Probabilistic Methods in Discrete Mathematics: Proceedings of the Fifth International Petrozavodsk Conference, Petrozavodsk, Russia, June 1–6, 2000 [Reprint 2020 ed.] 9783112314104, 9783112302835

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Probabilistic Methods in Discrete Mathematics

A L S O AVAILABLE:

Probabilistic Methods in Discrete Mathematics

Proceedings of the Fourth International Petrozavodsk Conference Editors:V.F. Kolchin,V.Ya. Kozlov.Yu.L Pavlov andYu.V. Prokhorov

1997; x+368 pages ISBN90-6764-245-2 (hardcover) Price: EUR 172 / US$ 201

Probabilistic Methods in Discrete Mathematics

Proceedings of the Third International Petrozavodsk Conference Editors.V.F. Kolchin.V.Ya. Kozlov,Yu.L Pavlov andYu.V. Prokhorov

1994; x+466 pages ISBN90-6764-158-8 (hardcover) Price: EUR 216 / USS 252

PROBABILISTIC METHODS IN DISCRETE MATHEMATICS PROCEEDINGS OF THE FIFTH INTERNATIONAL PETROZAVODSK CONFERENCE Petrozavodsk, RussiaJune I - 6,2000

EDITORS:

V.F. Kolchin, V.Ya.Kozlov, V.V. Mazalov, Yu.L. Pavlov and Yu.V. Prokhorov

III MSPIII Utrecht • Boston • Köln »Tokyo, 2002

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

Tel: +31 30 692 5790 Fax: +31 30 693 2081 [email protected] www.vsppub.com

© VSP BV 2002 First published in 2002 ISBN 90-6764-359-9

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CONTENTS Contents

v

Preface

ix

Injective mappings of words which do not multiply symbol skip and insertion errors M. M. Glukhov

1

Models for computer D. Bogdanovich, A. A.security Grusho, and E. E. Timonina

9

Probability distributions of the number of configurations and discordances of random permutations from regular cyclic classes V. N. Sachkov

23

Equilibrium in an arbitration game V. V. Mazalov, A. A. Zabelin, and A. S. Karpin

41

Dynamic games with random duration and uncertain payoffs L A. Petrosyan, M. Kultina, and D. Yeung

47

On stopping games when more than one stop is possible K. Szajowski

57

Local structure of a random polynomial over finite field G. I. Ivchenko and Yu. I. Medvedev

73

Galton-Watson forests Yu. L Pavlov and I. A. Cheplyukova On the existence of a giant component in schemes of allocating particles V. F. Kokhin Statistical estimation of distributions of sampling characteristics in the case of gamma families R. A. Abusev On estimation and group classification in the space of a sufficient statistic of the negative binomial distribution R. A. Abusev and N. V. Kolegova On the representation of bent functions by bent rectangles

93 99 105 113

S. V. Agievich

121

On of aD.lattice in local limit theorems T. A.destruction Aziarov and V. Akimov

137

V

Isoperiods of output sequences of automata A. V. Babash

147

On joint application of statistical tests S. P. Chistyakov

159

Chebyshev systems and generalised convex games versus nature A. Yu. Golubin

163

On the necessary number of observations needed for unique detection of insertions in the multinomial scheme A. V. Ivanov

179

Local limit theorems for an array scheme and Galton-Watson forests N. I. Kazimirov

189

Asymptotic behaviour of the waiting times in schemes of allocating particles in groups of random sizes E. R. Khakimullin and N. Yu. Enatskaya

197

Random partitions and their applications K. N. Khovanov, V. V. Kornikov, I. A. Seregin, and N. V. Khovanov

211

Random partitions of a set and the generalised allocation scheme A. V. Kolchin

215

On a problem of A. N. Kolmogorov M. M. Leri

219

Estimation of stochastic dependence and testing for the ^-dimensional uniformity by sample characteristic functions A. V. Martinevskii and Yu. S. Kharin

227

Cyclotomic integers and discrete logarithms in GF(p7) D. V. Matyukhin

237

Limit distribution of the number of leaves of a Galton-Watson forest T. B. Myllari

257

The Bayes risk asymptotics under testing composite hypotheses on Markov chains A. V. Nagaev

273

A generalised MTP 2 and a sequential stochastic model on a partially observable Markov process T. Nakai

291

vi

An optimal dichotomous search P. Neumann

303

Construction of the hedging strategies for one model of (S,S)-market I. V. Pavlov and N. P. Krasii

311

On application of statistical methods to authorship attribution A. A. Rogov and Yu. V. Sidorov

327

On the distribution of the number of occupied one-place cells by particles of two types P. N. Sapozhnikov

337

Characteristics of a random system of Boolean equations with non-regular left-hand side A. V. Shapovalov

345

On V. theSokolov problem of A. optimal stack control A. and A. Lemetti

351

On the dimension of Bayesian networks with latent variables S. V. Stafeev

367

On the asymptotics of the probability of large deviations in the equiprobable schemes of allocations A. N. Timashov

371

On asymptotic expansions of the number of allocations of particles to cells with restrictions on the sizes of cells A. N. Timashov

379

vii

Preface The Fifth International Petrozavodsk Conference 'Probabilistic Methods in Discrete Mathematics* was held at 1-6 June 2000 in Petrozavodsk in Russia. The conference was organized by the Steklov Mathematical Institute of the Russian Academy of Sciences, The Institute of Applied Mathematical Research of the Karelian Scientific Centre of the Russian Academy of Sciences, and the Petrozavodsk State University. The Organizing Committee of the Conference included V. Ya. Kozlov (the chairman, Moscow), V. F. Kolchin (vice-chairman, Moscow), V. V. Mazalov (vice-chairman, Petrozavodsk), Yu. V. Prokhorov (Moscow), B. A. Sevastyanov (Moscow), A. B. Zhizhchenko (Moscow), A. S. Fomin (Petrozavodsk), G. I. Ivchenko (Moscow), V. I. Khokhlov (Moscow), Yu. I. Medvedev (Moscow), V. G. MikhailoV (Moscow), Yu. L. Pavlov (Petrozavodsk), A. D. Sorokin (Petrozavodsk), V. A. Vatutin (Moscow), A. M. Zubkov (Moscow). During the last fifty years, the interest in probabilistic problems of discrete mathematics has grown steadily both in our country and abroad. The four previous conferences were held in Petrozavodsk in 1983, 1989, 1992, and 1996. The Fifth Petrozavodsk Conference received about 100 participants, including about 70 guests of Petrozavodsk, among which were participants from Germany, Japan, Lithuania, Poland, Belarus, the United States of America, Uzbekistan. The themes of the Petrozavodsk Conference cover almost all areas of probabilistic discrete mathematics. As usual, the lectures presented at the Fifth Petrozavodsk Conference were devoted to probabilistic problems of combinatorics, statistical problems of discrete mathematics, theory of random graphs, systems of random equations in finite fields, some questions of information security. At this time, a considerable part of the lectures concerned game theory and mathematical statistics. During the three working days of the Fifth Petrozavodsk Conference, 12 plenary and 60 sectional lectures were given. The following plenary lectures were presented (the lectures are listed in the order they were read at the conference): M. M. Glukhov (Moscow), Injective mappings of words which do not multiply the symbol skip and insertion errors; A. A. Nechaev (Moscow), Asymptotic properties of linear recurring sequences over commutative rings; A. A. Grusho, E. E. Timonina, and D. Bogdanovich (Moscow), Models for computer security; V. N. Sachkov (Moscow), Probability distributions of the number of configurations and discordances of random permutations from regular cyclic classes; V. V. Mazalov, A. A. Zabelin, and A. S. Karpin (Petrozavodsk), Equilibrium in an arbitration game; L. A. Petrosyan, M. Kultina (St. Petersburg), and D. Yeung (Hong-Kong), Dynamic games with random duration and uncertain payoffs; M. Tamaki (Japan), The optimal stop in a ballot problem; K. Szajowski (Poland), On a stopping game when more than one stop is possible;

ix

G. I. Ivchenko and Yu. I. Medvedev (Moscow), Local structure of random polynomials over finite fields; Yu. L. Pavlov and I. A. Cheplyukova (Petrozavodsk), Galton-Watson forests; V. F. Kolchin (Moscow), On the existence of a giant component in generalized schemes of allocating particles; V. I. Afanasyev (Moscow), Branching processes in a random environment. The plenary lectures suggested for publication are presented in the Proceedings in the above mentioned order. All sectional reports are arranged in the alphabetic order of the author names. The papers presented in the Proceedings reflect the current state of art in the probabilistic discrete mathematics and contain the information which is of interest to those who work in theoretical and applied areas of discrete mathematics. The Fifth International Petrozavodsk Conference was supported by the Russian Foundation for Basic Research. The conference was also supported by the Karelian Scientific Centre of the Russian Academy of Sciences, the Petrozavodsk State University, the Science Publishers TVP, and the Editorial board of Discrete Mathematics and Applications. The Organizing Committee has the pleasant duty to thank the sponsors of the Fifth Petrozavodsk Conference without whose support the conference could be not held. The Organizing Committee greatly appreciates the efforts of the Petrozavodsk colleagues who took up the organizing duties in Petrozavodsk. We are deeply indebted to VSP International Science Publishers who agreed to publish the Proceedings. We wish also to express our gratitude to all participants of the Conference and invite all of them to take part in the next Petrozavodsk Conference which will be held in 2004. V. F. Kolchin

X

Probabilistic Methods in Discrete Mathematics, pp. 1-7 V. F. Kolchin et al. (Eds.) © VSP 2002.

INJECTIVE MAPPINGS OF WORDS WHICH DO NOT MULTIPLY SYMBOL SKIP AND INSERTION ERRORS M. M. GLUKHOV Moscow,

RUSSIA

Abstract — We continue the investigations devoted to the description of injective mappings of words which do not multiply distortions of one kind or another (substitution, deletion or insertion of a symbol). In this paper, we give a complete description of all injective mappings of words over one alphabet into words over another alphabet which do not multiply symbol skips and insertions.

This paper continues the series of papers [1,2, 3] devoted to the description of injective mappings of words which do not multiply distortions of one kind or another (substitution, deletion, or insertion of a symbol). This property is of great interest in cryptology, where it is desirable to use mappings such that small distortions in a ciphered text imply only small changes in the deciphered information. In this paper, we give a complete description of all injective mappings of words over one alphabet into words over another alphabet which do not multiply symbol deletions and insertions. For a precise formulation of the problem we need the following notations. Let be a finite alphabet with n > 2 symbols, £2* be the set of all words over the alphabet P = Q denote the graphical equality of words P, Q, l(P) be the length of the word P, p(P, Q) be the Hamming distance between the words P, Q of equal lengths, S(P, Q) be the number of letters which have to be removed from the word P in order to obtain the word Q, S*(P, Q) = S(Q, P), d(P, Q) the minimal number of deletions and insertions of letters needed for obtaining the word Q from the word P, I(M, N) be the set of all injective mappings of the set M into the set N, Ps be the set obtained from P by replacing all the letters of P by their images under the mapping g € 1(0., and finally, let n(P) be the inversion of the word P, that is, H(aia2

...a„)

= a„...

0201.

The functions p, 8, and 5* characterise the number of errors such that substitutions, skips, and insertions of letters, respectively, and the function d(P, Q) characterises the number of insertions and skips of letters. Note that the functions p, 8, and 8* are partially defined. The function p is defined only on the pairs of words of equal lengths, the function 8 (8*) is defined only on the pairs of words such that the second word is obtained from the first by

M. M. Glukhov

2

deletion (insertion) of some letters. The function d is defined for all words and is a metric on the set This metric is widely used in various applications (see [4]). We say that a mapping • £2* does not increase the value of a function ij/ if the following condition is fulfilled: for any pair of words P, Q e £2*, for which the function

2, then inclusion (2) holds, that is, / ( n . O i . d ) C 7(i2,fti, 2 and n -*• 00, then according to Theorem 1 the random variable $nQ.\d\,... , dm) has in limit the Poisson distribution with parameter k = m. Consider now the case I = 2. Without loss of generality, assume that d\ = ... = dy = 2,

di > 2,

i = y + 1,...

, m.

36

V. N. Sachkov

Note that the transitions of the permutation s from [/]"/', which are common to transitions of permutations s i , . . . , s m , correspond to critical runs if and only if they correspond to common cycles of s and si,... ,sY. Any transition in a two element cycle determines uniquely the cycle. Therefore non-critical runs of transitions correspond to common transitions of the permutation s and permutations i y + i , . . . , sm. Thus, the transitions of s corresponding to critical runs are common transitions of s and the permutations i i , . . . , sY which are determined by the coinciding cycles corresponding to these transitions, and the transitions of s corresponding to non-critical runs are common transitions of s and the permutations 5y + l , . . . , sm. For I = 2, (23) yields the equality P{£«(2;

d\

dm)=r} r/2 = J^P{en(du...

,dy)

-

d

Y + u

... ,d

m

)=r-2v),

(55)

v=0

where r = 0 , 1 , . . . ,n and f i ^ ( 2 ; dy+1,... , dm) is the random variable equal to the number of transitions corresponding to non-critical runs of the random permutation s from [2]("/ 2 , which are common to transitions of the permutations , . . . , s m under the condition that v cycles in s are fixed. Let A (i, j ) be the event that a random permutation s from [2]"/ 2 with v fixed cycles has the transition (i, j ) common to the transition in the permutations sy+\,... , sm. Then P ( A ( i i , ; i ) n . . . n Aiik,

jk))

=

2 (n/2 V)

*

• oo, P{FN(1)(2; d\,...

, dm) =/•}—>

(61)

(m-rye-e»-r\

Now the assertion of Theorem 2 follows from (55) if we take into account (25)'and (61). The Corollary 2 follows directly from Theorem 2. The following assertions are also corollaries of Theorem 2. COROLLARY 4 . Under the conditions of Theorem 2, if y = 0 and n a n y â x e d r = 0, 1 , . . .

PTÈ»(2;DI

oo, then for

mr —e~m. r\

dm)=r)^

(62)

COROLLARY 5. Under the conditions of Theorem 2, if y = m and n -*• oo, then the random variable takes only even values and for any fixed r = 0 , 1 , . . . P{FN(2; dx COROLLARY 6 .

dm)

= 2r)

(63)

r\

If n -*• oo, then the limit distribution of £ „ ( 2 ; dY+\,...

composition of the limit distributions ofdn(2; dY+1,...

, dm) is the

, dm) andfi' ) (2; dY+1,...

, dm).

PROOF. Let P(t\m, y) be the generating function of the limit distribution of the random variable Ç„(2; dY+1,... , dm). It follows from (52) that Pit; m, y) = e('»->')('-i)+y(r2-i)/2

(64)

Let Pi (i ; m, y) and Pi(t\ m, y) be the generating functions of the limit distributions of 6n{2; dY+1,... , dm) and dY+\,... ,dm), then according to (25) and (55) Pl(f,m,y)=e^2-^2,

(65)

Pl(t\m, y) = e^-^H'-D.

(66)

It follows from (64), (65), and (66) that (67)

P(t\ m, y) = Pi(t;m,y)P2(t;m,y). Lemma 5 follows from the definition of the composition of distributions (see [7]). LEMMA 6. The kth binomial moment of Ç„(2; dY+\,...

B k ( m , y ) = J^J Y : Yv!(Jt u l —, 2v)!

, dm) is equal to

¿ = 0,1,...

(68)

38

V. N. Sachkov

The mathematical

expectation

E and the variance E - m ,

D are equal,

respectively,

to

(69)

D = m + y.

PROOF. Indeed, let B(f; m, y) be the generating function of the binomial moments of the limit distribution of £„(2; dY+\,... ,dm), then, as follows from (64), 00 B { f , m,y)

y)tk

= J ^ Bk(m,

=

e

m t +

^

2

.

(70)

*=o Hence, equation (68) is true. Formulas (69) follow from the equalities E = B\ = m,

B2 = (m2 + y)/2,

D = 2B2

+

fli -

B2.

PROOF OF COROLLARY 3. We prove formula (8) by induction on m. For m = 1 the formula is true. Suppose that it is true for m and prove that it is true for m + 1. If the adding permutation sm+\ belongs to the cyclic class [2]" / ' 2 , then the number of such permutations discordant to i i , . . . , sm is equal to e-(m-y/

2)

(l+o(l)) ^C+7

(71)

(n/dm+iV-

as n -¥• oo. Consequently, Um

+ 1, y + 1) = L(m, y)e~"-rm

wl

dnJ^\n!dm+,)!

(l+o(l)).

(72)

It follows from (8) and (63) that l ( m

+ 1 . X + » - - p {-(•

rH(T)| n

+» y, the parties appeal to an arbiter A to settle the conflict. There exist different arbitration schemes [1-5]. In this paper, we consider the so-called final-offer arbitration scheme [2-3]. Let the arbiter's judgement be denoted by a. Then the offer nearest to the point a is selected from the offers x and y. If a is fixed, then it is evident [2] that the equilibrium is the pair of strategies (a, a). If however the arbiter's judgement is flexible, then the task becomes non-trivial. Further we consider the case where the arbiter's solution is random and is distributed on two points a¡ and c¡22.

PROBLEM STATEMENT

Let us describe the formal statement of the problem. To simplify the expressions, suppose that a is a random variable that assumes the values a\ = — 1 and 0/2, x,

ifx




y.

ifx

>

a,

ifx

>

y.

y. 1* - a |

\y — a —a y. 1*

The equilibrium in the game belongs to the class of mixed strategies.

V. V. Mazalov, A. A. Zabelin, and A. S. Karpin

42

H

H = H{x,

F)

c +

Figure 1. 3.

THE SOLUTIONS

Note in the very beginning that by virtue of the symmetry the value of the game equals zero, and the optimal strategies must be symmetric about the origin of coordinates. Therefore it suffices to construct an optimal strategy for one of the players, for example, M. We denote a mixed strategy of the player M by F(y). Suppose that the support of the distribution F lies on the negative semiaxis. Then it follows from the conditions of the game that for all x < 0 the payoff of the player Z, is H(x, F) < 0, and for x > 0 his payoff is equal to OO

fOC

^

(2) / 2 ydF(y) + F(2-x)x + J'2—x ydF(y) - H We are looking for the distribution function F(y) such that its support is concentrated within the interval [—c — 4, — c], where 0 < c < 1, and the payoff function H(x, F) is a constant equal to zero in the interval [c, c + 4] and is negative for all other x (see Fig. 1). From (2), we find that in the interval [c, c + 2] H(x,

F)

F(—2

- x)x

+

~c 2—x

ydF(y)

+

1 l

~x,

X

6 [c, C + 2].

(3)

Under the assumption that H (x, F) is a constant in the interval [c, c+2], the distribution function F{y) is uniquely determined. Indeed, differentiating (3) and setting the derivative equal to zero, we obtain ^

= \ i - F \ - 2 - x)x + F ( — 2 - x) + ( - 2 - x)F\-2

- * ) ) + ! = ( ) .

Substituting — 2 — x = y in (4), we get the differential equation 2F'(y)(y

+ 1) = - F ( y ) + 1,

y e [ - c - 4, - c - 2],

(4)

43

Equilibrium in an arbitration game

whose solution gives the distribution function F(y) on the interval [—c —4,—c — 2] equal to F(y)

= - 1 +

, V - y

y

- 1

e [ - c - 4 , - c - 2 ] ,

where b is a constant. From the condition F(—c — 4) = 0 we obtain that b = v 3 + c and F(y)

= -1 +

y e [ - c - 4 , - c - 2 ] . V - y

(5)

- 1

The requirement that the function H{x, F) is a constant in the interval [c + 2, c + 4], which in this case is written as H{x,F)

= \ f C I J-c-4

y

d F ( y ) + ]-F(2-x)x •i

+ l- f ° ydF(y), 4 J2-x

x e [ c + 2,c

+ 4],

(6)

implies that ^

dx

= i(-F'(2

- x)x + F(2 - x ) + (2-

2

x)F\2

- x)) = 0.

Substituting 2 — x = y, we obtain the differential equation 2F\y)(y

-

1) = - F ( y ) ,

y e

[ - c - 2 , - c ] ,

and, taking into account the fact that F(—c) = 1, find that F(>>) = ^ 2 1 , VI - y

y

e i-c

-

2, -c].

(7)

We demand that the function F(y) is continuous and glue together functions (5) and (7) at the point y = — c — 2. Thus, we obtain the equation VI + c V3 + c ~

l

|

V3 + c Vl+c'

which is equivalent to the quadratic equation (l+c)(3 + c)=4,

(8)

whose solution can be represented as c = 2z - 1 « 0.236, where z is the golden section of the interval [0,1] (the solution of the quadratic equation z

2

+ z -

1 = 0).

Thus, we built a continuous distribution function F(y), y e [—c — 4, —c], such that the payoff function H(x, F) in the interval [c, c+4] is a constant. In order that it be the optimal strategy of the player M, it remains to prove that the function H(x, F) has the form shown in Fig. 1, that is, its plot lies below the abscissa.

44

4.

V. V. Mazalov, A. A. Zabelin, and A. S. Karpin

OPTIMAL STRATEGIES

The solution of the game problem is found in the following theorem. THEOREM 1. The optimal the arbitration

game

strategies

with payoff

function

G and F of the players (1) are of the

0, G(x)

=

x e (-oo,

where

c = V5

=

in

c],

1 - V l + c / V * + l,

xe(c,c

2 - V3 + c / V * - 1,

x e (c + 2 , c + 4], '

1,

x e (c

0, F(y)

L and M, respectively,

form

+ 2],

(9)

+ 4, oo),

y e (-oo,

-c

- 4],

- l + V3 + c / V - : v - 1, vT+c/yr^y,

ye(-c-4,-c-2], ye(-c-2,-c],

1,

ye(-c.oo),

(10)

— 2.

PROOF. TO prove the theorem, it is enough to show that H{x,F) < 0 for all X E R . For x < 0 this inequality is obvious, since y < 0 almost surely, and hence, by virtue of definition (1) H{x, F) is negative. As mentioned above, the function H(x, F) is a constant in the interval [c, c + 4]. Let us find this value. It follows from (3) that H{x,F)

= H{c

+ 2)+l-

+ 2) = ]-F{-c-A){c I

= \(y +

c

+

2).

C

[

ydF(y)

+ h c + 2)

I J—c—4

x e [c,c

I

+ 2],

(11)

where y is the mean value of the random variable y which has distribution (5), (7). A simple calculation yields -c-2 c-2

-c-4

yd

vT+c

n— = - c - 2.

v i -

y

It follows from (11) that H(x,F)=

jc e

0,

[c,c

+

2],

Similarly, in the interval [c + 2, c + 4], the function H(x, F) = 0. If x > c + 4, then the function H(x,F) has the form (6); its derivative after substituting 2 — x = y takes the form ^ dx

= U f \ 2 - x)(2 2

- 2x) + F(2

- x))

= \{F\y){2y 2

- 2) +

F(y)).

Substituting the expression for F from (5), we obtain dH dx

_

1 /

2y

2 \ J = J = \ y + l

\ )'

y

< - c - 2 .

(12)

45

Equilibrium in an arbitration game

Function (12) is monotone increasing in y from the interval [—c — 4, — c — 2] and attains its maximum at the point y — — c — 2. By virtue of (8) the maximum equals 1 A/3+72(-c-2) 2 ^yr+7 c i

1 (c + 3) 2 2(i+c)2

\ _ )

It follows from the above that the function H(x,F) its value in the point x = c + 4 is zero. Hence, H(x,F)
c

+ 4.

However,if* € [0, c], then H(x, F) is of form (3) and its derivative (4) after the substitution —2 — x — y takes the form ^ ax

= l(F\y)(2y

+ 2) + F(y)

2

+ l),

y > - c - 2.

(13)

Substituting the expression for F into (13), we obtain dH

^

1 / =

J T T c

, ^

m

, */T+c

2 ( T O ^ ?

,

V ^ y

j '

This function is monotone increasing in y from the interval [—c — 2, —2] and attains its minimum at the point y = —c — 2. The minimum is equal to l /

vTTc

, „

5 I t S t ^ - *

VTT7 ~

V Ï + c

\ +

)

1/2(1-c) =

2

(

(

W

\ +

0

and is positive. Hence, the function H{x,F) increases in the interval x e [0, c] and its value in the point JC = c equals zero. Therefore, the function H(x, F) < 0 for x 6 . Unfortunately, in (4) the non-negativity of Yi (&), which is very important in applications, cannot be guaranteed. For f T , x e [fo, oo) (£ r is d C V P in the subgame starting from i ( r ) ) , we define f by the formula

*-/.

p oc

fcr

2)) can be constructed by the recursive procedure as $\x,yuy2)

= Qn^yJ>2) (Sn(x, yi, y2)) are26 ® 2S ® ^-measurable (see [3]). If Player 1 is the first to accept x at moment n, then his expected gain is h(n,x,yuy2)

= E ^ i ^ , (*, yi, Xi),

(3)

63

On stopping games

for n = 0 , 1 , . . . , N — 1 and h(N, x,yi) expected gain of Player 1 is

= fo(x, >>i). When Player 2 is the first, then the

H(n, x, yu yi) = E ^ S ^ . l ^ i ' B . f o r n = 0, 1 , . . . ,N — \ and H(N,x,

y\, yi) = f i x , y\,yi).

(4) The functions h(n, x,

y\,yi)

and H ( n , x , y i , >2) are well defined. They are 28 ® 31 ® 38-measurable with respect to the second and third variable, h(n, X\, yi, >2) and H(n, x, yi, Xj) are integrable with respect toPx. Let An and MN be the sets of strategies in ^pj' for Player 1 and Player 2, respectively. For A. € An and ft e MN, we define the payoff function r (yi, X, fi), setting , . . ¡h(k,Xx,yuXx)I[xtí} *

10

'

iíX < N or fi < N, u •

otherwise,

(5) where is a characteristic function of a set A. As a solution of the game we search for equilibrium pair (A.*, /¿*) such that < Rm(x,yhX*,n*)

RV\X,yltk,li*)

< Rm(x,yx,\*,n)

(6)

for all x e E, where R(x, yi, X, ¡1) = Exr(y\, X, ¿¿)- By (5) we can observe that with the sets of strategies AN and MN is equivalent to the Neveu's stopping problem [10] considered by Yasuda [26] if the sets of strategies are extended to the set of stopping times not greater than N + 1 and the payoff function is (5). Because the Markov process is observed here, one can define a sequence lui^.yi),

n = 0,l,...,tf + l

on £ x £ by setting u;^.,(jc, _yi) = 0 and (l)/

^

V n \ x , y i ) = val

(h(n,x,y\,x) \H(n,x,yi,x)

h(n,x,y\,x)\ (1) y

Tw n^(x,yi)J

(7)

for n = 0 , 1 , . . . , N, where

and val A denotes the value of the two person zero-sum game with the payoff matrix A (see [8,26]). To prove the correctness of the construction, let us observe that the payoff matrix in (7) has the form (8) - C : )

where a, b, c are real numbers, and the rows and the columns are s (stop) and / (forward). By direct checking we obtain the following assertion.

64

K. Szajowski

LEMMA 2. The two person zero-sum game with payoff matrix A given by (8) has the equilibrium point (e, S) in pure strategies, where (s,s) ,

^

(£,d) =

ifa>b,

s

( >/ )

ifc, i = N, N — 1 , . . . , n + 1 are measurable. We introduce the notation 4? = {(*.yi)

eExE:

h(n,x,yi)

An={(x,yi)

eExE:

h(n,x,y

An

= {(*,

YI)

>

H(n,yi,x)},

i) < H(n,yux),h(n,x,yx)

€ E x E: h(n, x,yi)

< H(n, y\, x), H(n, yux)

> Tw^lx{x,y < T w ^ x ,

i)}, YI)J

and For the sets A*s, Asn(,

e 2ft ® 28, by Lemma 2, we have

yi) = h(n, x, yi)(IA»(x,

yi) + IA*(x, yi)) + H(n, y\,x)lAu(x,

yi)

Hence it follows that y\) is 28 ® 28-measurable. On the set [Xn = _yi}, we define v1n*=infUXk,yl)eA^UA% n