Chapter 17 Von Neumann-Morgenstern stable sets

Neumann and Morgenstern (1944) presented the first general model and solution concept for the multiperson cooperative theory. Stable set theory is viewed as only one of several approaches for analyzing coalitional games. This chapter focuses on the von Neumann-Morgenstern stable sets, describes their original model for the coalitional games along with some illustrations, analyzes the three-person case in detail, and discusses some of the mathematical properties of stable sets. Stable sets are defined in terms of two simple conditions: internal and external stability; along with a rather simple preference relation called domination. These stability concepts are rather basic and fundamental mathematical notions, and presume very little about the nature or structure of social institutions and interactions. Stable sets often predict likely social structures and how groups will organize themselves. They show the important role of minimal winning coalitions and minimal-sized veto (or blocking) coalitions. They often show how a game will decompose into subgames between critical coalitions. They exhibit a variety of standards of behavior and delineate bargaining ranges. They predict the formation of cartels and illustrate the stability of discrimination and its limits. Few assumptions can lead to many insights into coalition formation, competition, and distributions of wealth. There are also many situations where stable set theory matches well with experimental results. The chapter also discusses some highly undesirable properties of stable set theories.