CONJECTURAL VARIATIONS

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1 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Series on Mathematical Economics and Game Theory Vol.2 THEORY OF CONJECTURAL VARIATIONS Charles Figuieres University of Bristol, UK fllain Jean-Marie LIRMM, CNRS & University of Montpellier II, France Nicolas Querou INRA-LAMETA & University of Montpellier II, France Mabel Ticlball INRA-LAMETA, France World Scientific NEW JERSEY LONDON SINGAPORE SHANGHAI HONGKONG TAIPEI BANGALORE

2 Contents Preface vii 1. Static Conjectural Variations Equilibria: Initial Concepts Introduction Origin of the conjectural variations concept Definitions and characterisation of Conjectural Variations Equilibria Notation and assumptions Nash equilibrium, Pareto optimality Conjectures, reactions and consistency Conjectural Variations Equilibria with general conjectures (GCVE) Definitions Characterisation of GCVE Existence results Conjectural Variations Equilibria (CVE) Consistent General Conjectural Variations Equilibria (CGCVE) Definition Characterisation of CGCVE Existence results Consistent Conjectural Variations Equilibria (CCVE) Definition Characterisation of CCVE Existence results Equilibria with punctual consistency 19

3 xiv Theory of Conjectural Variations Definition Existence results Conjectures in many-player games Examples and illustrations Cournot's duopoly Bertrand's duopoly Voluntary contributions to a public good An attempt at generalisation Conclusion Conjectures as Reduced Forms for Dynamic Interactions Introduction Private provision of a public good One-shot simultaneous contributions Repeated contributions Private investment in a stock of public good Oligopoly Static Cournot oligopoly with constant conjectures A repeated linear oligopoly Dynamic duopoly with adjustment costs Public infrastructure competition Static infrastructure competition Dynamic infrastructure competition A class of state-space games and the associated static games with conjectural variations A linear-quadratic framework with two state variables Payoff structure and conjectures Conclusion Technical complements The feedback Nash equilibrium in the voluntary contribution game Proof of Theorem Proof of Theorem Consistent Conjectures in Dynamic Settings Introduction Conjectures for dynamic games, equilibria and consistency Principle 67

4 Contents xv Fershtman and Kamien: conjectures in differential games Laitner's discrete-time model with complete conjectures Friedman's dynamically consistent conjectures Feedback-consistency for linear-quadratic games Setting of the problem Optimal reaction Stationary and proportional conjectures Feedback-consistent conjectures Cournot's duopoly A distance game The model of Ba ar, Turnovsky and d'orey Conclusion Dynamic Conjectures, Incomplete Information and Learning Introduction Conjecture adjustment process Itaya and Dasgupta's conjecture adjustment process Principles Quadratic models The model of Friedman and Mezzetti A learning model for conjectures Principle General properties Results Cournot's oligopoly Bertrand's duopoly Comments and limitations Ill 4.5 Evolutionary games and consistent conjectures Conclusion Conclusion 115 Appendix A Properties of Conjectural Equilibria 119 A.I Iso-payoffs curves and conjectured reaction functions 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

5 xvi Theory of Conjectural Variations Appendix B Comparison Between Conjectural Equilibria, Nash Equilibria and Pareto-Efficient Outcomes 135 B.I Two-player games 136 B.I.I Main results 138 B.I.2 Discussion 144 B.2 Many players games 148 B.3 Consistent conjectures 150 Appendix C Examples and Illustrations 151 C.I 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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