  We study the liquid welfare in sequential first-price auctions with
budget-limited buyers. 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 buyers with budgets. 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
round. Under this assumption, 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 contrast to sequential 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 $2.41$; 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.
