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Characterizations of solutions for games with precedence constraints

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Abstract

We generalize the characterizations of the positive core and the positive prekernel to TU games with precedence constraints and show that the positive core is characterized by non-emptiness (NE), boundedness (BOUND), covariance under strategic equivalence, closedness (CLOS), the reduced game property (RGP), the reconfirmation property (RCP) for suitably generalized Davis–Maschler reduced games, and the possibility of nondiscrimination. The bounded positive core, i.e., the union of all bounded faces of the positive core, is characterized similarly. Just RCP has to be replaced by a suitable weaker axiom, a weak version of CRGP (the converse RGP) has to be added, and CLOS can be deleted. For classical games the prenucleolus is the unique further solution that satisfies the axioms, but for games with precedence constraints it violates NE as well as the prekernel. The positive prekernel, however, is axiomatized by NE, anonymity, reasonableness, the weak RGP, CRGP, and weak unanimity for two-person games (WUTPG), and the bounded positive prekernel is axiomatized similarly by requiring WUTPG only for classical two-person games and adding BOUND.

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Notes

  1. We mention here the dual view of distributive lattices, introduced by Gilles et al. (1992), and developed by, e.g., Derks and Gilles (1995) or van den Brink and Gilles (1996). There, hierarchies are interpreted as permission structures, in the sense that feasible coalitions are those who contain all superiors of its members, i.e., if \(i\) is present and \(i\prec j\), then \(j\) must be present. Under this viewpoint, players must have their superiors present, who permit them to act. By contrast, under our viewpoint, feasible coalitions \(S\) may be considered as teams to whom some task or project is assigned. The task can be achieved only if all subordinates are present in \(S\), and \(v(S)\) represents the benefit induced by the realization of the task by \(S\). This point of view is discussed by Grabisch and Xie (2007) in detail.

  2. A poset \((P,\preceq )\) is a lattice if for any \(x,y\in P\) their supremum, denoted \(x\wedge y\), and infimum, denoted \(x\vee y\), exist. A lattice is distributive if \(\wedge \) and \(\vee \) satisfy distributivity.

  3. When cooperation is not restricted, i.e., all coalitions are feasible, we call a game unrestricted (free).

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Acknowledgments

We are grateful to two anonymous referees of this journal for their remarks that helped to improve the writing of this paper. The first author thanks the Agence Nationale de la Recherche for financial support under contract ANR-13-BSHS1-0010. The second author acknowledges support from the Danish Council for Independent Research \(|\) Social Sciences under the FINQ project and from the Spanish Ministerio de Ciencia e Innovación under Project ECO2012-33618, co-funded by the ERDF

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Authors and Affiliations

  1. Paris School of Economics, University of Paris I, 106-112, Bd. de l’Hôpital, 75013, Paris, France

    Michel Grabisch

  2. Department of Business and Economics and COHERE, University of Southern Denmark, Campusvej 55, 5230, Odense M, Denmark

    Peter Sudhölter

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  1. Michel Grabisch
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  2. Peter Sudhölter
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Correspondence to Peter Sudhölter.

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Grabisch, M., Sudhölter, P. Characterizations of solutions for games with precedence constraints. Int J Game Theory 45, 269–290 (2016). https://doi.org/10.1007/s00182-015-0465-y

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