Journal of Optimization Theory and Applications

, Volume 180, Issue 3, pp 1027–1035 | Cite as

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

  • José M. Alonso-MeijideEmail author
  • Julián Costa
  • Ignacio García-Jurado


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.


Banzhaf value Shapley value Axiomatic characterizations Null agent Nullifying agent Necessary agent 

Mathematics Subject Classification




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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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Grupo MODESTYA, Departamento de Estatística, Análise Matemática e OptimizaciónUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Grupo MODES, Departamento de MatemáticasUniversidade da CoruñaA CoruñaSpain

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