On the Approximate Core and Nucleon of Flow Games with Public Arcs
Pengfei Liu, Han Xiao, Tianhang Lu, Qizhi Fang
TL;DR
The paper extends flow games to include public arcs and analyzes two relaxed stability concepts: the multiplicative $\epsilon$-approximate core and the nucleon. By leveraging a path-cycle decomposition and an auxiliary game $\widetilde{\Gamma}_D$, it derives a precise formula for the optimal core bound $\epsilon^*=\frac{\sigma_E}{\sigma_N}-1$ and a convex-hull characterization in terms of minimum $s$-$t$ cuts constrained to the private-arc set, while proving that the nucleon is computable in polynomial time. The results show a tight link between disjoint-path structure and core-like allocations, and establish the exact computational tractability of the nucleon in this setting. This yields complete structural and algorithmic insights for flow games with public arcs, enabling stable and fair resource-sharing in networks with shared infrastructure.
Abstract
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.