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The N-Dimensional N-Person Chesslike Game Strategy Analysis Model JYVÄSKYLÄ STUDIES in COMPUTING 250

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The N-Dimensional N-Person Chesslike Game Strategy Analysis Model JYVÄSKYLÄ STUDIES in COMPUTING 250 JYVÄSKYLÄ STUDIES IN COMPUTING 250 Jorma Kyppö The N-dimensional N-person Chesslike Game Strategy Analysis Model JYVÄSKYLÄ STUDIES IN COMPUTING 250 Jorma Kyppö The N-dimensional N-person Chesslike Game Strategy Analysis Model Esitetään Jyväskylän yliopiston informaatioteknologian tiedekunnan suostumuksella julkisesti tarkastettavaksi yliopiston Agora-rakennuksen auditoriossa 2 joulukuun 14. päivänä 2016 kello 12. Academic dissertation to be publicly discussed, by permission of the Faculty of Information Technology of the University of Jyväskylä, in building Agora, auditorium 2, on December 14, 2016 at 12 o’clock noon. UNIVERSITY OF JYVÄSKYLÄ JYVÄSKYLÄ 2016 The N-dimensional N-person Chesslike Game Strategy Analysis Model JYVÄSKYLÄ STUDIES IN COMPUTING 250 Jorma Kyppö The N-dimensional N-person Chesslike Game Strategy Analysis Model UNIVERSITY OF JYVÄSKYLÄ JYVÄSKYLÄ 2016 Editors Marja-Leena Rantalainen Department of Mathematical Information Technology, University of Jyväskylä Pekka Olsbo, Ville Korkiakangas Publishing Unit, University Library of Jyväskylä URN:ISBN:978-951-39-6878-6 ISBN 978-951-39-6878-6 (PDF) ISBN 978-951-39-6877-9 (nid.) ISSN 1456-5390 Copyright © 2016, by University of Jyväskylä Jyväskylä University Printing House, Jyväskylä 2016 But helpless Pieces of the Game He plays Upon this Chequer-board of Nights and Days; Hither and thither moves, and checks, and slays, And one by one back in the Closet lays... Omar Khayyam (Rubaiyat, translation by Edward Fitzgerald) ABSTRACT Kyppö, Jorma The N-dimensional N-person Chesslike Game Strategy Analysis Model Jyväskylä: University of Jyväskylä, 2016, 256 p. (Jyväskylä Studies in Computing ISSN 1456-5390; 250) ISBN 978-951-39-6877-9 (nid.) ISBN 978-951-39-6878-6 (PDF) In this research a mathematical, symmetric n-player game model, based on chess is designed. Symmetry in this context refers to players' positions with respect to each other. While the order of move naturally violates the symmetry, this prob- lem may also be solved. The motivation for building this kind of game model stems from the difficulty of finding mathematical solutions for multi-player games in general. The number of varying factors is so huge, that finding opti- mal strategies is mathematically almost impossible. The best way to attempt this is to use simulation. Once the model has been built, it can be applied in many ways by using computational algorithms based on the created model. Chess in this design is the basic structure around which the model is built. The players’ weighting values can later be changed, as well as the weighting values of the pieces, in order to better reflect the variety of real-life situations. While chess is a board game, it can mirror various types of interactions between a number of different participants. Thus the game, in a larger extent, may play a role in understanding such things as politics, ecology and weather forecasting. During this research a great number of spin-off results and observations were discovered. The main objective and result of this research was, however, to create a symmetric n- person strategy game, because currently there is no simple mathematical model for symmetric n-player, strategy games. Keywords: N-player strategy game, combinatorics, tiling, topology, chess, multinomial formula, tetrahedron, game theory, graph theory Author Jorma Kyppö Department of Computer Science and Information Systems University of Jyväskylä, Finland Email: [email protected] Supervisors Professor Pekka Neittaanmäki Department of Mathematical Information Technology University of Jyväskylä, Finland Professor Seppo Puuronen Department of Computer Science and Information Systems University of Jyväskylä, Finland Professor Frank Harary (1921 – 2005) Department of Computer Science New Mexico State University, USA Reviewers Professor Gennady Leonov St-Petersburg State University, Dept. of Mathematics and Mechanics, Russia Professor Hajo Broersma Department of Applied Mathematics, University of Twente, Netherlands Opponent Dr. Stephen Hedetniemi Professor Emeritus University of Clemson South Carolina, USA ACKNOWLEDGEMENTS This dissertation is the result of my research conducted entirely independently, nevertheless, there are several people who have had a significant influence on the process of conducting research and writing the thesis. Firstly, I would like to express my sincere gratitude to my advisor Prof. Pekka Neittaanmäki, Dean of the Faculty of Information Technology, for the continuous support of my PhD study and related research, for his patience, mo- tivation, and immense knowledge. His guidance helped me in all the time of research and writing this thesis. His support in final stage of my work is evi- dent, however, Pekka Neittaanmäki may not be aware of his influence already in the period of my basic studies of information technology. Besides my advisor, I would like to thank my colleague, friend and super- visor in the initial stage of my PhD. study, Prof. Frank Harary from the Univer- sity of New Mexico. After passing away in 2005, my research work was put aside for a while and my focus was turned over to teaching. Coincidentally, Frank Harary was born in the same year as my father Ensio Kyppö, whose mo- tivating and inspiring approach has created the overall basis for my research work. Moreover, a key role in the research work has Prof. Emeritus Seppo Puuronen from the Department of Computer Science and Information Systems at the University of Jyväskylä. I am thankful to him for his precious support and thorough reading of the thesis. In the beginning of my studies, he was also my teacher and thanks to him, I was recruited to the Department of Computer Science and Information Systems. I am also deeply thankful to Prof. Emeritus Vesa Savolainen from the De- partment of Computer Science and Information Systems at the University of Jyväskylä for getting me familiar with the Graph Theory and supervising my Licentiate Thesis. In the context of this dissertation, his influence has been ra- ther paradoxical – he introduced me to the classical, nevertheless, too addictive Four Color Map Problem, as a result of what I lost three years of my study. On the other hand, I found the way to the Graph Theory and my own research work. Prof. Savolainen, like my father, has always been very inspiring. He and my father believed in me more than it was necessary, what after all, was a good thing. My sincere thanks also go to Esko Peltonen, the Innovation Director and Business Development Director of the Jyväskylä Science Park. Thanks to him, my chess innovations were implemented and the Trichess became the focus of commerce and my research. I would also like to express the deepest appreciation to Prof. Jari Veijalainen, my colleague and discussion partner for his support and stimulat- ing discussions. I also thank my project mates from the MineSocMed project aimed at the exploration of social networks for fruitful and enjoyable work. In addition, I would like to thank the Nyyssönen Foundation for the stra- tegically important support in the initial stages of my research, as well as the Comas Graduate School of the University of Jyväskylä for granting me the trav- el funds for participation in the international conferences which evidently pro- moted my research work. My thanks also go to Steve Legrand, who has not only reviewed my thesis for the English spelling and grammar, but has also provided some good, de- tailed comments on my work. Another person who deserves thanks is our ‘sunny’ secretary Seija Paananen from the Department of Computer Science and Information Systems, who made sure that the external framework of my working environment has always been in good condition. And, of course, I would also like to thank Sami Kollanus, who took good care of all the practical arrangements related to the defense process. I would like to express my sincere gratitude to my excellent dissertation pre-examiners, Prof. Gennady Leonov from the St. Petersburg State University and Prof. Hajo Broersma from the University of Twente, and especially to the Opponent of the dissertation Prof. Emeritus Steve Hedetniemi from the Univer- sity of Clemson, for their valuable, insightful comments and encouragement. I also thank my fellows from the Department of Computer Science and In- formation Systems for the stimulating discussions in the Coffee Room. And finally, I would like to express my gratitude to my family for sup- porting me spiritually throughout writing this thesis and for creating favorable conditions for working on my thesis in the peace of the countryside without having to worry about any practical issues of everyday life. Jyväskylä 23.11.2016 Jorma Kyppö FIGURE Figure 1 The basic elements of a graph .................................................................... 24 Figure 2 The structure of the thesis ........................................................................... 26 Figure 3 The payoff matrix of zero-sum game ........................................................ 30 Figure 4 A Zero-sum game without a saddlepoint ................................................ 31 Figure 5 The Stone-, paper, scissors-game ............................................................... 32 Figure 6 A Constant-sum-game ................................................................................. 33 Figure 7 A Non-constant-sum game ......................................................................... 35 Figure 8 A Non-constant-sum-game seen in the coordinate system ..................
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