Inheritance of convexity for partition restricted games

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,vS) (for any 0̸≠S⊆N) defined for any A⊆N by vS(A)=|A∩S|−1 if A∩S≠0̸, and vS(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 Pmin correspondence (Pmin(A) is built by deleting edges of minimum weight in the subgraph GA of a weighted communication graph G), we show that inheritance of convexity for unanimity games already implies inheritance of convexity.