An_Introduction_to_Dynamics_Games-Haurie-Krawczyk.pdf

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An Introduction to Dynamic Games
A. Haurie
J. Krawczyk
March 28, 2000
2
Contents
1 Foreword
9
1.1 What are Dynamic Games? ....................... 9
1.2 Origins of these Lecture Notes ..................... 9
1.3 Motivation ............................... 10
I Elements of Classical Game Theory
13
2 Decision Analysis with Many Agents
15
2.1 The Basic Concepts of Game Theory .................. 15
2.2 Games in Extensive Form ........................ 16
2.2.1 Description of moves, information and randomness ...... 16
2.2.2 Comparing Random Perspectives ............... 18
2.3 Additional concepts about information ................. 20
2.3.1 Complete and perfect information ............... 20
2.3.2 Commitment .......................... 21
2.3.3 Binding agreement ....................... 21
2.4 Games in Normal Form ........................ 21
3
4
CONTENTS
2.4.1 Playing games through strategies ................ 21
2.4.2 From the extensive form to the strategic or normal form . . . 22
2.4.3 Mixed and Behavior Strategies ................. 24
3 Solution concepts for noncooperative games
27
3.1 introduction ............................... 27
3.2 Matrix Games .............................. 28
3.2.1 Saddle-Points .......................... 31
3.2.2 Mixed strategies ........................ 32
3.2.3 Algorithms for the Computation of Saddle-Points ....... 34
3.3 Bimatrix Games ............................. 36
3.3.1 Nash Equilibria ......................... 37
3.3.2 Shortcommings of the Nash equilibrium concept ....... 38
3.3.3 Algorithms for the Computation of Nash Equilibria in Bima-
trix Games ........................... 39
3.4 Concave m -Person Games ....................... 44
3.4.1 Existence of Coupled Equilibria ................ 45
3.4.2 Normalized Equilibria ..................... 47
3.4.3 Uniqueness of Equilibrium ................... 48
3.4.4 A numerical technique ..................... 50
3.4.5 A variational inequality formulation .............. 50
3.5 Cournot equilibrium ........................... 51
3.5.1 The static Cournot model .................... 51
CONTENTS
5
3.5.2 Formulation of a Cournot equilibrium as a nonlinear comple-
mentarity problem ....................... 52
3.5.3 Computing the solution of a classical Cournot model ..... 55
3.6 Correlated equilibria .......................... 55
3.6.1 Example of a game with correlated equlibria ......... 56
3.6.2 A general definition of correlated equilibria .......... 59
3.7 Bayesian equilibrium with incomplete information .......... 60
3.7.1 Example of a game with unknown type for a player ...... 60
3.7.2 Reformulation as a game with imperfect information ..... 61
3.7.3 A general definition of Bayesian equilibria .......... 63
3.8 Appendix on Kakutani Fixed-point theorem .............. 64
3.9 exercises ................................. 65
II Repeated and sequential Games
67
4 Repeated games and memory strategies
69
4.1 Repeating a game in normal form ................... 70
4.1.1 Repeated bimatrix games .................... 70
4.1.2 Repeated concave games .................... 71
4.2 Folk theorem .............................. 74
4.2.1 Repeated games played by automata .............. 74
4.2.2 Minimax point ......................... 75
4.2.3 Set of outcomes dominating the minimax point ........ 76
4.3 Collusive equilibrium in a repeated Cournot game ........... 77

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