Generalized Potentials, Value, and Core


Our objective is to analyze the relationship between the Shapley value and the core from the perspective of the potential of a game. To this end, we introduce a new concept, generalized HM-potential, which is a generalization of the potential function defined by Hart and Mas-colell (1989). We show that the Shapley value lies in the core if and only if the maximum of the generalized HM-potential of a game is less than a cutoff value. Moreover, we show that this is equivalent to the minimum of the generalized HM-potential of a game being greater than another, different cutoff value. We also provide a geometric characterization of the class of games in which the Shapley value lies in the core, which also shows the relationship with convex games and average convex games as a corollary. Our results suggest a new approach to utilizing the potential function in cooperative game theory.


Shapley value; Core; Potential; Cooperative game

JEL Classification



Takaaki ABE
Graduate School of Economics, Waseda University

Satoshi NAKADA
Department of Business Economics, Tokyo University of Science
1-11-2, Fujimi, Chiyoda-ku, Tokyo
102-0071, Japan

*This Discussion Paper won the Kanematsu Fellowship Prize (FY 2017).