# Path monotonicity, consistency and axiomatizations of some weighted solutions

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## Abstract

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).

## Keywords

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

C71 C78## Notes

### Acknowledgements

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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