Path monotonicity, consistency and axiomatizations of some weighted solutions

  • Pedro Calleja
  • Francesc LlerenaEmail author
Original Paper


On the domain of cooperative games with transferable utility, we introduce path monotonicity, a property closely related to fairness (van den Brink, in Int J Game Theory 30:309–319, 2001). The principle of fairness states that if a game changes by adding another game in which two players are symmetric, then their payoffs change by the same amount. Under efficiency, path monotonicity is a relaxation of fairness that guarantees that when the worth of the grand coalition varies, the players’ payoffs change according to some monotone path. In this paper, together with the standard properties of projection consistency (Funaki, in Dual axiomatizations of solutions of cooperative games. Mimeo, New York, 1998) and covariance, we show that path monotonicity characterizes the weighted surplus division solutions. Interestingly, replacing projection consistency by either self consistency (Hart and Mas-Colell, in Econometrica 57:589–614, 1989) or max consistency (Davis and Maschler, in Nav Res Logist Q 12:223–259, 1965) we obtain new axiomatic characterizations of the weighted Shapley values and the prenucleolus, respectively. Finally, by the duality approach we provide a new axiomatization of the weighted egalitarian non-separable contribution solutions using complement consistency (Moulin, in J Econ Theory 36:120–148, 1985).


Consistency Weighted surplus division solutions Weighted egalitarian non-separable contribution solutions Weighted Shapley values Prenucleolus 

JEL Classification

C71 C78 



The authors thank two anonymous referees and an associated editor for helpful and inspiring comments. We also acknowledge the support from research Grant ECO2016-75410-P (AEI/FEDER, UE) and ECO2017-86481-P (AEI/FEDER, UE) (Ministerio de Economía y Competitividad).


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Copyright information

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

Authors and Affiliations

  1. 1.Dep. de Matemàtica Econòmica, Financera i Actuarial and BEATUniversitat de BarcelonaBarcelonaSpain
  2. 2.Dep. de Gestió d’EmpresesUniversitat Rovira i Virgili and CREIPReusSpain

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