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 focus on first-price auctions, which are commonly used in many settings, and consider liquid welfare, a natural and well-studied generalization of social welfare for the case of budget-limited buyers. We use a behavioral model for the buyers, assuming a learning style guarantee: the resulting utility of each buyer is within a $\gamma$ factor (where $\gamma \ge 1$) of the utility achievable by shading her value with the same factor at each iteration. We show a $\gamma + 1/2 + O(1/\gamma)$ price of anarchy for liquid welfare assuming buyers have additive valuations. 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 maximum liquid welfare. We prove a lower bound of $\gamma$ on the liquid welfare loss under the above assumption in first-price auctions, making our bound asymptotically tight. For the case when $\gamma = 1$ our theorem implies a price of anarchy upper bound that is about $2.4$; we show a lower bound of $2$ for that case. We also give a learning algorithm that the players can use to achieve the guarantee needed for our liquid welfare result. Our algorithm achieves utility within a $\gamma = O(1)$ factor of the optimal utility even when a buyer's values and the bids of the other buyers are chosen adversarially, assuming the buyer's budget grows linearly with time. The competitiveness guarantee of the learning algorithm deteriorates somewhat as the budget grows slower than linearly with time. Finally, we extend our liquid welfare results for the case where buyers have submodular valuations over the set of items they win across iterations with a slightly worse price of anarchy bound of $\gamma + 1 + O(1/\gamma)$ compared to the guarantee for the additive case.