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.

翻译：我们研究了预算受限买家参与序贯第一价格拍卖中的流动性福利问题。聚焦于日益广泛使用的第一价格拍卖场景，并针对具有预算约束的买家，考虑社会福利的自然推广形式——流动性福利。我们采用买家行为模型，假设其满足学习风格保证：每位买家最终获得的效用，不超过每轮以相同因子γ（γ≥1）折让估值可获效用的γ倍。在此假设下，我们证明当买家具有可加估值时，流动性福利的无政府价格界为γ+1/2+O(1/γ)。这一正向结果与序贯第二价格拍卖形成鲜明对比——即便γ=1，后者的流动性福利可能远低于最大值。我们证明了第一价格拍卖在上述假设下流动性福利损失的下界为γ，使该界渐近紧确。当γ=1时，定理给出约2.41的无政府价格上界，而该情形下我们证明了下界2。我们还设计了一种学习算法，使参与者能够达到流动性福利结果所需的保证条件。即使买家的估值和其它竞拍者的出价由对手设定，只要买家预算随时间线性增长，该算法就能实现最优效用O(1)因子内的效用。当预算增速慢于线性时，学习算法的竞争力保证会有所下降。最后，我们将流动性福利结果推广至买家对跨轮竞得物品集合具有次模估值的情形，此时无政府价格界略微弱化为γ+1+O(1/γ)。