When the number of agents grows and becomes enormous, e.g., social network and the Internet, the finite network modeling capturing the explicit interactions between agents is inefficient and prohibitive. To this end, this chapter investigates decision-making on complex networks by proposing a new type of system framework. The focus of this chapter is to design an optimal strategy for controlling two competing epidemics spreading over complex networks. The designed strategy globally optimizes the trade-off between the control cost and the severity of epidemics in the network. We also provide structural results on the predictability of epidemic spreading by showing the existence and uniqueness of the solution. Finally, a gradient descent algorithm based on a fixed-point iterative scheme is proposed to find the optimal strategy.