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.