Title: Mechanism Design via the Interim Relaxation
Venue: NeurIPS 2025
Accepted: yes
Keywords: mechanism design

auctions

OCRS

CRS

revenue maximization

Bernoulli factories
Abstract: We study revenue maximization for agents with additive preferences, subject to downward-closed constraints on the set of feasible allocations. In seminal work,~\citet{alaei2014bayesian} introduced a powerful multi-to-single agent reduction based on an ex-ante relaxation of the multi-agent problem. This reduction employs a rounding procedure which is an online contention resolution scheme (OCRS) in disguise, a now widely-used method for rounding fractional solutions in online Bayesian and stochastic optimization problems. In this paper, we leverage our vantage point, 10 years after the work of Alaei, with a rich OCRS toolkit and modern approaches to analyzing multi-agent mechanisms; we introduce a general framework for designing non-sequential and sequential multi-agent, revenue-maximizing mechanisms, capturing a wide variety of problems Alaei's framework could not address. Our framework uses an \emph{interim} relaxation, that is rounded to a feasible mechanism using what we call a two-level OCRS, which allows for some structured dependence between the activation of its input elements. For a wide family of constraints, we can construct such schemes using existing OCRSs as a black box; for other constraints, such as knapsack, we construct such schemes from scratch. We demonstrate numerous applications of our framework, including a sequential mechanism that guarantees a $\frac{2e}{e-1} \approx 3.16$ approximation to the optimal revenue for the case of additive agents subject to matroid feasibility constraints. Finally, we show how our framework can be easily extended to multi-parameter procurement auctions, where we provide an OCRS for Stochastic Knapsack that might be of independent interest.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper suggests a new method, called two-level (O)CRS, that allows for \"Vertical-horizontal\" constraints (i.e., matroid constraints over both which items each agent gets, and which agents an item is allocated to). The paper shows how existing (O)CRSs can be combined into a t(O)CRS, and apply it to some settings, most notably improving [Cai & Zhao '17]'s approximation of 70 for the optimal revenue of an auction for constrained additive bidders, to around 3.16.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- The technical contribution is significant, and in some cases the results improve over the state of the art.\n\n- The paper is well-written\n\n- Personally I found Section B very helpful, as it provided a good introduction to the topic of Bernoulli factories. \n\nWeaknesses:\n\n- Since the topic is quite heavy and I'm not familiar with it, even though the paper is clearly written, there were still some things I couldn't figure out, which I detail here and in the questions:\n\nCorollary 1: It’s not clear from the statement of Theorem 2 that if you combine a CRS over items constraints with an OCRS over agent constraints you get a tOCRS (rather than tCRS). By the statement it seems like if you combine two CRSs, you get a tCRS, and if you combine two OCRSs, you get a tOCRS. \n\nNotice that in the proof of Theorem 1 you didn’t explicitly discuss the revenue approximation guarantee, but it’s pretty clear given the proof.\n\nQUESTIONS:\nQ1: I didn’t manage to understand the motivation for the vertical-horizontal constraints. I understand why the items allocated to an agent will have matroid constraints. But what does it mean that the agents that get item j satisfy matroid constraints? Isn’t it always Ex-post allocated to a single agent? Please explain the settings, and also references to works that previously discussed it before could be helpful. \nI see later that you mention [14] that restrict each item to be allocated to at most one agent: So maybe another way to ask my question is, so isn’t this guarantee already built into the CRS framework as the feasibility constraint (I guess not)? \n\nQ2: What is the argument for the last inequality in 1112? Why is the sum of probabilities for these elements to be active less than 4b (I can understand where the b is coming from, but why 4)? Same question for the last inequality of 1123, where it seems you use the fact that the sum of all preceding k_{i,j} are less than K/2, and again show a bound of 4 for the same sum of probabilities (but somehow here b is already outside the parantheses? so shouldn’t it be 2b^2/(1+4b)?)\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-09"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of designing revenue-maximizing mechanisms for selling m heterogeneous items to \nn additive agents, subject to downward-closed feasibility constraints. The authors propose a framework based on the following three steps:\n\n- Relaxing the feasibility constraints to allow interim allocation/payment rules that only satisfy the constraints in expectation.\n\n- Solving a linear program to compute an optimal Bayesian incentive-compatible (BIC) and individually rational (IR) interim solution.\n\n- Rounding the fractional solution using an extension of contention resolution scheme (OCRS) techniques.\n\nThe main contribution is the introduction of a new variant of OCRS that allows dependencies between the activations of elements, broadening the applicability of the framework. They also apply their framework to obtain approximation guarantees, including:\n\n- A 3.16-approximation sequential mechanism under “matroid Vertical-Horizontal” constraints (although the exact setting is not clearly defined—see comments).\n\n- A sequential, efficiently computable 10-approximation mechanism for additive agents under knapsack constraints.\n\n- A 9-approximation mechanism for agents with \"arbitrary\" valuations under knapsack constraints (this needs clarification—see comments).\n\nSTRENGTHS AND WEAKNESSES:\nI think the extension of the (O)CRS framework to handle dependencies among elements is technically nontrivial and represents an important contribution. The framework is flexible, applies to multiple constraint types, and in some cases improves over prior results. \n\nThe presentation of the paper can be significantly improved. Key definitions, intuition, and algorithmic descriptions are often compressed or interleaved, which can make the material hard to follow, especially for readers less familiar with the area. I understand that some of these issues are due to the lack of space, but the paper as it is now is hard to follow.\n\nQUESTIONS:\nIn the Applications section, when you say agents have arbitrary valuation functions, does that mean monotonicity, subadditivity, or other standard assumptions can be violated? Please clarify the assumptions.\n\nIn the first application (matroid VH constraints), can an item be allocated to multiple agents? Also, what does it mean for agents to \"form a matroid\"?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nBased on the influential work of Alaei, the authors introduce a novel, general framework for designing both non-sequential and sequential multi-agent, (approximately-optimal) revenue-maximizing mechanisms for agents with additive preferences, subject to downward-closed constraints on the set of feasible allocations. The framework relies on an interim relaxation, i.e., relaxation of the feasibility constraints, resulting in interim rules that are feasible in expectation. It also features two-level (online) contention resolution schemes (termed tOCRSs), which are variants of the classic (O)CRSs, and allow for dependencies between the activation of different elements. The framework takes as input such a BIC-IR interim allocation and a tCRS/tOCRS and outputs an approximately optimal mechanism for agents with constrained additive valuations. To find interim rules that are optimal and feasible in expectation, solving a (large) linear program is required. tCRSs and tOCRSs can be constructed in a black-box manner using existing CRSs/OCRSs for a general family of constraints (termed vertical-horizontal constraints). The authors also independently give a tOCRS for Knapsack constraints and a tOCRS for Multi-Choice Knapsack constraints. \n\nThey demonstrate how their framework can construct improved or new (sequential) mechanisms in three applications, which are natural and general enough, with one of them even allowing for arbitrary valuation functions for the agents (albeit computationally inefficient). For two of the applications, they use the novel tOCRS they construct for knapsack constraints. In the appendix, they also give an extension to procurement auctions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper makes for a nice read; the contributions (and limitations) are clearly stated and placed in the current OCRS and mechanism design literature, and the proofs are well-written.\n\nThe framework contains several nice components (finding the interim rules, defining and constructing the tCRS/tOCRS, the Bernoulli factory for the probabilities); it’s non-trivial to handle each of them separately and then bring them together. Also, the new tOCRS for knapsack constraints, can also handle mechanisms with arbitrary valuation functions, as demonstrated in Application 3, so it seems to have some nice generalization properties. \n\nWeaknesses:\n\nThe authors provide different results and applications, but I feel without a single point standing out. For instance, the framework is nice and presented as the main contribution of the paper, but then applied only to one of the three applications, which is an interesting enough one to improve upon the constant factor from the paper of Cai and Zhao. However, as mentioned in the submission, this was a subcase in their work, so they designed a broad enough framework that captured also this setting as a special case, and it probably not their main goal to optimize for the constant factor. Could you expand a bit on the importance of the framework and other well-known settings that it improves or provides first approximation guarantees? Or how the techniques used can provide new insights for either the construction of OCRSs or the design of (sequential) allocation mechanisms for other settings that are difficult to tackle? The other two applications are built on the specialized tOCRSs, which are a nice contribution, but maybe mainly of interest for the community working on CRS/OCRS problems for mechanism design. For these reasons, maybe the scope of the paper is a bit limited for NeurIPS and would fit better in either an EconCS or TCS conference.\n\nComputational concerns and applicability: As the authors state in 3.1, the LP is big (already the number of constraints is polynomial in the sum of the support sizes for BIC-IR + rest of polyhedral constraints). How can you deploy such schemes in practice? When all support sizes are small and there is an efficient representation of polytopes the LP step could be fast, does that occur often in practice? Could you expand a bit on how such a framework can be deployed in practice, in which situations, and if/how it can scale on its various parameters for large instances? Also, the construction of the tOCRS depends on the existence of the relevant OCRSs, which should also be easily implementable. The coin flipping processes for the non-constructive steps in the tOCRSs for knapsack might also be computationally demanding.In general, the principle is that when you go for approximately optimal mechanisms, you trade simplicity for approximation, which am not sure is happening here. For demonstrating some of the applicability of the framework and/or the tOCRSs built from scratch, it would be interesting to have some experimental results, e.g., for application 1, for the mechanism in this paper and that of Cai and Zhao (or other heuristics), to compare approximations and computational efficiency in real-world scenarios.\n\nQUESTIONS:\nSee questions under weaknesses and some smaller ones mentioned below.\n\nAre these constant-factor approximations for applications 2 and 3 the first for such settings in the literature? For application 2, it is mentioned that it has been studied only for the welfare objective. Are there lower bounds for these two settings? \n\nThe result for the arbitrary valuation functions is positive (despite not being computationally efficient); what makes it go beyond additive valuations and, apart from the tOCRS for knapsack, could the framework be used as a base, with new techniques on top, to go beyond additive valuations? \n\nI’m trying to understand what VH constraints exactly express. Is there some intuition behind the definition or is the definition mostly tailored to the specific application that the authors associate with the framework? What else do they capture in terms of known settings with combinatorial constraints?\n\nAs mentioned before, the computational efficiency also depends on whether the polytopes have efficient representations for solving the LP efficiently. From well-established mechanisms in the literature or other practical examples, when is this the case?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the online allocation problem, where agents report valuations for multiple heterogeneous products and make payments if a product is successfully allocated to them. The objective is to design a mechanism that guarantees Bayesian Incentive Compatibility and Individual Rationality (BIC-IR), while achieving approximate optimality up to a constant ratio. This extends existing work that simplifies the problem using single-agent reduction.\n\nThe key idea is to introduce a two-level interim relaxation that incorporates additional randomness into the selection of allocation rules, leading to the concepts of tOCRS and tCRS. Treating the relaxation (tOCRS or tCRS) as an offline oracle, the algorithm proceeds by querying the oracle to ensure that the selected allocation is feasible. It then decides whether to allocate or not using a second layer of Bernoulli sampling.\n\nThe paper establishes theoretical guarantees for the proposed algorithm by proving both the approximation ratio and the validity of the BIC and BIR properties. Finally, the paper concludes by interpreting these theoretical results in the context of the three motivating applications discussed earlier, demonstrating the practical relevance of the framework.\n\nSTRENGTHS AND WEAKNESSES:\nStrength \n\n- The paper is clearly written and enjoyable to read; a quite complicated problem is very well laid out \n\n- The algorithm is elegant and easy to interpret\n\n- The paper theoretically establishes the feasibility, and the approximation ratio of the algorithm \n\n- The application in Section 5 is very clear in supporting the insights of the theoretical results \n\nWeakness \n\n- I would appreciate highlighting the novelty more in the first few pages of the paper; the paper has not elaborated on the novel sections until page 6. I understand that it is for clarity, and it might be great if these two could be balanced. For example, the framework for\ntCRSs could be kept in the main body rather than in the appendix, since this is something new. \n\n- The description about flipping a coin with probability $p^{*}_{i,j}(v_i)$ without knowledge of this probability term might need further justification. In particular, why could this way get rid of the dependency on the knowledge of this probability?\n\n- The construction procedures of tOCRs is only illustrated for VH family or more specifically, Knapsack constraints. \n\n- Proposition 1 and 2 does not tell us how to implement such tOCRs given the constraints, which is of great importance to the algorithm, since part of the key factors of Algorithm 1 is the existence of such tOCRs. \n\n- I guess the main concern Is whether one can really implement this algorithm from scratch down to earth, based on the pseudo code. It would be very helpful if there is a case study containing certain numerical results to validate the practical use case of the proposed algorithm. \n\n- It is unclear whether the approximation ratio is ratio; a comparison with the related work would be helpful. \n\n- Would it be possible to remove the assumption that the valuations of different agents are independent of one another? Given the era of digitalization, it is possible that agents can communicate or gather information about one another. \n\n- It would be helpful to include a conclusion and future work section, summarizing the paper and inspiring subsequent study along this direction.\n\nQUESTIONS:\nPlease see the above weakness section.\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}