  Misinformation and rumor can spread rapidly and widely through online social
networks and therefore rumor controlling has become a critical issue. It is
often assumed that there is a single authority whose goal is to minimize the
spread of rumor by generating a positive cascade. In this paper, we study a
more realistic scenario when there are multiple positive cascades generated by
different agents. For the multiple-cascade diffusion, we propose the P2P
independent cascade (PIC) model for private social communications. The main
part of this paper is an analysis of the rumor blocking effect (i.e. the number
of the users activated by rumor) when the agents non-cooperatively generate the
positive cascades. We show that the rumor blocking effect provided by the Nash
equilibrium will not be arbitrarily worse even if the positive cascades are
generated non-cooperatively. In addition, we give a discussion on how the
cascade priority and activation order affect the rumor blocking problem. We
experimentally examine the Nash equilibrium of the proposed games by
simulations done on real social network structures.
