[Paper Review] Robust Temporal Guarantees in Budgeted Sequential Auctions

Giannis Fikioris, Robert Kleinberg+3 more|arXiv (Cornell University)|Feb 20, 2026

Advanced Bandit Algorithms Research0 citations

TL;DR

A simple primal learning algorithm for budgeted sequential auctions yields robust guarantees: a bidder with ρ fraction of the budget wins ≈ ρT rounds against adversaries, and in self-play achieves near-equal, low-discrepancy win distribution over time.

ABSTRACT

In modern advertising platforms, learning algorithms are deployed by budget-constrained bidders to maximize their accumulated value. These algorithms often offer classical utility guarantees like no-regret, i.e., the agent's utility is at least the utility achieved by some benchmark in which it is assumed that every other agent's bidding remains the same. These guarantees offer compelling properties: They are optimal against stationary competition distributions, and in unconstrained settings, the resulting empirical distribution of play induced by no-regret dynamics approximates a Coarse Correlated Equilibrium. However, no-regret algorithms are easily manipulable, and in budgeted settings, no stronger notion of regret (such as swap regret) is currently known that would limit such manipulation. We propose a very simple learning algorithm for budgeted sequential auctions where agents maximize their total number of wins and show that it has surprisingly appealing properties. We analyze this algorithm from two perspectives. First, we show that when an agent with a $ρ$ fraction of the total budget uses this algorithm, then she is guaranteed to win at least $ρT - O(\sqrt T)$ of the total $T$ rounds. This result holds for adversarial behavior by the other agents, as long as they respect their own budget restrictions. Second, we examine the scenario when all the agents follow our algorithm. By the first result, every agent's total wins are proportional to her budget, up to the additive $O(\sqrt T)$ term. In addition, we show that this result holds in a much stronger sense: after an initial period of $O(\sqrt T \log T)$ rounds, every agent gets the same guarantee over any time interval. For intervals of length $O(\sqrt T)$, we show that the deviation from the desired number of wins is an additive constant.

Motivation & Objective

• Motivate learning in auctions under global budget constraints beyond no-regret frameworks.
• Propose a primal, deterministic bid-update rule and analyze its budget-safety properties.
• Establish worst-case win guarantees against adversarial opponents respecting budgets.
• Demonstrate self-play properties: proportional win distribution and low discrepancy over time.

Proposed method

• Propose a deterministic bid update: b^{(t+1)} = b^{(t)} + η(ρ_i − p_i^{(t)}).
• Set η = 1/√T to obtain strong asymptotic guarantees.
• Show that bids remain non-negative and budgets are not exceeded (Lemma 2.1).
• Model optimizer behavior as an integer program and analyze via a Lagrangian relaxation to bound optimizer gains (Theorem 3.1).
• Characterize convergence: subgradient-descent interpretation on a convex function f(b) with unique minimum at b = 1 (Eq. 5, Propositions 4.3–4.5).
• Provide a multi-agent self-play analysis: after startup, each agent with share ρ_i wins ≈ ρ_iτ in any interval of length τ = Θ(√T).

Experimental results

Research questions

• RQ1 Can a simple primal bidding rule guarantee proportional win shares for budgeted bidders against adversaries with budgets?
• RQ2 Do such primal dynamics yield low temporal discrepancy in win distribution when all agents follow the rule?
• RQ3 How fast do bids converge to a stable interval, and what are the implications for interval-based win guarantees?
• RQ4 What are the theoretical limits of manipulation by an optimizer against budgeted learners using this rule?
• RQ5 How do results extend to self-play and multiple intervals under budget constraints?

Key findings

• An agent with budget ρ_i T wins at least ρ_i T − O(√T) rounds against any adversarial, budget-respecting opponents.
• If all agents use the algorithm, each agent’s total wins are approximately proportional to their budgets, with an additive O(√T) startup term.
• After an initial O(√T log T) rounds, every agent achieves near-equal guarantees over any time interval; over intervals of length O(√T), the discrepancy is O(1).
• In equal-budget settings, agents eventually win in a round-robin pattern, giving a discrepancy of at most (n−1)/n within any interval.
• Bids converge to an interval around 1 with width O(η) after appropriate rounds, enabling O(1) discrepancy bounds for small intervals.