Dynamic Budget Throttling in Repeated Second-Price Auctions

In today's online advertising markets, an important demand for an advertiser (buyer) is to control her total expenditure within a time span under some budget. Among all budget control approaches, throttling stands out as a popular one, where the buyer participates in only a part of auctions. This paper gives a theoretical panorama of a single buyer's dynamic budget throttling process in repeated second-price auctions, which is lacking in the literature. We first establish a lower bound on the regret and an upper bound on the asymptotic competitive ratio for any throttling algorithm, respectively, on whether the buyer's values are stochastic or adversarial. Second, on the algorithmic side, we consider two different information structures, with increasing difficulty in learning the stochastic distribution of the highest competing bid. We further propose the OGD-CB algorithm, which is oblivious to stochastic or adversarial values and has asymptotically equal results under these two information structures. Specifically, with stochastic values, we demonstrate that this algorithm guarantees a near-optimal expected regret. When values are adversarial, we prove that the proposed algorithm reaches the upper bound on the asymptotic competitive ratio. At last, we compare throttling with pacing, another widely adopted budget control method, in repeated second-price auctions. In the stochastic case, we illustrate that pacing is generally better than throttling for the buyer, which is an extension of known results that pacing is asymptotically optimal in this scenario. However, in the adversarial case, we give an exciting result indicating that throttling is the asymptotically optimal dynamic bidding strategy. Our results fill the gaps in the theoretical research of throttling in repeated auctions and comprehensively reveal the ability of this popular budget-smoothing strategy.