# Characterizations of weighted and equal division values

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## Abstract

New and recent axioms for cooperative games with transferable utilities are introduced. The *non-negative player axiom* requires to assign a non-negative payoff to a player that belongs to coalitions with non-negative worth only. The axiom of *addition invariance on bi-partitions* requires that the payoff vector recommended by a value should not be affected by an identical change in worth of both a coalition and the complementary coalition. The *nullified solidarity axiom* requires that if a player who becomes null weakly loses (gains) from such a change, then every other player should weakly lose (gain) too. We study the consequence of imposing some of these axioms in addition to some classical axioms. It turns out that the resulting values or set of values have all in common to split efficiently the worth achieved by the grand coalition according to an exogenously given weight vector. As a result, we also obtain new characterizations of the equal division value.

### Keywords

Equal division Weighted division values Non-negative player Addition invariance on bi-partitions Nullified solidarity## Notes

### Acknowledgments

The authors are grateful to an anonymous reviewer and participants and SING 9 conference for valuable comments. Financial support by the National Agency for Research (ANR)—research programs “DynaMITE: Dynamic Matching and Interactions: Theory and Experiments”, contract ANR-13-BSHS1-0010 —and the “Mathématiques de la décision pour l’ingénierie physique et sociale” (MODMAD) project is gratefully acknowledged by Sylvain Béal, Eric Rémila and Philippe Solal. Financial support by the German Research Foundation (DFG) is gratefully acknowledged by André Casajus (Grant CA 266/4-1) and Frank Huettner (Grant HU 2205/1-1).

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