First-price auctions are one of the most popular mechanisms for selling goods and services, with applications ranging from display advertising to timber sales. Unlike their close cousin, the second-price auction, first-price auctions do not admit a dominant strategy. Instead, each buyer must design a bidding strategy that maps values to bids—a task that is often challenging due to the lack of prior knowledge about competing bids. To address this challenge, we conduct a principled analysis of prior-independent bidding strategies for first-price auctions using worst-case regret as the performance measure. First, we develop a technique for evaluating the worst-case regret for (almost) any given value distribution and bidding strategy, reducing the complex task of ascertaining the worst-case competing-bid distribution to a simple line search. Next, building on our evaluation technique, we minimize worst-case regret and characterize a minimax-optimal bidding strategy for every value distribution. We achieve it by explicitly constructing a bidding strategy as a solution to an ordinary differential equation, and by proving its optimality for the intricate infinite-dimensional minimax problem underlying worst-case regret minimization. Our construction provides a systematic and computationally-tractable procedure for deriving minimax-optimal bidding strategies. When the value distribution is continuous, it yields a deterministic strategy that maps each value to a single bid. We also show that our minimax strategy significantly outperforms the uniform-bid-shading strategies advanced by prior work. Importantly, our result allows us to precisely quantify, through minimax regret, the performance loss due to a lack of knowledge about competing bids. We leverage this to analyze the impact of the value distribution on the performance loss, and find that it decreases as the buyer's values become more dispersed.

Learning to bid in repeated first-price auctions is a fundamental problem at the interface of game theory and machine learning, which has seen a recent surge in interest due to the transition of display advertising to first-price auctions. In this work, we propose a novel concave formulation for pure-strategy bidding in first-price auctions, and use it to analyze natural Gradient-Ascent-based algorithms for this problem. Importantly, our analysis goes beyond regret, which was the typical focus of past work, and also accounts for the strategic backdrop of online-advertising markets where bidding algorithms are deployed---we provide the first guarantees of strategic-robustness and incentive-compatibility for Gradient Ascent.

Major Internet advertising platforms offer budget pacing tools as a standard service for advertisers to manage their ad campaigns. Given the inherent non-stationarity in an advertiser’s value and also competing advertisers’ values over time, a commonly used approach is to learn a target expenditure plan that specifies a target spend as a function of time, and then run a controller that tracks this plan. This raises the question: how many historical samples are required to learn a good expenditure plan? We study this question by considering an advertiser repeatedly participating in $T$ second-price auctions, where the tuple of her value and the highest competing bid is drawn from an unknown time-varying distribution. The advertiser seeks to maximize her total utility subject to her budget constraint. Prior work has shown the sufficiency of $T\log T$ samples per distribution to achieve the optimal $O(\sqrt{T})$-regret. We dramatically improve this state-of-the-art and show that just one sample per distribution is enough to achieve the near-optimal $\tilde O(\sqrt{T})$-regret, while still being robust to noise in the sampling distributions.

Single-leg revenue management is a foundational problem of revenue management that has been particularly impactful in the airline and hotel industry: Given n units of a resource, e.g. flight seats, and a stream of sequentially-arriving customers segmented by fares, what is the optimal online policy for allocating the resource. Previous work focused on designing algorithms when forecasts are available, which are not robust to inaccuracies in the forecast, or online algorithms with worst-case performance guarantees, which can be too conservative in practice. In this work, we look at the single-leg revenue management problem through the lens of the algorithms-with-advice framework, which attempts to harness the increasing prediction accuracy of machine learning methods by optimally incorporating advice about the future into online algorithms. In particular, we develop online algorithms which optimally trade-off consistency (performance when advice is accurate) and competitiveness (performance when advice is inaccurate) for every advice. Our results extend to other unit-cost online allocations problems such as the display advertising and the multiple secretary problem together with more general variable-cost problems such as the online knapsack problem.

The internet advertising market is a multi-billion dollar industry, in which advertisers buy thousands of ad placements every day by repeatedly participating in auctions. In recent years, the industry has shifted to first-price auctions as the preferred paradigm for selling advertising slots. Another important and ubiquitous feature of these auctions is the presence of campaign budgets, which specify the maximum amount the advertisers are willing to pay over a specified time period. In this paper, we present a new model to study the equilibrium bidding strategies in standard auctions, a large class of auctions that includes first- and second-price auctions, for advertisers who satisfy budget constraints on average. Our model dispenses with the common, yet unrealistic assumption that advertisers' values are independent and instead assumes a contextual model in which advertisers determine their values using a common feature vector. We show the existence of a natural value-pacing-based Bayes-Nash equilibrium under very mild assumptions. Furthermore, we prove a revenue equivalence showing that all standard auctions yield the same revenue even in the presence of budget constraints. Leveraging this equivalence, we prove Price of Anarchy bounds for liquid welfare and structural properties of pacing-based equilibria that hold for all standard auctions. Our work takes an important step toward understanding the implications of the shift to first-price auctions in internet advertising markets.

Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a multiplier between 0 and 1, and her bid on each item is multiplicatively scaled down by the multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is the pacing equilibrium.