Foundations of Management, Vol. 5, No. 1 (2013), ISSN 2080-7279 DOI: 10.2478/fman-2014-0002 21 SEVERAL REMARKS ON BANACH–MAZUR GAMES AND ITS APPLICATIONS Ewa DRABIK Faculty of Management, Warsaw University of Technology, Warsaw, Poland e-mail: [email protected] Abstract: Certain type of perfect information games (PI-games), the so-called Banach–Mazur games, so far have not been applied in economy. The perfect information positional game is defined as the game during which at any time the choice is made by one of the players who is acquainted with the previous decision of his opponent. The game is run on the sequential basis. The aim of this paper is to discuss selected Banach–Mazur games and to present some applications of positional game. Key words: Banach–Mazur games, modelling of economic phenomena, Dutch auction, chess. JEL Classification: C72, D44. AMS Classification: 62P20, 91B26 1 Introduction Mazur, Stanisław Ulam and Hugo Steinhaus) for jotting down mathematical problems meant to be solved. The The most seriously played games of perfect informa- Scottish Book used to be applied for almost 6 years. Many tion (which we call PI–games) are chess. The principles problems presented therein were created in previous years of chess laid foundations for development of the latest and not all of them were solved. After the World War II, software, which process lasted many years. Perfect in- Łucja Banach brought the Book to Wrocław, where it formation means that at each time only one of the player was handwritten by Hugo Steinhaus and in 1956 sent to moves, which means the game depends only one of their Los Alamos (USA, Mexico) to Stanisław Ulam. Ulam choices, they remember the past, and in principle they translated it into English, copied at his own expense and know all possible future of the game. The first published dispatched to a variety of universities. The book in ques- paper devoted to general infinite PI–games is by Gale tion proved to enjoy such a great popularity that it was and Stewart (1953), but the first interesting theoretical soon published and edited – mainly in English [1]. The infinite PI-game was invented by S. Mazur about 1935 Scottish Book presents the following game no. 43 elabo- in the Scottish Book [7]. Positional games were created rated by Stanisław Mazur [7]. in 1940s by a prominent range of Polish mathematicians, Example 1. (Mazur) belonging to the Lwow School of Mathematics. Owing to the authors’ names they are otherwise known as Ba- Given is a set E of real numbers. A game between two nach–Mazur games. players I and II is defined as follows: player I selects an arbitrary interval d1, player II then selects an arbitrary This paper aims to address the most common versions segment (interval d2 contained in d1; then player I in turn of Banach–Mazur games, their modifications and their selects an arbitrary segment d3 contained in d2, and so on. possible applications. Player I wins if the intersection d1, d2, ..., dn,... contains a point of set E; otherwise he loses. If E is complement of a 2 The Banach-Mazur games and their appli- set of first category, there exists a method through which cations player I can win; if E is a set of first category, there exists a method through which player II will win. The relevant issue in the area of competitiveness is games Problem. It is true that there exists a method of winning displaying an infinite number of strategies. The over- for the player I only for those sets E whose complement whelming majority of dilemmas related to the above is, in certain interval, of first category; similarly, does a games were defined in the period from 1935 to 1941 method of win exist for player II of E is set of first cat- and incorporated into the so-called Scottish Book. The egory (see [5])? Scottish Book referred to a notebook purchased by a Addendum: Mazur’s conjecture is true. wife of Stefan Banach and used by mathematicians of the Lwow School of Mathematics (such as Stanisław Modifications of Mazur’s game are follows. Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw University of Technology 2021-09-30 22 Ewa Drabik Example 2. (Ulam) - player I chooses p0 ∈ P, next player II chooses ∈ ∈ There is given a set of real numbers E. Players I and II p1 P, than player I chooses p2 P, etc. give in turn the digits 0 or 1. Player I win if the number There is a function f : Pω → ℜ, such that the end player formed by these digits in given order (in the binary sys- II pays to player I the value f (p0, p1…) tem) belongs to E. For which E does there exists a method Definition 1. The triple〈 A,B,φ〉 is said to be game of per- of win for player I (player II)? fect information (PI-game) if there exists a set P such that Example 3. (Banach) A is set of all function A = a : P n , where P 0 = φ , There is given a set of real numbers E. The two players { } n<ω A = a : P n , where P 0 = φ , I and II in turn give real number which are positive and { } n<ω such that a player always gives a number smaller than the last one given. Player I wins if the sum of the given B = b : P n P , → series of numbers is an element of the set E. The same 0<n<ω B = b : P n P , → question as for Example 2. 0<n<ω and there exists a function f : Pω → ℜ such that φ (a, b) = Example 4. (some popular modification Banach–Ma- f(p)=0 if p/X f (p0, p1…) where p0∈ = a(φ), p1 = b(p0), p2 = a(p1), p3 = zur game) b(pf(,p p)=0)=1), p = if a(pp/p , p X). 0 2 4 1 ∈ 3 Two players choose alternatively one digit from the set 0, f(p)=1 if p X ∈ 1, …, 9. Through their choices they generate an infinite sequence of digits, e.g. 5, 7, 9, 1, … Such a sequence may be denoted by a number 0.5791...∈[0, 1]... Before the game begins, a subset X of the section [0, 1] is to be defined. Player I win provided that the mutually generated A = a : P n Figure, where 1. PI-gameP 0 = φ , { } number belongs to the set concerned. Player II wins if the n<ω ω A game 〈A,B,φ〉 defined in this way will be denoted〈 P , f〉 number at issue does not fall within the set in question. or 〈Pω, X〉. The conclusion seems inescapable that the above game The sequence p = (p , p ) be called a game, any finite B = b : P n 0 P1… , has a winning strategy. One may assume that at the begin- → n sequence q0 <n<ω= (p0,…, pn−1) ∈ P is called position. ning the players should establish the set X taking the fol- f is characteristic function of a set X ⊆ Pω, lowing form [0,1; 0,3]. Having arranged such a set, player I may initially select the digit 1 or 2, strategy makes him f(p)=0 if p/X ∈ . win the game automatically. The selection of any other f(p)=1 if p X ∈ digit will result in the win of player II. The player I wins the game if f(p) = 1 and player II wins Formally we can write PI games as follows: the game if f(p) = 0. Let A is called the set of strategies of player I, B be the Definition 2. [8]. A game〈 A,B,φ〉 is called determined if set of strategies of player II. inf sup ϕ(a, b)=v = sup inf ϕ(a, b) (1) b B a A a A b B φ: A×B → ℜ, where ℜ = ℜ ∪ {−∞,+∞} (ℜ is the set of ∈ ∈ ∈ ∈ real numbers). where v isinf valuesup ofϕ (thea, b )game<v< (commonsup inf ϕvalue(a, b )v of both b B a A a A b B This game is played as follows: sides of this∈ equation∈ is called the∈ value∈ of the game 〈A,B,φ〉). Player I chooses a ∈ A and player II chooses b ∈ B. Both n 2−Remark.where A gamep, q is Pdeterminedr such that if and 2n only+ k =if thep + gameq(∗) choices are made independently and without any knowlxn =- k ∃ ∈ 2− where k n p, q Pr such that 2n + k = p + q(∗) edge about the choice of the other player. Then player II has a value.inf sup≤ ϕ(¬∃a, b)=∈v = sup inf ϕ(a, b) b B a A a A b B pays to player I value φ (a, b). If φ (a, b) < 0 means that A game is not∈ determined∈ if ∈ ∈ n player II gets from player I the value φ (a, b). 2− where infp, q,sup t ϕ(Pa,r b)such<v< thatsup 2infn +ϕ(ka,= b)p + q +(2)t(∗∗) xn = k b∃ B a A∈ a A b B 2− where k∈ n∈ p, q, t P ∈such∈ that 2n + k = p + q + t(∗∗) Idea of an infinite game of perfect information is the fol- ≤ ¬∃ ∈ r lowing: Note. If the game is not determined, then the left–hand 2 n where p, q P such that 2n + k = p + q( ) −side of Equation (1) isr larger than the right–hand side∗ of xn = k ∃ ∈ - let ω ∈ {0,1,2…}, 2− where k n p, q P such that 2n + k = p + q(∗) Equation (1).