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 budgetbalanced 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.