Learning vs. Optimizing Bidders in Budgeted Auctions

TL;DR

This paper explores how budget constraints fundamentally change strategic interactions in repeated second-price auctions, introducing a new equilibrium concept and analyzing the robustness of common control heuristics.

Contribution

It generalizes the Stackelberg equilibrium to budgeted settings, revealing phase-based optimal strategies and demonstrating the strategic robustness of PID controllers in such auctions.

Findings

Optimal strategies decompose into up to k+1 phases for k-dimensional budgets.

Proportional controllers bound the optimizer's utility close to the baseline equilibrium.

Budget constraints significantly alter the strategic landscape in repeated auctions.

Abstract

The study of repeated interactions between a learner and a utility-maximizing optimizer has yielded deep insights into the manipulability of learning algorithms. However, existing literature primarily focuses on independent, unlinked rounds, largely ignoring the ubiquitous practical reality of budget constraints. In this paper, we study this interaction in repeated second-price auctions in a Bayesian setting between a learning agent and a strategic agent, both subject to strict budget constraints, showing that such cross-round constraints fundamentally alter the strategic landscape. First, we generalize the classic Stackelberg equilibrium to the Budgeted Stackelberg Equilibrium. We prove that an optimizer's optimal strategy in a budgeted setting requires time-multiplexing; for a $k$-dimensional budget constraint, the optimal strategy strictly decomposes into up to $k+1$ distinct phases, with each phase employing a possibly unique mixed strategy (the case of $k=0$ recovers the classic Stackelberg equilibrium where the optimizer repeatedly uses a single mixed strategy). Second, we address the intriguing question of non-manipulability. We prove that when the learner employs a standard Proportional controller (the "P" of the PID-controller) to pace their bids, the optimizer's utility is upper bounded by their objective value in the Budgeted Stackelberg Equilibrium baseline. By bounding the dynamics of the PID controller via a novel analysis, our results establish that this widely used control-theoretic heuristic is actually strategically robust.