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Chapter 1.Linear Programming: Examples, Definitions, and State——ments of the Principal Theorems1

1.Examples1

The diet problem1

The transportation problem4

Production to meet given demand at minimum cost6

Production to maximize income from given resources7

2.Duality and prices8

3.Further interpretation of duality13

4.Price equilibrium19

Bibliographical notes23

Exercises23

Chapter 2.Real Linear Algebra28

1.Vectors30

2.Scalar product, matrices, linear equations35

3.Real linear equations and inequalities42

4.Basic solutions of equations49

5.Geometry of linear inequalities Convex cones51

6.Extreme vectors and extreme solutions60

7.Convex sets and polytopes66

Bibliographical notes69

Exercises69

Chapter 3.The Theory of Linear Programming74

1.Definitions75

2.The duality theorems78

3.The equilibrium theorems82

4.Basiosolutions84

5.An application: allocation of resources in a competitive economy85

Bibliographical notes93

Exercises93

Chapter 4.Computation.The Simplex Method97

1.Solving simultaneous equations and inverting a matrix98

2.The simplex method for linear programming.Discussion105

3.Theory of the simplex method108

4.Some numerical examples113

5.Nonnegative solutions of linear equations119

6.Solving linear inequalities121

7.Degeneracy.The generalized simplex method123

Bibliographical notes128

Exercises129

Chapter 5.Integral Linear Programming132

1.Examples132

Transportation problem with indivisible commodity132

The optimal-assignment problem133

The loading problem134

2.Flows in networks134

3.The simple-assignment problem143

4.The transshipment problem148

5.The optimal-assignment problem155

6.A problem related to optimal assignment Price equilibrium160

7.The transportation problem162

8.Other examples: shortest route；the caterer170

9.Concluding remarks and open questions172

Bibliographical notes174

Exercises174

Chapter 6.Tuo-person Games: Examples, Definitions, and Ele-mentary Theory180

1.First examples and definitions182

Odds and evens(matching pennies)182

Morra183

2.Further examples of matrix games184

Goofspiel184

Bluffing186

A,B,C187

3.Solutions of games.Mixed strategies189

4.Value of a game and optimal strategies193

5.Some infinite games196

Continuous bluffing196

Duels198

The oil prospector (a game against nature)199

The bomber and the submarine201

High number201

Low number202

6.Saddle points and minimax202

7.Symmetric games204

8.Proof of the fundamental theorem207

Appendix to Chapter 6: A geometric “proof” of the fundamental theorem of game theory208

Bibliographical notes211

Exercises212

Chapter 7.Solutions of Matrix Games216

1.Relation between matrix games and linear programming216

2.Solving games by the simplex method220

3.Optimal strategies223

4.Solutions227

5.Examples231

6.The structure of symmetric games233

7.Constructing a game with prescribed solution235

8.Basic optimal strategies241

9.A method of “learning” a game246

10.Convergence of the learning method250

Bibliographical notes256

Exercises256

Chapter 8.Linear Models of Exchange260

1.Examples260

The simple exchange model The price problem260

The simple linear model of international trade263

2.Equilibrium for the exchange model264

3.Dynamics theory271

4.Dynamics in the reducible case278

5.Price equilibrium for linear exchange models281

6.An example of price equilibrium287

7.Uniqueness of equilibrium prices289

Bibliographical notes290

Exercises290

Chapter 9.Linear Models of Production294

1.The simple linear production model294

2.A dynamic property of the simple model299

3.The Leontief model301

4.The general linear production model Efficient points306

5.Von Neumann’s expanding model310

6.Some examples315

7.The expanding simple model317

Bibliographical notes318

Exercises319

Bibliography323

Index327