Minimax regret under adversarial winning-bid-only feedback in first-price auctions

Determine the minimax optimal regret rate achievable by any online bidding policy in repeated first-price auctions under winning-bid-only feedback when the sequence of highest competing bids is adversarial. Precisely characterize the optimal dependence on the time horizon T for this adversarial-feedback setting.

Background

The literature distinguishes several feedback models for repeated first-price auctions: binary feedback, winning-bid-only feedback (where the platform reveals only the winning bid), and full-information feedback (where the minimum bid to win is always revealed). Known minimax regret rates include: binary feedback with i.i.d. competition at order T{2/3} and with adversarial competition at order T{3/4}; winning-bid-only feedback with i.i.d. competition at order √T; and full-information feedback with adversarial competition at order √T.

However, the adversarial case under winning-bid-only feedback has not been characterized. The paper explicitly notes that the minimax optimal regret in this setting remains unknown, highlighting a gap between the i.i.d. and adversarial results across feedback regimes.

References

Although it remains unknown what the result would be when $m_t$ is adversarial under winning-bid-only feedback, studied the full-information feedback setting and showed that the minimax optimal regret of $\tilde{\Theta}(T{\frac{1}{2})$ can be achieved when $m_t$ is adversarial.

— Adaptive Bidding Policies for First-Price Auctions with Budget Constraints under Non-stationarity  (2604.03103 - Wang et al., 3 Apr 2026) in Related Literature — Adaptive Bidding in First-Price Auctions without Budget Constraints