This book provides an introduction to the mathematical theory of games using both classical methods and optimization theory. Employing a theorem-proof-example approach, the book emphasizes not only results in game theory, but also how to prove them.

Part 1 of the book focuses on classical results in games, beginning with an introduction to probability theory by studying casino games and ending with Nash's proof of the existence of mixed strategy equilibria in general sum games. On the way, utility theory, game trees and the minimax theorem are covered with several examples. Part 2 introduces optimization theory and the Karush–Kuhn–Tucker conditions and illustrates how games can be rephrased as optimization problems, thus allowing Nash equilibria to be computed. Part 3 focuses on cooperative games. In this unique presentation, Nash bargaining is recast as a multi-criteria optimization problem and the results from linear programming and duality are revived to prove the classic Bondareva–Shapley theorem. Two appendices covering prerequisite materials are provided, and a 'bonus' appendix with an introduction to evolutionary games allows an instructor to swap out some classical material for a modern, self-contained discussion of the replicator dynamics, the author's particular area of study.

Contents:

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  Classical Game Theory:

  
  
  
  
  • Games Against the House with an Introduction to Probability Theory
  
  
  • Elementary Utility Theory
  
  
  • Game Trees and Extensive Form
  
  
  • Games and Matrices: Normal and Strategic Forms
  
  
  • Saddle Points, Mixed Strategies, and Nash Equilibria
  
  
  
  


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  Optimization and Game Theory:

  
  
  
  
  • An Introduction to Optimization and the Karush–Kuhn–Tucker Conditions
  
  
  • Linear Programming and Zero-Sum Games
  
  
  • Quadratic Programs and General-Sum Games
  
  
  
  


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  Cooperation in Game Theory:

  
  
  
  
  • Nash's Bargaining Problem and Cooperative Games
  
  
  • An Introduction to N-Player Cooperative Games
  
  
  
  


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  Appendices:

  
  
  
  
  • Introduction to Matrix Arithmetic
  
  
  • Essential Concepts from Vector Calculus
  
  
  • Introduction to Evolutionary Games Using the Replicator Equation
  
  
  
  

Readership: This book is targeted towards advanced undergraduates and beginner graduates in mathematics and graduate students in other STEM disciplines. The book could also be used in an economics class as a secondary text.