  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.
