This book provides a novel and unifying perspective on the structural properties of fair solutions in optimization formulations.  The book also addresses a growing interest in incorporating fairness into the models that lie behind many business and public policy decisions.   Since there are several ways to formulate fairness mathematically, the authors characterize optimal solutions that result from different formulations with the aim of informing the choice of an appropriate model for a given application.  The focus is on fairness criteria that combine efficiency with fairness since typically both are important in practice.  Most of these results are new and do not appear in the current literature. The book is directed towards a wide range of audiences including practitioners, researchers in mathematical optimization, and welfare economists.   In addition, this book: Presents practical linear, nonlinear, or mixed integer programming formulations and a wide variety of fairness models Includes detailed proofs that provide insight into the properties of each criterion Provides guidelines for selecting a fairness model and the tendency to incentivize cooperation or competition  About the Authors Özgün Elçi, Ph.D., is a Research Scientist on the Modeling and Optimization team at Amazon. John Hooker, Ph.D. is University Professor of Operations Research and T. Jerome Holleran Professor of Business Ethics and Social Responsibility, Emeritus, at Carnegie Mellon University. Peter Zhang, Ph.D, is an Assistant Professor at Carnegie Mellon University’s Heinz College of Information Systems and Public Policy.

Table of Contents:
Chapter 1. Introduction.- A Generic Optimization Model with Fairness.- Chapter 2. Hierarchical Distribution.- Chapter 3. Incentives and Sharing.- Chapter 4. Axiomatic and Bargaining Arguments.- Inequality Metrics.- Chapter 5. Maximin and Leximax Criteria.- Chapter 6. Beta Fairness.- Chspter 7. Alpha Fairness and the Nash Bargaining Solution.- Chapter 8.The Kalai-Smorodinsky Bargaining Solution.- Chapter 9. Utility Threshold Criteria.- Chapter 10. Equity Threshold Criteria.- Chapter 11. Utility Threshold Criteria with Leximax.-  Chapter 12. Summary of Results.