von Neumann–Morgenstern Solution

This chapter discusses a von Neumann–Morgenstern solution. Von Neumann and Morgenstern realized that games with more than two players called for the introduction of new concepts and a new instrument in comparison with the two-person zero-sum ones. The characteristic function of the game and the imputations hold the central place in their theory. The von Neumann–Morgenstern theory assumes in general that the utilities can be distributed among the players. The possibilities of forming coalitions are peculiar to triads and, in general, to games with more than two players. It may happen in a triad, for instance, that player 2 is favored by a possible coalition with player 1, in which case he must pay, that is, transfer a part of his utility to player 1 so that the latter chooses a strategy suitable for player 2. Some of the players form a coalition if they can choose jointly the strategies they will use. A triad can involve the following three types of coalition: (1) a trivial coalition in which each player plays for himself, the triad being actually a three-person game; (2) a coalition proper, in which two players join forces while the third stays alone; and (3) a total coalition in which all three players unite and cooperate in choosing the strategies they will use in the interests of all.