# Spread, then Target, and Advertise in Waves: Optimal Budget Allocation Across Advertising Channels

## TL;DR

This paper develops a mathematical framework for optimal budget allocation across advertising channels over time, revealing that the best strategy involves waves of maximum effort followed by propagation, with the choice of channels evolving as the campaign progresses.

## Contribution

It introduces a bang-bang control model for influence strategies, characterizes the timing and switching of efforts, and provides explicit solutions for linear objectives in social influence campaigns.

## Key findings

- Optimal strategies are bang-bang, switching between maximum effort and no effort.
- Efficient algorithms are provided for computing optimal controls.
- Timing of channel investment shifts from broad reach to targeted efforts near the horizon.

## Abstract

We analyze optimal strategies for the allocation of a finite budget that can be invested in different advertising channels over time with the objective of influencing social opinions in a network of individuals. In our analysis, we consider both exogenous influence mechanisms, such as advertising campaigns, as well as endogenous mechanisms of social influence, such as word-of-mouth and peer-pressure, which are modeled using diffusion dynamics. We show that for a broad family of objective functions, the optimal influence strategy at every time uses all channels at either their maximum rate or not at all, i.e., a bang-bang strategy. Furthermore, we prove that the number of switches between these extremes is bounded above 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 of the campaign, the total cost-adjusted reach of an exogenous advertising channel determines its relative value. In contrast, as we approach our investment horizon (e.g., election day), the optimal strategy is to invest in channels able to target individuals instead of broad-reaching channels. We demonstrate that the optimal influence strategies are easily computable in several practical cases, and explicitly characterize the optimal controls for the case of linear objective functions in closed form. Finally, we see that, in the canonical example of designing an election campaign, identifying late-deciders is a critical component in the optimal design.