A Core Existence Theorem for Games Without Ordered Preferences

[Introduction] To a large extent the cooperative theory of games has an altogether different appearance from the noncooperative theory. The noncooperative theory generally deals with games in either extensive form or normal form, while the cooperative theory is usually described in characteristic function form. One of the central concepts in the cooperative theory is that of the core, which is the set of utility allocations which no coalition can improve upon. This notion of the core and of the characteristic function form of a game depends heavily on the existence of a utility representation for players' preferences. Recently Gale and Mas-Colell [3] and Shafer and Sonnenschein [6] have proven theorems on the existence of a Nash equilibrium for noncooperative games in normal form in which the players' preferences over strategy vectors are not necessarily complete or transitive and so may fail to have a utility representation. Thus it might appear that the noncooperative theory is applicable in environments where the cooperative theory is not. In order to formulate theorems in the cooperative theory of games which can be applied to environments in which players may have nonordered preferences, the characteristic function must be reformulated in terms of physical outcomes as opposed to utility outcomes. The players' preferences can then be expressed in terms of the physical outcomes without the use of a utility function.