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Core stability of the Shapley value for cooperative games

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Abstract

Our objective is to analyze the relationship between the Shapley value and the core of cooperative games with transferable utility. We first characterize balanced games, i.e., the set of games with a nonempty core, through geometric properties. We show that the set of balanced games generates a polyhedral cone and that a game is balanced if and only if it is a nonnegative linear combination of some simple games. Moreover, we show that the set of games whose Shapley value lies in the core also yields a polyhedral cone and that a game obeys this property if and only if it is a nonnegative linear combination of simple games satisfying certain properties. By-products, we also show that the number of games that correspond to the extreme rays of the polyhedron coincides with the number of minimal balanced collections.

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Notes

  1. See Young (1985), and, for recent studies, Casajus (2011, 2014), and Casajus and Yokote (2017) and references therein.

  2. Shapley and Shubik (1954) apply the Shapley value to evaluate the distribution of power among the members of a committee system. Hart and Moore (1990) use the Shapley value as each agent’s payoff to analyze the incomplete contract model. Gul (1989), Pérez-Castrillo and Wettstein (2001) and McQuillin and Sugden (2016) provide implementation procedures for obtaining the Shapley value as the subgame perfect equilibrium outcome of the game.

  3. Consistency properties play a central role in axiomatic characterizations of the core. Davis and Maschler (1965), Moulin (1985), Peleg (1986), and Tadenuma (1992) introduce different types of consistencies and axiomatize the core. Abe (2017) axiomatically characterizes the core for games with externalities. Perry and Reny (1994) offer a noncooperative game in which a core element is implemented.

  4. Average convexity is also analyzed by Sprumont (1990). He calls it quasiconvexity in his work. However, his approach is totally different from those of Inarra and Usategui (1993); Izawa and Takahashi (1998). He defines the Shapley value for every subset of the grand coalition and considers an allocation scheme for all possible coalitions. He shows that an allocation scheme is population monotonic for every quasiconvex game.

  5. We discuss their conditions in Sect. 4.

  6. Shapley (1971) provides the geometric characterization of the core in convex games. Marinacci and Montrucchio (2004) provide a similar characterization by means of the Choquet integral with respect to the underlying game.

  7. A solution \(f: {\mathcal {G}}_N \rightarrow {\mathbb {R}}^n\) is linear if for every \(c, c' \in {\mathbb {R}}\) and \(v, v' \in {\mathcal {G}}_N, f(cv+c'v')= cf(v)+c'f(v')\). Harsanyi’s dividend was proposed by Harsanyi (1959).

  8. The explicit description of each extreme point in general n-player games is still open. This is because it is generally difficult to construct extreme points of a convex polyhedron. Peleg (1965) provides an algorithm to calculate all extreme points.

  9. For every \(a, b \in {\mathbb {R}}^k\), \(a \cdot b=\sum ^k_{i=1}a_ib_i\) is a standard inner product in \({\mathbb {R}}^k\).

  10. In this result, (1) \(\Rightarrow\) (2) is known as Minkowski’s Theorem, and the converse, (2) \(\Rightarrow\) (1), is known as Weyl’s Theorem. For details, see Ziegler (1995).

  11. A game v is 0-normalized if \(v(\{i\})=0\) for all \(i \in N\).

  12. A game v is simple if \(v(S)=0\) or 1 for all \(S \subseteq N\). A player \(i \in N\) is a veto player in v if \(v(S)=0\) for every \(S \subset N \setminus \{i\}\). A game v is veto-controlled if there is a veto player in v. A game v is N-monotonic if \(v(S) \le v(N)\) for all \(S \subseteq N\). See also Derks (1987) and discussions in Sprumont (1990).

  13. In the decision theory literature, Dillenberger and Sadowski (2019) propose a similar concept, which they call generalized partition.

  14. Formally, a game \(v \in {\mathcal {G}}_{N}\) is convex if for every \(S, T \subseteq N\), \(v(S \cup T)+v(S \cap T) \ge v(S)+v(T)\), and is average convex if for every nonempty \(S,T\subseteq N\) with \(S\subseteq T\), \(\sum _{j\in S}(v(T)-v(T{\setminus } \{j\}))\ge \sum _{j\in S}(v(S)-v(S{\setminus } \{j\}))\). Note that convexity implies average convexity. For a counter-example of the opposite direction, see examples in Inarra and Usategui (1993) and Izawa and Takahashi (1998).

  15. We would like to thank an anonymous referee for suggesting that we consider this issue.

  16. Various linear solutions are intensively studied as a complement to or a counterpart of the Shapley value: for example, weighted Shapley values (Shapley 1953a; Chun 1988, 1991; Kalai and Samet 1987; Nowak and Radzik 1995; Yokote 2015), egalitarian Shapley values and their generalization (Joosten 1996; Casajus and Huettner 2013, 2014; van den Brink et al. 2013; Abe and Nakada 2019; Yokote and Funaki 2018), and the CIS/ENSC value (Driessen and Funaki 1991). See also Yokote et al. (2017) for other solutions.

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Authors and Affiliations

  1. School of Engineering, Tokyo Institute of Technology, 2-12-1, Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Takaaki Abe

  2. School of Management, Department of Business Economics, 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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Correspondence to Satoshi Nakada.

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This study was funded by the Japan Society for the Promotion of Science KAKENHI: Nos. 19K23206, 22K13362 (Abe) and No. 19K13651 (Nakada). The authors have no financial or proprietary interests in any material discussed in this article and there is no competing interest to declare.

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We thank an associate editor and an anonymous referee for their valuable comments. We are also grateful to Yukihiko Funaki, Toshiyuki Hirai, Stephen Morris, Takashi Ui, Jun Wako, three anonymous referees of the Kanematsu Fellowship, and participants of various seminars and conferences. This paper is a substantially revised version of our previous paper circulated as “Generalized Potentials, Value, and Core”, which received the Kanematsu Fellowship from the Research Institute for Economics and Business Administration of Kobe University in 2017. Abe and Nakada acknowledge the financial support from Japan Society for the Promotion of Science KAKENHI: Nos. 19K23206, 22K13362 (Abe), and No. 19K13651 (Nakada). All remaining errors are our own.

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Abe, T., Nakada, S. Core stability of the Shapley value for cooperative games. Soc Choice Welf 60, 523–543 (2023). https://doi.org/10.1007/s00355-022-01432-4

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