Abstract
One of the main issues in economic allocation problems is the trade-off between marginalism and egalitarianism. In the context of cooperative games this trade-off can be framed as one of choosing to allocate according to the Shapley value or the equal division solution. In this paper we provide three different characterizations of egalitarian Shapley values being convex combinations of the Shapley value and the equal division solution. First, from the perspective of a variable player set, we show that all these solutions satisfy the same reduced game consistency. Second, on a fixed player set, we characterize this class of solutions using monotonicity properties. Finally, towards a strategic foundation, we provide a non-cooperative implementation for these solutions which only differ in the probability of breakdown at a certain stage of the game. These characterizations discover fundamental differences as well as intriguing connections between marginalism and egalitarianism.
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Acknowledgements
The authors thank associate editor and a referee for useful comments on a previous version of the paper. Yukihiko Funaki thanks the financial support from the Netherlands Organization for Scientific Research, grant B 45-299. Yuan Ju thanks the financial support of the Anniversary Lectureship granted by the University of York.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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van den Brink, R., Funaki, Y. & Ju, Y. Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values. Soc Choice Welf 40, 693–714 (2013). https://doi.org/10.1007/s00355-011-0634-2
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DOI: https://doi.org/10.1007/s00355-011-0634-2
Keywords
- Cooperative Game
- Marginal Contribution
- Grand Coalition
- Subgame Perfect Equilibrium
- Payoff Vector