Weakly differentially monotonic solutions for cooperative games

  • André CasajusEmail author
  • Koji Yokote
Original Paper


The principle of differential monotonicity for cooperative games states that the differential of two players’ payoffs weakly increases whenever the differential of these players’ marginal contributions to coalitions containing neither of them weakly increases. Together with the standard efficiency property and a relaxation of the null player property, differential monotonicity characterizes the egalitarian Shapley values, i.e., the convex mixtures of the Shapley value and the equal division value for games with more than two players. For games that contain more than three players, we show that, cum grano salis, this characterization can be improved by using a substantially weaker property than differential monotonicity. Weak differential monotonicity refers to two players in situations where one player’s change of marginal contributions to coalitions containing neither of them is weakly greater than the other player’s change of these marginal contributions. If, in such situations, the latter player’s payoff weakly/strictly increases, then the former player’s payoff also weakly/strictly increases.


TU game Shapley value Differential marginality Weak differential marginality 

Mathematics Subject Classification


JEL Classification

C71 D60 



We are grateful to René van den Brink for valuable comments on this paper. André Casajus: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–388390901. Koji Yokote: Financial support by the Japan Society for the Promotion of Science (JSPS) is gratefully acknowledged.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.HHL Leipzig Graduate School of ManagementLeipzigGermany
  2. 2.Dr. Hops Craft Beer BarLeipzigGermany
  3. 3.Waseda Institute for Advanced Study, Waseda UniversityTokyoJapan

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