  In recent years, the growing adoption of autobidding has motivated the study
of auction design with value-maximizing auto-bidders. It is known that under
mild assumptions, uniform bid-scaling is an optimal bidding strategy in
truthful auctions, e.g., Vickrey-Clarke-Groves auction (VCG), and the price of
anarchy for VCG is $2$. However, for other auction formats like First-Price
Auction (FPA) and Generalized Second-Price auction (GSP), uniform bid-scaling
may not be an optimal bidding strategy, and bidders have incentives to deviate
to adopt strategies with non-uniform bid-scaling. Moreover, FPA can achieve
optimal welfare if restricted to uniform bid-scaling, while its price of
anarchy becomes $2$ when non-uniform bid-scaling strategies are allowed.
  All these price of anarchy results have been focused on welfare approximation
in the worst-case scenarios. To complement theoretical understandings, we
empirically study how different auction formats (FPA, GSP, VCG) with different
levels of non-uniform bid-scaling perform in an autobidding world with a
synthetic dataset for auctions. Our empirical findings include:
  * For both uniform bid-scaling and non-uniform bid-scaling, FPA is better
than GSP and GSP is better than VCG in terms of both welfare and profit;
  * A higher level of non-uniform bid-scaling leads to lower welfare
performance in both FPA and GSP, while different levels of non-uniform
bid-scaling have no effect in VCG.
  Our methodology of synthetic data generation may be of independent interest.
