  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 M\"obius 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 $k\geq 2$, and that it preserves the idea of
coalitional rationality. However, it produces $k$-imputations, that is,
imputations on individuals and coalitions of at most $k$ inidividuals, 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.
