Power in threshold network flow games

Yoram Bachrach*, Jeffrey S. Rosenschein

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

53 Scopus citations


Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much "real power" a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. We consider a particular multiagent domain, a threshold network flow game. Agents control the edges of a graph; a coalition wins if it can send a flow that exceeds a given threshold from a source vertex to a target vertex. The relative power of each edge/agent reflects its significance in enabling such a flow, and in real-world networks could be used, for example, to allocate resources for maintaining parts of the network. We examine the computational complexity of calculating two prominent power indices, the Banzhaf index and the Shapley-Shubik index, in this network flow domain. We also consider the complexity of calculating the core in this domain. The core can be used to allocate, in a stable manner, the gains of the coalition that is established. We show that calculating the Shapley-Shubik index in this network flow domain is NP-hard, and that calculating the Banzhaf index is #P-complete. Despite these negative results, we show that for some restricted network flow domains there exists a polynomial algorithm for calculating agents' Banzhaf power indices. We also show that computing the core in this game can be performed in polynomial time.

Original languageAmerican English
Pages (from-to)106-132
Number of pages27
JournalAutonomous Agents and Multi-Agent Systems
Issue number1
StatePublished - Feb 2009

Bibliographical note

Funding Information:
Acknowledgements Some preliminary material from this paper appeared at the Sixth International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS’07), in the paper “Computing the Banzhaf Power Index in Network Flow Games” [1]. This work was partially supported by grant #898/05 from the Israel Science Foundation.


  • Coalitional games
  • Network flow
  • Network reliability
  • Power index


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