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**Models in cooperative game theory.
2nd revised ed.**
*(English)*
Zbl 1142.91017

Berlin: Springer (ISBN 978-3-540-77953-7/hbk). xi, 203 p. (2008).

This is the second edition of the highly regarded monograph on cooperative games by Branzei, Dimitrov and Tijs. As in the first edition [Zbl 1079.91005], all games are transferable utility, coalition form \((N, v)\) with coalition function \(v\) and finite player set \(N\).

The monograph consists of 12 chapters, spread over 3 major classes of such games. The style is theorem-proof throughout. Many of the results and proofs are relatively new (the last decade). The first 5 chapters study crisp (as opposed to fuzzy) games, the sort introduced by von Neumann and Morgenstern. Besides standard results on core and Shapley value, this module sheds interesting light on the Weber set, clan games, and convex games. Chapters 6-9 are a very clear explication of fuzzy games, and contain most of what is known about such games. It is striking, the extent to which results for crips games can be extended to their fuzzy counterparts. Chapters 10-12 consider multi-choice games, essentially fuzzy games with a finite set of participation levels. The 128 references add to the value of the presentation.

This monograph will bring the reader to almost everything currently known about cooperative games with transferable utility in coalition form.

The monograph consists of 12 chapters, spread over 3 major classes of such games. The style is theorem-proof throughout. Many of the results and proofs are relatively new (the last decade). The first 5 chapters study crisp (as opposed to fuzzy) games, the sort introduced by von Neumann and Morgenstern. Besides standard results on core and Shapley value, this module sheds interesting light on the Weber set, clan games, and convex games. Chapters 6-9 are a very clear explication of fuzzy games, and contain most of what is known about such games. It is striking, the extent to which results for crips games can be extended to their fuzzy counterparts. Chapters 10-12 consider multi-choice games, essentially fuzzy games with a finite set of participation levels. The 128 references add to the value of the presentation.

This monograph will bring the reader to almost everything currently known about cooperative games with transferable utility in coalition form.

Reviewer: Roy Gardner (Bloomington)