Games with unique solutions that are nonconvex.
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In 1944 von Neumann and Morgenstern introduced a theory of solutions (stable sets) for ^-person games in characteristic function form. This paper describes an eight-person game in their model which has a unique solution that is nonconvex. Former results in solution theory had not indicated that the set of all solutions for a game should be of this nature. First, the essential definitions for an ^-person game will be stated. Then, a particular eight-person game is described. Finally, there is a brief discussion on how to construct additional games with unique and nonconvex solutions. The author [2] has subsequently used some variations of the techniques described in this paper to find a ten-person game which has no solution; thus providing a counterexample to the conjecture that every ^-person game has a solution in the sense of von Neumann and Morgenstern.
[1] Automorphisms of groups of similitudes over $F_{3}$. , 1969 .
[2] Donald B. Gillies,et al. 3. Solutions to General Non-Zero-Sum Games , 1959 .