Abstract
The paper is devoted to value concepts for cooperative games with a communication structure represented by a graph. Under assumptions that the players partition themselves into ‘components’ before realizing cooperation and the worth of the grand coalition not less than the sum of the worths of all components, the fair distribution of surplus solution and the two-step \(\tau \)-value are introduced as two efficient values for such games, each of which is an extension of the graph \(\tau \)-value. For the two efficient values, we discuss their special properties and we provide their axiomatic characterizations in views of those properties. By analysing an example applied to the two values, we conclude that the fair distribution of surplus solution allocates more surplus to the bigger coalitions and favors the powerful players, while the two-step \(\tau \)-value benefits the vulnerable groups and inspires to form small coalitions.
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Notes
Note that the minimal right property is weaker than the relative invariance under S-equivalence.
We could also consider such game as a graph game with no links, however, to keep the parameter \(\delta ^{v,L}\) is meaningful, we do not choose this way.
This property is an extension of restricted proportionality, because if \(b^w=0\), then it immediately reduces to the restricted proportionality.
This property is also an extension of the restricted proportionality, because if \(r^{v,L}=0\), then \(\delta ^{v,L}=\lambda ^{v,L}\), and so it reduced to the restricted proportionality.
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Acknowledgments
The authors are grateful to the referees for their valuable comments and carefully reading, which have led to improvements in the presentation of the paper. The work was supported in part by the NSFC (Grant Numbers 11571222 and 11471210).
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Zhang, G., Shan, E., Kang, L. et al. Two efficient values of cooperative games with graph structure based on \(\tau \)-values. J Comb Optim 34, 462–482 (2017). https://doi.org/10.1007/s10878-016-0081-1
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DOI: https://doi.org/10.1007/s10878-016-0081-1