  Throttling is a popular method of budget management for online ad auctions in
which the platform modulates the participation probability of an advertiser in
order to smoothly spend her budget across many auctions. In this work, we
investigate the setting in which all of the advertisers simultaneously employ
throttling to manage their budgets, and we do so for both first-price and
second-price auctions. We analyze the structural and computational properties
of the resulting equilibria. For first-price auctions, we show that a unique
equilibrium always exists, is well-behaved and can be computed efficiently via
tatonnement-style decentralized dynamics. In contrast, for second-price
auctions, we prove that even though an equilibrium always exists, the problem
of finding an equilibrium is PPAD-complete, there can be multiple equilibria,
and it is NP-hard to find the revenue maximizing one. We also compare the
equilibrium outcomes of throttling to those of multiplicative pacing, which is
the other most popular and well-studied method of budget management. Finally,
we characterize the Price of Anarchy of these equilibria for liquid welfare by
showing that it is at most 2 for both first-price and second-price auctions,
and demonstrating that our bound is tight.
