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.

中文翻译：

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传播，然后定位，并在波浪中做广告：跨广告渠道的最佳预算分配

我们分析了分配有限预算的最佳策略，这些预算可以随着时间的推移投资于不同的广告渠道，目的是影响个人网络中的社会观点。在我们的分析中，我们考虑了外生影响机制，如广告活动，以及社会影响的内生机制，如口碑和同伴压力，它们使用扩散动力学建模。我们表明，对于广泛的目标函数系列，最佳影响策略每次都以最大速率或根本不使用所有渠道，即 bang-bang 策略。此外，我们证明了这些极端之间的切换次数受到一个通常远小于代理数量的项的限制。这意味着最佳影响策略是在每个通道的波浪中施加最大努力，然后停止努力并让效果传播。我们还表明，在活动开始时，外生广告渠道的总成本调整后覆盖率决定了其相对价值。相比之下，当我们接近我们的投资期限（例如选举日）时，最佳策略是投资于能够针对个人的渠道而不是广泛的渠道。我们证明了最佳影响策略在几个实际案例中很容易计算，并明确表征了封闭形式的线性目标函数情况下的最佳控制。最后，我们看到，在设计竞选活动的典型示例中，识别迟到的决策者是优化设计的关键组成部分。