In many online platforms, customers' decisions are substantially influenced by product rankings as most customers only examine a few top-ranked products. Concurrently, such platforms also use the same data corresponding to customers' actions to learn how these products must be ranked or ordered. These interactions in the underlying learning process, however, may incentivize sellers to artificially inflate their position by employing fake users, as exemplified by the emergence of click farms. Motivated by such fraudulent behavior, we study the ranking problem of a platform that faces a mixture of real and fake users who are indistinguishable from one another. We first show that existing learning algorithms---that are optimal in the absence of fake users---may converge to highly sub-optimal rankings under manipulation by fake users. To overcome this deficiency, we develop efficient learning algorithms under two informational environments: in the first setting, the platform is aware of the number of fake users, and in the second setting, it is agnostic to the number of fake users. For both these environments, we prove that our algorithms converge to the optimal ranking, while being robust to the aforementioned fraudulent behavior; we also present worst-case performance guarantees for our methods, and show that they significantly outperform existing algorithms. At a high level, our work employs several novel approaches to guarantee robustness such as: (i) constructing product-ordering graphs that encode the pairwise relationships between products inferred from the customers' actions; and (ii) implementing multiple levels of learning with a judicious amount of bi-directional cross-learning between levels.

We consider a dynamic pricing problem for repeated contextual second-price auctions with strategic buyers whose goals are to maximize their long-term time discounted utility. The seller has very limited information about buyers' overall demand curves, which depends on $d$-dimensional context vectors characterizing auctioned items, and a non-parametric market noise distribution that captures buyers' idiosyncratic tastes. The noise distribution and the relationship between the context vectors and buyers' demand curves are both unknown to the seller. We focus on designing the seller's learning policy to set contextual reserve prices where the seller's goal is to minimize his regret for revenue. We first propose a pricing policy when buyers are truthful and show that it achieves a $T$-period regret bound of $\tilde{\mathcal{O}}(\sqrt{dT})$ against a clairvoyant policy that has full information of the buyers' demand. Next, under the setting where buyers bid strategically to maximize their long-term discounted utility, we develop a variant of our first policy that is robust to strategic (corrupted) bids. This policy incorporates randomized "isolation" periods, during which a buyer is randomly chosen to solely participate in the auction. We show that this design allows the seller to control the number of periods in which buyers significantly corrupt their bids. Because of this nice property, our robust policy enjoys a $T$-period regret of $\tilde{\mathcal{O}}(\sqrt{dT})$, matching that under the truthful setting up to a constant factor that depends on the utility discount factor.