Convexity in Stochastic Cooperative Situations

This paper introduces a new model concerning cooperative situations in which the payoffs are modeled by random variables. First, we study adequate preference relations of the agents. Next, we define corresponding cooperative games and we introduce and study various basic notions like an allocation, the core and marginal vectors. Furthermore, we introduce three types of convexity, namely coalitional-merge, individual-merge and marginal convexity. The relations between these definitions are studied and in particular, as opposed to the deterministic counterparts for TU games, we show that these three types of convexity are not equivalent. However, all types imply that the core of the game is nonempty and the first two types even imply that each subgame has a nonempty core. In particular, we show that the Shapley value, the average of the marginal vectors, belongs to the core of the convex game for certain types of preferences and for any type of convexity. Journal of Economic Literature ClassificationNumber: C71. 1991 Mathematics Subject ClassificationNumber: 90D12.