Statistical Cost Sharing

We study the cost sharing problem for cooperative games in situations where the cost function C is not available via oracle queries, but must instead be learned from samples drawn from a distribution, represented as tuples (S,C(S)), for different subsets S of players. We formalize this approach, which we call STATISTICAL COST SHARING, and consider the computation of the core and the Shapley value. Expanding on the work by Balcan et al. [2015], we give precise sample complexity bounds for computing cost shares that satisfy the core property with high probability for any function with a non-empty core. For the Shapley value, which has never been studied in this setting, we show that for submodular cost functions with bounded curvature κ it can be approximated from samples from the uniform distribution to a √ 1− κ factor, and that the bound is tight. We then define statistical analogues of the Shapley axioms, and derive a notion of statistical Shapley value and that these can be approximated arbitrarily well from samples from any distribution and for any function.