Multi-person games consist of three or more players and differ theoretically from single- and two-person games because they potentially involve coalitions. If the interests of the players coincide exactly, so that coalitions are unnecessary or meaningless, then the games are ones of pure coordination and reduce to the case of two-person cooperative games discussed already in Chapter 4. In such cases, the only possible coalition is the grand coalition, which involves all players acting in unison, and coordination is effected either by explicit communication or by informal expectation.

Zero-sum multi-person games, on the other hand, are radically affected by the possibility of coalition, since they introduce the potential for cooperation into a game that would otherwise not have any. These non-cooperative multi-person games use an approach which is an extension of the saddle/equilibrium point approach.

Partially cooperative and mixed-motive games come somewhere between the two extremes of purely cooperative and zero-sum games. Partially cooperative and mixed-motive games have more realistic solutions than those arising from completely non-cooperative games, although some have approaches which tend towards obscurity (von Neumann & Morgenstern, 1953).

Following a brief discussion on non-cooperative multi-person games, this chapter begins by extending some concepts and definitions to mixed-motive and partially cooperative multi-person games. Theories such as the minimal winning coalition theory and the minimum resource theory are discussed as useful predictors of coalition forming on committees. The bulk of the chapter is devoted to developing methods for analysing the distribution of power among factions on a committee.