Convexity of Network Restricted Games Induced by Minimum Partitions

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Working paper
Alexandre Skoda
Issue number: 
CES Working Papers
Centre d’Economie de la Sorbonne
We consider restricted games on weighted communication graphs associated with minimum partitions. We replace in the classical definition of Myerson’s graph-restricted game the connected components of any subgraph by the sub-components corresponding to a minimum partition. This minimum partition Pmin is induced by the deletion of the minimum weight edges. We provide necessary conditions on the graph edge-weights to have inheritance of convexity from the underlying game to the restricted game associated with Pmin. Then we establish that these conditions are also sufficient for a weaker condition, called Fconvexity, obtained by restriction of convexity to connected subsets. Moreover we show that Myerson’s game associated to a given graph G can be obtained as a particular case of the Pmin-restricted game for a specific weighted graph G′. Then we prove that G is cycle-complete if and only if a specific condition on adjacent cycles is satisfied on G′.
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