  We study the efficiency of non-truthful auctions for auto-bidders with both
return on spend (ROS) and budget constraints. The efficiency of a mechanism is
measured by the price of anarchy (PoA), which is the worst case ratio between
the liquid welfare of any equilibrium and the optimal (possibly randomized)
allocation. Our first main result is that the first-price auction (FPA) is
optimal, among deterministic mechanisms, in this setting. Without any
assumptions, the PoA of FPA is $n$ which we prove is tight for any
deterministic mechanism. However, under a mild assumption that a bidder's value
for any query does not exceed their total budget, we show that the PoA is at
most $2$. This bound is also tight as it matches the optimal PoA without a
budget constraint. We next analyze two randomized mechanisms: randomized FPA
(rFPA) and "quasi-proportional" FPA. We prove two results that highlight the
efficacy of randomization in this setting. First, we show that the PoA of rFPA
for two bidders is at most $1.8$ without requiring any assumptions. This
extends prior work which focused only on an ROS constraint. Second, we show
that quasi-proportional FPA has a PoA of $2$ for any number of bidders, without
any assumptions. Both of these bypass lower bounds in the deterministic
setting. Finally, we study the setting where bidders are assumed to bid
uniformly. We show that uniform bidding can be detrimental for efficiency in
deterministic mechanisms while being beneficial for randomized mechanisms,
which is in stark contrast with the settings without budget constraints.
