Abstract
The main goal of this paper is to understand the reasons driving the coincidence of different allocation rules for different classes of games. We define a new symmetry property, reverse symmetry, and study its geometric and game theoretic implications. In particular, we show that most classic allocation rules satisfy it. Then, we introduce and study a notion of orthogonality between TU-games, which allows to establish a restricted additivity property for the nucleolus. Also, in our analysis we identify different classes of games for which all allocation rules satisfying some sets of basic properties coincide. These properties are satisfied, among others, by the Shapley value and the nucleolus.
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Notes
For a couple of exceptions see Brink (2007), where the author obtains characterizations of the equal division and equal surplus division solutions in which, essentially, the null player property in Shapley’s characterization is replaced with a nullifying player property (and translation covariance for equal surplus).
We use \(\subset \) for strict set inclusions and \(\subseteq \) for weak set inclusions.
Although throughout the paper we use the core to illustrate and motivate our analysis, in most of the figures we could have also used the Weber set or the core cover.
The statement for the core-center only applies to games where the core is nonempty. Since \(v\) is strongly reverse symmetric, nonemptyness of the core is equivalent to require that, for each \(S \subset N,\,v(S) \le 0\).
It is worth noting that orthogonality is related to the notion of decomposability introduced in Shapley (1971). Roughly speaking, a convex game is decomposable if and only if it is the sum of orthogonal games. Further, it is not hard to check that two zero-normalized games that are weakly orthogonal are also disjoint in the sense of van den Brink et al. (2006) and, therefore, orthogonal additivity will typically be weaker than disjoint additivity.
If the core is a singleton we define its (0-dimensional) volume to be 1.
The operation \(A+B\) denotes the Minkowski sum of the sets \(A\) and \(B\), i.e., \(A+B:=\{a+b: a\in A, b\in B\}\).
This version is just the adaptation of the approach taken in Kohlberg (1971) for the nucleolus.
We could as well rely on the less standard strong null player property (see, for instance, Peleg and Sudhölter 2003).
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Acknowledgments
We are grateful to Mikel Álvarez-Mozos, Gustavo Bergantiños, Juan Vidal, and two anonymous referees for helpful comments. We acknowledge the financial support of the Spanish Ministry for Science and Innovation through Projects ECO2008-03484-C02-02, MTM2011-27731-C03, and from the Xunta de Galicia through Project INCITE09-207-064-PR.
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González-Díaz, J., Sánchez-Rodríguez, E. Understanding the coincidence of allocation rules: symmetry and orthogonality in TU-games. Int J Game Theory 43, 821–843 (2014). https://doi.org/10.1007/s00182-013-0406-6
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DOI: https://doi.org/10.1007/s00182-013-0406-6