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A Shapley value representation of network potentials

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Abstract

We consider network formation games by Jackson and Wolinsky (J Econ Theory 71:44–74, 1996) and characterize the class of games that have a network potential. We show that there exists a network potential if and only if each player’s payoff function can be represented as the Shapley value of a special class of cooperative games indexed by the networks. We also show that a network potential coincides with a potential of the same class of cooperative games.

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Notes

  1. Examples include supply chain networks among firms (Palsule-Desai et al. 2013), FTAs among several countries (Goyal and Joshi 2006; Furusawa and Konishi 2007), and social network services (Farrell and Fudge 2013). See Jackson (2008) for other examples.

  2. Ui (2000) shows that there exists a potential of noncooperative games (Monderer and Shapley 1996) if and only if each player’s payoff function is represented as the Shapley value of a particular class of cooperative games indexed by strategy profiles.

  3. Note that the value of a coalition \(S \in 2^N\) is determined by the network structure in S and not by the network structure of \(N \backslash S\).

  4. This result is similar to that of Slade (1994) and Facchini et al. (1997) in noncooperative games.

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Authors and Affiliations

  1. Graduate School of Economics, Hitotsubashi University, 2-1, Naka, Kunitachi, Tokyo, 186-8601, Japan

    Satoshi Nakada

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  1. Satoshi Nakada
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Correspondence to Satoshi Nakada.

Additional information

I am most grateful to my advisor Takashi Ui for his guidance and invaluable comments that have significantly improved this paper. I also thank an associate editor, two anonymous referees, Yukihiko Funaki, Ryota Iijima, Akifumi Ishihara, Atsushi Kajii, Kazuya Kikuchi, Takashi Kunimoto, Shintaro Miura and seminar participants in The Summer Meeting on the Game Theory 2015, The 21st Decentralization Conference in Japan and Waseda University for helpful comments. I acknowledge the financial support from Japan Society for Promotion of Science. All remaining errors are my own.

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Nakada, S. A Shapley value representation of network potentials. Int J Game Theory 47, 1151–1157 (2018). https://doi.org/10.1007/s00182-017-0605-7

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