Characterizing distribution rules for cost sharing games

We consider the problem of designing the distribution rule used to share “welfare” (cost or revenue) among individually strategic agents. There are many distribution rules known to guarantee the existence of a (pure Nash) equilibrium in this setting, e.g., the Shapley value and its weighted variants; however a characterization of the space of distribution rules that yield the existence of a Nash equilibrium is unknown. Our work provides a step towards such a characterization. We prove that when the welfare function is strictly submodular, a budget-balanced distribution rule guarantees equilibrium existence for all games (i.e., all possible sets of resources, agent action sets, etc.) if and only if it is a weighted Shapley value.