Games with No Solutions and Empty Cores

Rather general existence theorems are known for two of the three classical models in game theory. These include the theorems by J. von Neumann and J. F. Nash on the existence of equilibria for 2-person, zero-sum and n-person, general-sum games in normal (strategic) form; as well as the theorems on information and the nature of optimal strategies by E. Zermelo, H. W. Kuhn, N. N. Vorob'ev and G. L. Thompson for games in extensive (tree) form. This paper discusses the lack of a general existence theorem for the von Neumann-Morgenstern theory of solutions (stable sets) for their multiperson cooperative games in characteristic function (coalitional) form. In particular, it demonstrates that there are games of 14 or more players for which no such solution exists, and for which the core (an alternate solution concept) is the empty set.