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The weighted Shapley-egalitarian value for cooperative games with a coalition structure

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Abstract

In this paper, we study a solution, the weighted Shapley-egalitarian value, for transferable utility cooperative games with a coalition structure. It first distributes the worth of the grand coalition among a priori unions of the coalition structure with the weighted Shapley value, with the sizes of unions acting as weights. And then, it allocates the payoff of every union among its players with the equal division value, thus players inside a union exhibit a greater degree of solidarity than players in different unions. We give several equivalent definitions of the value, particularly show its relationship with the collective value. We apply the value to the recent field of airport cost pooling game, and find that the Shapley value therein can be viewed as a special case of our value, if we view an airport cost pooling game as a transferable utility cooperative game with a coalition structure. Finally, to further highlight the differences between our value and the collective value, we provide parallel axiomatizations of the two values, by replacing the collective balanced contributions axiom with two intuitive axioms.

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Notes

  1. Given two values \(\phi \) and \(\psi \) for TU games, an Owen-type value (respectively, a collective-type value) first distributes the worth of the grand coalition among unions with \(\phi \), and then allocates the payoff of every union (respectively, the worth of every union) among its players with \(\psi \). For more information about these two types of values, see Béal et al. (2018).

  2. Besides this kind of restriction, another kind of restriction, \({\mathcal {C}}|_S\equiv \{C\in {\mathcal {C}}\mid C\subseteq S\}\), is widely used in the context of TU games with a hypergraph. For example, see Shan et al. (2018).

  3. Strictly, to distinguish a player of the quotient game and a coalition of the original game, we should use different notations. For example, as the same with Winter (1989), for every \(C\in {\mathcal {C}}\), [C] represents a player in \(({\mathcal {C}},v^{{\mathcal {C}}})\), and C represents a coalition in (N, v); And \([{\mathcal {C}}]\) represents the grand coalition of the quotient game, and \({\mathcal {C}}\) represents the coalition structure. However, for notational conveniences, we use the present form, which is also a common way in the literature. For example, see Kamijo (2013).

  4. In fact, for every \(C'\in {\mathcal {U}}{\setminus }{\mathcal {C}}\), \(\omega _{C'}\) can be any positive number, as it is irrelevant to Eq. (4).

  5. Levy and McLean (1989) define another weighted induced game, which requires a subunion to keep the weight of the union. In this paper, as you will see, both weighted-induced games lead to the weighted Shapley-egalitarian value.

  6. In this paper, \(\lfloor S\rfloor \) is a singleton. However, to make the definition general enough to include the corresponding definition in Kamijo (2009), we use the present definition of coalition restricted game.

  7. Kamijo (2013) calls it balanced contributions, which may be easily confused with the balanced contributions axiom of Myerson (1980).

  8. The essence of the symmetry axiom in Hou et al. (2018) is egalitarian for the grand coalition.

  9. Assume that \(\sum _{i\in C}f_i(N{\setminus } C,v|_{N\setminus C},{\mathcal {C}}|_{N{\setminus } C})=0\).

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  1. School of Management, Guangzhou University, 230 Wai Huan Xi Road, Guangzhou Higher Education Mega Center, Guangzhou, 510006, China

    Xun-Feng Hu

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This work was supported by the National Social Science Fund of China (18ZDA043) and the National Nature Science Foundation of China (71901076, 71671053, 71841024).

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Hu, XF. The weighted Shapley-egalitarian value for cooperative games with a coalition structure. TOP 28, 193–212 (2020). https://doi.org/10.1007/s11750-019-00530-4

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