Self-duality of solutions in cooperative games

In this paper we extend the self-duality of a rule in a bankruptcy problem to a property (we call this the self-duality) of a solution-concept in coalitional games. First, we define a new coalitional game, corresponding to an original coalitional game. When the original game is associated with a bankruptcy problem, the new coalitional game corresponds to the bankruptcy problem which appears in the dual rule. Then we define the dual of a solution-concept, using the new coalitional game. Then we define the self-duality of a solution-concept. When original games are associated with bankruptcy problems, the dual solution and the self-duality correspond to the dual rule and self-duality of rules in bankruptcy problems. We show both the Shapely value and the prenucleolus have the self-duality. Furthermore, we see the self-duality is closely related to the negative duality of a solution-concept.-p.1.