Efficiency of the First-Price Auction in the Autobidding World

We study the price of anarchy of first-price auctions in the autobidding world, where bidders can be either utility maximizers (i.e., traditional bidders) or value maximizers (i.e., autobidders). We show that with autobidders only, the price of anarchy of first-price auctions is 1/2, and with both kinds of bidders, the price of anarchy degrades to about 0.457 (the precise number is given by an optimization).