Theory of Conjectural Variations

This book presents, for the first time, a comprehensive account of conjectural variations equilibria in their static inceptions, featuring new comparative results of equilibria with regard to efficiency. It then describes several advances in Dynamic Game Theory, allowing to understand Conjectural Variations Equilibria as dynamic equilibria. The question of how conjectures evolve in strategic and learning situations with boundedly rational agents is also discussed.

Preface

vii

Static Conjectural Variations Equilibria: Initial Concepts

1

(32)

Introduction

1

(1)

Origin of the conjectural variations concept

2

(5)

Definitions and characterisation of Conjectural Variations Equilibria

7

(15)

Notation and assumptions

7

(1)

Nash equilibrium, Pareto optimality

8

(1)

Conjectures, reactions and consistency

8

(2)

Conjectural Variations Equilibria with general conjectures (GCVE)

10

(1)

Definitions

10

(2)

Characterisation of GCVE

12

(1)

Existence results

13

(1)

Conjectural Variations Equilibria (CVE)

14

(1)

Consistent General Conjectural Variations Equilibria (CGCVE)

15

(1)

Definition

15

(1)

Characterisation of CGCVE

15

(1)

Existence results

16

(1)

Consistent Conjectural Variations Equilibria (CCVE)

16

(1)

Definition

17

(1)

Characterisation of CCVE

17

(1)

Existence results

18

(1)

Equilibria with punctual consistency

19

(1)

Definition

19

(2)

Existence results

21

(1)

Conjectures in many-player games

21

(1)

Examples and illustrations

22

(6)

Cournot's duopoly

23

(2)

Bertrand's duopoly

25

(1)

Voluntary contributions to a public good

26

(2)

An attempt at generalisation

28

(3)

Conclusion

31

(2)

Conjectures as Reduced Forms for Dynamic Interactions

33

(32)

Introduction

33

(1)

Private provision of a public good

34

(10)

One-shot simultaneous contributions

34

(2)

Repeated contributions

36

(4)

Private investment in a stock of public good

40

(4)

Oligopoly

44

(5)

Static Cournot oligopoly with constant conjectures

44

(1)

A repeated linear oligopoly

45

(2)

Dynamic duopoly with adjustment costs

47

(2)

Public infrastructure competition

49

(4)

Static infrastructure competition

49

(1)

Dynamic infrastructure competition

50

(3)

A class of state-space games and the associated static games with conjectural variations

53

(6)

A linear-quadratic framework with two state variables

53

(4)

Payoff structure and conjectures

57

(2)

Conclusion

59

(1)

Technical complements

60

(5)

The feedback Nash equilibrium in the voluntary contribution game

60

(1)

Proof of Theorem 2.1

61

(2)

Proof of Theorem 2.2

63

(2)

Consistent Conjectures in Dynamic Settings

65

(26)

Introduction

65

(1)

Conjectures for dynamic games, equilibria and consistency

66

(20)

Principle

67

(3)

Fershtman and Kamien: conjectures in differential games

70

(1)

Laitner's discrete-time model with complete conjectures

71

(2)

Friedman's dynamically consistent conjectures

73

(1)

Feedback-consistency for linear-quadratic games

74

(1)

Setting of the problem

75

(1)

Optimal reaction

76

(2)

Stationary and proportional conjectures

78

(4)

Feedback-consistent conjectures

82

(1)

Cournot's duopoly

83

(2)

A distance game

85

(1)

The model of Basar, Turnovsky and d'Orey

86

(1)

Conclusion

87

(4)

Dynamic Conjectures, Incomplete Information and Learning

91

(24)

Introduction

91

(1)

Conjecture adjustment process

92

(8)

Itaya and Dasgupta's conjecture adjustment process

93

(2)

Principles

95

(2)

Quadratic models

97

(3)

The model of Friedman and Mezzetti

100

(3)

A learning model for conjectures

103

(9)

Principle

104

(2)

General properties

106

(3)

Results

109

(1)

Cournot's oligopoly

109

(1)

Bertrand's duopoly

110

(1)

Comments and limitations

111

(1)

Evolutionary games and consistent conjectures

112

(1)

Conclusion

113

(2)

Conclusion

115

(4)

Appendix A Properties of Conjectural Equilibria

119

(16)

A.1 Iso-payoffs curves and conjectured reaction functions

119

(4)

A.2 Families of payoff functions with consistent CVE

123

(7)

A.3 Polynomial consistent conjectures

130

(2)

A.4 Nash equilibria, Pareto optima and consistency

132

(3)

Appendix B Comparison Between Conjectural Equilibria, Nash Equilibria and Pareto-Efficient Outcomes

135

(16)

B.1 Two-player games

136

(12)

B.1.1 Main results

138

(6)

B.1.2 Discussion

144

(4)

B.2 Many players games

148

(2)

B.3 Consistent conjectures

150

(1)

Appendix C Examples and Illustrations

151

(12)

C.1 Cournot's duopoly

151

(2)

C.2 Voluntary contributions to a public good

153

(4)

C.3 A model of competition between regions

157

(2)

C.4 A model of aggregate-demand externalities

159

(4)

Bibliography

163

(4)

Index

167

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Theory of Conjectural Variations

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