From No-Regret to Strategically Robust Learning in Repeated Auctions

TL;DR

We address repeated single-item auctions with $n$ bidders whose strategies are no-regret learners and an auctioneer who can adapt the format to maximize revenue. The authors introduce a meta-algorithm that feeds the gradient of a bidder’s quantile-utility with respect to a quantile strategy into any no-regret learner, yielding strategic robustness under formats that satisfy allocation monotonicity and voluntary participation. They prove that the auctioneer’s cumulative revenue is bounded by $\textnormal{Mye}(\mathcal{D})\cdot T + o(T)$ (and by $O(n\sqrt{T\log K})$ in the MWU setting), while bidders incur no regret, linking Myerson’s auction theory with standard online-learning guarantees. This framework translates regret guarantees into strategic robustness without explicit swap-regret minimization and shows that MWU can achieve both optimal regret and strong robustness in this setting.

Abstract

In Bayesian single-item auctions, a monotone bidding strategy--one that prescribes a higher bid for a higher value type--can be equivalently represented as a partition of the quantile space into consecutive intervals corresponding to increasing bids. Kumar et al. (2024) prove that agile online gradient descent (OGD), when used to update a monotone bidding strategy through its quantile representation, is strategically robust in repeated first-price auctions: when all bidders employ agile OGD in this way, the auctioneer's average revenue per round is at most the revenue of Myerson's optimal auction, regardless of how she adjusts the reserve price over time. In this work, we show that this strategic robustness guarantee is not unique to agile OGD or to the first-price auction: any no-regret learning algorithm, when fed gradient feedback with respect to the quantile representation, is strategically robust, even if the auction format changes every round, provided the format satisfies allocation monotonicity and voluntary participation. In particular, the multiplicative weights update (MWU) algorithm simultaneously achieves the optimal regret guarantee and the best-known strategic robustness guarantee. At a technical level, our results are established via a simple relation that bridges Myerson's auction theory and standard no-regret learning theory. This showcases the potential of translating standard regret guarantees into strategic robustness guarantees for specific games, without explicitly minimizing any form of swap regret.

Figures (2)

• Figure 1: This figure illustrates a monotone (i.e., non-decreasing) bidding strategy that bids $0$ for values in $[0,v_1]$, bids $\frac{1}{2}$ for values in $(v_1,v_2]$, and bids $1$ otherwise. Suppose that $0,\frac{1}{2},1$ are the only possible bids, and let $F$ denote the CDF of the bidder's value distribution. KSS24 represent a monotone bidding strategy using the quantiles of the value thresholds at which the bid increases ($F(v_1)$ and $F(v_2)$ in this case). We refer to this quantile-based representation as a quantile strategy. In this paper, we instead use the probability masses of the value intervals corresponding to distinct bids ($F(v_1)$, $F(v_2)-F(v_1)$ and $1-F(v_2)$ in this case) to represent the same quantile strategy, for a reason discussed in Section \ref{['section:technical_overview']}.
• Figure 2: This figure illustrates an example of an auxiliary auction $\tilde{\mathcal{M}}$ in the single-bidder case. On the left is the allocation function $x$ for the bidder in the original auction $\mathcal{M}$. On the right is the allocation function $\tilde{x}$ for the bidder in the auxiliary auction $\tilde{\mathcal{M}}$, where $\tilde{x}(v)=x(s(v))$ is obtained by composing $x$ with a monotone bidding strategy $s(v):=\frac{1}{2}\cdot\mathds{1}(v\ge\frac{1}{2})$.

Theorems & Definitions (36)

• Theorem 1.1: Informal restatement of Theorem \ref{['thm:meta_strategic_robustness']} and Proposition \ref{['prop:meta_no_regret']}
• Corollary 1.2: Informal restatement of Corollary \ref{['cor:MWU+meta']}
• Lemma 2.1: Myerson's lemma
• Definition 2.5
• proof : Assumption on auction formats
• proof : Randomness from private values
• proof : Consistency with Myerson's setting
• Definition 3.1
• Lemma 3.2
• Definition 3.3