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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 mostly focused on auctions which are truthful (such as SPA) since uniform bidding is optimal in such auctions, which makes it manageable to reason about equilibria. A tantalizing question is whether one can obtain more efficient outcomes by leaving the realm of truthful auctions. This is the first paper to study non-truthful auctions in the prior-free auto-bidding setting. Our first result is that non-truthfulness provides no benefit when one considers deterministic auctions. Any deterministic mechanism has a price of anarchy (PoA) of at least $2$, even for $2$ bidders; this matches what can be achieved by deterministic truthful mechanisms. In particular, we prove that the first price auction has PoA of exactly $2$. For our second result, we construct a randomized non-truthful auction that achieves a PoA of $1.8$ for $2$ bidders. This is the best-known PoA for this problem. The previously best-known PoA for this problem was $1.9$ and was achieved with a truthful mechanism. Moreover, we demonstrate the benefit of non-truthfulness in this setting by showing that the truthful version of this randomized auction also has a PoA of $1.9$. Finally, we show that no auction (even randomized, non-truthful) can improve upon a PoA bound of $2$ as the number of advertisers grow to infinity.