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.

可转移效用博弈的解决方案是自对偶的，如果它为每个游戏分配一组它分配给具有相反符号的对偶游戏的收益分配。自对偶解的著名例子是核心、Shapley 值、前核和 Dutta-Ray 解。为了评估解违反自对偶的程度，本注释定义了它的最小自对偶扩展，即包含它的最小自对偶解。类似地，定义了最大自对偶子解，即解包含的最大自对偶解。我们表明，每个解的最小自对偶扩展和最大自对对子解都是唯一存在的。作为一个应用，我们研究了插补解的自对偶性。