On Superadditivity and Convexity for Cooperative Games with Random Payoffs

In this paper we study the relations between superadditivity and several types of convexity for cooperative games with random payoffs. The types of convexity considered are marginal convexity (all marginal vectors belong to the core), individual-merge convexity (any individual player is better off joining a larger coalition) and coalitional-merge convexity (any coalition of players is better off joining a larger coalition). In particular, in this work we answer two open questions in the literature. The first question is whether a marginal convex game is always superadditive. In general, the answer is negative as is shown by two counterexamples. However, for some type of games marginal convexity does imply superadditivity. The second question is whether individual-merge convexity implies coalitional-merge convexity for games with at least four players. An example of a four-player game that is individual-merge convex but not coalitional merge convex shows that this is not the case in general. But also here we show that for some type of games the implication does hold. 2000 AMS Subject classification: 91A12.