We study the efficiency of non-truthful auctions for auto-bidders with both return on spend (ROS) and budget constraints. The efficiency of a mechanism is measured by the price of anarchy (PoA), which is the worst case ratio between the liquid welfare of any equilibrium and the optimal (possibly randomized) allocation. Our first main result is that the first-price auction (FPA) is optimal, among deterministic mechanisms, in this setting. Without any assumptions, the PoA of FPA is $n$ which we prove is tight for any deterministic mechanism. However, under a mild assumption that a bidder's value for any query does not exceed their total budget, we show that the PoA is at most $2$. This bound is also tight as it matches the optimal PoA without a budget constraint. We next analyze two randomized mechanisms: randomized FPA (rFPA) and "quasi-proportional" FPA. We prove two results that highlight the efficacy of randomization in this setting. First, we show that the PoA of rFPA for two bidders is at most $1.8$ without requiring any assumptions. This extends prior work which focused only on an ROS constraint. Second, we show that quasi-proportional FPA has a PoA of $2$ for any number of bidders, without any assumptions. Both of these bypass lower bounds in the deterministic setting. Finally, we study the setting where bidders are assumed to bid uniformly. We show that uniform bidding can be detrimental for efficiency in deterministic mechanisms while being beneficial for randomized mechanisms, which is in stark contrast with the settings without budget constraints.

翻译：我们研究了对同时具有投资回报率（ROS）和预算约束的自动竞标者的非真实拍卖的效率。机制效率通过无政府价格（PoA）衡量，即任意均衡的流动福利与最优（可能为随机）分配之间的最坏情况比率。我们的第一个主要结果是，在此设定下，一级价格拍卖（FPA）在确定性机制中是最优的。在无任何假设条件下，FPA的PoA为$n$，我们证明这对任何确定性机制都是紧的。然而，在一个温和假设（即竞标者对于任何查询的价值不超过其总预算）下，我们证明PoA至多为$2$。该界也是紧的，因为它与无预算约束时的最优PoA相匹配。接下来我们分析两种随机机制：随机化FPA（rFPA）和“准比例”FPA。我们证明了两个结果，突显了在此设定中随机化的有效性。首先，我们证明在无任何假设条件下，两个竞标者的rFPA的PoA至多为$1.8$，这扩展了之前仅关注ROS约束的研究。其次，我们证明在无任何假设条件下，准比例FPA对任意数量的竞标者均具有PoA为$2$。这两者均规避了确定性设定中的下界。最后，我们研究了假设竞标者统一出价的设定。我们证明统一出价在确定性机制中可能损害效率，而在随机机制中却有益，这与无预算约束的设定形成鲜明对比。