🤖 AI Summary
This paper investigates stability and computational aspects of cooperative solutions in flow games featuring private arcs (owned by individual players) and free public arcs (shared by the entire coalition). To address the frequent emptiness of the classical core in this setting, the authors introduce two novel solution concepts: (1) the *approximate core*, which permits bounded relative deviations in coalition payoffs, and (2) the *nucleon*, a multiplicative generalization of Schmeidler’s nucleolus defined via lexicographic maximization of the nondecreasingly ordered vector of relative payoff deviations. The work provides the first necessary and sufficient characterization of the approximate core for flow games with public arcs. It also devises a polynomial-time algorithm for exact nucleon computation. Furthermore, it reveals how public arcs structurally influence coalition stability and payoff allocation, yielding a computationally tractable and interpretable allocation paradigm for cooperative games with shared resources.
📝 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.