The Berge Equilibrium: A Game-Theoretic Framework for the Golden Rule of Ethics (Static & Dynamic Game Theory: Foundations & Applications)

The goal of this book is to elaborate on the main principles of the theory of the Berge equilibrium by answering the following two questions: What are the basic properties of the Berge equilibrium? Does the Berge equilibrium exist, and how can it be calculated? The Golden Rule of ethics, which appears in Christianity, Judaism, Islam, Buddhism, Confucianism and other world religions, states the following: “Behave towards others as you would like them to behave towards you." In any game, each party of conflict seeks to maximize some payoff. Therefore, for each player, the Golden Rule is implemented through the maximization of his/her payoff by all other players, which matches well with the concept of the Berge equilibrium. The approach presented here will be of particular interest to researchers (including undergraduates and graduates) and economists focused on decision-making under complex conflict conditions. The peaceful resolution of conflicts is the cornerstone of the approach: as a matter of fact, the Golden Rule precludes military clashes and violence. In turn, the new approach requires new methods; in particular, the existence problems are reduced to saddle point design for the Germeier convolution of payoff functions, with further transition to mixed strategies in accordance with the standard procedure employed by E. Borel, J. von Neumann, J. Nash, and their followers. Moreover, this new approach has proven to be efficient and fruitful with regard to a range of other important problems in mathematical game theory, which are considered in the Appendix. Biography of Mindia E. Salukvadze Preface Basic Notations Introduction Contents 1 What Is the Golden Rule of Ethics? 1.1 Scribitur ad narrandum, non ad probandum 1.2 World Religions About the Golden Rule 1.3 The Golden Rule and Philosophy 1.4 What Does the Golden Rule Suggest? 1.5 The Golden Rule as the Key Principle of Social Life 1.6 Moral Decline of Modern Society 1.7 The Golden Rule and Policy 1.8 Is Ethical Policy Possible? 2 Static Case of the Golden Rule 2.1 What is the Content of the Golden Rule? 2.2 Main Notions 2.2.1 Preliminaries 2.2.2 Elements of the Mathematical Model 2.2.3 Nash Equilibrium 2.2.4 Berge Equilibrium 2.3 Compactness of the Set XB 2.4 Internal Instability of the Set XB 2.5 No Guaranteed Individual Rationality of the Set XB 2.6 Two-Player Game 2.7 Comparison of Nash and Berge Equilibria 2.8 Sufficient Conditions 2.8.1 Continuity of the Maximum Function of a Finite Number of Continuous Functions 2.8.2 Reduction to Saddle Point Design 2.8.3 Germeier Convolution 2.8.3.1 Necessary and Sufficient Conditions 2.8.3.2 Geometrical Interpretation 2.9 Mixed Extension of a Noncooperative Game 2.9.1 Mixed Strategies and Mixed Extension of a Game 2.9.2 Préambule 2.9.3 Existence Theorem 2.10 Linear-Quadratic Two-Player Game 2.10.1 Preliminaries 2.10.2 Berge Equilibrium 2.10.3 Nash Equilibrium 2.10.4 Auxiliary Lemma 2.10.5 Concluding Remarks 3 The Golden Rule Under Uncertainty 3.1 Uncertainty and Types of Uncertainty 3.1.1 Conceptual Meaning of Uncertainty 3.1.2 Uncertainty in Economic Systems 3.1.3 Uncertainty in Mechanical Control Systems 3.1.4 Uncertainty in Decision-Making 3.1.5 Classification of Uncontrolled Factors 3.1.6 Classification of Uncertainty 3.2 General Notions and Obtained Results 3.2.1 Saddle point and maximin 3.2.2 Auxiliary Results from Operations Research, Theory of Multicriteria Choice and Game Theory 3.3 Balanced Equilibrium as an Analog of Saddle Point 3.3.1 Analogs of Saddle Point: The Idea and Formalization 3.3.2 Pro et contra of Balanced Equilibrium 3.3.3 Games with Separated Payoff Functions 3.3.4 Existence in Mixed Strategies and One Remark 3.4 Strongly-Guaranteed Berge Equilibrium 3.4.1 Introduction 3.4.2 Maximin and Its Interpretation Using Two-Level Game 3.4.3 Drawback of Balanced Equilibrium as Solution of Noncooperative Game Under Uncertainty 3.4.4 Formalization 3.4.5 Existence in Mixed Strategies 3.4.6 Linear-Quadratic Setup of Game 3.5 Slater-Guaranteed Equilibria 3.5.1 Definition and Properties 3.5.2 Existence of Guaranteed Equilibrium in Mixed Strategies 3.5.3 Existence Theorem 4 Applications to Competitive Economic Models 4.1 The Cournot Oligopoly Model 4.1.1 Introduction 4.1.2 Basic Notations and Definitions 4.1.3 The Cournot Oligopoly and Equilibrium Strategies 4.1.4 Comparison of Payoffs: Berge Equilibrium vs. Nash Equilibrium 4.2 The Cournot Duopoly with Import 4.2.1 Mathematical Model 4.2.2 Strongly-Guaranteed Equilibrium 4.2.3 Pareto-Guaranteed Equilibrium 4.2.3.1 Design Algorithm for Pareto-Guaranteed Equilibrium 4.2.3.2 Pareto Inner Minimum Calculation 4.2.3.3 Nash Equilibrium Calculation 4.3 The Bertrand Duopoly Model 4.3.1 Mathematical Model 4.3.2 Main Notions 4.3.3 Explicit Design of Berge and Nash Equilibria 4.3.3.1 Berge Equilibrium 4.3.3.2 Nash Equilibrium 4.3.4 Use of Berge Equilibrium 4.3.4.1 First Application 4.3.5 Choice of Appropriate Equilibrium on the Boundaries of the Constructed Domains 4.3.5.1 Boundary l1=0 4.3.5.2 Boundary l1=l2>0 4.3.5.3 Boundary l2=l1-q/c 4.3.5.4 Boundary l2=0 4.3.5.5 Subcase q>cl1 4.3.5.6 Subcase q