Efficiency of Non-Truthful Auctions in Auto-bidding: The Power of Randomization

Auto-bidding is now widely adopted as an interface between advertisers and internet advertising as it allows advertisers to specify high-level goals, such as maximizing value subject to a value-per-spend constraint. Prior research has mainly focused on auctions that are truthful (such as a second-price auction) because these auctions admit simple (uniform) bidding strategies and are thus simpler to analyze. The main contribution of this paper is to characterize the efficiency across the spectrum of all auctions, including non-truthful auctions for which optimal bidding may be complex. For deterministic auctions, we show a dominance result: any uniform bidding equilibrium of a second-price auction (SPA) can be mapped to an equilibrium of any other auction – for example, first price auction (FPA) – with identical outcomes. In this sense, SPA with uniform bidding is an instance-wise optimal deterministic auction. Consequently, the price of anarchy (PoA) of any deterministic auction is at least the PoA of SPA with uniform bidding, which is known to be 2. We complement this by showing that the PoA of FPA without uniform bidding is 2. Next, we show, surprisingly, that truthful pricing is not dominant in the randomized setting. There is a randomized version of FPA that achieves a strictly smaller price of anarchy than its truthful counterpart when there are two bidders per query. Furthermore, this randomized FPA achieves the best-known PoA for two bidders, thus showing the power of non-truthfulness when combined with randomization. Finally, we show that no prior-free auction (even randomized, non-truthful) can improve on a PoA bound of 2 when there are a large number of advertisers per auction. These results should be interpreted qualitatively as follows. When the auction pressure is low, randomization and non-truthfulness is beneficial. On the other hand, if the auction pressure is intense, the benefits diminishes and it is optimal to implement a second-price auction.