Title: The Price of Opportunity Fairness in Matroid Allocation Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: Bipartite Matching

Opportunity Fairness

Matroids

Price of Fairness
Abstract: We consider matroid allocation problems under \textit{opportunity fairness} constraints: resources need to be allocated to a set of agents under matroid constraints (which includes classical problems such as bipartite matching). Agents are divided into $C$ groups according to a sensitive attribute, and an allocation is opportunity-fair if each group receives the same share proportional to the maximum feasible allocation it could achieve in isolation. We study the Price of Fairness (PoF), i.e., the ratio between maximum size allocations and maximum size opportunity-fair allocations. We first provide a characterization of the PoF leveraging the underlying polymatroid structure of the allocation problem. Based on this characterization, we prove bounds on the PoF in various settings from fully adversarial (worst-case) to fully random. Notably, one of our main results considers an arbitrary matroid structure with agents randomly divided into groups. In this setting, we prove a PoF bound as a function of the (relative) size of the largest group. Our result implies that, as long as there is no dominant group (i.e., the largest group is not too large), opportunity fairness constraints do not induce any loss of social welfare (defined as the allocation size). Overall, our results give insights into which aspects of the problem's structure affect the trade-off between opportunity fairness and social welfare.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper focuses on a fair allocation problem subject to matroid constraints where (1)there is a set E of agents each of which either gets allocated an item or does not get; (2) a set I of agent subsets is specified as a set of “feasible allocations” and such agent set system (E,I) forms a matroid; (3) a feasible allocation is optimal if it contains the maximum number of agents among all feasible allocations and the number of agents in an allocation is defined as social welfare; (4)the agent set are assumed to already be partitioned into several groups according to a sensitive attribute and an allocation x is considered to be “opportunity fair” (newly proposed by the paper) if the ratio: “# of agents from group c in x / maximum number of agents in a feasible allocation containing only agents from group c” are all the same for each group c. The paper studies the price of opportunity fairness defined as the ratio between the cardinality of an optimal allocation and the cardinality of a fair allocation.\nIn four contexts the paper gives bounds of PoF. First is normal worst case bound which depends on a specific “polarized” matroid structure; secondly the paper uses a parameter measuring the degree of balancedness of matroid to bound the PoF; thirdly the paper studies the semi-random scenario where each element in the matroid is assigned a color under some probability distribution; lastly the paper studies the complete random scenario in the sense that both matroid and color are constructed randomly. This is achieved based on a graph matroid of a random graph.\n\nSTRENGTHS AND WEAKNESSES:\nStrength\n- The four contexts under which PoF is measured are rich and representative, offering a comprehensive picture of its bounds. The logic flow between them is intuitive and makes good sense.\n- The proposed opportunity fairness notion generalizes and complements proportionality fairness and is suitable for matroid setting, well considering and eliminating the effect of unrelated agents.\n- As far as I know, there is little literature studying PoF for problems of this kind. This paper compensates for that gap.\n\nWeakness\n- As the paper itself points out, it uses fractional fair allocation to derive bounds on PoF, which introduces negligible inaccuracy when the smallest group is large but may not be accurate otherwise. But this weakness is acceptable and understandable to me.\n\nQUESTIONS:\nWeakness on Writing\nThe writing in both main body and the proofs is a bit too technical and could be better organized to improve the readability, making readers more efficiently get the exact settings of the problem and the logic of the proofs. The writing is a bit abstract and not easy to follow in some places with many jargons. Adding more intuitive explanations can be helpful.\n- For example, the paper names the problem they study as “matroid allocation problem” and uses that phrase in the title. This is somewhat vague and potentially misleading. In the literature such problems are usually referred to as “selection problems” rather than “allocation problems”. Usually I expect that allocation problems involve more items than agents, where each agent may receive multiple items and preferences are involved. But in this paper’s setting, each agent can only get at most one item, and the goal is to select a bunch of agents who can get something simultaneously. Additionally, the fairness notion the paper introduces is a bit vague in the introduction. These factors left me very confused about the problem after reading the introduction. No concrete example is provided to aid understanding.\n- The paper is proof-heavy, but in many instances, it jumps directly into details without offering an overview or explaining the intent, making the proofs difficult to follow.\n- The model section (Section 2) is loosely organized. For example, the not-short comparison of fairness definitions inserted into the model section is somewhat distracting, though it does offer insights. Since the introduction did not give me a clear view of the problem, I hoped the model section would, but it took many words and effort to piece together the full setting. I had to summarize four pages myself to grasp the core idea.\n\nMinor errors in spelling and format: \n(1) Between line 58 and 59, should add “of” after # on numerator.\n(2) Line 1012: “sub-addivitiy” should be “sub-additivity”.\n(3) Line 168: min_{c\\in[C]} r_c, r_c should be r(c) since you didn’t define r_c before.\n(4) Line 188: the range of submodular function should be $\\mathbb{R}_+$ instead of $\\mathbb{R}_+^C$.\n(5) Line 978: the second r([C]) should be r(c).\n(6) In appendix E.10 proof of Theorem 4.8 and Lemma E.2, the proof environments are missed. In Line 1136 “consists on” should be “consists of”.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the matroid allocation problem with the fairness constraint - opportunity fairness, and focuses on the price of fairness (PoF) to capture the loss when the opportunity fairness is added. First, they build the link between fractional allocation and integral allocation when considering the opportunity fairness in the large market, which gives us sufficient reasons to consider fractional allocation. Then, they give the characterization of the PoF for a polymatroid, which is the foundation of the following analysis. Next, they bound the PoF in adversarial, parametric, semi-random, and random graph settings and present a series of results.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\nThey address matroid allocation problems under a novel group-fairness notion—opportunity fairness—and prove tight bounds for the price of fairness. They give insights into which aspects of the problem’s structure affect the trade-off between opportunity fairness and social welfare. It is meaningful. There are many positive results in multiple settings from adversarial to fully random. \n\n\nWeakness:\n\nThe model description is not clear enough. For example, the expression for social welfare considered in the article is not indicated in the model introduction. Lack of explanation of the definition when proposing the new concept of fairness.\n\nQUESTIONS:\nQ1: In the abstract, the authors said “matroid constraints include classical problems such as bipartite matching”, but in my personal opinion, bipartite graph matching is not a matroid because it does not meet the augmentation property.\n\nQ2: This paper proposed the concept of opportunity fair. Can you give an example to illustrate this fairness? Is it related to the “($\\alpha$, $\\beta$) – balance” mentioned in reference [17]? I haven't seen any papers related to this concept in the related work.\n\nQ3: Can the “C-colored matroid” be understood as the intersection of two matroids?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies a new notion of fairness, which they term \"opportunity fairness\", for the problem of selecting a maximal independent set in a matroid.  In this setup, the elements $E$ of the universe are partitioned into $C$ groups.   Given an independent set $I$, they say that this set is \"opportunity fair\"  if for every two groups $c_1, c_2$, it is true that $|I \\cap c_1| / r(c_1) = |I \\cap c_2| / r(c_2)$.  In other words, if a group $c$ has large rank, then it should accordingly be highly represented in the chosen independent set.  On the other hand, if $c$ has low rank, then it should not appear as much in the independent set.  By assuming that every group has large rank, this paper extends this definition to the fractional setting with only negligible loss, and then mainly focuses on the fractional setting.  \n\nThe obvious comparison for this notion of fairness is to proportional fairness, which looks at the ratios $|I \\cap c_1| / |c_1|$ rather than $|I \\cap c_1 / r(c_1)|$.  As the authors point out, proportional fairness has the bad property that if we simply add lots of loops to the matroid (elements e where $\\{e\\}$ is not an independent set) in group $c$ then we can force $|c|$ to be large and hence force $|I \\cap c| / |c|$ to be small, which then would require for fairness that all other groups also get a small number of elements into $I$.  While this can of course be easily avoided by preprocessing to remove loops, this issue is more general -- it is trivial to construct matroids where a group has large size but low rank, and this would then require all other groups to have small representation in $I$ in order to achieve proportional fairness.  This new definition gets around this problem by normalizing by the rank.\n\nThis studies the \"price of fairness\" for opportunity fairness, and proves a number of results:\n- In the worst case the PoF is extremely large: $C-1$.\n- If all groups have about the same rank then the PoF is more like $C/4$\n- More refined bounds on the PoF in terms of the \"independence index\" of the matroid\n- In a semi-random setting where the adversary picks the matroid but the groups are random (there is a distribution over groups and each element draws from this distribution to determine its group), there is a very complex bound on the PoF, which (as far as I can tell) gives nontrivial bounds whenever the max probability from this distribution is reasonably small.\n- If we also make the matroid random, then for ranom graphic and transversal matroids (defined by Erdos-Renyi random graphs or bipartite versions thereof) the authors prove even stronger bounds.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- I like this notion of fairness -- it feels very natural, and well-motivated at least for some settings (e.g., the job allocation problem that they mention is their motivating example).  And once there's a nice notion of fairness, studying the PoF is a natural question.\n- The mathematical depth and rigor is above the bar.\n- The paper is quite well-written, with very nice and helpful figures.\n\nWeaknesses:\n- I don't quite see the motivation for the general version of this problem that they've stated (As opposed to for specific matching problems like job allocation).  I wrote the whole summary without using the word \"allocation\", because as far as I can tell nothing in the paper is actually about allocations.  They just assume that the feasible allocations form a matroid, and from then on we don't have to think about allocations at all.  This is a reasonable thing to do, and in similar situations has led to lots of super interesting work (e.g., on matroid secretary problems).  But in the context of fairness, I wasn't actually convinced by any of their other examples.  As far as I can tell, their examples are the following (see top of page 2):\n\t- Matching markets, which correspond basically to transversal matroids.  This motivation makes sense to me, and as the authors point out, there has been previous work on fairness in matching markets.\n\t- Communication networks between cities, where we want to ensure fairness in order to have competition between providers.  They say that this corresponds to graphic matroids for their problem, but I don't really see this.  Presumably  with this motivation we want to select not an independent set, but rather a full-rank subset (including a base) since we don't want a non-connected forest!  So the problem doesn't seem to fit.  And I'm not sure how fairness would correspond to competition -- that seems like quite a leap economically.\n\t- \"the selection of members for a constitutional commission that must satisfy parity requirements such as gender and ethnic representation\", which they say corresponds to a uniform matroid.  But in a uniform matroid the rank of each group is the minimum of the rank of the matroid and the size of the group, so under their large-market assumption the ranks would all be the rank of the matroid itself and so this just boils down to the question of selecting an equal number from each group (which is trivial).  \n- Similarly, as the authors state in their limitations section, they cannot handle the weighted scenario.  This feels to me like an extremely important limitation -- in the vast majority of economic situations that I'm aware of (where one would talk about allocations like this) there are weights involved (see, for example, the matroid secretary literature mentioned above).  \n- The semi-random model feels very strange to me.  Why would the assignment to groups be IID?  The authors don't provide any example or justifications (as far as I can tell).  \n  - The technical setting for the semi-random model is also very strange to me.  Consider equation (2), where they require that a limit exists as the size of the matroid goes to infinity.  But this means that (as they say) they have an infinite series of matroids.  Given an arbitrary infinite set of matroids, why would one expect the normalized rank to be at all stable?  Maybe the matroid of size 1000 is a low-rank uniform matroid, the matroid of size 1001 is a high-rank transversal matroid, the matroid of size 1002 is a random paving matroid, the matroid of size 1003 is a gammoid, etc.  I don't see why one would expect a limit for the ranks to exist, when the matroids are chosen adversarially!  It seems to me like the much more natural semi-random setting would be for there to be a single adversarial matroid and then a random coloring of it.  Why is there an infinite sequence of adversarially chosen matroids that has to have a limit?\n- The random graphical and transversal matroids are fun mathematical exercises, but again, I'm not sure why they're important or what the example is where one would actually expect to see such a random graph.\n\nQUESTIONS:\n- What are other examples where this setup makes sense?  Or is it really only matching markets?\n- Similarly, can you provide examples or justification for the semi-random model?\n- In the semi-random model, why do you have an infinite series of matroids?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies matroid allocation problems under opportunity fairness constraints, a novel group-fairness notion, which states that each group gets the same share proportional to the max it could achieve in isolation. The quality of the allocation under such constraints is measured via the well-established Price of Fairness (PoF), which quantifies the social welfare loss due to the fairness constraints. The authors work with fractional allocations, since they argue that we can always get a random allocation that produces an integral allocation at each realization, with very small loss in the PoF and violation of the fairness constraints under a large market assumption. \n\nThe authors prove tight PoF bounds in a variety of regimes:\ni) For adversarial matroid selection and group partition, they prove a tight bound that depends linealry on the number of groups, and is independent of the other parameters of the problem. For obtaining potentially better bounds, they work with specific parameters that relate to the shape of the underlying polymatroid and provide parametric bounds; one bound has to do with the ranks of the groups, and the other with the level of competition among groups for the same resources. \nii) Moving beyond worst-case, the authors first consider the regime in which there is an adversarial matroid choice with a random group partition of the agents. Here, the main message is that when there is no “dominant” group, there is no social welfare loss. When that’s not the case, the worst-case PoF bound is still linear in the number of groups. \niii) Finally, for random groups, and for random Erdos-Renyi graphs for graphic and traversal matroids, under some conditions on the randomness of the graph, they recover the result on PoF of 1 while being opportunity fair, without now the assumption that there cannot be any dominant group.\n\nThe results are based on the important observation that the set of feasible per-group allocations can be represented as a polymatroid, which gives a new characterization for the price of opportunity, and subsequently enables/simplifies the corresponding proofs.\n\nThey also show lower and upper bounds for $\\gamma$-opportunity fairness, so solving for perfect fairness can automatically give bounds for approximate fairness as well.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nI like the generality of the results, i.e., that for the adversarial (resp. the semi-random setting), the results apply for any type of matroid and any partition into groups (resp. distribution that randomly divides agents into groups). I also like the fact that most bounds are tight, and that the authors put the effort to construct tight examples for the bounds, as e.g., in Theorem 4.1 and Proposition 4.6.\n\nThe paper is carefully written, with clear statements, clean notation and precise proofs, some of which have technical depth. For instance, I found interesting the steps of reducing the infinite-dimensional combinatorial optimization problem to a one-dimensional optimization problem, which then leads to the convergence to optimal social welfare under certain conditions. This, and other results, certainly require a very good understanding of the underlying structure.\n\nAlthough the writing is quite dense at times, it’s easy to follow the contributions and the reasoning behind the results, how it places in the literature and the challenges/limitations, two of which are clearly stated in the conclusion.\n\nWeaknesses:\n\nWhat happens in the computational front? There are two points with potential computational hurdles. The first is in the optimization problem in corollary 3.3, where things have to be restricted to specific subclasses of matroids for things to potentially be computed efficiently. However, I think that’s fine since some of the most common matroids have poly-time separation oracles. Do you happen to know how it performs and scales in practice (there can be problems in practice with the ellipsoid method..)? The second point comes from the fact the optimization problem computes a fractional allocation, and then we still have to go back to an integer one. How do we do that? Is proposition 2.4 merely for existence, or we know that we can easily obtain these $p_i$’s for drawing the integral allocation? \nExperiments would help address some of the practical aspects and computational concerns, but I understand that the paper is already quite heavy in its theoretical contribution.\n\nI think most of the motivating problems mentioned in the intro (and other well-known ones) are integer allocation problems, and not necessarily always for large markets? I’m not sure that we will be getting always very good approximations efficiently, when calculating the fractional allocations and moving to the integer ones. Also, in general, the deviation ex-post is not great when the number of groups is large, which can be, depending on the application. Do you have any example where the ex-post deviation is (almost) matched? So that we understand that ex-post we can’t do much better than this bound. Also, is it always immediate in large markets, that each group has large enough rank, so that the min is large as the authors state? It’s also that $\\min r_c$ is also quite big compared to $C$, for the constraint violations to be small. \n\nThis is not exactly a weakness, but the justification of opportunity fairness could be justified a bit better. An important part of believing in the strengths of the paper is believing in the definition of opportunity fairness, since the results and techniques are tailored to it. I appreciate the comparison with proportional fairness and other fairness notions in the appendix, but I don’t know if I’m entirely convinced yet that the potential presence of irrelevant agents is enough justification by itself for disregarding the other fairness notions? Why not make some assumption of the sort that there are no irrelevant agents, i.e., any agent in the problem can be allocated at least some small amount of resources? To me, this (or some other assumption on irrelevant agents) doesn’t sound much different in nature to the large markets assumption, or that there is no dominant group for the result in the semi-random setting.\n\nQUESTIONS:\nSee the main questions under weaknesses, and some smaller ones for clarifications below: \n\nDo you have to assume in proposition 4.6 that all groups have equal ranks? Can the results extend? Does this imply any limitation in the result and the message the paper tries to convey with this parametric bound?\n\nCould you provide some intuition on Proposition 2.5, how we ended up with this upper bound and how the gap in the bounds changes as we vary $C$ and $\\gamma$?\n\nIn the semi-random setting without the no-dominant-group assumption, this $C-2$ bound is worse than the adversarial setting because the analysis here is non-tight, or what am I missing?\n\nA small suggestion: For subsection 4.4, either allocate a bit more space to explain the results and the assumptions, or move all technical statements to the appendix, and provide only intuition and key takeaways in the main body.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}