Auctions without commitment in the auto-bidding world

Advertisers in online ad auctions are increasingly using auto-bidding mechanisms to bid into auctions instead of directly bidding their value manually. One of the prominent auto-bidding formats is that of target cost-per-acquisition (tCPA) which maximizes the volume of conversions subject to a return-ofinvestment constraint. From an auction theoretic perspective however, this trend seems to go against foundational results that postulate that for profit-maximizing (aka quasi-linear) bidders, it is optimal to use a classic bidding system like marginal CPA (mCPA) bidding rather than using strategies like tCPA. In this paper we rationalize the adoption of such seemingly sub-optimal bidding within the canonical quasi-linear framework. The crux of the argument lies in the notion of commitment. We consider a multistage game where first the auctioneer declares the auction rules; then bidders select either the tCPA or mCPA bidding format (and submit bids accordingly); and then, if the auctioneer lacks commitment, it can revisit the rules of the auction (e.g., may readjust reserve prices depending on the bids submitted by the bidders). Our main result is that so long as a bidder believes that the auctioneer lacks commitment to follow the rule of the declared auction then the bidder will make a higher profit by choosing the tCPA format compared to the classic mCPA format. We then explore the commitment consequences for the auctioneer. In a simplified version of the model where there is only one bidder, we show that the tCPA subgame admits a credible equilibrium while the mCPA format does not. That is, when the bidder chooses the tCPA format the auctioneer can credibly implement the auction rules announced at the beginning of the game. We also show that, under some mild conditions, the auctioneer's revenue is larger when the bidder to uses the tCPA format rather than mCPA. We further quantify the value for the auctioneer to be able to commit to the declared auction rules.