Considering two decision-making tasks $A$ and $B$, each of which wishes to compute an effective decision $Y$ for a given query $X$, can we solve task $B$ by using query-decision pairs $(X, Y)$ of $A$ without knowing the latent decision-making model? Such problems, called inverse decision-making with task migrations, are of interest in that the complex and stochastic nature of real-world applications often prevents the agent from completely knowing the underlying system. In this paper, we introduce such a new problem with formal formulations and present a generic framework for addressing decision-making tasks in social contagion management. On the theory side, we present a generalization analysis for justifying the learning performance of our framework. In empirical studies, we perform a sanity check and compare the presented method with other possible learning-based and graph-based methods. We have acquired promising experimental results, confirming for the first time that it is possible to solve one decision-making task by using the solutions associated with another one.