We consider a dynamic learning and ranking problem of a digital platform. Uninformed of the products' intrinsic qualities, the platform strives to design a ranking rule that learns from historical traffic data while accounting for potential manipulation by sellers through "brushing" activities, such as fake orders or sales. How does the ranking manipulation disrupt ranking efficiency under various market conditions? Are there effective yet simple ranking algorithms to combat ranking manipulation?
Under an Experiment-Then-Commit (ETC) policy framework, we formulate an $N$-player-$T$-period dynamic "brushing war" game for the sellers. We provide a (static) budget-competition equilibrium characterization and study its asymptotic behavior when $T$ is large. For a small market with two sellers, we show the nonexistence of pure strategy equilibria and identify a mixed-strategy equilibrium, shedding light on the possibility of efficiency loss. For a large market, we formulate a novel non-atomic game with a continuum of sellers as a limiting case where $N$ is large. We characterize a "self-reinforcing" market equilibrium, where the seller's brushing amount increases in the product's quality. In other words, the sellers' strategic responses "reinforce" complete learning of the platform.