Abstract
This paper extends the notion of individual minimal rights for a transferable utility game (TU-game) to coalitional minimal rights using minimal balanced families of a specific type, thus defining a corresponding minimal rights game. It is shown that the core of a TU-game coincides with the core of the corresponding minimal rights game. Moreover, the paper introduces the notion of the \(k\)-core cover as an extension of the core cover. The \(k\)-core cover of a TU-game consists of all efficient payoff vectors for which the total joint payoff for any coalition of size at most \(k\) is bounded from above by the value of this coalition in the corresponding dual game, and from below by the value of this coalition in the corresponding minimal rights game. It is shown that the core of a TU-game with player set \(N\) coincides with the largest integer below or equal to \(\frac{|N|}{2}\)-core cover. Furthermore, full characterizations of games for which a \(k\)-core cover is nonempty and for which a \(k\)-core cover coincides with the core are provided.
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Notes
In a similar spirit, Grabisch and Miranda (2008) introduced the \(k\)-additive core, where coalitions of at most size \(k\) pay an important role. It turns out that the \(k\)-additive core in Grabisch and Miranda (2008) has no general relation with neither the \(k\)-core cover, nor the \(k\)-core introduced in this paper. Moreover, unlike the \(k\)-core cover and \(k\)-core here defined, \(k\)-additive cores need not be linear and coalitions of size at least \(|N|-k\) do not have an influential role in their definition.
\(\lfloor \frac{|N|}{2}\rfloor \) is the integer part of \(\frac{|N|}{2}\)
In general, it is computationally hard to obtain \(v_m(S)\) even for coalitions \(S\) of size \(1\). In case of convex games, \(v_m=v\) as we show subsequently.
For each \(r\in \mathbb {R}\), \(\lfloor r\rfloor \) denotes the largest integer below or equal to \(r\).
Given a finite set \(A \subseteq \mathbb {R}^N\), \(con(A)\) denotes the convex hull of \(A\).
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Authors acknowledge the financial support of Ministerio de Ciencia, MTM2011-27731-C03.
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Sánchez-Rodríguez, E., Borm, P., Estévez-Fernández, A. et al. \(k\)-core covers and the core. Math Meth Oper Res 81, 147–167 (2015). https://doi.org/10.1007/s00186-014-0490-9
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DOI: https://doi.org/10.1007/s00186-014-0490-9