Learning to Bid in Discriminatory Auctions with Budget Constraints

2026-06-28

AI Summary

This study addresses the problem of maximizing cumulative utility for a single bidder in a multi-unit discriminatory-price auction over $T$ rounds, under time-varying valuations, an exponentially large action space, and a global budget constraint. The work proposes a scalable algorithm based on shortest paths in directed acyclic graphs, decomposing utility per unit and incorporating a primal-dual mechanism to handle budget constraints. By dynamically adjusting edge weights via online gradient descent on dual variables, the algorithm achieves polynomial-time learning with regret bounds independent of the number of contexts, under both full-information and bandit feedback. It enables full cross-context generalization and attains a $\rho$-approximate sublinear regret bound under budget limitations, with time and space complexity suitable for large-scale or even infinite context spaces.

Abstract

We study repeated bidding in multi-unit discriminatory (pay-as-bid) auctions for a single bidder with per-round utility equal to value minus $α$ times payment, where $α\in[0,1]$ is a cost-of-capital parameter. The bidder aims to maximize cumulative utility over $T$ rounds subject to a total budget $B$. The problem is challenging even without budgets: the action space is exponential in $M$, the maximum demand of the bidder and the valuation vector (context) varies over time. Exploiting a decomposition of utility across units, we develop polynomial-time learning algorithms based on shortest paths in a directed acyclic graph, obtaining sublinear regret under both full-information and bandit feedback. In the bandit setting, the regret is independent of the number of contexts due to complete cross-learning: observing the utility of the chosen action under the realized context reveals the utility for the same action under all counterfactual contexts. With budget constraints, when the average normalized per-round budget $ρ=\frac{B}{MT}<1$, we design a coupled primal-dual algorithm in which the DAG-based procedure uses dual-adjusted edge weights for primal updates, while online gradient descent updates the dual variable, yielding $ρ$-approximate sublinear regret. Finally, we give implementations whose per-round time and space are independent of the number of contexts, enabling scalability to large or even infinite context spaces.

Problem

• discriminatory auctions
• budget constraints
• online learning
• repeated bidding
• utility maximization

Innovation

• discriminatory auctions
• budget-constrained bidding
• cross-learning
• primal-dual algorithm
• sublinear regret