Inheritance of convexity for partition restricted games

Alexandre Skoda (1)

1 CES - Centre d'économie de la Sorbonne

A correspondence P associates to every subset A ⊆ N a partition P ( A ) of A and to every game ( N , v ) , the P -restricted game ( N , v ¯ ) defined by v ¯ ( A ) = ∑ F ∈ P ( A ) v ( F ) for all A ⊆ N . We give necessary and sufficient conditions on P to have inheritance of convexity from ( N , v ) to ( N , v ¯ ) . The main condition is a cyclic intersecting sequence free condition. As a consequence, we only need to verify inheritance of convexity for unanimity games and for the small class of extremal convex games ( N , v S ) (for any 0̸ ≠ S ⊆ N ) defined for any A ⊆ N by v S ( A ) = | A ∩ S | − 1 if A ∩ S ≠ 0̸ , and v S ( A ) = 0 otherwise. In particular, when ( N , v ¯ ) corresponds to Myerson’s network-restricted game, inheritance of convexity can be verified by this way. For the P min correspondence ( P min ( A ) is built by deleting edges of minimum weight in the subgraph G A of a weighted communication graph G ), we show that inheritance of convexity for unanimity games already implies inheritance of convexity

Mots clés: Cooperative game; Convex game; Restricted game; Partitions; Supermodularity; Intersecting subsets

Domaines: Sciences de l'Homme et Société; Economies et finances

Alexandre Skoda. Inheritance of convexity for partition restricted games. Discrete Optimization, 2017, 25, pp.6-27. ⟨10.1016/j.disopt.2017.01.004⟩. ⟨halshs-01487381⟩