Stable sets in matching problems with coalitional sovereignty and path dominance

We study von Neumann Morgenstern stable sets for one-to-one matching problems under the assumption of coalitional sovereignty (C), meaning that a deviating coalition of players does not have the power to arrange the matches of agents outside the coalition. We study both the case of pairwise and coalitional deviations. We argue further that dominance has to be replaced by path dominance (P) along the lines of van Deemen (1991) and Page and Wooders (2009). This results in the pairwise CP vNM set in the case of pairwise deviations and the CP vNM set in the case of coalitional deviations. We obtain a unique prediction for both types of stable sets: the set of matchings that belong to the core.