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On the core: complement-reduced game and max-reduced game

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Abstract

This paper presents two characterizations of the core on the domain of all NTU games. One is based on consistency with respect to “complement-reduced game” and converse consistency with respect to “max-reduced game”. The other is based on consistency with respect to “max-reduced game” and weak converse consistency with respect to “complement-reduced game”. Besides, we introduce an alternative definition of individual rationality, we name conditional individual rationality, which is compatible with non-emptiness. We discuss axiomatic characterizations involving conditional individual rationality for the core.

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Notes

  1. We follow the terminology of Thomson (1995).

  2. Of course, it is redundant on the domain of games with a nonempty core.

  3. For \(A\subseteq I\!\!R^S\), we say that x is an interior point of A if there is a \(y \in A\) such that \(y \gg x\). But, under the assumption of the non-levelness condition, the strict notion “\(\gg \)” can be replaced by the weak notion of a blocking coalition “\(>\)”. That is, x is an interior point of A if there is a \(y \in A\) such that \(y > x\). In Definition 2, we state that x is in the core of \((N,V)\) if \(x^S\) is not an interior point of \(V(S)\) for all \(S\subseteq N\). Therefore, under the assumption of the non-levelness condition, the weak notion of a blocking coalition (“\(>\)”) can be adopted instead. Here, we prefer the strict notion “\(\gg \)” because it is commonly used to define the core in the literature.

  4. I acknowledge that it was the AE who suggested CIR and raised the questions about its implications.

  5. For the definitions of the prenucleolus and the nucleolus of a game, see Orshan and Sudhölter (2010).

  6. For the definition of the kernel of a game, see Davis and Maschler (1965).

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Acknowledgments

The author is very grateful to the AE and anonymous referees for valuable comments which much improved the paper.

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  1. Department of Applied Mathematics, National Dong Hwa University, Hualien, Taiwan

    Yan-An Hwang

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  1. Yan-An Hwang
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Hwang, YA. On the core: complement-reduced game and max-reduced game. Int J Game Theory 42, 339–355 (2013). https://doi.org/10.1007/s00182-013-0372-z

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