**To my wife Zinaida I. Model**

**To my daughter Svetlana B. Model and my son Dmitri B. Model**

**To my brother Lazar I. Model**

**In memory of my mom Freida L. Dobrovinskaia and my dad Izrail Z. Model**

**TO THE THEORY OF INFINITE STEP PROCESSES OF SEQUENTIAL DECISION MAKING**

** or**

** MATHEMATICS HAS BEEN MY LIFE**

** A Collection of Remembrances**

** Boris I. Model**

**Toronto (Ontario, Canada), Fremont (California, USA)**

** Oct. 18, 2018 — Nov. 21, 2018**

**Instead of introduction**

Everyone has his or her own remembrances. And I am not an exclusion.

I love mathematics since my early childhood from the time when I did not know yet the word «mathematics» itself. I remember when I was a small child I told once my dad that I love to calculate. I think I said «to calculate» because I did not know then another word to express what I love.

**Cubic Equetion Solution**

For the first time I challenged myself in 1965, in ninth grade of Moscow (Russia) mathematical high school.

I remember once when I studied in ninth grade of the school I thought that if someone had solved cubic equation and had done it already a long ago so why don’t I solve it also. And indeed after about two weeks of continuous effort I succeeded to find general solution of cubic equation.

**Solution of Dolichobrachistochrone Differential Game of Rufus Isaacs**

When I studied in second grade of Faculty of Mechanics and Mathematics of Moscow State University (Russia) I got involved in Differential Games.

In third grade (1968 — 1969) I wholly solved Dolichobrachistochrone Differential Game of Rufus Isaacs which was only partly solved by Rufus Isaacs.

My academic adviser, a wonderful mathematician and a man, Genrikh K. Pozharitsky suggested me to prepare my solution as paper for some mathematical journal.

When he read my draft of the paper he told me that a statement of the problem was missing.

Apparently he meant then that a statement of this specific Dolichobrachistochrone Differential Game was missing in my draft. But I understood his remark so that a statement of Differential Games in general was missing.

I stopped to prepare the paper and left my solution of Dolichobrachistochrone Differential Game of Rufus Isaacs in the form of unpublished manuscript at the Department of Theoretical Mechanics (of Faculty of Mechanics and Mathematics of Moscow State University).

Starting from that moment, from 1970, and till I now I have been keen on different topics of actually the same subject which I have been calling

THE THEORY OF INFINITE STEP PROCESSES OF SEQUENTIAL DECISION MAKING.

By the way in about six years I was told that solution of Dolichobrachistochrone Differential Game of Rufus Isaacs was published recently in a paper. As far as I remember it was actually the same solution as mine at least technically. There was no reference to my solution in that paper.

I don’t know whether the author of that paper used my unpublished solution but I believe that I can make an educated guess.

**Differential Games as Infinite Step Processes of Sequential Decision Making**

The most difficult problem in a Differential Games statement I have seen in a deep internal contradiction between a nature of a game itself and its differential description.

Let us consider the following example.

Let us assume that on plane R^2 a cat (that has velocity C=(c1,c2)) is running after a mouse (that has velocity M=(m1,m2) and that is running from the cat). They begin at a moment t=0 and are to stop at the moment t=1. The aim of the cat is to catch the mouse as soon as possible or at least to be as close as possible to the mouse at the moment t=1. The aim of the mouse is quite opposite. A “solution” of the game is quite obvious: the cat is to run directly to the mouse while the mouse is to run directly from the cat. But the word Solution is written especially in quotation marks, because the main question (or at least one of the main questions) is not only to find a solution, but TO UNDERSTAND first of all what the movements themselves of the cat and of the mouse are. From the point of view of the game itself it can be suggested that the mouse (and the cat also) can choose its velocity at any moment of time. So let us assume, for example that the mouse is choosing some vector M*=(m*1,m*2) at any moment of time, belonging to non-measurable Vitali set V and is choosing another vector M**=(m**1,m**2) (different from M*) at any other moment of time. It’s clear that in this case mouse’s movement is not understandable at all??! So above suggestion (that velocity could be chosen at any moment of time) can make impossible description of the game by means of differential equations!!!

This deep internal contradiction between Game Nature and its Differential Description can be solved, for example by means of assuming that the cat and the mouse can choose their decisions not just at any moment of time but choosing a decision at a moment t(i) the mouse can choose a next one at any moment t(i+1)>t(i) and does not change a previous decision (made at t(i)) till t(i+1) (the same is assumed for the cat also). So time interval [0,1] is divided into countable set of subintervals (ordinal of this partition [0,1] can be any ordinal number from the set of ordinals of countable sets). Such approach seems to be very natural, because it is consistent with the Game Nature and also gives possibility to determine very easily the game by means of Differential Equations (because on each subinterval velocities are constant). The same approach can be used not only in the above game but also in posing Differential Games in general and leads us from Differential Games to a broad class of Infinite Step Processes of Sequential Decision Making.

These results were published in my works:

B. I. Model’, The existences of an overall έ-optimal strategy and validity of Bellman’s functional equation in an extended class of dynamic processes. I; II, Engineering Cybernetics, No.5, 1975, pp. 13 – 19; No. 6, 1975, pp. 12 – 19.

B. I. Model’, A certain class of differential games, Engineering Cybernetics, No.2, 1978, pp. 32 – 38.

Questions arising in such class of well-ordered processes are connected with Set Theory.

On 2003 Dr. Rene Schipperus had made a talk “Differential games on Cardinal numbers” in Seminar of Kurt Gödel Research Center for Mathematical Logic at the University of Vienna.

**Infinite Step Processes of Sequential Decision Making**

On the border of Set Theory and Game Theory there is a broad class of Infinite Step Processes of Sequential Decision Making that can be characterized by the main following property: a Future development of the process depends on the process Present state and does not depend directly on the process Past (these processes have the same nature as for example chess and checkers have: a game Future depends on the game Present state and does not depend directly on the game Past).

Some examples of such Infinite Step Processes give us Differential Games in certain posing and Games

of Search and Completion (subsequently also called Boris Model Games).

For these Processes with the use of Axiom of Choice some generalizations of basic results, which are well known facts in the case of Finite Step Decision Making Processes (for example, existence of a uniformly optimal strategy), can be proved.

These results were published (in Russian) in my book: B. I. Model, Elements of the theory of multistep

processes of sequential decision making, Nauka, Moscow, 1985 (Модель, Борис Израилович. Элементы теории многошаговых процессов последовательного выбора решений — Москва: Наука, 1985)

This book can be found in libraries in different countries or can be read online for free.

Main results of the book are presented in English in the articles:

B. I. Model’, The existences of an overall έ-optimal strategy and validity of Bellman’s functional equation in an extended class of dynamic processes. I; II, Engineering Cybernetics, No.5, 1975, pp. 13 – 19; No. 6, 1975, pp. 12 – 19

and

B. I. Model’, A certain class of differential games, Engineering Cybernetics, No.2, 1978, pp. 32 – 38.

**Games of Search and Completion**

At the beginning I considered Games of Search and Completion only as an example of Infinite Step Processes of Sequential Decision Making. But after some time I understood that these games are extremely interesting by themselves also.

These games can be represented as follows. Suppose we are given the interval (0,1). The points of (0,1) are divided during the game into labeled and unlabeled. Before the game, m points from (0,1) are labeled (m is a natural number). Two players take part in the game, making moves in turn. On the first move, the first player searches through the labeled points, adding k of them to the unlabeled ones, k < m. On the second move, the second player augments the set of remaining m — k labeled points, adding k unlabeled points to it, etc. (the move number runs through the set of natural numbers). We say that a point remains labeled after the end of the game if it remains labeled starting from some move. The first player tries to guarantee the least number of points that can remain labeled after the end of the game, whereas the goal of the second player is quite the opposite.

We consider the most difficult case where the players know when choosing a move the current state and the number of the move. With the use of the continuum hypothesis the following unexpected statement can be proved: the first player can guarantee that for all m and k at most one point will remain labeled after the end of the game.

The games of search and completion played jointly as simultaneous parties are of particular interest. We also consider the most difficult case where the players know when choosing a move the current state and the number of the move. With the use of the continuum hypothesis it is possible to reveal the unexpected possibilities of the first player, namely he can guarantee that not more than one point will remain labeled after the end of the simultaneous parties. Thus, the least guaranteed result of the simultaneous parties turns out to be not worse than the least guaranteed result of any constituent game.

These games have the same nature as for example chess and draughts have: a Future depends on the Present and does not depend directly on the Past.

But the least guaranteed result of these simultaneously played and completely independent games turns out to be less than the sum of the least guaranteed results of constituent games (and not equal to this sum as it could be supposed and as it is in the case of chess or draughts!).

These results were published in my book: B. I. Model’, Games of search and completion, Journal of Mathematical Sciences, Vol. 80, No 2, 1996, pp. 1699 — 1744, Plenum Publishing Corporation, New York.

Games of search and completion (they were subsequently also called Boris Model Games) have been cited and received further investigation in:

U. Abraham, R. Schipperus, Infinite games on finite sets, Israel Journal of Mathematics, 2007. (We study a family of infinite games with imperfect information introduced by B. Model, Boris Model Games on Cardinals.)

D. H. Fremlin, Give a penny, take a penny, University of Essex, UK, 2009.

F. Piccoli, Partially Ordered Sets and Their Invariants, University of East Anglia, UK, 2014. (We investigate Model Games. Model games are set theoretical games on infinite sets introduced by B. Model.)

These further investigations were reported in a number of talks in different seminars, among them:

Uri Abraham. Boris Model Games on Cardinals. Ernest Schimmerling Department of Mathematical Sciences. Carnegie Mellon University. Pittsburgh, Pennsylvania, USA. 2004, January 20.

J. Pochrybniak. On the work of Rene Schipperus and Uri Abraham «Boris Model games on cardinals». Faculty Mathematics, Informatics and Mechanics. University of Warsaw. Warsaw, Poland. 2005, November 10.

Rene Schipperus. The Boris Model Game. Department of Mathematics and Statistics. University of Calgary. Calgary, Alberta, Canada. 2007, October 1.

Earlier Games of search and completion were published in russian in my books:

B. I. Model, Elements of the theory of multistep processes of sequential decision making, Nauka, Moscow, 1985 (Модель, Борис Израилович. Элементы теории многошаговых процессов последовательного выбора решений — Москва: Наука, 1985)

and

B. I. Model, Multistep processes of search with augmentation, 1990 (Модель, Борис Израилович. Многошаговые процессы перебора с пополнением — М. : Наука, 1990).

Some part of this book was also published in Hebrew in:

Dr. Boris Model, Processes of search with augmentation in infinite step games, Mathematical Enrichment in Various Topics, Booklet No 31, The Weizmann Institute of Science in Rehovot, 1996

ד׳׳ר בוריס מודל, תהליכי הוצאה-הכנסה) במשחק בעל מספר אינסופי של צעדים, העשרה מתמטית בנושאים שונים,

חוברת מס׳ 31

.(1996 ,מכון ויצמן למדע — רחובות,

**An open question**

The question when information of process Past history additional to information of process Present

state, for example, perfect information, does not improve and when such information does improve best

guaranteed result in considered infinite stage processes (like chess, checkers, Games of Search and Completion) comparing with best guaranteed result provided by information of only process Present state has remained an open question.

For example, for chess and checkers (even without rules of stopping) information of game Past history additional to information of game Present state does not improve best guaranteed result but for Infinite Stage Games of Search and Completion, having just the same structure as chess and checkers, information of game Past history additional to information of game Present state does improve best guaranteed result.

So the question is: what criteria distinguish between processes for which information of process Past history in addition to information of process Present state does not improve best guaranteed result, and processes for which information of process Past history additional to information of process Present state does improve best guaranteed result?

See:

B. I. Model’, The existences of an overall έ-optimal strategy and validity of Bellman’s functional equation in an extended class of dynamic processes. I; II, Engineering Cybernetics, No.5, 1975, pp. 13 – 19; No. 6, 1975, pp. 12 – 19.

B. I. Model’, Games of search and completion, Journal of Mathematical Sciences, Vol. 80, No 2, 1996, pp. 1699 — 1744, Plenum Publishing Corporation, New York.

Оригинал материала находится **здесь.**