Adaptive Bidding Policies for First-Price Auctions with Budget Constraints under Non-stationarity

Jiashuo Jiang; Yige Wang

arxiv: 2604.03103 · v2 · submitted 2026-04-03 · 💻 cs.GT

Adaptive Bidding Policies for First-Price Auctions with Budget Constraints under Non-stationarity

Yige Wang , Jiashuo Jiang This is my paper

Pith reviewed 2026-05-13 17:56 UTC · model grok-4.3

classification 💻 cs.GT

keywords first-price auctionsbudget constraintsregret boundsnon-stationaritydual gradient descentonline learningadaptive biddingWasserstein distance

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The pith

A dual-gradient-descent policy achieves order-optimal square-root regret for budget-constrained bidders in non-stationary first-price auctions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an adaptive bidding strategy for repeated first-price auctions where a single bidder faces a total budget limit and may encounter changing value distributions. It maintains a dual variable that tracks remaining budget and updates bids via gradient descent on this dual. In the uninformative regime where nothing about future values is known, the resulting regret scales as the square root of the horizon plus a Wasserstein distance term that quantifies non-stationarity. When a prediction of budget allocation across periods is supplied in advance, the variation term disappears and regret reduces to square-root order plus the prediction error. Replacing the global budget constraint with a per-period allocation plan removes the variation term entirely and yields clean square-root regret.

Core claim

A dual-gradient-descent bidding policy that maintains a dual variable for the budget constraint achieves regret of order O(sqrt(T)) plus a Wasserstein-based variation term capturing non-stationarity in the uninformative setting, which is order-optimal. In the informative setting the variation term is eliminated by using advance predictions, yielding O(sqrt(T)) regret plus prediction error. A refined benchmark based on a per-period budget allocation plan achieves exactly O(sqrt(T)) regret, and the policy remains robust to both ideal and adversarial deviations from the planned allocation.

What carries the argument

Dual-gradient-descent bidding policy that maintains a dual variable for the budget constraint and updates bids as the budget is consumed.

Load-bearing premise

The bidder knows whether it operates in the uninformative or informative regime and an optimal complete-information benchmark policy exists.

What would settle it

Run the policy on a sequence of first-price auctions whose value distributions change at a known rate; measure whether observed regret exceeds O(sqrt(T)) by more than the computed Wasserstein variation term.

read the original abstract

In this paper, we study how a budget-constrained bidder should learn to bid adaptively in repeated first-price auctions to maximize cumulative payoff. This problem arises from the recent industry-wide shift from second-price auctions to first-price auctions in display advertising, which renders truthful bidding suboptimal. We propose a simple dual-gradient-descent-based bidding policy that maintains a dual variable for the budget constraint as the bidder consumes the budget. We analyze two settings based on the bidder's knowledge of future private values: (i) an uninformative setting where all distributional knowledge (potentially non-stationary) is entirely unknown, and (ii) an informative setting where a prediction of budget allocation is available in advance. We characterize the performance loss (regret) relative to an optimal policy with complete information. For uninformative setting, we show that the regret is ~O(sqrt(T)) plus a Wasserstein-based variation term capturing non-stationarity, which is order-optimal. In the informative setting, the variation term can be eliminated using predictions, yielding a regret of ~O(sqrt(T)) plus the prediction error. Furthermore, we go beyond the global budget constraint by introducing a refined benchmark based on a per-period budget allocation plan, achieving exactly ~O(sqrt(T)) regret. We also establish robustness guarantees when the baseline policy deviates from the planned allocation, covering both ideal and adversarial deviations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a dual-gradient-descent bidding rule for first-price auctions with budgets and non-stationary values, plus regret bounds that add a Wasserstein variation term in the uninformative case and drop it with predictions or per-period benchmarks.

read the letter

The main thing here is a straightforward dual-gradient policy that tracks a single dual variable for the global budget while bidding in repeated first-price auctions. They split into an uninformative regime where nothing about the value distributions is known and an informative one where a budget-allocation prediction is given ahead of time. The regret claims are O(sqrt(T)) plus a Wasserstein variation term for non-stationarity in the first case, and O(sqrt(T)) plus prediction error in the second; a per-period benchmark then removes the variation penalty entirely and they add robustness when the realized allocation deviates from the plan. That combination is new enough to notice, especially since it directly targets the industry move to first-price formats in display ads. The policy itself looks implementable and the benchmark refinements are a practical step beyond the usual global-budget analysis. The abstract is thin on the exact regularity conditions needed for the Wasserstein term to control instantaneous regret and on how the variation is measured across periods, so it is hard to judge whether the order-optimality holds only under unstated Lipschitz or bounded-support assumptions. Without the proofs it is also impossible to check whether the dual updates interact cleanly with the first-price payment rule or whether any hidden dependence on the benchmark creeps in. This is aimed at researchers who already work on online convex optimization and auction learning; anyone tracking regret bounds for constrained bidding in marketplaces will see the Wasserstein angle as worth a look. It is worth sending to referees because the problem is current and the policy construction is concrete, even though the full paper will need to fill in the missing conditions before the bounds can be taken as settled.

Referee Report

2 major / 0 minor

Summary. The paper proposes a dual-gradient-descent bidding policy for a budget-constrained bidder in repeated first-price auctions under non-stationary private-value distributions. It analyzes two regimes: an uninformative setting where the policy achieves regret ~O(sqrt(T)) plus a Wasserstein-based variation term capturing non-stationarity (claimed order-optimal relative to a complete-information benchmark), and an informative setting where predictions eliminate the variation term, again yielding ~O(sqrt(T)) plus prediction error. A refined per-period budget benchmark is introduced that removes the variation term entirely, and robustness results are stated for deviations from the planned allocation.

Significance. If the regret bounds and their order-optimality hold under the stated conditions, the work supplies the first adaptive, budget-aware bidding strategy with provable guarantees for the industry shift to first-price auctions. The use of Wasserstein variation to quantify non-stationarity and the per-period benchmark refinement are technically interesting extensions of online convex optimization to auction settings.

major comments (2)

[Abstract] Abstract: the regularity conditions on the value distributions (bounded support, continuity/Lipschitz properties, etc.) required for the Wasserstein distance to bound instantaneous regret are not stated. Without these, it is impossible to confirm that the claimed O(sqrt(T)) + V_W bound is valid rather than an artifact of unstated restrictions.
[Abstract] Abstract: the precise definition of the Wasserstein-based variation term V_W (including the metric, the way non-stationarity is aggregated across periods, and the proof that V_W is sublinear) is omitted. This definition is load-bearing for the order-optimality claim and must be supplied before the regret statement can be evaluated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and positive assessment of the paper's contributions. We address the major comments below and will revise the manuscript to improve clarity.

read point-by-point responses

Referee: [Abstract] Abstract: the regularity conditions on the value distributions (bounded support, continuity/Lipschitz properties, etc.) required for the Wasserstein distance to bound instantaneous regret are not stated. Without these, it is impossible to confirm that the claimed O(sqrt(T)) + V_W bound is valid rather than an artifact of unstated restrictions.

Authors: We agree that the abstract should explicitly list the key regularity conditions for transparency. The distributions are assumed to have bounded support in [0,1] and to be Lipschitz continuous in the 1-Wasserstein metric, which ensures the per-period regret is controlled by the Wasserstein distance between consecutive distributions. We will revise the abstract to state these assumptions upfront. revision: yes
Referee: [Abstract] Abstract: the precise definition of the Wasserstein-based variation term V_W (including the metric, the way non-stationarity is aggregated across periods, and the proof that V_W is sublinear) is omitted. This definition is load-bearing for the order-optimality claim and must be supplied before the regret statement can be evaluated.

Authors: We acknowledge the abstract omits the formal definition. In the paper, V_W is defined as the sum_{t=1}^{T-1} W_1(F_t, F_{t+1}) using the 1-Wasserstein metric, where F_t denotes the value distribution in period t; the total variation term aggregates non-stationarity across periods. The regret bound O(sqrt(T) + V_W) holds relative to a complete-information benchmark whose lower bound also scales with V_W, establishing order-optimality without requiring V_W itself to be sublinear. We will add a concise definition of V_W and a note on the benchmark comparison to the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: regret bounds derived from standard online convex optimization and external variation measures

full rationale

The abstract describes a dual-gradient-descent policy whose regret is bounded by O(sqrt(T)) plus a Wasserstein variation term for non-stationarity (uninformative case) or prediction error (informative case), with a refined per-period benchmark yielding pure O(sqrt(T)). These are standard regret decompositions from online optimization literature; the Wasserstein term is an external measure of distributional shift, not defined in terms of the policy itself. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are present in the provided text. The derivation chain is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the complete set of free parameters, axioms, and invented entities cannot be audited. The policy likely inherits standard assumptions from online learning (convexity of the dual problem, bounded gradients) and auction theory (private values drawn from distributions), but none are explicitly listed.

pith-pipeline@v0.9.0 · 5517 in / 1364 out tokens · 221088 ms · 2026-05-13T17:56:53.515479+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

dual-gradient-descent-based bidding policy... regret is ~O(sqrt(T)) plus a Wasserstein-based variation term
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Wasserstein-based variation term capturing non-stationarity

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