Title: Procurement Auctions with Predictions: Improved Frugality for Facility Location
Venue: NeurIPS 2025
Accepted: yes
Keywords: Algorithm with predictions

Uncapacited facility locations

mechanism design
Abstract: We study the problem of designing procurement auctions for the strategic uncapacitated facility location problem: a company needs to procure a set of facility locations in order to serve its customers and each facility location is owned by a strategic agent. Each owner has a private cost for providing access to their facility (e.g., renting it or selling it to the company) and needs to be compensated accordingly. The goal is to design truthful auctions that decide which facilities the company should procure and how much to pay the corresponding owners, aiming to minimize the total cost, i.e., the monetary cost paid to the owners and the connection cost suffered by the customers (their distance to the nearest facility). We evaluate the performance of these auctions using the \emph{frugality ratio}.


We first analyze the performance of the classic VCG auction in this context and prove that its frugality ratio is exactly $3$. We then leverage the learning-augmented framework and design auctions that are augmented with predictions regarding the owners' private costs. Specifically, we propose a family of learning-augmented auctions that achieve significant payment reductions when the predictions are accurate, leading to much better frugality ratios. At the same time, we demonstrate that these auctions remain robust even if the predictions are arbitrarily inaccurate, and maintain reasonable frugality ratios even under adversarially chosen predictions. We finally provide a family of ``error-tolerant'' auctions that maintain improved frugality ratios even if the predictions are only approximately accurate, and we provide upper bounds on their frugality ratio as a function of the prediction error.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work examines a variant of the learning-augmented facility location auction problem, where each facility must be explicitly rented or purchased to be utilized. This introduces incentives not for the facility users—as in prior work—but also for the facility owners. The main performance metric studied is the *frugality ratio*.\n\n**Key results**:\n\n- Without predictions: The classic VCG auction is shown to achieve a frugality ratio of exactly **3**.\n\n- With predictions:\n  1. An auction that guarantees a $(1+\\varepsilon, \\max\\{5, \\frac{2}{\\varepsilon}\\})$ consistency-robustness trade-off.\n  2. A smoother alternative trade-off is also given:\n     - If $(\\eta \\le \\lambda)$, the frugality is bounded by $\\eta(1+\\lambda) + 2\\varepsilon$.\n     - Otherwise, the bound becomes $\\max $ {$\\{ 2\\lambda^4 + 3\\lambda^2, \\ 3 + \\frac{2}{\\varepsilon} \\}$ }.\n\n     where $\\varepsilon \\in (0,2]$ and $\\lambda > 1$ are tunable parameters.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n* The VCG no-predictions frugal analysis is simple (elegant), yet tight.\n\n* The paper achieves  consistency-robustness results for the settings with predictions. The Error Tolerant Scaled VCG auction allows smooth consistency-robust results with tunable parameters $\\lambda$, $\\varepsilon$.\n\nWeaknesses:\n\n* The problem is not about strategic facility location as much as it is about auctions. That is, the users are not strategic and all of their locations are known (the user locations only affect the cost of each solution). This reduces to an auction where the auction designer attempts to buy/rent the facility locations from the owners. Hence, the standard techniques of VCG and Myersons lemma apply. In terms of the combination with predictions, the techniques seem similar to the ones given for auctions with predictions by Xu and Lu 2022 - the $\\lambda$ parameter seems (roughly) to play the role of the $\\gamma$ parameter of Xu and Lu 2022 (their Theorem 2,5).\n\n* The PredictedLimits auction is really only useful if the optimal solution is exactly correct, which seems unrealistic (a problem which the \"ErrorTolerant\" auction attempts to solve).\n\n* While the last (\"ErrorTolerant\") auction is more robust to small errors than the PredictedLimits one, its performance of the Error Tolerant auction only seems to work well for very good predictions. $\\lambda$ seems to roughly be the user's estimate of $\\eta$. Say the auctioneer chooses a reasonable estimate - that the worst of all predictions is off by 50 percent ($\\lambda = 1.5$). And say that in reality $\\eta = 1.2$ (the worse prediction is off by 20 percent). Then the algorithm will get a frugality of $3$ which is what the VCG auction without predictions achieves. On the other hand, the robustness guarantees are not really small either ($16.875$ for $\\lambda = 1.5$).\nNote that also the $\\varepsilon$ parameter must be small enough to allow the $\\eta(1+\\lambda)+2\\varepsilon$ term be smaller than $0.5$ to allow any better than $3$ result in the best-case, but this results in a large robustness guarantee. Say for example, the user chooses reasonable $\\varepsilon = \\frac{1}{3}$ and $\\lambda = 1.15$. So the robustness is $9$ (high), but, the consistency is around $3$ for reasonable error of $\\eta = 1.1$, which is again what we would get with the no-predictions regime.\n**To summarize this point**: it seems hard to choose reasonable $\\lambda,\\varepsilon$ parameters that will make the given auction with predictions more attractive than the no-predictions one.\n\nQUESTIONS:\n* Are there negative results about strategic UFL for the approximation ratio objective? The paper would benefit from listing impossibilities or hardness result of achieving a constant approximation ratio (with and without predictions).\n\n\n* In line 169 $U_{l,f}$ is used even though it was not defined anywhere. From the context, it seems like $U_{l,f}$ is defined to be the set of users that are connected to both facilities $l \\in OPT$ in the optimal solution and to $f \\in F$ in the frugal solution. Is this true? \n\n* You have shown a lower bound of $3$ for the tree metric for the frugal ratio. Are you aware if the results are different for a Euclidean metric space (the real line, the plane, $R^d$ Eucledian space)?\n\n* In Appendix B: line 457 states \"Figure B\" but the caption says \"Figure 2\". Please choose one of these (probably 2). Also in the same line, I think the edges should be $(l_i,u_i)$ instead of $(l,u_i)$. Line 458: \"distance of from\" - remove the \"of\". Line 459: $d(l,l_j)$ should be $d(l_i, l_j)$. The same goes for the end of the sentence.\n\n* Putting the ErrorTolerant auction in the paper body rather than the appendix (in the version where there are $\\ge$ 10 pages to the paper body rather than $9$) might be a good idea.\n\n* What happens if there's one erroneous facility cost prediction (say one predicted cost is far from the actual cost) but all the rest are not? Does the ErrorTolerant auction get any guarantee? It is fine if it is unknown to you.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-15"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study a procurement version of the strategic facility location problem where facilities report their operating costs, and a central planner decides on which facilities to open to minimize operating costs plus the costs of serving a fixed set of users. The planner has access to predictions about the facilities' operating costs. The work contributes to a recent active line of study on mechanism design with predictions, so it is a timely contribution. The main performance metric is the frugality ratio, which compares a mechanisms cost to the cost of the second-best solution. The authors first improve a result on the frugality ratio of VCG (establishing a tight result), then present a mechanism attaining (1+eps)-consistency and max(5, 3+2/eps)-robustness, and then present a error-tolerant mechanism with consistency and robustness ratios depending on the accuracy of the cost predictions.\n\n(Prior work on strategic facility location assumes strategic agents who can misreport their true locations. The present work considers agents with fixed locations, and the facilities are the strategic entities that need to report their operating costs.)\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n-------------------\n\nI think the problem and model studied are nice. Frankly, I think the model studied here is a more meaningful reflection of real-world procurement than anything modeled by prior work on strategic facility location :)\n\nThe improvement to the old result of Talwar on VCG's frugality ratio is a nice standalone contribution as well.\n\nThe paper is well written.\n\nWeaknesses\n-------------------\n\nThe robustness guarantees seem a bit weak since regardless of epsilon the frugality ratio is >= 5 which is worse than the VCG benchmark. Some result showing tightness would be nice here.\n\nI know that frugality is a standard notion and has been studied before, but its motivation is a bit unclear to me. A sentence or two giving the intuition behind why the second best solution is the right measure would be helpful. To me it seems like a needless weakening of the strongest benchmark which is OPT.\n\nIt would be nice to include a few lines of intuitive description of how Theorem 5.1 is proved in the main body.\n\nQUESTIONS:\nWhat is the equivalent notion of frugality for a forward auction (1 seller, many buyers)? Why not just compare directly to OPT instead---that seems like a stronger benchmark and is what the literature on learning-augmented mechanism design (e.g. Xu and Lu [IJCAI'22], Balcan et al. [NeurIPS'23]) has typically focused on. My guess is that this is a feature specific to the facility-location/procurement setting, but some clarification would be helpful.\n\nRelated to the above, if one receives perfect predictions of the opening costs, isn't it possible to obtain cost = OPT while remaining truthful? (In contrast to the weaker result presented here that achieves a frugality ratio of 1.) Along that vein, consider the randomized mechanism that w.p. p trusts the predictions completely and implements OPT by paying each facility exactly the predicted cost, and w.p. 1-p discards the predictions and uses VCG. The consistency (I'm not sure about robustness) of this mechanism can be compared directly to OPT instead of the second-best. Where does this fit into the picture with the mechanisms in the present work?\n\nI am a bit confused about the explanation in lines 211-213. The authors say that scaling up under-estimated costs is to deter overbidding. My understanding is that incentive compatibility has nothing to do with the predictions/the facilities do not see the predictions. Could the authors clarify?\n\nWhat is the computational complexity of Auction 1? Vanilla facility location can at least be solved efficiently in practice via integer programming---is there a concise integer programming formulation for Auction 1?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies procurement auctions for the strategic uncapacitated facility location (UFL) problem, where facilities are owned by strategic agents with private opening costs. The goal is to open facilities and determine payments to minimize total cost, balancing opening and user connection costs. The authors evaluate auction performance using the frugality ratio, which compares auction cost to a “second-best” benchmark. They first show that the classical VCG auction has a tight frugality ratio of 3. Then, they introduce learning-augmented auctions that use potentially inaccurate cost predictions to significantly reduce payments when predictions are accurate, while still maintaining bounded frugality under adversarial errors. They also propose an error-tolerant auction variant that adapts to approximately accurate predictions, achieving improved frugality guarantees as a function of prediction error.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\nThe paper addresses an important and timely problem---the design of procurement auctions for strategic uncapacitated facility location---by exploring how predictive information can improve frugality in mechanism design.\n\n**Weaknesses:**\nThe paper is poorly written and lacks clarity in both exposition and notation. Key concepts, such as the VCG auction, are not properly defined when first introduced. The proofs are difficult to follow, partly due to inconsistent and undefined notation. For example, the term $U_{\\ell, f}$ appears in key arguments without any formal definition, and the notation for the auction mechanism $\\mathcal{M}$ is used inconsistently throughout the paper with varying input sets, without acknowledgment. Similarly, $\\mathsf{OPT}$, a central object in the definition of the frugality ratio, is used before being clearly defined. Additionally, the motivation for focusing on the VCG auction and for using the frugality ratio as the primary performance metric is not clearly justified. \n\nFurthermore, the prediction-augmented algorithm may have limited practical applicability, as it penalizes certain agents by scaling up their costs based on potentially inaccurate predictions. The design appears somewhat single-minded, focusing narrowly on optimizing a specific objective (i.e., frugality) while overlooking broader consequences of the mechanism, such as fairness or unintended distortions in participation incentives.\n\nFinally, the paper completely overlooks the computational aspects of the optimization problems involved in the proposed mechanisms. These problems are often NP-hard, and the paper provides no discussion on tractability or practical solvability.\n\n\nThese are just some example problems. The paper has some potential. But, in its current form, it is not ready for publication.\n\nQUESTIONS:\nJust wondering if the authors have thought about this comment:\n\n**Furthermore, the prediction-augmented algorithm may have limited practical applicability, as it penalizes certain agents by scaling up their costs based on potentially inaccurate predictions. The design appears somewhat single-minded, focusing narrowly on optimizing a specific objective (i.e., frugality) while overlooking broader consequences of the mechanism, such as fairness or unintended distortions in participation incentives.**\n\nAnd if they have done some simulations to see how biases in prediction can hurt some of the players?\n\nLIMITATIONS:\nSee my earlier comments.", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the strategic uncapacitated facility location problem (UFL). In UFL, an agency wants to procure a set of facilities among a set of potential locations to provide service to its users. Each facility location comes with an individual cost. The goal is to choose a set of facility locations that minimizes the sum of the distances from users to facilities and the total payment to the facility owners.\n\nIn strategic UFL, each facility owner has a private cost, and the agency does not know these private values. Instead, it runs an incentive-compatible auction to determine which locations to procure and how much to pay. In the literature, there has been some work on analyzing the worst-case guarantees of IC auctions, comparing them to the total cost when the location costs are public. More formally, the benchmark is the frugality ratio, which is the worst-case ratio between the total cost of a truthful auction and the cost of the second-best solution when the agency knows the true costs.\n\nThe first result of the paper is a tight analysis of the frugality ratio for the VCG auction. Previously, the best known upper bound for VCG was 4. This work improves it to 3, with a matching lower bound instance.\n\nThe second contribution is that while most previous works focus only on worst-case guarantees, this paper designs and analyzes new auction frameworks that take into account predictions on the cost of each location. These mechanisms not only perform well when the predictions are accurate, but also retain worst-case guarantees when the predictions are wrong.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1.The problem is well-motivated, and the authors provide a tight analysis of the VCG auction, improving upon previous results in the literature.\n2. The paper goes beyond traditional worst-case analysis. It designs algorithms that achieve strong guarantees when given accurate predictions, while still providing baseline performance guarantees even when the predictions are completely inaccurate.\n\nWeaknesses:\n1. The presentation can be improved—some proofs are hard to follow, with a missing definition that makes them difficult to parse.\n2. The model is somewhat limited: it focuses on the case where the agency has accurate predictions for the cost of each location. It would be more interesting to consider Bayesian models, where the true costs are drawn from known prior distributions.\n\nQUESTIONS:\nThe notation $U_{\\ell, f}$ is not defined. Is it the set of users that are assigned to $\\ell$ in OPT and assigned to $f$ when the facility set is F?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}