Title: Comparing Uniform Price and Discriminatory Multi-Unit Auctions through Regret Minimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Auction

Online Learning
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 $\tilde{\Theta} ( \sqrt{T}  )$ and $\tilde{\Theta} ( T^{2/3}  )$, respectively. However, analysis beyond worst-case regret reveals structural differences:  uniform-price auctions may admit faster learning rates, with regret scaling as $\tilde{\Theta} ( \sqrt{T}  )$ in settings where discriminatory auctions remain at $\tilde{\Theta}  ( T^{2/3}  )$. 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study a \"learning to bid\" problem in a repeated auction of several identical units of an indivisible item. They study two possible welfare-maximizing auctions that differ in their payment rules: uniform price and discriminatory price. This problem, and the derivation of regret rates for learning to bid, has been studied recently by Branzei et al. and Galgana and Golrezaei. The key conceptual contribution of the present work is to consider a stochastic model of utilities, where the bidder of interest evaluates her utilities in a Bayesian fashion over the draw of the other competing bids. Regret is also evaluated with respect to the stochasticity in the other bids. This allows the authors to transfer over many of the previous regret rates. They also show the existence of settings where learning to bid in uniform price auctions is easier than in discriminatory price auctions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n---\n\nMulti-unit auctions are prevalent in the real world, and learning to bid in these repeated auctions is an important problem that has received study in recent years. The authors contribute to that literature with a model of stochastic adversaries, showing that many of the regret bounds transfer over to their setting, and derive improved results in some cases.\n\nThe standalone improvement (Lemma 3) to the regret lower bound of Galgana+Golrezaei is nice, and the setting showing that learning to bid in uniform price auctions is strictly easier (O(T^{1/2}) vs. Omega(T^{2/3})) than in discriminatory price auctions, is fairly interesting.\n\nWeaknesses\n---\n\nI found it difficult to assess the contribution over the prior works of Branzei et al. and Galgana and Golrezaei. The authors should improve the presentation to emphasize the main differences---both conceptual and technical. From what I could gather, the model is identical to those two papers, and the only difference is in how the other competing bidders are treated. In the present work, the competing bidders are treated as stochastic, and the main bidder's objective is to maximize her expected utility over the draw of the competing bids. The main techniques appeal to standard CDF estimation concentration bounds. Here, some differentiation over the techniques from prior work would help the reader substantially. Also, as  I mention below in the comments section, some further exposition clarifying the purpose of instance-dependent guarantees (and what exactly they mean mathematically) would be helpful.\n\nThe proof sketch that involves multiclass classification/Natarajan dimension sounds quite interesting to me and I would have liked to see a slightly more-expanded intuitive explanation in the main body. Where/why does multiclass classification come into the picture?\n\nQUESTIONS:\nLine 15: Perhaps that claim would have stronger support if the authors included a reference that is more recent than 2009.\n\nLines 61-71: regret rates are presented with dependence on K, but K has not once been defined before that.\n\nLine 99: I think you should explicitly state immediately that the K items are K identical copies of the same indivisible item. \n\nLine 104: If the \\\\(v_{\\ell}\\\\) are marginal valuations, is there an assumption that \\\\(v_1\\ge v_2\\ge\\cdots\\ge v_K\\\\)? In Line 110, are the beta's also drawn from a distribution such that the beta's have decreasing marginal utilities?\n\nLine 119: I believe that the decreasing marginal returns assumption above is required for this to be the correct allocation function that implements the efficient/welfare-maximizing allocation. The authors should make that clear. Also, it might help to (i) use the words \"efficient\" or \"welfare-maximizing\" instead of \"standard\" (line 118), and (ii) include a one line explanation why the allocation function has that particular form. \n\nLines 124-126: \"(the bid that won its emitter this item)\" I don't understand what this means. Based on eqn (3), the discriminatory price auction sets the price of the kth item to b_k (the bidder's marginal value for the kth unit), regardless of who wins. Is that correct? I guess the b_k's for k >= x(b, beta) don't impact the bidder's utility.\n\nRemark 1: is this well-known? If so, include a citation/reference. Otherwise a few lines of intuition would be good. I don't think the authors should include precise claims without proof.\n\nSection 5: I was a bit confused as to how this is \"beyond worst-case\" as opposed to the previous section. My understanding is that this section is the same bandit-feedback setting as Section 4, and the authors are just proposing and analyzing a better algorithm (basically doing a more refined CDF estimation). (OK, I see now that the main application is for instance-dependent bounds. It would help if you included a formal discussion of the benefits instance-dependent bounds provide. I know that it's a standard thing in this literature, but putting things into context would help.)\n\nDid Branzei et al. and Galgana+Golrezaei study separations in learning complexity between the two auction formats?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies multi-unit auctions (MUAs), where several identical items are sold in each round. It focuses on two common formats: uniform-price and discriminatory-price auctions. While both allocate items to the highest bidders, they differ in how winning bidders are charged. The central question explored is how a single bidder, participating in repeated instances of such auctions, can learn to optimize their strategy when the aggregated bids of their opponents follow a known probability distribution.\n\nThe analysis is framed through the lens of regret minimization, evaluating the bidder’s ability to learn over time. Two feedback settings are considered: full information and bandit feedback. In the full information setting, both auction types pose similar learning challenges. However, under bandit feedback, the paper shows that uniform-price auctions allow for a smaller instance-dependent regret than discriminatory auctions, making them more favorable for learning in practice. While most of the novel theoretical results concern uniform-price auctions, the key contribution of the paper lies in the comparative perspective on the learning complexity in the two auction formats.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths :** The motivation is clear and straightforward. The topic itself is timely and interesting, addressing a well-established yet practically significant problem in auction theory. The paper is well-written, with a coherent structure and precise explanations that guide the reader through many different instances of the problem.. The results presented are thorough and cover a wide range of scenarios. Moreover, the contributions are not only substantial in number but also of high quality.\n\n**Weaknesses :** While the paper is strong, there are a few weaknesses that could be improved. Some parts of the general setting are not entirely clear. For example, it is not stated whether the bids $\\beta_1^t, \\ldots, \\beta_n^t$ are ordered in increasing value, which can affect the interpretation. The issue of non-truthfulness is also not discussed in depth. This is important because, in single-unit auctions, uniform-price formats are known to be truthful. The special case where K=1 is not clearly separated from the general case, which makes the analysis feel less complete. For the instance-dependent regret results, it would have been helpful to study how the regret depends on specific parameters of the instance, to better understand which situations are easier or harder. Finally, there are no experiments included in the paper, but this is acceptable given the theoretical nature of the contributions.\n\nQUESTIONS:\nCan you comment on the dependence of the regret on specific parameters of the instance for the instance-dependent bound ?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study multi-unit uniform price and discriminatory price auctions from an online learning perspective. At each time step $t$, the learner participates in either a uniform (respectively, discriminatory) auction with the goal of winning up to $K$ identical items by submitting a vector of bids $(b_1^t, …, b_K^t)$. It is assumed that the learner’s utility is additive and that the utility associated with winning the $k$-th item for any $k \\in [K]$ is known and fixed over time.\nThe learner’s bids compete with the top $K$ bids submitted by other participants. The learner wins exactly $k$ items if precisely $k$ of their bids are among the $K$ highest overall. The payment scheme is as follows: in uniform price auctions, the learner pays $k \\cdot p$, where $p$ is the value of the first rejected bid; in discriminatory price auctions, the learner pays the sum of their $k$ winning bids.\nThe feedback model can be either full-information—where the learner observes all the top $K$ competing bids—or bandit feedback, where the learner only observes the number of items won and the corresponding price. The competing bids are modeled as independent and identically distributed (i.i.d.) random variables.\nThe learner’s objective is to minimize regret with respect to a benchmark bidder who knows the distribution of the competing bids and bids optimally to maximize expected reward.\nAs far as I can tell (despite several typos in the paper that make this somewhat unclear) these appear to be the authors’ main contributions:\n1.\tFull-feedback: The authors claim to improve the previously known upper bound for uniform price auctions by a $\\sqrt{K}$  factor. However, I believe there may be a typo here. In the related work section, they state that Branzei et al. achieve a regret rate of $O(K \\sqrt{T})$, which is exactly the rate the authors obtain in this paper. \nIt is unclear whether the authors provide any additional improvements for the full-feedback setting. They state: \"We show that both auction formats admit tight worst-case regret rates of $\\tilde{O}(K\\sqrt{T}))$ under the full-information setting (improving the best known rates for uniform price auctions by a factor of $\\sqrt{K}$).\" This wording suggests they also contribute something for discriminatory price auctions, but I did not manage to find any explicit contribution in this sense, leaving the reader uncertain about what has actually been improved.\n2.\tBandit Feedback – Upper and Lower Bounds: For the bandit feedback setting, the authors claim: \"Our analysis also yields regret rates of $\\tilde{O}(K^{5/3} T^{2/3})$ for uniform price auctions, and we show a matching lower bound.\"\nHowever, this seems problematic. Based on the context, this result should only hold in the stochastic i.i.d. setting. Furthermore, the authors reference Potfer et al., who achieve a regret rate of $\\tilde{O}(K^{4/3} T^{2/3})$, which seems actually better than the upper bound presented in this paper. If Potfer et al.'s result is correct, it would contradict the claim that the authors' lower bound is tight. In fact, I could only find a $\\Omega(T^{2/3})$ lower bound in the paper, not the $\\Omega(K^{5/3} T^{2/3})$ bound they seem to suggest with the phrasing above. Overall, this part of the contribution is unclear.\n3.\tInstance-Dependent Rates: The authors also claim to provide improved instance-dependent regret rates for the uniform price auction, specifically for one-unit and two-unit demand cases, along with matching lower bounds in the time horizon.\n\nSTRENGTHS AND WEAKNESSES:\nI believe that understanding the regret rates in stochastic i.i.d. uniform and discriminatory is an interesting contribution.\nHowever, as already pointed out in the summary above, the related work and contribution section left me perplexed about how to contextualize the contributions provided by this paper.\n\nQUESTIONS:\nCould you please clarify what are the contributions relative to the literature answering to the point raised in the summary?\nAlso, the summary table 1 should report also the dependences on K, not just on T.\nFinally, a \"techniques and challenges\" section would be beneficial for the reader to understand better what are the non-standard technical contributions with respect to the existing literature.\n\nTypo: line 123, diplayed formula: extra ]\n\nI am willing to reconsider my score depending on the authors response.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 1, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-30"}

comments: SUMMARY:
The paper compares two common multi-unit auction formats—uniform-price and discriminatory auctions—from the perspective of a bidder using online learning to optimize strategies against stochastic adversaries. It analyzes regret bounds under full-information and bandit feedback settings, highlighting both worst-case and instance-dependent learning difficulties.

STRENGTHS AND WEAKNESSES:
Strengths:
1. This paper provides concrete and thorough theoretical proof, showing the regret bounds for uniform and discriminatory multi-unit auctions with various formats.
2. The paper is well-written and properly organized.
3. It is interesting to consider the so-called "instance-independent" regret, providing insights how the algorithms can be improved beyond worse-case.

Weaknesses:
1. The comparison results in some sense are not surprising.
2. The assumption that the bids of the adversary are drown from fixed distribution is kind of strong.

QUESTIONS:
NA

LIMITATIONS:
NA
TITLE: Official Review
CLARITY: 4
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-04