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：我们研究了同时具有广告支出回报率（ROS）和预算约束的自动竞价者在非真实拍卖中的效率。机制效率通过"无政府价格"（PoA）来衡量，该指标定义为任意均衡下的液体福利与最优（可能为随机化）分配之间的最坏情况比率。我们的第一个主要结论是，在此设定下，第一价格拍卖（FPA）是确定性机制中的最优选择。在没有假设的情况下，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"竞价者对任意查询的价值不超过其总预算"这一温和假设下，我们证明PoA至多为$2$。该界限也是紧的，因为它与无预算约束下的最优PoA相匹配。接下来我们分析了两种随机化机制：随机化FPA（rFPA）和"准比例"FPA。我们通过两个结果凸显随机化在此设定中的有效性。首先，我们证明在两个竞价者且无需任何假设的情况下，rFPA的PoA至多为$1.8$，这拓展了仅关注ROS约束的先前工作。其次，我们证明在无需任何假设的情况下，准比例FPA对任意数量竞价者的PoA均为$2$。这两个结果均突破了确定性机制中的下界。最后，我们研究了假设竞价者进行统一竞价的情形。结果表明，统一竞价在确定性机制中可能损害效率，但在随机化机制中却能提升效率，这与无预算约束下的情形形成鲜明对比。