Liquid Welfare guarantees for No-Regret Learning in Sequential Budgeted Auctions

We study the liquid welfare in repeated first-price auctions with budget limited buyers. We use a behavioral model for the buyers, assuming a learning style guarantee on the utility each achieves. We focus on first-price auctions, which are increasingly commonly used in many settings, and consider liquid welfare, a natural and well-studied generalization of social welfare for the case of budget-constrained buyers. We show a $\gamma+O(\sqrt{\gamma})$ price of anarchy for liquid welfare assuming buyers have additive valuations and the utility of each buyer is within a $\gamma$ factor of the utility achievable by shading her value with the same factor each iteration. This positive result is in stark contrast to repeated second-price auctions, where even with $\gamma=1$, the resulting liquid welfare can be arbitrarily smaller than the optimal one. We prove a lower bound of $\gamma$ on the liquid welfare loss under the above assumption, making our bound asymptotically tight. For the case when $\gamma = 1$ our theorem proves a price of anarchy upper bound that is about $3.18$; we prove a lower bound of $2$ for that case. We also offer a learning algorithm that achieves utility of at least a $\gamma = O(\log T)$ fraction of the optimal utility even when a buyer's values and the bids of the other buyers are chosen adversarially, offering a possible algorithm they can use to achieve the guarantee needed for our liquid welfare result. Finally, we extend our liquid welfare results for the case where buyers have submodular valuations with a slightly worse constant in the big $O(.)$ of the guarantee for the linear case.