Potentials and solutions of cooperative games with a fixed player set

This paper considers the solutions of cooperative games with a fixed player set that admit a potential function. We say that a solution admits a potential function if the solution is given as the marginal contribution according to the potential function. Hart and Mas-Colell (Econometrica 57(3):589–614, 1989) show that the Shapley value is the only solution that is efficient and admits the HM potential function for games with variable player sets. First, we argue that various solutions admit a potential function if we remove efficiency. Second, we define a potential function for games with a fixed player set and characterize the class of the solutions that admit a potential function by providing their general functional form. Finally, we associate a potential function with the axioms that the Shapley value obeys, which uncovers why the efficiency requirement uniquely pins down the Shapley value in the class of solutions.