Self-antidual extensions and subsolutions

A solution for transferable utility games is self-antidual if it assigns to each game the set of payoff allocations that it assigns to the antidual game with opposite sign. Well-known examples of self-antidual solutions are the core, the Shapley value, the prenucleolus, and the Dutta–Ray solution. To evaluate the extent to which a solution violates self-antiduality, this note defines its minimal self-antidual extension, i.e. the smallest self-antidual solution that contains it. Similarly, the maximal self-antidual subsolution is defined, i.e. the largest self-antidual solution that the solution contains. We show that both the minimal self-antidual extension and the maximal self-antidual subsolution uniquely exist for each solution. As an application, we study self-antiduality of the imputations solution.