Equal Treatment for Both Sides of Assignment Games in the Modified Least Core

In contrast to the least core and the nucleolus, the modified least core and the modified nucleolus take into account both, the "power", i.e. the worth, and the "blocking power" of a coalition, i.e. the amount which the coalition cannot be prevented from by the complement coalition. The least core and nucleolus respectively minimizes the highest excess and successively minimizes the nonincreasingly ordered vector of excesses respectively. The modified solutions arise from an analogous procedure in which excesses are replaced by differences of excesses. In this paper it is shown that both sides of an assignment game are treated equally from every element of the modified least core. Moreover, a similar property is shown for games which can be written as a minimum of two additive games and are called M2-games. Both classes of games share a common property. Every assignment and every M2-game is a member of the class of complementary concave games. With the help of complementary convexity it is shown that the modified least core of an assignment and M2-game is contained in the least core of its dual game.