Comparing Uniform Price and Discriminatory Multi-Unit Auctions through Regret Minimization

TL;DR

This paper compares uniform price and discriminatory multi-unit auctions through regret minimization in a repeated setting with stochastic opposing bids. By estimating the opponents’ marginal CDFs and optimizing estimated utilities, it shows that both formats share the same worst-case learning difficulty: $ ilde{Θ}(√T)$ under full-information and $ ilde{Θ}(T^{2/3})$ under bandit feedback, with order-statistic–based concentration tools enabling continuous-action analysis. Beyond worst-case scenarios, the authors identify a regime where uniform pricing admits faster learning, achieving $ ilde{Θ}(√T)$ regret while discriminatory pricing can remain at $ ilde{Θ}(T^{2/3})$, particularly for unit-demand and Δ-separated distributions, and in symmetric I.I.D. adversary settings. They introduce algorithms such as Explore-Then-Commit and a bandit K+1 approach, along with a novel concentration inequality for partially observed order statistics, providing both upper and lower bounds and clarifying when learning-to-bid is easier under uniform pricing. The results offer insights for mechanism design in repeated auctions, highlighting how feedback richness and unit-demand structure can influence bidding strategies and convergence rates.

Abstract

Repeated multi-unit auctions, where a seller allocates multiple identical items over many rounds, are common mechanisms in electricity markets and treasury auctions. We compare the two predominant formats: uniform-price and discriminatory auctions, focusing on the perspective of a single bidder learning to bid against stochastic adversaries. We characterize the learning difficulty in each format, showing that the regret scales similarly for both auction formats under both full-information and bandit feedback, as $\tildeΘ ( \sqrt{T} )$ and $\tildeΘ ( T^{2/3} )$, respectively. However, analysis beyond worst-case regret reveals structural differences: uniform-price auctions may admit faster learning rates, with regret scaling as $\tildeΘ ( \sqrt{T} )$ in settings where discriminatory auctions remain at $\tildeΘ ( T^{2/3} )$. Finally, we provide a specific analysis for auctions in which the other participants are symmetric and have unit-demand, and show that in these instances, a similar regret rate separation appears.

Theorems & Definitions (58)

• Lemma 1
• Theorem 1
• proof : Proof Sketch
• Remark 1
• Lemma 2
• Lemma 3
• Theorem 2
• proof : Proof Sketch
• Theorem 3
• proof : Proof idea