The probabilistic games and the Shapley function

Cooperative games with transferable utilities are studied in a probabilistic framework: call them probabilistic games. In this setup, each coalition has some probability of formation, and the worth of the grand coalition is the expectation over its sub-coalitions due to this probability distribution. We propose the Shapley function for the class of probabilistic games. A special subclass of probabilistic games is studied, and the Shapley function for this subclass is characterized. In this special subclass, players make coalitions in a sequential manner, where prior knowledge about their compatibility with one another in a preceding coalition is used to predict the worth of the succeeding coalitions. This is a natural assumption and is never being studied in the literature.