The Shapley Dividend: A Stability-Oriented Solution Concept for Superadditive Games

In superadditive games, one of the primary goals is to identify allocation mechanisms to maintain the stability of the grand coalition. This paper explores stability concepts including the Core as well as its extensions, the ε-Core and the Δ-Core, addressing the potential emptiness issues. This paper further evaluates allocation methods such as the Shapley Value under stability concept of the Core and the Nucleolus under stability concept of the ε-Core. Under stability concept of the Δ-Core, we propose the Shapley Dividend as a sufficient solution concept to ensure the stability, derived from a modification of the Shapley Value calculating formula. Bounds for the Δ*of the Shapley Dividend are provided, and we highlight that it satisfies both the symmetrical property and the linear property, similar to the Shapley Value. Additionally, we provide theoretical analysis and numerical example to show that the Shapley Dividend satisfies the win-win property, benefiting the evolution of cooperation. To further demonstrate the win-win property, we refer to the SQDDPG algorithm and apply the Shapley Dividend to cooperative multi-agent reinforcement learning. The preliminary result indicates that the resulting SD3PG algorithm achieves superior experimental performance compared to SQDDPG.