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We consider the problem of designing distribution rules to share "welfare"(cost or revenue) among individually strategic agents. There are many knowndistribution rules that guarantee the existence of a (pure) Nash equilibrium inthis setting, e.g., the Shapley value and its weighted variants; however, acharacterization of the space of distribution rules that guarantee theexistence of a Nash equilibrium is unknown. Our work provides an exactcharacterization of this space for a specific class of scalable and separablegames, which includes a variety of applications such as facility location,routing, network formation, and coverage games. Given arbitrary local welfarefunctions W, we prove that a distribution rule guarantees equilibrium existencefor all games (i.e., all possible sets of resources, agent action sets, etc.)if and only if it is equivalent to a generalized weighted Shapley value on some"ground" welfare functions W', which can be distinct from W. However, ifbudget-balance is required in addition to the existence of a Nash equilibrium,then W' must be the same as W. We also provide an alternate characterization ofthis space in terms of "generalized" marginal contributions, which is moreappealing from the point of view of computational tractability. A possiblysurprising consequence of our result is that, in order to guarantee equilibriumexistence in all games with any fixed local welfare functions, it is necessaryto work within the class of potential games.

Potential Games are Necessary to Ensure Pure Nash Equilibria in Cost Sharing Games