^) + ie]dmj0;)+ X у)а'и

j<=B2 Uj нв2

=

Z y'jaij+ Z ^ «y + ie X y'j

j e B 2 # B 2 j e B 2

^ Z

Ÿja'ij + i 5 ^v's + %£ < v'E +s.

j - k

b) Let x = т,- for i e B2. Then we have

K(x, v) = K(xit v) = X y'j J K(xt, y)dnij(y)+ X ÿjK(xti x;)

j e B 2 U j H B 2

= X У} J К(т4, y )^ .(y ) + /i f L (if, y)dmf(y)+ X

У)аи

j * i U j U i Ц В 2

j e B 2

< x У}

S tK (Ti’

^{x j )}

+

³

£]àmj(y)

⁺

y,i

j [L(xf, Tf)+i£]dmiOO+ X

У)аа

j * i U j U i j $ B 2

j e B 2

= Z

yjK^t’

^ ) + ^^ф'(^т,-)+ X ^ ^j+ ^зё Z ^у;

i * i j $ B 2 j e B 2

j e B 2

= Z У}ау + У*«к + Z У;ау + з£ Z y}

J * i m 2 j e B 2

j e B 2

= Z yjfly+ie Z У;

J = j e B 2

r

< X yjfly + i e < ^г+з£ < Уе + £.

j = fc

266 A. C e g i e l s k i

c) Let хе(т,-, ^t1+1) for i — k, 1 and i$ B 2. Then we have K (x,

v) = X

У)

J

K (x > y)dmj(y)+

Z

y'jM (x ’tj

)+ Z

У'М Х’

Ti)

J 6B 2 Uj j ^ i j > i

Л В2 j $ B 2

< Z

y'j

I

K (ro y)dmj(y)+

Z

Л м (ъ> zj)+

Z ^Т(тг, ту) + зе

j e B 2 Uj j ^ i j > i

H B2 j i B 2

< Z

у

Л k (t«>

y)dmj(y)+

Z У;х '(ь

7'бВг Uj j $ B 2

Now if we proceed as in a) we obtain

K (x, v) ^ «é+ie + Je < v’e + e.

d) Let x e ( if, Ti+1) for ie B 2. Since for хе(т,-, xi+1) and ye(xt, x) M (x, у) ^ M(y, у) + |г ^ L{y, j) + ^e < L{xt, y)+fe

we have

v) = £ /у J X(x, j)dWyO;) + ^ [ f M (x, y)dmi(y)

j¥=i Uj Vi

j e B 2

+ f L(x, y)dmi{y)]+ £ у'-К(х, Ту)

x j $ B 2

< Z У} f [№»> y) + i e№ j(y)+ /»{]" [T(b ^) + Se]dm,.(y)

j^i Uj Vi

j e B 2

+ J [L(t{, y) + ifi]dmf();)}+ £ Ty) + Je]

X j4B2

^ Z

у

Н ^(Ti’ y)^(y) + yî J L (T;> У)<Ч(у)

t/j t/i

j e B 2

+ Z

у

}

к

(

т

„

tj

)+3£ Z

y'j+hy'i-

НВ2 }Ф1

Now if we proceed as after (*) in b) we obtain

K(x, v) < < + i e + i e Z Vj+hy'i < v'B + e.

' j*i

e) Let x e [0, xk). From (i) and from the proof in a) and b) it follows that K(x, v) < K(xk, v) < y'+ e since the support of v is disjoint from [0, xk).

In a similar way it can be proved that

VMo,i] K{p, y ) ^ v'B- e .

Next arguing as in the proof of Theorem 1 we show that the game Г has a value and that x'e and y'e are e-optimal strategies of players in this game.

4. A generalization. Observe that we did not use the continuity of the functions L and M on the whole triangles A1 and A2. We only used the continuity of L on ([т4, тг+1) x [t,., tj + 1 )) n A1 and the continuity of M on ([t£, t, + 1) x [ij, iy+i)) n A2. Hence the following theorem holds:

Theorem 3. I f the functions L and M are continuous on the triangles A t and A2, respectively, except possibly on a finite number of lines x = tl t ..., x= tm and у = мх, у = un where they are right-continuous, and if the conditions (i), (ii) and (iv) are satisfied then the game Г has a value and for every s > 0 there exist e-optimal strategies for both players.

References

[1] S. K a r lin , Mathematical Methods and Theory in Games, Programming and Economics, Addison-Wesley, Reading, Mass., 1959.

[2] J. S z é p and F. F o r g o , Einfiihrung in die Spieltheorie, Akadémiai Kiadô, Budapest 1983.

DEPARTMENT OF APPLIED MATHEMATICS HIGHER COLLEGE O F ENGINEERING PODGÔRNA 50, 65-246 ZIELONA GÔRA, POLAND