Decision Analysis through Modeling and Game Theory

Dr. William P. Fox is currently a visiting professor of Computational Operations Research at the College of William and Mary. He is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School and teaches a three-course sequence in mathematical modeling for decision making. He received his Ph.D. in Industrial Engineering from Clemson University. He has taught at the United States Military Academy for twelve years until retiring and at Francis Marion University where he was the chair of mathematics for eight years. He has many publications and scholarly activities including twenty plus books and one hundred and fifty journal articles. Books by William P. Fox from CRC Press: Probability and Statistics for Engineering and the Sciences with Modeling using R (w/Rodney X. Sturdivant, 2023, CRC Press Mathematical Modeling in the Age of the Pandemic, 2021, CRC Press. Advanced Problem Solving Using Maple: Applied Mathematics, Operations Research, Business Analytics, and Decision Analysis (w/William Bauldry), 2020, CRC Press. Mathematical Modeling with Excel (w/Brian Albright), 2020, CRC Press. Nonlinear Optimization: Models and Applications, 2020, CRC Press. Advanced Problem Solving with Maple: A First Course (w/William Bauldry), 2019. CRC Press. Mathematical Modeling for Business Analytics, 2018, CRC Press.

Chapter 1: Introduction to Decision Models

1.1 Overview of Decision Making

1.2 Decision Theory

1.3 Game Theory: Total Conflict

Example 1.5: A Total Conflict Game with Pure Strategies

1.4 Game Theory: Partial Conflict

1.5 Mathematical Modeling of Decisions

1.4 ILLUSTRATE EXAMPLES

1.5 Technology

Summary

Chapter 2 Decision Theory and Expected Value

2.1 Introduction

2.2 Expected Value

2.3 Decisions Under Risk: Probabilities are known or estimated in advance

2.4 Decisions under Uncertainty: Probabilities are not known nor can they be estimated

2.5 Decision Trees
2.6 Sequential Decisions and Conditional Probability (from Fox, Mathematical Modeling for Business Analytics, Taylor and Francis, 2018)

Chapter 3 Decisions under certainty: Mathematical Programming Modeling: Linear, Integer, and Mixed Integer Optimization

3.1 Introduction
3.2 Formulating Linear Programming Problems
3.3 Graphical Linear Programming
3.4 Linear Programming with Technology
3.5 Case Studies in Linear Programming

Projects
3.5.1 Modeling of Ranking Units using Data Envelopment Analysis (DEA) as a LP

3.5.2 Recruiting Raleigh Office (modified from McGrath, 2007)

Exercises

References and Suggested Further Readings

Chapter 4 Multi-Attribute Decision Making using weighting schemes with SAW, AHP and TOPSIS

4.1 Weighting Methods

4.1.1 Rank Order Centroid (ROC)

4.1.2 Ratio Method for Weights

4.1.3 Pairwise Comparison (AHP)

4.1.4 Entropy Method:

4.2 Simple Additive Weights (SAW) Method

4.3 Weighted Product Method

4.4 Analytical Hierarchy Process

4.5 Technique of Order Preference by Similarity to the Ideal Solution

Methodology

Normalization

Additional Reading and References

EXERCISES Chapter 4


CHAPTER 5 Game Theory: Total Conflict

5.1 Introduction to Total Conflict Games

5.2 Models with Pure Strategy Solutions

5.2.1 Movement Arrows with two players and a payoff matrix:

5.2.2 Saddle Point Method

5.3 Dominance and Dominated strategies

Exercises Section 5.1 Pure Strategy Games

5.3 Mixed Strategy in two player 2 strategy games

5.3 Linear Programming and Total Conflict Games

Summary

Chapter 6 Partial Conflict Games: The Classical Two-Player Games. Error! Bookmark not defined.

6.1 Partial Conflict Simultaneous Games Introduction

6.2 The Prisoner’s Dilemma

6.3 The Game of Chicken

Reference and Further Readings

Chapter 7 Utility Theory
7.1 Introduction
7.2 Ordinal Numbers
7.3 Cardinal numbers
7.4 Utility
7.4 Von Neumann-Morgenstern Utilities Applied to Game Theory.
7.5 An alternative approach to the lottery method in utility theory for game theory

7.5.1 Lottery Method Illustrated
7.5.2 AHP Method

7.5.3 AHP Example in Game Theory

7.6 Summary and Conclusions

References

Chapter 8. Nash Equilibrium and Non-Cooperative Solutions in Partial Conflict Games

8.1 Introduction
8.2 Pure Strategies and Dominance review in symmetric games

8.3 Equalizing Strategies

8.4 Prudential Strategies with LP
8.5 Applications

EXERCISES

Chapter 9 Evolutionary stable Strategies

9.1 Introduction

Summary
Exercises Chapter 9
Reference

Chapter 10 Communications
10.1 Introduction
10.2 The Game of Chicken Without Communication
10.3 The Game of Chicken With Communication

10.3.1 Moving First or Committing to Move First

10.3.2 Issuing a Threat

10.3.3 Issuing a Promise

10.4 Credibility

Classical Game Theory and the Missile Crisis (from Brahm ,1994)

Theory of Moves and the Missile Crisis

Chapter 10 Exercises

References and Further Reading

Chapter 11 Nash Arbitration Method
11.1 Introduction to Nash Arbitration
11.2 Methods without calculus
11.3 More than two strategies
11.4 Writer’s Guild Strike example with cardinal numbers

Introduction

Nash Arbitration Scheme

Chapter 12 Three Person Games
12.1 Three Person Zero-Sum games
12.2 Three-Person Partial Conflict Game ( Non-Zero Sum Game).
12.4 NON-ZERO Sum (non-constant sum) with no pure strategies.
12.5 3-Person game with Technology
Exercises

Chapter 13 Extensive Form Games

13.1 Introduction

Example 1. Kidnapping for ransom
Applying TOM
Exercises Chapter 13