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Theory of Conjectural Variations (Series on Mathematical Economics and Game Theory - Vol. 2)
By Alain Jean-Marie, Nicolas Querou, Mabel Tidball
Publisher: World Scientific Publishing Company
Number Of Pages: 184
Publication Date: 2004-04
ISBN-10 / ASIN: 9812387366
ISBN-13 / EAN: 9789812387363
Preface vii
1. Static Conjectural Variations Equilibria: Initial Concepts 1
1.1 Introduction 1
1.2 Origin of the conjectural variations concept 2
1.3 Definitions and characterisation of Conjectural Variations
Equilibria 7
1.3.1 Notation and assumptions 7
1.3.2 Nash equilibrium, Pareto optimality 8
1.3.3 Conjectures, reactions and consistency 8
1.3.4 Conjectural Variations Equilibria with general conjectures
(GCVE) 10
1.3.4.1 Definitions 10
1.3.4.2 Characterisation of GCVE 12
1.3.4.3 Existence results 13
1.3.5 Conjectural Variations Equilibria (CVE) 14
1.3.6 Consistent General Conjectural Variations Equilibria
(CGCVE) 15
1.3.6.1 Definition 15
1.3.6.2 Characterisation of CGCVE 15
1.3.6.3 Existence results 16
1.3.7 Consistent Conjectural Variations Equilibria (CCVE) 16
1.3.7.1 Definition 17
1.3.7.2 Characterisation of CCVE 17
1.3.7.3 Existence results 18
1.3.8 Equilibria with punctual consistency 19
2. Conjectures as Reduced Forms for Dynamic Interactions 33
2.1 Introduction 33
2.2 Private provision of a public good 34
2.2.1 One-shot simultaneous contributions 34
2.2.2 Repeated contributions 36
2.2.3 Private investment in a stock of public good 40
2.3 Oligopoly 44
2.3.1 Static Cournot oligopoly with constant conjectures . 44
2.3.2 A repeated linear oligopoly 45
2.3.3 Dynamic duopoly with adjustment costs 47
2.4 Public infrastructure competition 49
2.4.1 Static infrastructure competition 49
2.4.2 Dynamic infrastructure competition 50
2.5 A class of state-space games and the associated static games
with conjectural variations 53
2.5.1 A linear-quadratic framework with two state variables 53
2.5.2 Payoff structure and conjectures 57
2.6 Conclusion 59
2.7 Technical complements 60
2.7.1 The feedback Nash equilibrium in the voluntary contribution
game 60
2.7.2 Proof of Theorem 2.1 61
2.7.3 Proof of Theorem 2.2 63
3. Consistent Conjectures in Dynamic Settings 65
3.1 Introduction 65
3.2 Conjectures for dynamic games, equilibria and consistency . 66
3.2.1 Principle 67
4. Dynamic Conjectures, Incomplete Information and Learning 91
4.1 Introduction 91
4.2 Conjecture adjustment process 92
4.2.1 Itaya and Dasgupta's conjecture adjustment process 93
4.2.2 Principles 95
4.2.3 Quadratic models 97
4.3 The model of Friedman and Mezzetti 100
4.4 A learning model for conjectures 103
4.4.1 Principle 104
4.4.2 General properties 106
4.4.3 Results 109
4.4.3.1 Cournot's oligopoly 109
4.4.3.2 Bertrand's duopoly 110
4.4.4 Comments and limitations Ill
4.5 Evolutionary games and consistent conjectures 112
4.6 Conclusion 113
5. Conclusion 115
Appendix A Properties of Conjectural Equilibria 119
A.l Iso-payoffs curves and conjectured reaction functions . . . . 119
A.2 Families of payoff functions with consistent CVE 123
A.3 Polynomial consistent conjectures 130
A.4 Nash equilibria, Pareto optima and consistency 132
Appendix B Comparison Between Conjectural Equilibria,
Nash Equilibria and Pareto-Efficient Outcomes 135
B.l Two-player games 136
B.l.l Main results 138
B.1.2 Discussion 144
B.2 Many players games 148
B.3 Consistent conjectures 150
Appendix C Examples and Illustrations 151
C.l Cournot's duopoly 151
C.2 Voluntary contributions to a public good 153
C.3 A model of competition between regions 157
C.4 A model of aggregate-demand externalities 159
Bibliography 163
Index 167
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