  We study the problem of auction design for advertising platforms that face
strategic advertisers who are bidding across platforms. Each advertiser's goal
is to maximize their total value or conversions while satisfying some
constraint(s) across all the platforms they participates in. In this paper, we
focus on advertisers with return-over-investment (henceforth, ROI) constraints,
i.e. each advertiser is trying to maximize value while making sure that their
ROI across all platforms is no less than some target value. An advertiser
interacts with the platforms through autobidders -- for each platform, the
advertiser strategically chooses a target ROI to report to the platform's
autobidder, which in turn uses a uniform bid multiplier to bid on the
advertiser's behalf on the queries owned by the given platform.
  Our main result is that for a platform trying to maximize revenue,
competition with other platforms is a key factor to consider when designing
their auction. While first-price auctions are optimal (for both revenue and
welfare) in the absence of competition, this no longer holds true in
multi-platform settings. We show that there exists a large class of advertiser
valuations over queries such that, from the platform's perspective, running a
second price auction dominates running a first price auction.
  Furthermore, our analysis reveals the key factors influencing platform choice
of auction format: (i) intensity of competition among advertisers, (ii)
sensitivity of bid landscapes to an auction change (driven by advertiser
sensitivity to price changes), and (iii) relative inefficiency of second-price
auctions compared to first-price auctions.
