Credible Threats in Negotiations: A Game-theoretic Approach

329 Pages·2002·26.127 MB·English

by  Harold Houba, Wilko Bolt (auth.)

CREDIBLE THREATS IN NEGOTIATIONS
THEORY AND DECISION LIBRARY
General Editors: W. Leinfellner(Vienna) and G. Eberlein(Munich)
Series A:Philosophy and Methodology of the Social Sciences
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SERIES C: GAME THEORY, MATHEMATICAL PROGRAMMING
AND OPERATIONS RESEARCH
VOLUME 32
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The titles published in this series are listed at the end of this volume.
CREDIBLE THREATS
IN NEGOTIATIONS
A Game-theoretic Approach
by
HAROLD HOUBA
Centre for World Food Studies,
Vrije Universiteit, Amsterdam,
The Netherlands
and
WILKO BOLT
Research Department,
De Nederlandsche Bank, Amsterdam,
The Netherlands
KLUWER ACADEMIC PUBLISHERS
NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 0-306-47539-1
Print ISBN: 1-4020-7183-3
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Contents
List of Figures
xi
List of Tables
xiii
Preface
xv
Acknowledgments
xix
On the authors
xxi
1. THE ESSENCE OF NEGOTIATION 1
1.1. Introduction 1
1.2. Real life negotiations: motivating examples 3
1.3. Outline of the book 7
2. A BARGAINING MODEL WITH THREATS 15
2.1. Introduction 15
2.2. The bargaining problem 16
2.2.1 The contract space 16
2.2.2 Disagreement actions 18
2.2.3 Utility functions 19
2.2.4 Mutual and conflicting interests 20
2.2.5 Pareto efficiency 21
2.2.6 Individual rationality 23
2.2.7 Generic uniqueness 23
2.2.8 Utility representation 26
2.3. A bargaining game with threats 28
2.3.1 The order of moves 29
2.3.2 The players’ information 31
2.3.3 Information sets and strategies 32
2.3.4 Outcomes and utilities 35
2.3.5 Reinterpretation as expected utilities 36
v
vi CREDIBLE THREATS IN NEGOTIATIONS
2.3.6 An appropriate equilibrium concept 38
2.3.7 Limit sets of equilibria 40
2.3.8 Markov strategies 41
2.3.9 Strategies represented by tables 42
2.4. Related Literature 44
Part I Exogenous disagreement outcomes
3. THE ALTERNATING OFFERS PROCEDURE 49
3.1. Introduction 49
3.2. Alternating offers 50
3.3. Markov perfect equilibrium 51
3.3.1 An important fixed point problem 51
3.3.2 Dynamic programming 56
3.3.3 Optimal response 57
3.3.4 Optimal proposals 60
3.3.5 Characterization in utility representation 62
3.3.6 Characterization in the contract space 64
3.4. Subgame perfect equilibrium 67
3.4.1 The method of Shaked and Sutton 67
3.4.2 Characterization of the SPE 69
3.4.3 First-mover advantage 70
3.4.4 Computation of the SPE contract 71
3.5. Applications 72
3.5.1 Dividing a dollar 73
3.5.2 A barter economy 75
3.6. Related literature 77
4. THE NASH PROGRAM 81
4.1. Introduction 81
4.2. Nash’s bargaining solution 82
4.2.1 Utility representation 82
4.2.2 Two geometrical properties 85
4.2.3 Bargaining in the contract space 88
4.2.4 Computation of axiomatic contracts 93
4.2.5 Two critical remarks 94
4.2.6 A reinterpretation 97
4.2.7 Alternative axioms for Nash’s bargaining solution 99
Contents vii
4.2.8 Alternative axiomatic solutions 101
4.3. Strategic bargaining and Nash’s bargaining solution 102
4.3.1 Nash’s demand game 103
4.3.2 Interpretation of demands 105
4.3.3 Convergence in alternating offers 107
4.3.4 Convergence in the contract space 110
4.4. The two approaches are complementary 112
4.5. Related Literature 114
5. COMPREHENSIVE BARGAINING PROBLEMS 117
5.1. Introduction 117
5.2. Comprehensive bargaining problems 118
5.3. Markov perfect equilibrium 120
5.3.1 The fixed point problem 120
5.3.2 MPE in utility representation 122
5.4. Subgame perfect equilibrium 124
5.4.1 Bounds for SPE utilities 124
5.4.2 Equilibrium switching 126
5.4.3 SPE with equilibrium switching 127
5.4.4 SPE with delay 130
5.5. Nash program 132
5.5.1 Generalized Nash’s solutions 132
5.5.2 Limit set of SPE utility pairs 134
5.5.3 Convergence or nonconvergence, that’s the question 137
5.6. Contract space 138
5.6.1 SPE contracts 138
5.6.2 Sufficient conditions for uniqueness 139
5.7. Related Literature 141
6. COMPARATIVE STATICS 145
6.1. Introduction 145
6.2. Utility functions and the contract space 146
6.2.1 Mutual and conflicting interests 146
6.2.2 Imperfectly divisiblegoods 149
6.3. Nonstationary bargaining problems 152
6.3.1 Alternating probabilities of breakdown 152
6.3.2 Alternating disagreement points 155
6.4. Alternative bargaining procedure 160
viii CREDIBLE THREATS IN NEGOTIATIONS
6.4.1 Markovprocess 160
6.4.2 Strategic timing of proposals 163
6.5. Related Literature 167
Part II Endogenous Threats
7. COMMITMENT AND ENDOGENOUS THREATS 175
7.1. Introduction 175
7.2. Optimal threats with commitment 177
7.2.1 Nash’s original variable-threat game 177
7.2.2 The variable-threat game with alternating offers 180
7.3. Credible threats without commitment 184
7.3.1 Nash’s variable-threat game: no commitment 185
7.3.2 Variable threats with alternating offers:
no commitment 188
7.3.3 Bounds for SPE utilities 190
7.3.4 The set of SPE utility pairs 192
7.3.5 SPE with delay 196
7.3.6 A comparison between models 197
7.4. Numerical examples 198
7.5. Related literature 200
8. BARGAINING OVER WAGES 203
8.1. Introduction 203
8.2. A model of wage negotiations 204
8.2.1 Wage bargaining: some facts and assumptions 205
8.2.2 The wage bargaining model 207
8.3. Wage bargaining with efficient holdouts 209
8.3.1 Markov perfect equilibrium 210
8.3.2 The minimum-wage and maximum-wage contract 210
8.3.3 Equilibria with lengthy strikes 213
8.3.4 Intermezzo: unequal discount factors 214
8.4. Dutch wage bargaining: an application 217
8.4.1 Work-to-rule as a substitute for strike 218
8.4.2 Equilibria with lengthy work-to-rule 220
8.4.3 Backdating 223
8.5. Related literature 225
Contents ix
9. THE POLICY BARGAINING MODEL 229
9.1. Introduction 229
9.2. Subgame perfect equilibria 230
9.2.1 Markov perfectequilibrium 230
9.2.2 Worst SPE strategies: Example 231
9.2.3 Worst SPE strategies: General case 233
9.2.4 Optimal disagreement actions 237
9.2.5 Conditions for uniqueness 241
9.2.6 Characterization of SPE utilities 242
9.3. Policy Bargaining 244
9.3.1 The policy bargaining model 245
9.3.2 Characterization of SPE utilities 246
9.3.3 Renegotiationof Agreements 247
9.3.4 Nonbinding agreements 248
9.4. Numerical Examples 251
9.5. Related literature 253
10.DESTRUCTIVE THREATS 257
10.1. Introduction 257
10.2. Difference games 258
10.2.1 General framework 259
10.2.2 The great fish war 261
10.2.3 Linear-quadratic differencegames 263
10.2.4 Pareto efficient joint policies 266
10.3. Negotiations for quota 268
10.3.1 Optimal disagreement catches 268
10.3.2 Optimal proposals 270
10.3.3 Numerical Solutions 271
10.3.4 The set of SPE utility pairs 273
10.4. Multiple state variables 275
10.4.1 Motivating Example 276
10.4.2 Optimal disagreement actions 278
10.4.3 Optimal proposals 279
10.4.4 Breakdown quadratic value functions 280
10.4.5 To negotiate or not? 281
10.5. Concluding remarks 283
10.6. Related literature 286
Appendices 291
x CREDIBLE THREATS IN NEGOTIATIONS
Proofs of Selected Theorems 291
References 311
Index
317