Title: Multidimensional Bayesian Utility Maximization: Tight Approximations to Welfare
Venue: NeurIPS 2025
Accepted: yes
Keywords: ordeals

money burning

utility maximization

approximation
Abstract: We initiate the study of multidimensional Bayesian utility maximization, focusing on the unit-demand setting where values are i.i.d. across both items and buyers. The seminal result of Hartline and Roughgarden '08 studies simple, information-robust mechanisms that maximize utility for $n$ i.i.d. agents and $m$ identical items via an approximation to social welfare as an upper bound, and they prove this gap between optimal utility and social welfare is $\Theta(1+\log{n/m})$ in this setting. We extend these results to the multidimensional setting.  To do so, we develop simple, prior-independent, approximately-optimal mechanisms, targeting the simplest benchmark of optimal welfare.  We give a $(1-1/e)$-approximation when there are more items than buyers, and a $\Theta(\log{n/m})$-approximation when there are more buyers than items, and we prove that this bound is tight in both $n$ and $m$ by reducing the i.i.d. unit-demand setting to the identical items setting.  Finally, we include an extensive discussion section on why Bayesian utility maximization is a promising research direction. In particular, we characterize complexities in this setting that defy our intuition from the welfare and revenue literature, and motivate why coming up with a better benchmark than welfare is a hard problem itself.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies utility maximization mechanism design in the multidimensional setting. Specifically, they consider the specific sub-case of unit-demand buyers, whose value towards each item is drawn i.i.d. from the same distribution $\\mathcal{D}$. Their benchmark is against the maximum welfare of the instance. It is worth noting that this is different (albeit similar) to a previous model by [Hartline, Roughgarden '2008], in which each buyer has a single value for all the items.\n\nThey developed a simple and prior-independent class of mechanism, called \"Favorites\" mechanism, that first bucket the buyers by their favorite items, then run a prior-free mechanism $\\mathcal{M}$ per each item. They showed that:\n- For the case where there are more items than buyers, the mechanism $\\mathcal{M}$ that randomly allocates the item to a buyer that has it as their favorite guarantees a $1 - 1/e$ approximation.\n- For the case where there are more buyers than items, applying the mechanism introduced in [Hartline, Roughgarden '2008] as $\\mathcal{M}$ guarantees a $O(\\log n/m)$ approximation.\n\nFurthermore, they demonstrate that these approximations are tight, due to a lower bound of $\\Omega(1 + \\log n/m)$ derived similarly to that in [Hartline, Roughgarden '2008].\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n- The paper is well-written, and the relationship of the set of results of the paper against other related results are properly discussed (especially to those of [Hartline, Roughgarden '2008] and [Ezra et. al '2025]).\n- The mechanisms introduced are very simple and easy to implement yet optimal, and I believe that would be incredibly practical if being used in applications. The proofs of the technical results are also well-written and clear.\n\n**Weakness**:\n- This set of results is superseded by that of [Ezra et. al '2025], who considered the case of unit-demand non-identical buyers, and the values of the items for each buyer can be correlated. The other paper also recovers the results of this paper (and beyond), albeit with a more complicated mechanism. In that sense, I believe the main contribution of this paper is proving the existence of **simple** mechanisms that gets the job done for the i.i.d. case.\n- This paper also relies heavily on the previous result of [Hartline, Roughgarden '2008] (especially tightness result and the $n > m$ case), to the extent that I believe the i.i.d. model of this paper can simply be reduced to the identical-item model of [Hartline, Roughgarden '2008].\n\nQUESTIONS:\n**Questions**\n\n1. (Related to weakness 2 above) I believe there is a direct reduction from the i.i.d. model in this paper to the identical-item model in the [Hartline, Roughgarden '2008] paper as follows: for each buyer, replace all item values with the highest item value for this buyer. In fact, this is exactly what Definition 3 is doing for the tightness result, and (I believe) also exactly what the Favorites mechanism class is doing: it is ignoring all non-favorite items for all buyers. Can you provide evidence that this is (or is not) the case?\n2. What can you do against a different benchmark, for example the optimal utility achievable? I find comparing utility to welfare is a bit like comparing rotten apples to apples, but I understand that this has been the common benchmark for utility maximizing results.\n\n**Comments for authors**:\n1. It is minor, but in the proof for the $1-1/e$ mechanism, you should have stated that the probability that each agent is allocated is $1-1/e$ **independent** of how their item values are drawn (otherwise, you can imagine an argument that says that they have higher probability of being allocated if their value for their favorite item is smaller, which is not the case here).\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of Bayesian utility maximization in the setting with n unit-demand agents and m items, where the valuation  of an agents for an item is drawn independently and identically (so that an agent could have different valuations for different items). This generalize the classic results with n unit-demand agents and m identical items (where an agent has the same valuation across items). \n\nThis paper proposes two simple mechanisms with provable approximation guarantees, Random-Favorites for the case with m >= n and Prior-Free-Favorites in the n > m case. Both mechanisms ask the agents to report their favorite items first. Random-Favorites simply allocates the item to a random agent who indicates the item as favorite while Prior-Free-Favorites applies the mechanism from the setting with n unit-demand agents and m identical items.\n\nThe authors further demonstrate tightness of their results.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n1. The paper is generally well-written, clear, and easy to follow. The problem studied in this paper is theoretically interesting, and of interest in the AGT community.\n2. This paper initiates a new research direction and the collection of results it provided is technically non-trivial.\n\nWeakness:\n\n1. (This might not be a weakness) This paper is not a standard submission since there is a follow-up paper with stronger results (which cover all results in this paper as special cases) in STOC'25.\n\nQUESTIONS:\nMinor comment:\n\n1. It is unclear whether this paper is using an approximation ratio that is smaller than 1 (i.e., A smaller number indicates a worse approximation) or larger than 1 (i.e., A larger number indicates a worse approximation). In particular, (1 - 1/e) is a number smaller than 1. But log(n/m) when n > m can go from a number smaller than 1 to a number larger than 1, while when n gets reasonably large, log(n/m) is larger than 1.\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors consider the problem of Bayesian utility maximization for $n$ unit-demand buyers and $m$ heterogeneous items. The goal of the designer is to maximize the utility (which is welfare minus payments). This setting generalize the prior work of Hartline and Roughgarten(2008) who considered the case of $m$ identical items. The authors propose a simple scheme called M-favorites and characterize the bound on the ratio of utility over welfare when $m \\geq n$ (many items regime) and $n > m$ (many bidders regime).\n\nSTRENGTHS AND WEAKNESSES:\nStrengths \n\n* I think the paper is really well written. I do not work in this area of research, and still I was able to follow the construction and the results. \n\n* The M-favorites scheme seems quite elegant as it reduces the multidimensional problem to several single-item problems that are addressed via either random or the prior-free mechanism of Hartline-Roughgarden.\n\nWeaknesses \n\n* I did not observe any obvious weaknesses, other than the fact that some of the assumptions seem quite strong (i.i.d. valuations across items and bidders, unit demand etc.). It would be helpful if the authors include some discussion on possible relaxation of these assumptions in the main text.\n\nQUESTIONS:\n**Q1:** I wonder if the i.i.d. assumption across items and bidders necessary for proving Lemma 1? Can we relax it to something more general, like exchangeability across rows and identical across bidders? \n\n**Q2:** Can you discuss the computational cost of implementing the scheme in the $m \\geq n$ and $n>m$ cases? \n\n\n\nMinor Questions / Comments \n\n* Line 168: \"it is without loss to restrict attention to allocation rules that allocate at most one item to each agent, that is to have feasibility constraint $\\sum_{j} x_{ij}(v) \\leq 1$\". But isn't this feasibility constraint in expectation? Because in the previous paragraph it is mentioned that $x_{ij}(v)$ is the probability that bidder $i$ receives item $j$. \n\n* Line 179: The beginning of the sentence seems to be missing?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers multidimensional Bayesian utility maximization, with an emphasis on a unit-demand environment where values are independently and identically distributed (i.i.d.) over items and buyers alike. As such, it seeks to extend the foundational work of Hartline and Roughgarden (2008) from a single-dimensional environment to a multidimensional one. The authors propose simple, prior-independent mechanisms that yield approximations to optimal social welfare.\n\nThe paper contains two main results: For the case with more goods than agents (m≥n), their mechanism achieves a $(1−1/e)$-approximation of utility to maximum welfare.\n\nFor the cases with more agents than goods $(n>m)$, it achieves an $O(\\log n / m)$-approximation.\n\nThe authors establish these bounds are tight in both n and m, which leads to an overall $\\Theta(1+log n/m)$ gap between optimal utility and social welfare.\n\nTo achieve these guarantees, they introduce \"Favorites Mechanisms,\" where each buyer indicates their favorite good, and allocation progresses item-by-item utilizing specific single-item mechanisms for each item-buyer ratio. The paper also includes a precise description of the difficulties of Bayesian utility maximization in multiple dimensions, explaining why known techniques in revenue maximization literature are not directly applicable. One counter-intuitive finding is that an optimal utility mechanism in some cases has to leave a good intentionally unallocated in order to be optimal.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nThe authors obtain tight and near-optimal approximation guarantees for utility relative to social welfare, taking a first step to generalising a classic result in single-dimensional problems to the more complicated multidimensional case.\n\nInnovative Problem Formulation: The paper formulates a serious research on Bayesian maximization of multidimensional utility, which is a less mature field than revenue or welfare maximization. It highlights the relevance in social service allocation where non-monetary \"payments\" (ordeals) are provided.\n\nThe \"Favorites Mechanisms\" proposed are prior-independent (i.e., do not require complete knowledge of the distributions beneath), which makes them more practical.\n\nI think the paper is well written, the main contributions are presented in a clear manner. \n\nWeaknesses\n\nThe key results are heavily reliant upon the assumption that item values are i.i.d across items as well as buyers. The authors note that relaxing this assumption, e.g., to non-i.i.d. bidders, can make basic properties like incentive compatibility or the analysis invalid.\n\nDespite the general simplicity of the overall architecture, the \"Prior-Free-Favorites\" mechanism for the $n>m$ case draws on the one-dimensional prior-free mechanism of Hartline and Roughgarden (2008) as a subroutine. The implementation details in practice and computational expense of these subroutines are not fully detailed.\n\nQUESTIONS:\nThe paper highlights that effective utility mechanisms can \"throw away\" objects. Could the authors give some more discussion of the real-world implications of this finding in social service allocation problems, possibly in the form of further concrete examples beyond the thought experiment argument?\n\nAside from theoretical lower bounds, are there specific empirical conditions or problem structures under which the current mechanisms might hit performance limits, even in the i.i.d. case?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}