On the Approximate Core and Nucleon of Flow Games with Public Arcs

We investigate flow games featuring both private arcs owned by individual players and public arcs accessible cost-free to all coalitions. We explore two solution concepts within this framework: the approximate core and the nucleon. The approximate core relaxes core requirements by permitting a bounded relative payoff deviation for every coalition, and the nucleon is a multiplicative analogue of Schmeidler's nucleolus which lexicographically maximizes the vector consisting of relative payoff deviations for every coalition arranged in a non-decreasing order. By leveraging a decomposition property for paths and cycles in a flow network, we derive complete characterizations for the approximate core and demonstrate that the nucleon can be computed in polynomial time.