Robust Temporal Guarantees in Budgeted Sequential Auctions

Abstract: In modern advertising platforms, learning algorithms are deployed by budget-constrained bidders to maximize their accumulated value. These algorithms often offer classical utility guarantees like no-regret, i.e., the agent's utility is at least the utility achieved by some benchmark in which it is assumed that every other agent's bidding remains the same. These guarantees offer compelling properties: They are optimal against stationary competition distributions, and in unconstrained settings, the resulting empirical distribution of play induced by no-regret dynamics approximates a Coarse Correlated Equilibrium. However, no-regret algorithms are easily manipulable, and in budgeted settings, no stronger notion of regret (such as swap regret) is currently known that would limit such manipulation. We propose a very simple learning algorithm for budgeted sequential auctions where agents maximize their total number of wins and show that it has surprisingly appealing properties. We analyze this algorithm from two perspectives. First, we show that when an agent with a $ρ$ fraction of the total budget uses this algorithm, then she is guaranteed to win at least $ρT - O(\sqrt T)$ of the total $T$ rounds. This result holds for adversarial behavior by the other agents, as long as they respect their own budget restrictions. Second, we examine the scenario when all the agents follow our algorithm. By the first result, every agent's total wins are proportional to her budget, up to the additive $O(\sqrt T)$ term. In addition, we show that this result holds in a much stronger sense: after an initial period of $O(\sqrt T \log T)$ rounds, every agent gets the same guarantee over any time interval. For intervals of length $O(\sqrt T)$, we show that the deviation from the desired number of wins is an additive constant.