Contents
Preface
Chapter1 Introduction 1
Chapter2 Zero-Sum Game 6
2.1 Two-person zero-sumgame 6
2.2 Minimax and maxmin 7
2.3 Saddle point 11
2.4 Matrixgamein pure strategies 13
2.5 Matrixgamein mixed strategies 15
2.6 Reductionofgame theoryto linear programming 16
Chapter3 Maxmin and MinimaxProblems 19
3.1 Maxmin problem 19
3.2 Optimality conditionsfor maxmin problem 21
3.3 Optimality conditions for minimax problem 25
Chapter4 Non-Zero Sum Game 31
4.1 Two-person non-zero sumgame 31
4.1.1 Bimatrixgame 31
4.1.2 Nash equilibrium 33
4.1.3 Berge equilibrium 38
4.2 Non-zero sum three-persongame 43
4.3 Non-zero sum four-persongame 48
4.4 Non-zero sum n-persongame 55
Chapter5 Anti-Nash and Anti-Berge Equilibriumin Bimatrix Game 59
5.1 Anti-Nash equilibriumin bimatrixgame 59
5.2 Anti-Berge equilibriumin bimatrixgame 63
Chapter6 Polymatrix Game 65
6.1 Three-sidedgame 65
6.1.1 Main propertiesof thegame Γ(A, B,C) 66
6.1.2 Optimization formulationof three-sidedgame 69
6.2 Four-players triplegame 72
6.3 Game of N-players 79
6.3.1 Nash theorem and the optimization problem 81
Chapter7 N-Players Non-Cooperative Games 85
7.1 Non-cooperativegames 85
7.2 Generalized Nash equilibrium problems 87
7.3 Someequivalent approachto generalizedNash equilibrium problems 89 89
7.3.1 Variational inequality approach
7.3.2 Nikaido-Isoda function based approach 90
7.3.3 Karush-Kuhn-Tucker conditions approach 92
7.4 Global optimization D.Capproach to quadratic nonconvex generalized Nash equilibrium problems 94
7.4.1 Generalized Nash equilibrium problem and equivalent optimization formulation 94
7.4.2 Quadratic nonconvexgame andgap function 96
7.4.3 D.Coptimization approachto non-cooperativegame 100
7.5 Generalized Nash equilibrium problem based on Malfatti’s problem 103
7.5.1 Malfatti’s problemand convex maximization 104
7.5.2 Generalized Nash equilibrium problems 105
7.6 Aglobal optimization approach to Berge equilibrium based on a regularized function 109
7.6.1 Existence of Berge equilibrium and constrained optimization reformulations 110 Chapter8 Game Theory and Hamiltonian System 116
8.1 Hamiltonian system 116
8.2 Evolutionarygames and Hamiltonian systems.119
8.3 Optimal controltheoryandthe Hamiltonian operator 121
8.4 Differentialgames and the Hamilton-Jacobi-Bellman (HJB) Principle 122
8.4.1 The relationship betweengame theoryandthe Hamiltonian operator 124
8.4.2 Two-person zero-sum differentialgames 125
8.4.3 Two-person non-zero sum differentialgames 127
Chapter9 Computational Methods and Algorithmsfor Matrix Game 132
9.1 D.Cprogramming approachtoBerge equilibrium 132
9.1.1 Local search method 133
9.1.2 Global search method 134
9.1.3 Numerical results for D.Cprogramming approach to Berge equilibrium 137
9.2 Global search method curvilinear algorithm forgame 142
9.2.1 The curvilinear global search algorithm 142
9.2.2 Numerical results for three-persongame 145
9.2.3 Numerical results for four-persongame 147
9.2.4 Numerical results N-persongame 149
9.3 The numerical approach for anti-Nash equilibrium search 152
9.3.1 The modi.ed Rosenbrock algorithm 153
9.3.2 Theunivariate global search procedure 154
9.3.3 Numerical results for anti-Nash equilibriumby Rosenbrock algorithm 156
9.4 Modi.ed parallel tangent algorithm for anti-Berge equilibrium 159
9.4.1 The modi.ed parallel tangent algorithm 160
9.4.2 Theunivariate global search procedure 161
9.4.3 Numerical results for anti-Berge equilibrium by modi.ed tangent algorithm 162
9.5 The curvilinear multistart algorithmfor polymatrixgame 166
9.5.1 Numericalexperimentof polymatrixgame 169
9.6 Numerical resultsfor non-cooperativegame 171
9.7 Numerical resultsforMalfati’s problem 175
Bibliography 182