In 1950-53 two solution concepts for cooperative games were proposed, both of which (unlike the core) provide unique solutions that are never null. These were due, respectively, to Nash and Shapley. Both were derived from systems of axioms that describe properties of a solution that might be considered reasonable or appropriate. Luce and Raiffa (1956) suggested that they might be interpreted as frameworks for arbitration, in that an arbiter would consider the properties of the decision in deciding on the distribution of payoffs among the group. (For more detail see McCain 2013, Chapters 1, 2; 2014b, Chapter 17 section 5.)
Nash begins by assuming that the payoffs to two interdependent decision makers must fall within a feasible set that is convex and compact.5 These properties assure that the assignment of payoffs to the two persons will be unique and are trivially satisfied for games in coalition function form. (Nash did not assume transferable utility). He assumes that the decision will have the properties of
(1) Individual rationality, that is, each agent receives at least as much as he could obtain if there is no agreement.
- (2) Pareto-optimality, that is, the decision cannot be improved on for both decision makers simultaneously.
- (3) Independence of irrelevant alternatives (see Chapter 7, section 7.2.1).
- (4) Solutions are covariant under strategic equivalence.
- (5) Symmetry. That is, if the two bargainers are interchanged, the chosen payoffs are interchanged accordingly.6
Given these assumptions Nash shows that the vector of chosen payoffs can be computed as the solution to a maximum problem. Let x*, y* be the payoffs to the two bargainers if there is no agreement and x, y be their payoffs from the agreement. Then x and y will be chosen so that the product (x - x*)(y - y*) is maximized among all the feasible pairs (x,y). This solution assigns unique payoffs to the bargainers and is never null. However, it directly applies only to two-person games and makes no allowance for differences in bargaining power. There are several proposals of extensions to more than two players, but none seems widely accepted (for example Harsanyi, 1956, 1963; Roth, 1979; McCain, 2013, Chapters 2, 6).