In this work, we consider dynamic influence maximization games over social networks with multiple players (influencers). The goal of each influencer is to maximize their own reward subject to their limited total budget rate constraints. Thus, influencers need to carefully design their investment policies considering individuals' opinion dynamics and other influencers' investment strategies, leading to a dynamic game problem. We first consider the case of a single influencer who wants to maximize its utility subject to a total budget rate constraint. We study both offline and online versions of the problem where the opinion dynamics are either known or not known a priori. In the singe-influencer case, we propose an online no-regret algorithm, meaning that as the number of campaign opportunities grows, the average utilities obtained by the offline and online solutions converge. Then, we consider the game formulation with multiple influencers in offline and online settings. For the offline setting, we show that the dynamic game admits a unique Nash equilibrium policy and provide a method to compute it. For the online setting and with two influencers, we show that if each influencer applies the same no-regret online algorithm proposed for the single-influencer maximization problem, they will converge to the set of $\epsilon$-Nash equilibrium policies where $\epsilon=O(\frac{1}{\sqrt{K}})$ scales in average inversely with the number of campaign times $K$ considering the average utilities of the influencers. Moreover, we extend this result to any finite number of influencers under more strict requirements on the information structure. Finally, we provide numerical analysis to validate our results under various settings.

翻译：本文研究了多参与者（影响力者）在社交网络上进行的动态影响力最大化博弈问题。每位影响力者的目标是在其有限总预算速率约束下最大化自身收益。因此，影响力者需要根据个体的观点动态及其他影响力者的投资策略，审慎设计其投资策略，由此构成动态博弈问题。我们首先考虑单一影响力者在其总预算速率约束下最大化效用的情况。我们研究了该问题的离线与在线两种版本，其中观点动态要么已知，要么事先未知。在单一影响力者情形下，我们提出了一种在线无遗憾算法，这意味着随着活动机会数量的增加，离线解与在线解获得的平均效用趋于一致。随后，我们考虑了离线与在线设定下具有多个影响力者的博弈模型。对于离线设定，我们证明了该动态博弈存在唯一的纳什均衡策略，并给出了一种计算方法。对于在线设定且包含两个影响力者的情形，我们证明若每位影响力者均采用针对单一影响力者最大化问题提出的同一无遗憾在线算法，则他们将收敛至一组$\epsilon$-纳什均衡策略，其中$\epsilon=O(\frac{1}{\sqrt{K}})$ 随活动轮次数$K$的增加而平均递减（基于影响力者的平均效用衡量）。此外，在更严格的信息结构要求下，我们将该结果推广至任意有限数量的影响力者。最后，我们提供了数值分析以验证不同设定下的研究结果。