Spread, then target, and advertise in waves: Optimal capital allocation across advertising channels

We obtain optimal strategies for the allocation of influence budget across multiple channels and across time for an external influencer, e.g., a political campaign, seeking to maximize its effect on an election given a network of agents with linear consensus-seeking opinion dynamics. We show that for a general set of objective functions, the optimal influence strategy at every time uses all channels at either their maximum rate or not at all. Furthermore, we prove that the number of switches between these extremes is bounded above both by a term that is typically much smaller than the number of agents. This means that the optimal influence strategy is to exert maximum effort in waves for every channel, and then cease effort and let the effects propagate. We also show that at the beginning, the total cost-adjusted reach of a channel determines its relative value, while targeting matters more closer to election time. We demonstrate that the optimal influence structures are easily computable in several practical cases. We explicitly characterize the optimal controls for the case of linear objective functions via a closed form. Finally, we see that in the canonical election example, identifying late-deciders approximately determines the optimal campaign resource allocation strategy.