Pacing Equilibrium in First Price Auction Markets

Abstract

Mature internet advertising platforms offer high-level campaign management tools to help advertisers run their campaigns, often abstracting away the intricacies of how each ad is placed and focusing on aggregate metrics of interest to advertisers. On such platforms, advertisers often participate in auctions through a proxy bidder, so the standard incentive analyses that are common in the literature do not apply directly. In this paper, we take the perspective of a budget management system that surfaces aggregated incentives—instead of individual auctions—and compare first and second price auctions. We show that theory offers surprising endorsement for using a first price auction to sell individual impressions. In particular, first price auctions guarantee uniqueness of the steady-state equilibrium of the budget management system, monotonicity, and other desirable properties, as well as efficient computation through the solution to the well-studied Eisenberg–Gale convex program. Contrary to what one can expect from first price auctions, we show that incentives issues are not a barrier that undermines the system. Using realistic instances generated from data collected at real-world auction platforms, we show that bidders have small regret with respect to their optimal ex post strategy, and they do not have a big incentive to misreport when they can influence equilibria directly by giving inputs strategically. Finally, budget-constrained bidders, who have significant prevalence in real-world platforms, tend to have smaller regrets. Our computations indicate that bidder budgets, pacing multipliers, and regrets all have a positive association in statistical terms. This paper was accepted by Gabriel Weintraub, revenue management and market analytics.

Stability and Learning in Strategic Queuing Systems

“…Another important repeated game setting with such carryover effect is the repeated ad-auction game with limited budgets. The papers of [4,7,8] consider such games and offers results on convergence to equilibrium as well as understanding equilibria in the first-price auction settings under a particular behavioral model of the agents. Analyzing such systems for the more commonly used second-price auction system is an important open problem.…”

“…and over w steps of our process the sum of the geometric variables of subsequent packet arrivals, and the sum of the Bernoulli server successes concentrate around their expectation with an error probability of at most η/256 with the above values of δ, ϵ i , λ, and µ. 5 (See the required inequalities at (7) and (8).) Then, if each queue satisfies Assumption 3.1 on each consecutive time interval of length w with probability at least 1 − η 256n , then the random process T t under these dynamics is strongly stable.…”

“…This simply follows from our assumption that w was chosen large enough so that the first two lines hold simultaneously with probability at least 1 − η/256 via Corollary A.7 and Lemma A.8 in the Appendix, the fact that the no-regret bound held with probability at least 1 − η/(256n) for each queue, and taking a union bound. Notice that (7) asserts that every prefix of each of the geometric ensembles is additively not too far from the expectation, relative to w.…”

Budget Pacing in Repeated Auctions: Regret and Efficiency without Convergence

“…the pacing game, along with the appropriate equilibrium concept was introduced and studied in Conitzer et al [2021] in the context of second-price auctions with quasilinear utilities. These results were later extended by Conitzer et al [2019] to first-price auctions, although the restriction to pacing strategies is not quite as well-motivated theoretically as in the second-price case. Equilibria efficiency guarantees for such games were established in Aggarwal et al [2019] under more general payment constraints, as well as in Babaioff et al [2021] for more general utility measures.…”

“…The benchmark therefore tracks the best performance that one could achieve in hindsight with a manual bid given variation in click rates. Second, linear pacing is commonly used in practice as an algorithmic bidding tool even for non-truthful auctions [Conitzer et al, 2019], so from a practical perspective it is useful to focus attention on linear pacing as a goal unto itself. Third, linear pacing is reasonable from the online learning perspective: it is very common to focus on a particular subclass of policies in the face of infeasibility, and linear policies is a common and natural class to consider.…”