  Auto-bidding has recently become a popular feature in ad auctions. This
feature enables advertisers to simply provide high-level constraints and goals
to an automated agent, which optimizes their auction bids on their behalf. In
this paper, we examine the effect of different auctions on the incentives of
advertisers to report their constraints to the auto-bidder intermediaries. More
precisely, we study whether canonical auctions such as first price auction
(FPA) and second price auction (SPA) are auto-bidding incentive compatible
(AIC): whether an advertiser can gain by misreporting their constraints to the
autobidder.
  We consider value-maximizing advertisers in two important settings: when they
have a budget constraint and when they have a target cost-per-acquisition
constraint. The main result of our work is that for both settings, FPA and SPA
are not AIC. This contrasts with FPA being AIC when auto-bidders are
constrained to bid using a (sub-optimal) uniform bidding policy. We further
extend our main result and show that any (possibly randomized) auction that is
truthful (in the classic profit-maximizing sense), scalar invariant and
symmetric is not AIC. Finally, to complement our findings, we provide
sufficient market conditions for FPA and SPA to become AIC for two advertisers.
These conditions require advertisers' valuations to be well-aligned. This
suggests that when the competition is intense for all queries, advertisers have
less incentive to misreport their constraints.
  From a methodological standpoint, we develop a novel continuous model of
queries. This model provides tractability to study equilibrium with
auto-bidders, which contrasts with the standard discrete query model, which is
known to be hard. Through the analysis of this model, we uncover a surprising
result: in auto-bidding with two advertisers, FPA and SPA are auction
equivalent.
