Null, Nullifying, and Necessary Agents: Parallel Characterizations of the Banzhaf and Shapley Values

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Abstract

In a cooperative game, we consider three special kinds of agents: null, nullifying, and necessary agents. A coalition with a null agent receives the same payoff if this agent leaves the coalition, a coalition with a nullifying agent receives nothing, and a coalition without a necessary agent also receives nothing. The null agent property proposes zero payoff to any null agent. We introduce new properties that propose payoffs for nullifying and necessary players. With these three properties and additivity, we obtain new characterizations of the Banzhaf and Shapley values.

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Notes

1. Let S be a finite set. In short notation, given $$i\in S$$, we will write $$S\setminus i$$ instead of $$S\setminus \left\{ i\right\} ,$$ and given $$i\notin S$$, we will write $$S\cup i$$ instead of $$S\cup \left\{ i\right\} .$$ We denote by s the cardinality of the set S and by n the cardinality of the grand coalition.

2. Different versions of the 2-Efficiency property can be observed in [3].

3. Kamijo and Kongo [9] proposed the property of Invariance from nullifying player deletion: this property states that the deletion of a nullifying player does not affect the other players’ payoffs.

4. Casajus and Huettner [10] provided a characterization of the Equal Surplus Division value (Driessen and Funaki [11]) with a modification of the Nullifying Agents Get Nothing property.

References

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Acknowledgements

This work has been supported by the ERDF, the MINECO/AEI grants MTM2017-87197-C3-1-P, MTM2017-87197-C3-1-P, and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2016-015 and ED431C-2017/38 and Centro Singular de Investigación de Galicia ED431G/01).

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Correspondence to José M. Alonso-Meijide.

Irinel Chiril Dragan.

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Alonso-Meijide, J.M., Costa, J. & García-Jurado, I. Null, Nullifying, and Necessary Agents: Parallel Characterizations of the Banzhaf and Shapley Values. J Optim Theory Appl 180, 1027–1035 (2019). https://doi.org/10.1007/s10957-018-1403-5

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• DOI: https://doi.org/10.1007/s10957-018-1403-5