Given a network N and a set of nodes that are the starting point for the spread of misinformation across N and an integer k, in the influence blocking maximization problem the goal is to find k nodes in N as the starting point for a competing information (say, a correct information) across N such that the reach of the misinformation is minimized. In this paper, we deal with a generalized version of this problem that corresponds to a more realistic scenario, where different nodes have different costs and the counter strategy has a “budget” for picking nodes for a solution. Our experimental results show that the success of a given strategy varies substantially depending on the cost function in the model. In particular, we investigate the cost function implicitly used in all previous works in the field (i.e., all nodes have cost 1), and a cost function that assigns higher costs to higher-degree nodes. We show that, even though strategies that perform well in these two diverse cases are very different from each other, both correlate well with simple (but different) strategies, greedily choose high-degree nodes and choose nodes uniformly at random. Furthermore, we show properties and approximations results for the influence function in several diffusion models.