"This book synthesizes the game-theoretic modeling of decision-making processes and an ancient moral requirement, called the Golden Rule of ethics (GR). This rule states, "Behave to others as you would like them to behave to you." The GR is one of the oldest, most widespread and specific moral requirements that appear in Christianity, Islam, Judaism, Buddhism, and Confucianism. The book constructs and justifies mathematical models of dynamic socio-economic processes and phenomena that reveal the mechanism of the GR and are based on the concept of Berge equilibrium. The GR can be naturally used for resolving or balancing conflicts, and its "altruistic character" obviously excludes wars, blood-letting, and armed clashes. The previous book of the authors, The Berge Equilibrium: A Game-Theoretic Framework for the Golden Rule of Ethics, covers the static case of the GR. In this book, the dynamic case of the GR is investigated using the altruistic concept of Berge equilibrium and three factors as follows: a) amodification of N.N. Krasovskii's mathematical formalization of differential positional games (DPGs), in view of the counterexamples given by A.I. Subbotin and A.F. Kononenko; b) the method of guiding control, proposed by N.N. Krasovskii; c) the Germeierconvolution of the payoff functions of different players. Additionally, the book features exercises, problems, and solution tips collected together in Appendix 1, as well as new approaches to conflict resolution as presented in Appendices 2-4. The book will be of use to undergraduate and graduate students and experts in the field of decision-making in complex control and management systems, as well as anyone interested in game theory and applications"--
This book synthesizes the game-theoretic modeling of decision-making processes and an ancient moral requirement called the Golden Rule of ethics (GR). This rule states "Behave to others as you would like them to behave to you." The GR is one of the oldest, most widespread, and specific moral requirements that appear in Christianity, Islam, Judaism, Buddhism, and Confucianism.
This book constructs and justifies mathematical models of dynamic socio-economic processes and phenomena that reveal the mechanism of the GR and are based on the concept of Berge equilibrium. The GR can be naturally used for resolving or balancing conflicts, and its "altruistic character" obviously excludes wars, blood-letting, and armed clashes.
The previous book by the authors, The Berge Equilibrium: A Game-Theoretic Framework for the Golden Rule of Ethics, covers the static case of the GR. In this book, the dynamic case of the GR is investigated using the altruistic concept of Berge equilibrium and three factors as follows: 1) a modification of N.N. Krasovskii’s mathematical formalization of differential positional games (DPGs), in view of the counterexamples given by A.I. Subbotin and A.F. Kononenko; 2) the method of guiding control, proposed by N.N. Krasovskii; and 3) the Germier convolution of the payoff functions of different players. Additionally, this book features exercises, problems, and solution tips collected together in Appendix 1, as well as new approaches to conflict resolution as presented in Appendices 2 to 4.
This book will be of use to undergraduate and graduate students and experts in the field of decision-making in complex control and management systems, as well as anyone interested in game theory and applications.
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xi | |
| Abstract |
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xiii | |
| Biography of Mindia E. Salukvadze |
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xv | |
| Introduction |
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1 | (2) |
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3 | (60) |
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1.1 What is the meaning of the Golden Rule of ethics? |
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3 | (6) |
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1.1.1 The Golden Rule as a basis of world religions |
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3 | (5) |
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1.1.2 The Golden Rule and philosophy |
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8 | (1) |
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1.2 Static case of the Golden Rule: sources and components of mathematical theory |
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9 | (5) |
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1.2.1 Abstract of the book the Berge Equilibrium: a Game-Theoretic Framework for the Golden Rule of Ethics |
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9 | (3) |
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1.2.2 Berge equilibrium as a fundamental concept of mathematical theory of the Golden Rule |
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12 | (2) |
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1.3 Uncertainty and maximin |
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14 | (36) |
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14 | (1) |
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1.3.2 Uncertainty and its types |
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15 | (6) |
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1.3.3 Maximin in static case |
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21 | (5) |
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1.3.4 Maximin in dynamic case: linear-quadratic positional control problem |
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26 | (5) |
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31 | (10) |
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1.3.6 Discrete-time model of optimal advertising |
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41 | (9) |
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50 | (13) |
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1.4.1 Ad narrandum, non ad probandum |
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51 | (3) |
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1.4.2 Guaranteed solution in risks |
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54 | (2) |
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1.4.3 Properties of risk function |
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56 | (3) |
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1.4.4 Types of risk functions in noncooperative game under uncertainty (NGU) |
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59 | (4) |
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2 Mathematical model of the Golden Rule of ethics in form of differential positional game |
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63 | (48) |
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Part I Games with separated dynamics |
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63 | (1) |
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64 | (10) |
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2.1.1 Noncooperative positional differential game of N players |
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64 | (1) |
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64 | (1) |
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2.1.3 Strategies and strategy profiles |
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65 | (1) |
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2.1.4 Piecewise continuous step-by-step quasimotions |
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66 | (8) |
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2.1.5 Terminal payoff functions |
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74 | (1) |
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2.2 Alternative and saddle point |
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74 | (7) |
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74 | (1) |
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2.2.2 Theorem of alternative |
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75 | (1) |
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2.2.3 Minimax, maximin and saddle point |
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76 | (2) |
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2.2.4 Properties of saddle point |
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78 | (1) |
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2.2.5 Corollary of theorem of alternative |
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79 | (2) |
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2.3 Differential game with separated dynamics |
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81 | (6) |
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81 | (2) |
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83 | (2) |
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2.3.3 Auxiliary noncooperative game |
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85 | (2) |
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87 | (1) |
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Part II Linear-quadratic game with small influence of one player on the rate of change of state vector |
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87 | (1) |
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88 | (9) |
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88 | (2) |
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2.4.2 Explicit solution of Riccati matrix differential equation |
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90 | (2) |
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92 | (5) |
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2.5 Formalization of equilibria and sufficient conditions |
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97 | (2) |
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97 | (1) |
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2.5.2 Sufficient conditions |
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97 | (2) |
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2.6 Explicit form of equilibria |
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99 | (5) |
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2.7 Application of small parameter method |
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104 | (3) |
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104 | (1) |
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105 | (1) |
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106 | (1) |
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2.8 La fin couronne l'oeuvre |
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107 | (4) |
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3 Berge equilibria in multistage games |
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111 | (74) |
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3.1 Mathematical model of multistage conflict under uncertainty |
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112 | (12) |
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113 | (2) |
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3.1.2 Elements of the mathematical model |
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115 | (7) |
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3.1.3 Multistage noncooperative game |
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122 | (2) |
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3.2 Pro et contra of Berge equilibrium |
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124 | (13) |
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3.2.1 Classical noncooperative game as single-stage statement of game |
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125 | (1) |
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3.2.2 Positive properties of Berge equilibrium |
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126 | (4) |
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3.2.3 Negative properties of Berge equilibrium |
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130 | (7) |
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3.3 Sufficient conditions |
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137 | (18) |
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3.3.1 Multistage two-player game |
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137 | (3) |
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3.3.2 Brief introduction to dynamic programming |
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140 | (8) |
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3.3.3 Bellman's principle of optimality |
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148 | (7) |
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3.4 Sufficient conditions of Berge equilibrium |
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155 | (12) |
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155 | (2) |
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3.4.2 Sufficient conditions |
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157 | (10) |
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3.5 Berge equilibrium in Cournot duopoly |
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167 | (18) |
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3.5.1 Static case of the model |
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167 | (1) |
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3.5.2 Dynamic case of the model |
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168 | (3) |
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3.5.3 Berge equilibrium design |
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171 | (2) |
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3.5.4 Controlled Cournot duopoly with import |
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173 | (12) |
| Appendix 1 Exercises, problems, and solution tips |
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185 | (38) |
| Appendix 2 Pareto equilibrium in threats and counter-threats for a differential three-player game |
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223 | (24) |
| Appendix 3 Guaranteed solution for risk-neutral decision-maker: an analog of maximin in single-criterion choice problems |
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247 | (18) |
| Appendix 4 The concept of strong coalitional equilibrium |
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265 | (14) |
| Bibliography |
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279 | (22) |
| Subject Index |
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301 | (2) |
| Author Index |
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303 | |
Vladislav Iosifovich Zhukovskiy was born in Odessa (Ukrainian SSR) on April 30, 1937, in a student family. At the beginning of the Great Patriotic War, was evacuated to the Urals in the city of Chelyabinsk. In 1954, graduated from the men's secondary school no. 48 of Chelyabinsk. In 1959, graduated from the Ural State University named after A.M. Gorky, Faculty of Physics and Mathematics, with specialization in mechanical engineering. In 1966, defended his Candidates Dissertation in Mathematics and Physics, entitled "Some problems of conditional stability," at the Institute of Mechanics of Moscow State University. In 1989, defended his Doctoral Dissertation in Mathematics and Physics, entitled "Equilibrium in positional dynamic systems," at the Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg. Has academic title of Professor. Since 1989, has been working at the position of full professor at Moscow State University. Honored Scientist of the Russian Federation (1998), Honorary Member of the Academy of Nonlinear Sciences (2000), and Foreign member of the Georgian Academy of Sciences. A member of the local executive committee in the Russian branch of the International Society of Dynamic Games (ISDG), a member of the International Society on Multiple Criteria Decision Making (MCDM), and a member of the editorial boards of four scientific journals. Prepared over 40 Candidates and Doctors of Sciences. Published over 300 scientific works, including 29 monographs and four textbooks.
Mindia E. Salukvadze was born in 1933 in Tbilisi. In 1955, he received his Diploma with honors from the Physics Faculty of Tbilisi State University. His postgraduate studies were continued at the Institute of Electronics, Automation and Remote Control of the USSR Academy of Sciences (presently, Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences, Moscow), where he defended first the Candidates Dissertation (1963) and then the Doctors Dissertation (1974) in Engineering. In 1983, he was elected a Corresponding Member of the Academy of Sciences of the Georgian SSR; in 1993, a Full Member of the Georgian National Academy of Sciences. Later on, he became the Academician-secretary of the Department for Applied Mechanics, Machine Building, Energy and Control Processes of the Academy and also a Member of the Academys Presidium. In 1996, he was awarded the Nikoladze Prize of the Academy, which was established as far back as 1973 for the best scientific works in engineering. In 1996 and again in 2004, he was awarded the State Prize of Georgia in the field of science and technology. In 2014, he and Vladislav I. Zhukovskiy became the winners of the International Contest for the Best Scientific Book, held in Russia. For many years, Salukvadze was the Head of the Georgian Section of the International Federation of Automatic Control (IFAC) as well as a member of the editorial boards of several scientific journals such as Moambe (Bulletin of the Georgian National Academy of Sciences), International Journal of Information Technology and Decision Making, and Automation and Remote Control (Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences). In addition, he was a member of several scientific councils and organizations. He had close cooperation with a series of research centers all over the world and participated in leading international conferences and symposia. He was a Member of different international academies of sciences, including the New York Academy of Sciences (since 1994). For 25 years, Academician Salukvadze was Director of the Eliashvili Institute of Control Systems (19812006), and then Chairman of the Institutes Scientific Council. The research interests of Salukvadze covered the stability of control systems and the theory of optimal control. He authored over 140 scientific papers, 14 monographs and six textbooks, known in Georgia and also abroad. Salukvadzes method, Salukvadzes solution, Salukvadzes principlethese terms were introduced by American and Russian investigators. In 1975, Metsniereba Press (Tbilisi) published Salukvadzes well-known monograph Zadachi vektornoi optimizatsii v teorii upravleniya, which was translated into English under the title Vector-Valued Optimization Problems in Control Theory and published by Academic Press in 1979. Another prominent monograph by Salukvadze, Vector-Valued Maximin (in co-authorship with Vladislav V. Zhukovskiy), was published by Academic Press in 1994. Salukvadze was a fruitful educator, holding the position of Professor at Tbilisi State University. He supervised a series of Doctors and Candidates Dissertations. He passed away late 2018.
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