On the set of imputations induced by the k-additive core

An extension to the classical notion of core is the notion of k-additive core, that is, the set of k-additive games which dominate a given game, where a k-additive game has its Möbius transform (or Harsanyi dividends) vanishing for subsets of more than k elements. Therefore, the 1-additive core coincides with the classical core. The advantages of the k-additive core is that it is never empty once k ≥ 2, and that it preserves the idea of coalitional rationality. However, it produces k-imputations, that is, imputations on individuals and coalitions of at most k individuals, instead of a classical imputation. Therefore one needs to derive a classical imputation from a k-order imputation by a so-called sharing rule. The paper investigates what set of imputations the k-additive core can produce from a given sharing rule.