We study a game played between advertisers in an online ad platform. The platform sells ad impressions by first-price auction and provides autobidding algorithms that optimize bids on each advertiser's behalf. Each advertiser strategically declares a budget constraint (and possibly a maximum bid) to their autobidder. The chosen constraints define an "inner" budget-pacing game for the autobidders, who compete to maximize the total value received subject to the constraints. Advertiser payoffs in the constraint-choosing "metagame" are determined by the equilibrium reached by the autobidders. Advertisers only specify budgets and linear values to their autobidders, but their true preferences can be more general: we assume only that they have weakly decreasing marginal value for clicks and weakly increasing marginal disutility for spending money. Our main result is that despite this gap between general preferences and simple autobidder constraints, the allocations at equilibrium are approximately efficient. Specifically, at any pure Nash equilibrium of the metagame, the resulting allocation obtains at least half of the liquid welfare of any allocation and this bound is tight. We also obtain a 4-approximation for any mixed Nash equilibrium, and this result extends also to Bayes-Nash equilibria. These results rely on the power to declare budgets: if advertisers can specify only a (linear) value per click but not a budget constraint, the approximation factor at equilibrium can be as bad as linear in the number of advertisers.

翻译：我们研究在线广告平台中广告商之间的一场博弈。该平台通过第一价格拍卖出售广告展示机会，并提供代表广告商优化出价的自动竞价算法。每个广告商策略性地向自动竞价器声明预算约束（并可能声明最高出价）。所选约束定义了自动竞价器之间的一场“内部”预算调控博弈：自动竞价器在约束条件下竞争以最大化获得的总价值。广告商在约束选择“元博弈”中的收益由自动竞价器达到的均衡决定。广告商仅向自动竞价器指定预算和线性价值，但其真实偏好可以更一般化：我们仅假设他们对点击的边际价值递减，对支出的边际负效用递增。我们的主要结果是：尽管存在这种一般偏好与简单自动竞价器约束之间的差距，均衡时的分配近似有效。具体而言，在元博弈的任何纯纳什均衡处，所得分配至少获得任何分配的流动福利的一半，且该界是紧的。对于任何混合纳什均衡，我们获得4倍近似，并且该结果也扩展到贝叶斯-纳什均衡。这些结果依赖于声明预算的能力：如果广告商只能指定（线性）每次点击价值而不能指定预算约束，则均衡时的近似因子可能差到与广告商数量成线性关系。