  We study a game played between advertisers in an online ad platform. The
platform sells ad impressions by first-price auction and provides autobidding
algorithms that optimize bids on each advertiser's behalf, subject to
advertiser constraints such as budgets. Crucially, these constraints are
strategically chosen by the advertisers. The chosen constraints define an
"inner'' budget-pacing game for the autobidders. Advertiser payoffs in the
constraint-choosing "metagame'' are determined by the equilibrium reached by
the autobidders.
  Advertiser preferences can be more general than what is implied by their
constraints: we assume only that they have weakly decreasing marginal value for
clicks and weakly increasing marginal disutility for spending money.
Nevertheless, we show that at any pure Nash equilibrium of the metagame, the
resulting allocation obtains at least half of the liquid welfare of any
allocation and this bound is tight. We also obtain a 4-approximation for any
mixed Nash equilibrium or Bayes-Nash equilibria. These results rely on the
power to declare budgets: if advertisers can specify only a (linear) value per
click or an ROI target but not a budget constraint, the approximation factor at
equilibrium can be as bad as linear in the number of advertisers.
