Abstract
This paper discusses games where cooperation is restricted by a hierarchical structure. The model assumes that there is a hierarchy between certain coalitions given by a partition. Face games (González-Díaz and Sánchez-Rodríguez in Games Econ Behav 62:100–105, 2008) play a central role in the definition of the proposed hierarchical game. It turns out that the Shapley value of the hierarchical game coincides with the Faigle and Kern precedence constraint value (Faigle and Kern in Int J Game Theory 21:249–266, 1992) and with some particular weighted Shapley value (Kalai and Samet in Int J Game Theory 16:205–222, 1987). Two new characterizations of this hierarchical value are given. Finally, an application of the model is given for bankruptcy problems.
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Notes
Given a finite set \(A \subseteq \mathbb {R}^n, con(A)\) denotes the convex hull of \(A\).
In that paper the lemma is stated for convex games, but the proof do not use convexity.
Geometrically, it means that the hierarchical value of a convex game is at the boundary of the core. It is located in the intersection of all the hyperplanes defined by the hierarchical efficiency conditions.
Observe that hierarchical efficiency is stronger than efficiency.
Relative efficiency was already proposed in Sprumont (1990) as one of the requirements of the population monotonic allocation scheme (PMAS).
If the game \((N,v_{P_>})\) is decomposable with respect to \(P_>\), then \(\sum _{i\in P_j}Sh_i(N,v_{P_{>}})=v_{P_>}(P_j)\).
Players of \(N \!\backslash \! P_l\) are null players in \((N,v_{P_>}^l)\).
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We are grateful to two anonymous referees for helpful comments. Authors acknowledge the financial support of the Spanish Ministerio de Ciencia e Innovación, MTM2011-27731-C03-03.
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Fiestras-Janeiro, M.G., Sánchez-Rodríguez, E. & Schuster, M. A precedence constraint value revisited. TOP 24, 156–179 (2016). https://doi.org/10.1007/s11750-015-0380-1
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DOI: https://doi.org/10.1007/s11750-015-0380-1