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Potentials and solutions of cooperative games with a fixed player set

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Abstract

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.

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Notes

  1. Hart and Mas-Colell (1989) define \(P(\emptyset ,v)=0\) in their work. However, this difference does not matter in general: The HM-potential uniquely exists up to constant \(P(\emptyset ,v)=0\). For details, see (Casajus and Huettner 2018). We will briefly revisit this in Lemma 2 in Sect. 3.

  2. Sánchez (1997) and Calvo and Santos (1997) show that f admits a potential if and only if

    $$\begin{aligned} f_i(N, v)=Sh_i(N,v^f) \end{aligned}$$

    for all \((N, v) \in \mathcal {G}\) and \(i \in N\), where \(v^f(S)=\sum _{j\in S}f_j(v|_S)\) for every \(S\subseteq N\), and \(v|_S(T)=v(T)\) for every \(T\subseteq S\). In this sense, a solution that admits a potential is almost equivalent to the Shapley value. However, this consistency property does not insist that f is exactly equal to the Shapley value even if f admits a potential. See also Theorem 4 in Casajus and Huettner (2018).

  3. See, for example, Owen (1975) for their mathematical properties.

  4. Strictly speaking, the original result by Shapley does not fix the player set and considers career axiom on games with a variable player set. For detail, see (Winter 2002).

  5. This game is also called a nullified game (Béal et al. 2014, 2016). Specifically, game \(v^i\) is a nullified game with player i if \(v^i(S)=v(S \setminus \{i\})\) for all \(S \subseteq N\). By successive elimination of players in \(N {\setminus } T\), we can see that \(v_T=v^{N {\setminus } T}\).

  6. A solution f is decomposable if there is a solution \(\psi\), called a decomposer, such that

    $$\begin{aligned} f_i(N, v)=\psi _i(N,v)+\sum _{j \in zN \setminus \{i\}}\bigl [ \psi _j(N, v)-\psi _j(N \setminus \{i\}, v) \bigr ] \end{aligned}$$

    for all \((N, v) \in \mathcal {G}\) and \(i \in N\). If a decomposer \(\psi\) of f is also decomposable, then \(\psi\) is said to be a decomposable decomposer. Casajus and Huettner (2018) show that f is decomposable if and only if it admits a potential and, for each decomposable solution f, there is a unique decomposable decomposer. Note that (Casajus and Huettner 2018) define the decomposability and provide their results for all games \(\mathcal {G}\), that is, the games with a variable player set.

  7. An interaction potential is introduced by Ui (2000) in the noncooperative games to characterize the class of potential games (Monderer and Shapley 1996). Nakada (2018) adopts this concept to network formation games to characterize the class of games that has a network potential function (Chakrabarti and Gilles 2007).

  8. This condition is first introduced by Hart and Mas-Colell (1989). They call it the summability condition and note that the Shapley value satisfies it. Ortmann (1998) calls this condition path-independence, which is the same name as the corresponding concept used in Physics.

  9. This condition is introduced by Myerson (1980). Béal et al. (2016) call this axiom balanced contributions under nullification in games with a fixed player set.

  10. f is linear if for any \(v, v' \in \mathcal {G}_N\) and \(c, c' \in \mathbb {R}\), \(f(cv+c'v')= cf(v)+c'f(v')\).

  11. The former result shows that the unique efficient solution that admits a potential is the Shapley value, and the latter shows that the unique efficient solution that satisfies balanced contribution is the Shapley value.

  12. Béal et al. (2016) additionally impose that \(P({\textbf {0}})\) as a normalization and argue that this property is related to the null player property. However, our Lemma 2 suggests that these restrictions are unnecessary and the null player property holds for every potential with \(P({\textbf {0}})\). See the proof of Lemma 3.

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Authors and Affiliations

  1. School of Economics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Takaaki Abe

  2. Department of Business Economics, School of Management, Tokyo University of Science, 1-11-2, Fujimi, Chiyoda-ku, Tokyo, 102-0071, Japan

    Satoshi Nakada

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  1. Takaaki Abe
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  2. Satoshi Nakada
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Correspondence to Satoshi Nakada.

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We thank an associate editor and two anonymous referees for valuable comments that have significantly improved our paper. We also thank the seminar participants of the Summer Workshop on Game Theory 2018 at Fukuoka for their helpful comments. Abe and Nakada acknowledge the financial support from Japan Society for the Promotion of Science KAKENHI: No.19K23206 (Abe) and No.19K13651 (Nakada). All remaining errors are our own.

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Abe, T., Nakada, S. Potentials and solutions of cooperative games with a fixed player set. Int J Game Theory 52, 757–774 (2023). https://doi.org/10.1007/s00182-023-00839-2

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