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Several Remarks on the Role of Certain Positional Foundations of Management, Vol. 8 (2016), ISSN 2080-7279 DOI: 10.1515/fman-2016-0022 289 SEVERAL REMARKS ON THE ROLE OF CERTAIN POSITIONAL AND SOCIAL GAMES IN THE CREATION OF THE SELECTED STATISTICAL AND ECONOMIC APPLICATIONS Ewa DRABIK Warsaw University of Technology, Faculty of Management, Warsaw, Poland e-mail: [email protected] Abstract: The game theory was created on the basis of social as well as gambling games, such as chess, poker, baccarat, hex, or one-armed bandit. The aforementioned games lay solid foundations for analogous mathematical models (e.g., hex), artificial intelligence algorithms (hex), theoretical anal- ysis of computational complexity attributable to various numerical problems (baccarat), as well as illustration of several economic dilemmas particularly in the case where the winner takes every- thing (e.g., noughts and crosses). A certain gambling games, such as a horse racing, may be successful- ly applied to verify a wide spectrum of market mechanism, for example, market effectiveness or customer behavior in light of incoming information regarding a specific product. One of a lot appli- cations of the slot machine (one-armed bandit) is asymptotically efficient allocation rule, which was as- signed by T.L. Lai and H. Robbins (1985). In the next years, the rule was developed by another and was named a multi-armed. The aim of the paper is to discuss these social games along with their potential mathematical models, which are governed by the rules predominantly applicable to the social and natural sciences. Key words: positional games, social games, gambling games, one-armed bandit, chess. 1 Introduction Melvin Dresher working at RAND in 1950), gender discrimination game, chicken game, or Colonel Blot- The humanity has been accompanied by gambling to game were created. and gambling games for ages, as they entertained This paper aims to address that social games people, provoked excitement, and gave an illusory and gambling played a key role in the development hope for the better life. According to a representative of the game theory, statistics, and their applications. number of scientists, the behavior of gamblers very often reflects the conduct of market players. In fact, gambling may be defined as a permanent pursuit 2 Some remarks about positional games of win, which in itself eventually becomes the prize. The overwhelming majority of dilemmas related The decision-making process under high-risk cir- to the perfect information (PI) games were defined cumstances pushes the arguments of rationality in the period from 1935 to 1941 and incorporated to the background. The fact in question has been into the so-called Scottish Book. The Scottish Book included in the modern finance theory, for example, referred to a notebook purchased by a wife of Stefan the prospect theory developed by Daniel Kahneman Banach and used by mathematicians of the Lwów and Amos Tversky. The attention was particularly School of Mathematics (such as Stanisław Mazur, drawn to the market players who while making Stanisław Ulam, and Hugo Steinhaus) for jotting decisions under risk were more prone to follow down mathematical problems meant to be solved. herd and suffer from panic attacks. Their behavior The Scottish Book used to be applied for almost six very often proves to be irrational. In the ensuing years. Many problems presented therein were created years, the problems revolving around the game theo- in previous years and not all of them were solved. ry were becoming more and more complex. In order After the World War II, Łucja Banach brought the to provide a reasonable solution as well as to de- Book to Wrocław, where it was handwritten by Hu- scribe a sequence of social and economic phenome- go Steinhaus and sent in 1956 to Los Alamos (USA) na, a variety of new games such as a prisoner’s to Stanisław Ulam. Ulam translated it into English, dilemma (originally framed by Merrill Flood and Brought to you by | Politechnika Warszawska - Warsaw University of Technology Authenticated Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw UniversityDownload Dateof Technology | 1/13/17 12:21 2021-09-30 PM 290 Ewa Drabik copied at his own expense, and dispatched to a varie- where ty of universities. The book in question proved p0 = a(∅), p1 = b(p0), p2 = a(p1), to enjoy such a great popularity that it was soon pub- p3 = b(p0, p2), p4 = a(p1, p3) lished and edited mainly in English (Duda, 2010; Definition 2: Mauldin, 1981). Formally, we can write PI games as follows. A game A, B,φ is called determined if Let A be the set of strategies of player I and B be the inf sup φ(a, b) v sup inf φ(a, b) (1) bB bB set of strategies of player II. aA aA Remark: A game is determined if and only if the : A B , game has a value. where A game is not determined if , inf supφ(a,b) v supinf φ(a,b) (2) bB aA aA bB ( is the set of real numbers). Note. If the game is not determined, then the left- This game is played as follows: hand side of (1) is larger than the right-hand side Player I chooses a A , and player II chooses b B of (1). Both choices are made independently and without If the game has a value v and there exists an a0 such any knowledge about the choice of the other player. that (a0,b) v for all b, then a0 is called an optimal Then player II pays to player I the value (a, b). strategy for player I. If (a,b0) v for all a, the b0 is If (a, b)<0 it means that player II gets from player I called an optimal strategy for player II. the value (a, b). We will say that P ω ,f is a win for player I or The following is the idea of an infinite game of PI (perfect information): a win for the player II if P ω , f has value 1 or 0, let {0,1, 2,...} , respectively. If f : Pω has the property that there exists an n such that f(p ,p ...) does not de- there is a set P called the set of choices, 0 1 ω pend on the choice pi with i < n, then P ,f is player I chooses p0 P , next player II chooses called a finite game. p1 P, then player I chooses p 2 P , and so on. The following theorem is true. There is a function f : Pω such that the end Theorem 1: player II pays to player I the value f(p0, p1,…). Every finite game has a value. Definition 1: Proof (Mycielski, 1992). The triple A,B,φ is said to be the PI-game if there exists a set P such that A is set of all function: 3 The game of Hex and some infinity games n 0 with perfect information A a : P P , where P {} , nω One of the simplest finite PI-games is Hex. Hex is n played as follows. B b : P P 0nω We use a board with a ‘honeycomb’ with 122 cells, and we use 61 white and 61 black pieces (Fig. 1). and there exists a function ω f : P such that φ(a,b) = f(p0, p1, …), Brought to you by | Politechnika Warszawska - Warsaw University of Technology Authenticated Pobrano z http://repo.pw.edu.pl / Downloaded from Repository of Warsaw UniversityDownload Dateof Technology | 1/13/17 12:21 2021-09-30 PM Several Remarks on the Role of Certain Positional and Social Games in the Creation . 291 Figure 1. The board of the hex (source: Pijanowski, 1972) The players alternatively put white or black stones A function φ : E R is given, and a point pfirst P is on the hexagon board. White begins and he/she wins fixed. The player I and player II pick alternately if the white stones connect the top of the board with p0 pfirst , q0 Q, p1 P, q1 Q , … such that the bottom. Black wins if the black stones connect (p ,q ) E and (q , p ) E , thereby defining the left edge with the right edge. i i i i1 a zigzag path composed of arrows. We define three Theorem 2: PI-games. (i) When the board is filled with stones, then one G : player II pays to player I the value of the players has won and the other has lost. 1 n1 (ii) White has a winning strategy. 1 lim sup (φ(pi ,qi ) φ(qi , pi1 )) (3) n 2n Proof (Mycielski, 1992). i0 Hex has a relative called Bridge_it. But for G2: player II pays to player I the value Bridge_it, a useful description of winning strategy 1 n1 lim inf (φ(p i , q i ) φ(q i , p i1 )) (4) for player I (white) has been used. However, this n 2n i0 does not seem to solve for the problem on hex. G3: the game ends as soon as a closed loop arises Some games of the some type are difficulty com- in the path defined by the players, that is, as soon pared in the sense that the problem of deciding as player I picks any p {p ,..,p } or player II if a position is a win for player I or for player II is n 0 n1 complete (see Theorem 1). picks any qn {q0 ,..,qn1}. The more general games are the infinity games with Then player II pays to player I the ‘loop average’ PI. v defined as follows.
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