Set Functions, Games and Capacities in Decision Making

1. Front Matter

   Pages i-xvi

   PDF

2. Introduction

   • Michel Grabisch

   Pages 1-23

3. Set Functions, Capacities and Games

   • Michel Grabisch

   Pages 25-144

4. The Core and the Selectope of Games

   • Michel Grabisch

   Pages 145-187

5. Integrals

   • Michel Grabisch

   Pages 189-279

6. Decision Under Risk and Uncertainty

   • Michel Grabisch

   Pages 281-323

7. Decision with Multiple Criteria

   • Michel Grabisch

   Pages 325-375

8. Dempster-Shafer and Possibility Theory

   • Michel Grabisch

   Pages 377-437

9. Back Matter

   Pages 439-473

   PDF

The book provides a thorough treatment of set functions, games and capacities as well as integrals with respect to capacities and games, in a mathematical rigorous presentation and in view of application to decision making. After a short chapter introducing some required basic knowledge (linear programming, polyhedra, ordered sets) and notation, the first part of the book consists of three long chapters developing the mathematical aspects. This part is not related to a particular application field and, by its neutral mathematical style, is useful to the widest audience. It gathers many results and notions which are scattered in the literature of various domains (game theory, decision, combinatorial optimization and operations research). The second part consists of three chapters, applying the previous notions in decision making and modelling: decision under uncertainty, decision with multiple criteria, possibility theory and Dempster-Shafer theory.

• Capacities
• Choquet integral
• Cooperative Games
• Decision making
• Decision under uncertainty
• Decision with multiple criteria
• Dempster-Shafer theory
• Possibility theory
• Pseudo-Boolean function
• Set functions

“Michel Grabisch has fully succeeded in transforming his deep knowledge of the field into a very rich and quite readable text.” (Professor Denis Bouyssou, CNRS and Université Paris Dauphine)

“This monograph provides a much-needed perspective on a rich collection of structural properties of set functions which have been investigated for more than 60 years, but which have never been collected in a single volume. By establishing these results in a clear and rigorous mathematical framework, the author makes them accessible to a broad audience. His book should rapidly become a classic reference in the field.” (Professor Yves Crama, University of Liège and HEC Liège)

“This book is an excellent compendium of technical notions pertaining to set functions, that are relevant in uncertainty management, game theory, and multicriteriadecision-making. It will be a very useful companion to any student or researcher willing to investigate these fields beyond standard probability theory and simple additive methods.” (Professor Didier Dubois, IRIT, Université Paul Sabatier, Toulouse)

“In this book, written by a leading scholar, the readers can finally find a unified and comprehensive study of set functions. It is a much needed, beautiful, book that will be most useful to the cognoscenti, who will appreciate its depth and breadth, as well as to the readers who desire to learn about this fundamental topic.” (Professor Massimo Marinacci, Università Bocconi)

“Nonadditive set functions are used in many optimization and decision theories (uncertain, fuzzy, multi-criteria, strategic, AI, combinatorial, statistical, …), many developing the same results independently.  The only person knowing all these fields brought it all together and improved it brilliantly in this well-focused book. It is a must for anyone using set functions.” (Professor Peter P. Wakker, Erasmus University Rotterdam)

• Paris School of Economics, Université Paris I Panthéon-Sorbonne, Paris, France

  Michel Grabisch

Policies and ethics