Investment under Uncertainty, Coalition Spillovers and Market Evolution in a Game Theoretic Perspective (eBook)

Contents 6
List of Figures 10
Acknowledgments 12
Chapter 1 INTRODUCTION 13
1. Rational Behaviour and Economics 13
2. Introduction to Part I 17
3. Introduction to Part II 22
4. Introduction to Part III 25
Chapter 2 MATHEMATICAL PRELIMINARIES 32
1. Basic Concepts from Topology and Calculus 32
2. Convex Analysis 35
3. Triangulations 36
4. Game Theory 37
4.1 Non-Cooperative Theory 38
4.2 Cooperative Theory 40
5. Basic Concepts from Probability Theory 43
6. Markov Chains 46
6.1 Stochastic Stability 47
6.2 Nearly-Complete Decomposability 50
7. Stochastic Processes in Continuous Time 54
7.1 Basic Stochastic Processes 59
7.2 Optimal Stopping 61
Chapter 3 THE EFFECT OF INFORMATION STREAMS ON CAPITAL BUDGETING DECISIONS 64
1. Introduction 64
2. The Model 67
3. The Optimal Investment Decision 69
4. Error Analysis 74
5. Economic Interpretation 75
6. Conclusions 81
Appendix 82
A. Proof of Proposition 3.1 82
B. Proof of Proposition 3.2 83
Chapter 4 SYMMETRIC EQUILIBRIA IN GAME THEORETIC REAL OPTION MODELS 86
1. Introduction 86
2. The General Model 89
2.1 The Equilibrium Concept 90
2.2 Preemption Games 97
3. The Standard Model 99
4. The Interaction of Competition and Uncertainty 101
4.1 Equilibrium Selection 104
4.2 The Impact of Uncertainty on Equilibria 107
4.3 The E.ect of Competition on the Value of Waiting 108
4.4 The E.ect of Uncertainty on Welfare Predictions 109
5. Conclusion 112
Appendix 113
A. Proof of Theorem 4.1 113
B. Some Integrals 117
Chapter 5 THE EFFECTS OF INFORMATION ON STRATEGIC INVESTMENT 119
1. Introduction 119
2. The Model 123
3. Exogenous Firm Roles 127
4. Endogenous Firm Roles 128
4.1 Dominant First Mover Advantage 129
4.2 Dominant Second Mover Advantage 131
5. Welfare Analysis 135
6. Conclusions 140
Appendix 141
A. Proof of Proposition 5.1 141
B. Proof of Lemma 5.1 143
C. Proof of Proposition 5.2 144
D. Proof of Proposition 5.3 145
Chapter 6 SPILLOVERS AND STRATEGIC COOPERATIVE BEHAVIOUR 149
1. Introduction 149
2. The Model 151
3. The Core 154
4. Public-Private Connection Problems 156
5. Government Situations 160
5.1 The Power Index 161
6. Mergers in Oligopolistic Markets 163
6.1 The Compromise Value 165
Appendix 169
A. Proof of Theorem 6.1 169
B. Constructing a Spillover Game from a Non- Cooperative Game 170
Chapter 7 THE TIMING OF VERTICAL MERGERS UNDER UNCERTAINTY 171
1. Introduction 171
2. The Model 173
3. The Effect of the Stand-Alone Option on the Timing of Mergers 176
4. Implications 180
Appendix 181
A. Proof of Proposition 7.1 181
Chapter 8 MULTI-LEVEL EVOLUTION IN COURNOT OLIGOPOLY 184
1. Introduction 184
2. The Model 187
2.1 The Quantity Setting Level 188
2.2 The Behavioural Level 189
3. Analysis 192
4. Conclusion 198
Appendix 199
A. Proof of Lemma 8.1 199
B. Proof of Lemma 8.2 200
Chapter 9 EVOLUTION OF CONJECTURES IN COURNOT OLIGOPOLY 201
1. Introduction 201
2. The Model 205
3. A Simulation Study 208
4. A Theoretical Analysis 211
5. Discussion 218
Appendix 219
A. Proof of Lemma 9.1 219
B. Proof of Lemma 9.2 220
Chapter 10 BOUNDED RATIONALITY IN A FINANCE ECONOMY WITH INCOMPLETE MARKETS 221
1. Introduction 221
2. The Finance Economy 224
3. Existence of Equilibrium 226
4. A Boundedly Rational Path Towards Equilibrium 232
5. The Algorithm 235
References 241
Author Index 251
Subject Index 253
About the Author 255
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Chapter 4 SYMMETRIC EQUILIBRIA IN GAME THEORETIC REAL OPTION MODELS (p. 77-78)

1. Introduction

The timing of an investment project is an important problem in capital budgeting. Many decision criteria have been proposed in the literature, the net present value (NPV) rule being the most famous one. In the past twenty years the real options literature emerged (cf. Dixit and Pindyck (1996)) in which uncertainty about the pro.tability of an investment project is explicitly taken into account. In the standard real option model the value of the investment project is assumed to follow a geometric Brownian motion. By solving the resulting optimal stopping problem one can show that it is optimal for the .rm to wait longer with investing than when the firm uses the NPV approach.

A natural extension of the one firm real option model is to consider a situation where several firms have the option to invest in the same project. Important fields of application of the game theoretic real options approach are R&D competition, technology adoption and new market models. Restricting ourselves to a duopoly framework, the aim of this chapter is to propose a method that solves the coordination problems which arise if there is a .rst mover advantage that creates a preemptive threat and, hence, erodes the option value. In the resulting preemption equilibrium, situations can occur where it is optimal for one firm to invest, but not for both. The coordination problem then is to determine which firm will invest. This problem is particularly of interest if both firms are ex ante identical. In the literature, several contributions (see Grenadier (2000) for an overview) solve this coordination problem by making explicit assumptions which are often unsatisfactory. In this chapter, we propose a method, based on Fudenberg and Tirole (1985), to solve the coordination problem endogenously.

The basic idea of the method is that one splits the game into a timing game where the preemption moment is determined and a game that is played either as soon as the preemption moment has been reached, or when the starting point is such that it is immediately optimal for one firm to invest but not for both. The outcome of the latter game determines which firm is the .rst investor. The .rst game is a game in continuous time where strategies are given by a cumulative distribution function. The second game is analogous to a repeated game in which firms play a fixed (mixed) strategy (invest or wait) until at least one firm invests.

As an illustration a simplified version of the model of Smets (1991) that is presented in Dixit and Pindyck (1996, Section 9.3), is analysed.

In the preemption equilibrium situations occur where it is optimal for one firm to invest, but at the same time investment is not beneficial if both firms decide to do so. Nevertheless, contrary to e.g. Smets (1991) and Dixit and Pindyck (1996), we .nd that there are scenarios in which both firms invest at the same time, which leads to a low payo. for both of them. We obtain that such a coordination failure can occur with positive probability at points in time where the payoff of the first investor, the leader, is strictly larger than the payo. of the other firm, the follower. From our analysis it can thus be concluded that Smets’ statement that "if both players move simultaneously, each of them becomes leader with probability one half and follower with probability one half" (see Smets (1991, p. 12) and Dixit and Pindyck (1996, p. 313)) need not be true.

The point we make here extends to other contributions that include the real option framework in duopoly models. These papers, such as Grenadier (1996), Dutta et al. (1995), and Weeds (2002), make unsatisfactory assumptions with the aim to be able to ignore the possibility of simultaneous investment at points of time that this is not optimal. Grenadier (1996, pp. 1656-1657) assumes that "if each tries to build first, one will randomly (i.e., through the toss of a coin) win the race", while Dutta et al. (1995, p.568) assume that "If both [firms] i and j attempt to enter at any period t, then only one of them succeeds in doing so".