The cost of getting local monotonicity
  Manfred Holler introduced the Public Good index as a proposal to divide a public good among players. In its unnormalized version, i.e., the raw measure, it counts the number of times that a player belongs to a minimal winning coalition. Unlike the Banzhaf index, it does not count the remaining winning coalitions in which the player is crucial. Holler noticed that his index does not satisfy local monotonicity, a fact that can be seen either as a major drawback or as an advantage. 
Josep Freixas and Sascha Kurz
cs.GT math.OC