# References for Game theory with Operations Research Orientation

I am studying the book by Martin Osborne "An Introduction to Game Theory". My background is IE and I am more familiar with OR. Is there any book that teaches Game theory in Operations Research framework? If its concentration is on revenue management, it will be better.

Don't know whether this is stating the obvious, but Algorithmic Game Theory by Nisan, Roughgarden, Tardos, and Vazirani has to be in the mix here.

Major content would be: (Complexity of) Finding (pure) Nash equilibria, Algorithmic Mechanism Design, the Price of Anarchy (ratio of efficiency between equilibria and optima), Cooperative Games.

There is a new book Bilevel Optimization Advances and Next Challenges which has a great deal of material on game theory, and how it can be handled with Bilevel Optimization.

Chapters:

•   Interactions Between Bilevel Optimization and Nash Games
•   On Stackelberg–Nash Equilibria in Bilevel Optimization Games
•   A Short State of the Art on Multi-Leader-Follower Games
•   Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization
•   Applications of Bilevel Optimization in Energy and Electricity Markets
•   Bilevel Optimization of Regularization Hyperparameters in Machine Learning
•   Bilevel Optimization and Variational Analysis
•   Constraint Qualifications and Optimality Conditions in Bilevel Optimization
•   Algorithms for Simple Bilevel Programming
•   Algorithms for Linear Bilevel Optimization
•   Global Search for Bilevel Optimization with Quadratic Data
•   MPEC Methods for Bilevel Optimization Problems
•   Approximate Bilevel Optimization with Population-Based Evolutionary Algorithms
•   Methods for Pessimistic Bilevel Optimization
•   Methods for Multiobjective Bilevel Optimization
•   Bilevel Optimal Control: Existence Results and Stationarity Conditions
•   Bilevel Linear Optimization Under Uncertainty
•   A Unified Framework for Multistage Mixed Integer Linear Optimization
•   BOLIB: Bilevel Optimization LIBrary of Test Problems
•   Bilevel Optimization: Theory, Algorithms, Applications and a Bibliography


I'm not sure exactly what you mean by Operations Research framework, but I'll interpret a mathematical treatment, heavy on O.R. and optimization material, as fitting the bill.

The book Mathematical Methods and Theory in Games, Programming, and Economics, 1st Edition: Matrix Games, Programming, and Mathematical Economics, by Samuel Karlin was published in 1959. An oldie but goodie. This is not a book on "Game Theory applied to Revenue Management", however. But it should provide a solid foundation which you can supplement with more recent material.

When I was taking a course in Total Positivity from Karlin in Fall 1980, somehow the topic of Game Theory came up.
Karlin: "Game Theory. You have two nonnegative sigma-finite measures ..." Then he mentioned this book, which was 21 years old at the time. "It should be out of date, but it isn't."

Description

Matrix Games, Programming, and Mathematical Economics deals with game theory, programming theory, and techniques of mathematical economics in a single systematic theory. The principles of game theory and programming are applied to simplified problems related to economic models, business decisions, and military tactics. The book explains the theory of matrix games and some of the tools used in the analysis of matrix games. The text describes optimal strategies for matrix games which have two basic properties, as well as the construction of optimal strategies. The book investigates the structure of sets of solutions of discrete matrix games, with emphasis on the class of games whose solutions are unique. The examples show the use of dominance concepts, symmetries, and probabilistic arguments that emphasize the principles of game theory. One example involves two opposing political parties in an election campaign, particularly, how they should distribute their advertising efforts for wider exposure. The text also investigates how to determine an optimal program from several choices that results with the maximum or minimum objective. The book also explores the analogs of the duality theorem, the equivalence of game problems to linear programming problems, and also the inter-industry nonlinear activity analysis model requiring special mathematical methods. The text will prove helpful for students in advanced mathematics and calculus. It can be appreciated by mathematicians, engineers, economists, military strategists, or statisticians who formulate decisions using mathematical analysis and linear programming.

Introduction. the Nature of the Mathematical Theory of Decision Processes
1 The Background
2 The Classification of the Mathematics of Decision Problems
3 The Main Disciplines

Part I. The Theory of Matrix Games

Chapter 1. The Definition of a Game and the Min-Max Theorem 1.1 Introduction. Games in Normal Form
1.2 Examples
1.3 Choice of Strategies
1.4 The Min-Max Theorem for Matrix Games
1.5 General Min-Max Theorem

Chapter 2. The Nature of Optimal Strategies for Matrix Games 2.1 Properties of Optimal Strategies
2.2 Types of Strict Dominance
2.3 Construction of Optimal Strategies
2.4 Characterization of Extreme-Point Optimal Strategies
2.5 Completely Mixed Matrix Games
2.6 Symmetric Games

Chapter 3. Dimension Relations for Sets of Optimal Strategies
3.1 The Principal Theorems
3.2 Proof of Theorem 3.1.1
3.3 Proof of Theorem 3.1.2
3.4 The Converse of Theorem 3.1.2
3.5 Uniqueness of Optimal Strategies

Chapter 4. Solutions of Some Discrete Games
4.1 Colonel Blotto Game
4.2 Identification of Friend and Foe (I.F.F. Game)
4.3 Poker Game
4.5 a Bargaining Example

Part II. Linear and Nonlinear Programming and Mathematical Economics

Chapter 5. Linear Programming
5.1 Formulation of the Linear Programming Problem
5.2 The Linear Programming Problem and its Dual
5.3 The Principal Theorems of Linear Programming (Preliminary Results)
5.4 The Principal Theorems of Linear Programming (Continued)
5.5 Connections Between Linear Programming Problems and Game Theory
5.6 Extensions of the Duality Theorem
5.7 Warehouse Problem
5.8 Optimal Assignment Problem
5.9 Transportation and Flow Problem
5.10 Maximal-Flow, Minimal-Cut Theorem
5.11 The Caterer's Problem
5.12 Price Speculation Model

Chapter 6. Computational Methods for Linear Programming and Game Theory
6.1 The Simplex Method
6.2 Auxiliary Simplex Methods
6.3 An Illustration of the Use of the Simplex Method
6.4 Computation of Network Flow
6.5 A Method of Approximating the Value of a Game
6.6 Proof of the Convergence
6.7 A Differential-Equations Method for Determining the Value of a Game

Chapter 7. Nonlinear Programming
7.1 Concave Programming
7.2 Examples of Concave Programming
7.4 The Vector Maximum Problem
7.5 Conjugate Functions
7.6 Composition of Conjugate Functions
7.7 Conjugate Concave Functions
7.8 A Duality Theorem of Nonlinear Programming
7.9 Applications of the Theory of Conjugate Functions to Convex Sets

Chapter 8. Mathematical Methods in the Study of Economic Models
8.1 Open and Closed Linear Leontief Models
8.2 The Theory of Positive Matrices
8.3 Applications of the Theory of Positive Matrices to the Study of Linear Models of Equilibrium and Exchange
8.4 The Theory of Production
8.5 Efficient Points of a Leontief-Type Model
8.6 The Theory of Consumer Choice
8.7 Nonlinear Models of Equilibrium
8.8 The Arrow-Debreu Equilibrium Model of a Competitive Economy

Chapter 9. Mathematical Methods in the Study of Economic Models (Continued)
9.1 Welfare Economics
9.2 The Stability of a Competitive Equilibrium
9.3 Local Stability
9.4 Global Stability of Price Adjustment Processes
9.5 Global Stability (Continued)
9.6 A Difference-Equations Formulation of Global Stability
9.7 Stability and Expectations (Model I)
9.8 Stability and Expectations (Model II)
9.9 The Von Neumann Model of an Expanding Economy
9.10 A General Model of Balanced Growth

Appendix A. Vector Spaces and Matrices
A.1 Euclidean and Unitary Spaces
A.2 Subspaces, Linear Independent, Basis, Direct Sums, Orthogonal Complements
A.3 Linear Transformations, Matrices, and Linear Equations
A.4 Eigenvalues, Eigenvectors, and the Jordan Canonical Form
A.5 Transposed, Normal, and Hermitian Matrices; Orthogonal Complement
A.7 Matrix-Valued Functions
A.8 Determinants; Minors, Cofactors
A.9 Some Identities
A.10 Compound Matrices

Appendix B. Convex Sets and Convex Functions B.1 Convex Sets in En
B.2 Convex Hulls of Sets and Extreme Points of Convex Sets
B.3 Convex Cones
B.4 Convex and Concave Functions

Appendix C. Miscellaneous Topics
C.1 Semicontinuous and Equicontinuous Functions
C.2 Fixed-Point Theorems
C.3 Set Functions and Probability Distributions

• Is the copy-and-pasting of the entirety of the contents really needed? Dec 30 '20 at 15:02
• @TheSimpliFire In my opinion, yes. It shows people what they're getting, including how it aligns with the specific request. I am more than happy to defer to your judgment for your posts, and i would prefer to use my judgment for mine. Dec 30 '20 at 15:22