We study the price of anarchy of the generalized second-price auction where bidders are value maximizers (i.e., autobidders). We show that in general the price of anarchy can be as bad as $0$. For comparison, the price of anarchy of running VCG is $1/2$ in the autobidding world. We further show a fined-grained price of anarchy with respect to the discount factors (i.e., the ratios of click probabilities between lower slots and the highest slot in each auction) in the generalized second-price auction, which highlights the qualitative relation between the smoothness of the discount factors and the efficiency of the generalized second-price auction.

翻译：我们研究了竞拍者为价值最大化者（即自动竞价者）时广义第二价格拍卖的无序代价。研究表明，一般情况下无序代价可能低至0。作为对比，在自动竞价环境下运行VCG拍卖的无序代价为1/2。我们进一步刻画了广义第二价格拍卖中无序代价与折扣因子（即每次拍卖中较低广告位与最高广告位点击概率的比值）之间的精细关联，揭示了折扣因子的平滑程度与广义第二价格拍卖效率之间的定性关系。