The Owen and Banzhaf–Owen values revisited

Abstract

In this work, we consider games with coalitional structure. We afford two new parallel axiomatic characterizations for the well-known Owen and Banzhaf–Owen coalitional values. Two properties are common to both characterizations: a property of balanced contributions and a property of neutrality. The results prove that the main difference between these two coalitional values is that the former is efficient, while the latter verifies a property of 2-efficiency.

Keywords:

• cooperative game
• Shapley value
• Banzhaf value
• coalition structure
• coalitional value
• balanced contributions

AMS Subject Classifications:

• 91A12
• C71

Acknowledgements

We also thank two anonymous referees for some thoughtful and constructive comments.

Notes

No potential conflict of interest was reported by the authors.

1 In the section of Final Remarks, we discuss in more detail the properties of delegation neutrality and neutrality for the reduced game, as well as 2-efficiency within unions and the so-called delegation transfer.

2 This notation corresponds to the number of iterations used to obtain a coalitional game with all the unions formed by isolated players.

3 We say that two players are symmetric in a TU game if for all .

4 We say that a player is a null player in a TU game if for all .

5 If , is the TU-game with set of players such that for each and is the coalition structure over such that if and if for each .

Additional information

Funding

Financial support from Ministerio de Ciencia y Tecnología, Ministerio de Economía y Competitividad, and FEDER through [grant number MTM2011-27731-C03-02], [grant number ECO2011-23460], [grant number MTM2014-53395-C3-1-P] and [grant number MTM2014-53395-C3-2-P].