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Generalized Potentials, Value, and Core

Takaaki Abe and Satoshi Nakada
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Takaaki Abe: Graduate School of Economics, Waseda University
Satoshi Nakada: Department of Business Economics, Tokyo University of Science

No DP2018-19, Discussion Paper Series from Research Institute for Economics & Business Administration, Kobe University

Abstract: 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.

Keywords: Shapley value; Core; Potential; Cooperative game (search for similar items in EconPapers)
JEL-codes: C71 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-des and nep-gth
Date: 2018-08
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