Inheritance of convexity for partition restricted games

A correspondence P associates to every subset A subset of N a partition P(A) of A and to every game (N, v), the P-restricted game (N, (v) over bar) defined by (v) over bar (A)Sigma(F is an element of P(A)) (v(F)) for all A subset of N. We give necessary and sufficient conditions on P to have inheritance of convexity from (N, v) to (N, (V) over bar). 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 empty set not equal S subset of N) defined for any A subset of N by v(S) (A) = vertical bar A boolean AND S vertical bar - 1 if A boolean AND S not equal empty set, and v(S) (A) = 0 otherwise. In particular, when (N, (v) over bar) corresponds to Myerson's network-restricted game, inheritance of convexity can be verified by this way. For the Pmin 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. (C) 2017 Elsevier B.V. All rights reserved.