# Interpretations of the Solution Concepts

The solution concepts are usually presented as mathematical forms, with their interpretation largely left open. Indeed, they may be susceptible of at least two interpretations, and it may be that more than one solution concept might be adopted for different interpretations.

## Stability interpretation

Returning to Game 2.5, we have said that imputation xa = 5, xb = 5, xc = 10 should be excluded, since xa and xb could then secede and earn 12. But what are we to make of this argument? Critics of the core concept have questioned this criterion along the following lines: if a and b are members of the grand coalition, then they have committed themselves to it, and to some prearranged imputation. For them to opportunistically abandon the coalition to increase their payoffs as a group is then seen as inconsistent for a cooperative-game analysis. The cooperative game solution should represent a binding contract. This binding contract interpretation might support Nash bargaining, the Shapley value, or the nucleolus, against the core and related concepts.

Indeed, we might say that the core concept is based on coalitional egoism. But isn’t coalitional egoism something we are likely to see in the real world, sometimes?

One possible response to the criticism is that commitments, however binding, are not forever. Thus, even if a and b were to remain with the grand coalition for a time, eventually their commitments would expire and they would be likely to make other arrangements. On this sort of interpretation, the core is a concept of stability. The empirical prediction would then be that imputations outside the core are unstable, and therefore relatively unlikely to be observed, since they will be short-lived in case they do occur.

## Rhetorical interpretation

Another possible interpretation is that the abandonment of the grand coalition by {a,b} might not literally take place, but might be a threat made in the course of bargaining over the division of the value created by the grand coalition. This interpretation has been primarily associated with other solution concepts, such as Aumann and Maschler’s (1964) bargaining sets (which will be beyond the scope of this chapter) and the nucleolus. Both Nash and Shapley referred to bargaining power as motivation for their models. But the rhetorical interpretation could be applied to the core as well. The empirical prediction then would be that imputations outside the core would be rarely or never observed, since they would be rejected in the bargaining that precedes the formation of coalitions.