Strategizing against No-Regret Learners in First-Price Auctions

TL;DR

This work analyzes repeated Bayesian games where a no-regret learner interacts with a strategically informed optimizer in first-price auctions. It shows that mean-based learners are exploitable in Bayesian first-price auctions, enabling the optimizer to exceed the Stackelberg utility, while no-polytope-swap-regret learners can robustly cap the optimizer in general Bayesian settings under a no-negligible-context condition. The paper also provides algorithmic refinements for Bayesian first-price auctions that reduce polytope swap regret by exploiting auction structure, and proves theoretical sufficiency and (under reasonable assumptions) necessity results for no-polytope-swap-regret to prevent exploitation. A key contribution is the construction of a Bayesian game instance where the optimizer attains a substantial fraction of the best possible utility over time, illustrating the gap between Stackelberg and achievable utilities against mean-based learners. The results have implications for robust mechanism design, highlighting when simple learning dynamics can be safely employed and when stronger regressor classes are required to prevent exploitation.

Abstract

We study repeated first-price auctions and general repeated Bayesian games between two players, where one player, the learner, employs a no-regret learning algorithm, and the other player, the optimizer, knowing the learner's algorithm, strategizes to maximize its own utility. For a commonly used class of no-regret learning algorithms called mean-based algorithms, we show that (i) in standard (i.e., full-information) first-price auctions, the optimizer cannot get more than the Stackelberg utility -- a standard benchmark in the literature, but (ii) in Bayesian first-price auctions, there are instances where the optimizer can achieve much higher than the Stackelberg utility. On the other hand, Mansour et al. (2022) showed that a more sophisticated class of algorithms called no-polytope-swap-regret algorithms are sufficient to cap the optimizer's utility at the Stackelberg utility in any repeated Bayesian game (including Bayesian first-price auctions), and they pose the open question whether no-polytope-swap-regret algorithms are necessary to cap the optimizer's utility. For general Bayesian games, under a reasonable and necessary condition, we prove that no-polytope-swap-regret algorithms are indeed necessary to cap the optimizer's utility and thus answer their open question. For Bayesian first-price auctions, we give a simple improvement of the standard algorithm for minimizing the polytope swap regret by exploiting the structure of Bayesian first-price auctions.

Theorems & Definitions (53)

• Theorem 1.1: Informal restatement of Theorem \ref{['thm:robust-standard-mean-based']} and Theorem \ref{['thm:exploit-bayesian-mean-based']}
• Theorem 1.2: Informal restatement of Theorem \ref{['thm:exploitable-poly-swap-regret']}
• Definition 2.1
• Definition 2.5: external regret
• Definition 2.6: mean-based learner
• Definition 2.7: polytope swap regret
• Lemma 2.8: mansour2022strategizing
• Theorem 3.1
• proof : High-level proof sketch
• proof : Proof of Theorem \ref{['thm:robust-standard-mean-based']}