An Introductory Course on Mathematical Game Theory
Julio González-Díaz, Ignacio García-Jurado, M Gloria Fiestras-Janeiro
180 x 240 mm
Year of Publishing
Territorial Rights
American Mathematical Society

Game theory provides a mathematical setting for analyzing competition and cooperation in interactive situations. The theory has been famously applied in economics, but is relevant in many other sciences, such as political science, biology, and, more recently, computer science. This book presents an introductory and up-to-date course on game theory addressed to mathematicians and economists, and to other scientists having a basic mathematical background. The book is self-contained, providing a formal description of the classic game-theoretic concepts together with rigorous proofs of the main results in the field. The theory is illustrated through abundant examples, applications, and exercises.

The style is distinctively concise, while offering motivations and interpretations of the theory to make the book accessible to a wide readership. The basic concepts and results of game theory are given a formal treatment, and the mathematical tools necessary to develop them are carefully presented. Cooperative games are explained in detail, with bargaining and TU-games being treated as part of a general framework. The authors stress the relation between game theory and operations research. The book is suitable for a graduate or an advanced undergraduate course on game theory.

Julio González-Díaz, Universidade de Santiago de Compostela, Spain, Ignacio García-Jurado, Universidad de Coruña, Spain, and M. Gloria Fiestras-Janeiro, Universidade de Vigo, Spain


Chapter 1. Introduction to Decision Theory

Ordinal Utility
Linear Utility

Chapter 2. Strategic Games

Introduction to Strategic Games
Nash Equilibrium in Strategic Games
Two-Player Zero-Sum Games
Mixed Strategies in Finite Games
Bimatrix Games
Matrix Games
Algorithms for Matrix Games
Matrix Games and Linear Programming
Refinements of Nash Equilibrium in Finite Games
A Basic Model of Knowledge
Correlated Equilibrium
On the Epistemic Foundations of the Different Solution
Concepts for Strategic Games
Fixed-Point Theorems
On Extreme Points and Convex Sets: Krein-Milman Theorem
Exercises of Chapter

Chapter 3. Extensive Games

Introduction to Extensive Games
Strategies in Extensive Games: Mixed Strategies vs. Behavior Strategies
Nash Equilibrium in Extensive Games
Subgame Perfect Equilibrium
Sequential Equilibrium
Further Refinements
Repeated Games
Exercises of Chapter 3

Chapter 4. Games with Incomplete Information

 Incomplete Information: Introduction and Modeling
Bayesian Games and Bayesian Nash Equilibrium
The Chain Store Paradox in Perspective
A First Application of Bayesian Games: Auctions
A Second Application of Bayesian Games: Mechanism Design and the Revelation Principle Extensive Games with Incomplete Information: Multistage Games and Perfect Bayesian               Equilibrium
An Outline of Harsanyi’s Approach
Exercises of Chapter 4

Chapter 5. Cooperative Games

 Introduction to Cooperative Games
Nontransferable Utility Games
Transferable Utility Games
The Core and Related Concepts
The Shapley Value
The Nucleolus
Convex Games
Noncooperative Models in Cooperative Game Theory: Implementation Theory
Airport Problems and Airport Games
Bankruptcy Problems and Bankruptcy Games
Voting Problems and Voting Games: Power Indices
Cooperation in Operations Research Models
Exercises of Chapter
Index of Authors
Index of Solution Concepts
Subject Index

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