Marginality and convexity in partition function form games

In this paper an order on the set of embedded coalitions is studied in detail. This allows us to define new notions of superaddivity and convexity of games in partition function form which are compared to other proposals in the literature. The main results are two characterizations of convexity. The first one uses non-decreasing contributions to coalitions of increasing size and can thus be considered parallel to the classic result for cooperative games without externalities. The second one is based on the standard convexity of associated games without externalities that we define using a partition of the player set. Using the later result, we can conclude that some of the generalizations of the Shapley value to games in partition function form lie within the cores of specific classic games when the original game is convexThis work has been supported by FEDER/Ministerio de Ciencia, Innovación y Universidades – Agencia Estatal de Investigación/MTM2017-87197-C3-2-P, /MTM2017-87197-C3-3-P,/ PID2020-113110GB-L00, /MTM2017-83455-P, by the Generalitat de Catalonia through grant 2017-SGR-778, by the Junta de Andalucía through grant FQM237, and by the Xunta de Galicia through the European Regional Development Fund (Grupos de Referencia Competitiva ED431C-2016-040 and ED431C-2017/38)S