Subsolutions and the Supercore of Cooperative Games

1. Solutions. When von Neumann and Morgenstern first defined the solution of a cooperative game, they did so in the context of characteristic function games with side payments. They defined a solution to be a set S of imputations such that: (a) " N o / contained in S is dominated by an x contained in S". (b) "Every/ not contained in S is dominated by some x contained in S". Condition (a) is called internal stability, while condition (b) is called external stability. The above definition does not involve any of the special structure associated with characteristic function games with side payments, and has subsequently been used in precisely the same form to define solutions for more general kinds of games.' Here, we will require very little structure of any sort and, following von Neumann and Morgenstern,^ we will direct our attention to abstract games, which we define to be pairs {X, >), where X is an arbitrary set whose members are called the outcomes of the game, and > is an arbitrary binary relation defined on X and called domination. Following Gillies [1959], we will define the dominion of a point x e A' to be the set D(x) = {y ^ X \ x> y). That is, the dominion of an outcome x consists of all those outcomes that are dominated by it. In the same way, we define the dominion of a set of outcomes S c X to be D{S)=\j^^D{x), the set of outcomes dominated by some point in 5. Finally, we denote the complement of D{S), the set of outcomes undominated by any outcome in the set S, by U{S)^ X D{S), and denote \ We are now in a position to rewrite the conditions of internal and external stability in terms of the function U, and define a solution to be a set S c X such that S c U{S) and S D U{S). These two conditions are precisely the same as conditions (a) and (b), and they make it apparent that 5* is a solution if and only if̂ (c) 5 = U{S). In a similarly compact way, the core of an abstract game is defined to be the set of outcomes undominated by any other outcome, i.e., the set C = U(X). von Neumann and Morgenstern interpreted a solution as a 'standard of behavior', which once generally accepted by the players of the game, would become selfenforcing. They justified this by arguing that outcomes outside of the solution would