Incentive-aware Contextual Pricing with Non-parametric Market Noise

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

Online Learning in Multi-unit Auctions

We consider repeated multi-unit auctions with uniform pricing, which are widely used in practice for allocating goods such as carbon licenses. In each round, $K$ identical units of a good are sold to a group of buyers that have valuations with diminishing marginal returns. The buyers submit bids for the units, and then a price $p$ is set per unit so that all the units are sold. We consider two variants of the auction, where the price is set to the $K$-th highest bid and $(K+1)$-st highest bid, respectively. We analyze the properties of this auction in both the offline and online settings. In the offline setting, we consider the problem that one player $i$ is facing: given access to a data set that contains the bids submitted by competitors in past auctions, find a bid vector that maximizes player $i$'s cumulative utility on the data set. We design a polynomial time algorithm for this problem, by showing it is equivalent to finding a maximum-weight path on a carefully constructed directed acyclic graph. In the online setting, the players run learning algorithms to update their bids as they participate in the auction over time. Based on our offline algorithm, we design efficient online learning algorithms for bidding. The algorithms have sublinear regret, under both full information and bandit feedback structures. We complement our online learning algorithms with regret lower bounds. Finally, we analyze the quality of the equilibria in the worst case through the lens of the core solution concept in the game among the bidders. We show that the $(K+1)$-st price format is susceptible to collusion among the bidders; meanwhile, the $K$-th price format does not have this issue