Abstract
We axiomatize the set of affine combinations between the Shapley value, the equal surplus division value, and the equal division value in cooperative games with transferable utilities. The set is characterized by efficiency, linearity, the balanced contributions property for equal contributors and outsiders, and the differential null player out property. The balanced contributions property for equal contributors and outsiders requires the balance of contributions between two players who contribute the same amount to the grand coalition and whose singleton coalitions earn the same worth. The differential null player out property requires that an elimination of a null player affects the other players identically. These two relational axioms are obtained by investigating Myerson’s (Int J Game Theory 9:169–182, 1980) balanced contributions property and Derks and Haller’s (Int Game Theory Rev 1:301–314, 1999) null player out property, respectively, from the perspective of a principle of Aristotle’s distributive justice, whereby “equals should be treated equally”.
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19 July 2019
We are grateful to Zhengxing Zou
Notes
More precisely, here, we also observe another principle that unequals should be treated unequally; the two principles together imply “The just, then, is a species of the proportionate” (p.113 in Aristotle et al. 1980).
See also Moulin (2003)
Thomson (2012) categorizes axioms into punctual and relational as follows: “... a relational axiom relates choices made across problems that are related in certain ways.”
The corresponding conditions on the class of games on a fixed player set are examined by Ferrières (2016), Ferrières (2017) and Kongo (2018) under the name of nullified equal loss or equal effect of a player’s nullification on others. Their properties focus on the effect of a player’s nullification (i.e., a player becoming a null player) on others, instead of that of a null player’s departure because the player set is fixed.
This axiom is also known as weak null player out in van den Brink and Funaki (2009).
While Sh and ED satisfy BCEC, ESD does not. These facts enable us to obtain a characterization of the affine combination between Sh and ED by replacing BCECO with a stronger BCEC.
A game (N, v) is monotonic if \(v(S)\ge v(T)\) for all \(S, T\subseteq N\) such that \(S\supseteq T\).
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Acknowledgements
The authors are grateful to an associate editor and two anonymous referees for their comments on the previous version of our paper. This work was supported by JSPS KAKENHI grant numbers 17H02503, 24220033, and 26380247.
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Yokote, K., Kongo, T. & Funaki, Y. Relationally equal treatment of equals and affine combinations of values for TU games. Soc Choice Welf 53, 197–212 (2019). https://doi.org/10.1007/s00355-019-01180-y
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DOI: https://doi.org/10.1007/s00355-019-01180-y