Title: On Hierarchies of Fairness Notions in Cake Cutting: From Proportionality to Super Envy-Freeness
Venue: NeurIPS 2025
Accepted: yes
Keywords: fair division

cake cutting

fairness
Abstract: We consider the classic cake-cutting problem of producing fair allocations for $n$ agents, in the Robertson–Webb query model. In this model, it is known that: (i) proportional allocations can be computed using $O(n \log n)$ queries, and this is optimal for deterministic protocols; (ii) envy-free allocations (a subset of proportional allocations) can be computed using $O\left( n^{n^{n^{n^{n^{n}}}}} \right)$ queries, and the best known lower bound is $\Omega(n^2)$; (iii) perfect allocations (a subset of envy-free allocations) cannot be computed using a bounded (in $n$) number of queries.

In this work, we introduce two hierarchies of new fairness notions: \newnotioninverse \,(\newnotioninverseabbrev) and \newnotionlinear \,(\newnotionlinearabbrev). An allocation is \newnotioninverseabbrev-$k$ if the allocation is complete and, for any subset of agents $S$ of size at most $k$, every agent $i \in S$ believes the value of all pieces allocated to agents in $S$ to be at least $\frac{1}{n-|S|+1}$, making the union of all pieces allocated to agents not in $S$ at most $\frac{n-|S|}{n-|S|+1}$; for \newnotionlinearabbrev-$k$ allocations, these bounds become $\frac{|S|}{n}$ and $\frac{n-|S|}{n}$, respectively. Intuitively, these notions of fairness ask that, for every agent $i$, the collective value (from the perspective of agent $i$) that a group of agents receives is limited. If the group includes $i$, its value is lower-bounded, and if the group excludes $i$, it is upper-bounded, thus providing the agent some protection against the formation of coalitions.

Our hierarchies bridge the gap between proportionality, envy-freeness, and super envy-freeness.
\newnotioninverseabbrev-$k$ and \newnotionlinearabbrev-$k$ coincide with proportionality for $k=1$. For all $k \leq n$, \newnotioninverseabbrev-$k$ allocations are a superset of envy-free allocations (i.e., easier to find). On the other hand, for $k \in [2, \lceil n/2 \rceil - 1]$, \newnotionlinearabbrev-$k$ allocations are incomparable to envy-free allocations. For $k \geq \lceil n/2 \rceil$, \newnotionlinearabbrev-$k$ allocations are a subset of envy-free allocations (i.e., harder to find), while \newnotionlinearabbrev-$n$ coincides with super envy-freeness: the value of each agent for their piece is at least $1/n$, and their value for the piece allocated to any other agent is at most $1/n$.

 We prove that \newnotioninverseabbrev-$n$ allocations can be computed using $O(n^4)$ queries in the Robertson–Webb model. On the flip side, finding \newnotioninverseabbrev-$2$ (and therefore all \newnotioninverseabbrev-$k$ for $k \geq 2$) allocations requires $\Omega(n^2)$ queries, while \newnotionlinearabbrev-$2$ (and therefore all \newnotionlinearabbrev-$k$ for $k \geq 2$) allocations cannot be computed using a bounded (in $n$) number of queries. 
 Our results reveal that envy-free allocations occupy a curious middle ground, between a computationally impossible notion of fairness, \newnotionlinearabbrev-$\lceil n/2 \rceil$, and a computationally ``easy'' notion, \newnotioninverseabbrev-$n$.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the cake cutting problem. It introduces two new families of fairness notions, which both conceptually lie in between the known fairness notions of proportionality and envy-freeness. The paper gives upper and lower bounds on the query complexity of finding outcomes satisfying these fairness notions and relate them to previously defined ones. Most importantly, the paper gives an algorithm which finds a CHB-n outcome (which is one of the defined fairness notions) with O(n^4) queries.\n\nSTRENGTHS AND WEAKNESSES:\nIn general, cake cutting (or fair division in total) is a suitable topic for NeurIPS. The paper is well-written and for the most part easy to follow, the introduced fairness notions are interesting (in particular, since they manage to slot in somewhere between the almost \"untouchable\" envy-freeness and the easy-to-achieve proportionality) and I can see this paper sparking some follow-up in the fair division community. \n\nYet, there is something I am struggling with in this paper and which makes me question its suitability for NeurIPS. \nIn essence, I do not really find the introduced fairness notions to be entirely convincing and I do not think the authors really motivate them well-enough, besides the fact that they are (let's say) curious fairness notions, that lie somewhere in-between the standard ones of envy-freeness and proportionality. In particular, I am missing some motivation and discussion on why one might be interested in the utility of bundles assigned to other agents. Why do I as an agent do not just care about my own bundle and maybe the bundle of other agents in comparison to mine, but also about the total welfare (according to my own subjective opinion) of every subgroup I could possibly be a part of? Similarly, I am missing some motivation for the actual form the fairness notions take. CLB is easy to understand, but what is the motivation behind CHB (except that it works out with the algorithm and that envy-freeness implies it). I think especially for a conference like NeurIPS (which is not just purely a theory conference) a discussion on things like this is important. To add on to this, I think these prior points also lead to \"weirdly-behaving\" algorithms. Consider for instance the case of two agents, where the first agent uniformly values the interval [0,1/2] and does not value the rest, while the second agent only values [1/2, 1]. Here, Algorithm 1, (due to its requirement of having every agent value every other agents bundle positively) gives agent 1 some part of [1/2, 1] and vice versa for agent 2, now this is not forced by CHB, but still due to the insistence on giving every subset of agents something (in the eyes of every agent) it seems somewhat inevitable that something like this would happen in a simple algorithm satisfying CHB. Still, this makes the algorithm seem more like a theoretical novelty, and not like a \"rule\" or an actual algorithm which you would want to use for an actual cake cutting instance --- which in my mind would greatly improve its suitability for NeurIPS. \n\nMinor comments: \n- I am not a big fan of citing Wikipedia. For this particular use case there are better/more scientific sources on Steinhaus\n- Line 169: Missing a space after \"queries\"\n- Line 263: Might be worth making a brief comment why this is non-negative\n- Line 269: should be \"for finding *an* epsilon-perfect\" (also I prefer the $\\varepsilon$ over $\\epsilon$, but this is personal preference)\n- I am personally missing a discussion section in this paper\n- Line 824: Sometimes you write V_i(\\overline{S'}) instead of V_i(A_\\overline{S'})\n\nQUESTIONS:\nI am very much willing to increase my score, given a better intuitive motivation of (in particular) CHB and the accompanying algorithm answering (a subset of) the following questions: \n\nWhat is your motivation behind CHB? Why is the CHB family of fairness notions natural and worth achieving over just proportionality? Why does an agent (implicitly) care about what other agents receive?\n\nI might also be misunderstanding something, so I am quite open-minded regarding the rebuttal :)\n\nLIMITATIONS:\n/", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the cake-cutting problem, where a cake, modelled by [0, 1], is to be allocated to $n$ agents, each of whom admits an unknown distinct preference, and considers the popular RW query model. It was previously known that a proportional (PROP) allocation can be found in $O(n \\log n)$ queries, an envy-free (EF) allocation can be found using $O(n^{n^{n^{n^{n^n}}}})$ queries, and a perfect allocation cannot be computed within a bounded number of queries.\n\nThis paper proposes new fairness notions that interpolate between PROP and EF/perfection, which are respectively named Complement Harmonically Bounded (CHB) and Complement Linearly Bounded (CLB). An allocation satisfies CHB-$k$ (resp. CLB-$k$) for some $k \\in [n]$ if for all subsets $S$ with $|S| \\leq k$ and $i \\in S$, the value of $i$ for the union of pieces allocated to agents not in $S$ is at most $\\frac{n - |S|}{n - |S| + 1}$ (resp. $\\frac{n - |S|}{n}$), assuming each agent's total value for the cake is normalized to be $1$. CHB interpolates between PROP and EF with CHB-$1$ coinciding with PROP and CHB-$n$ being strictly weaker than EF. CLB interpolates between PROP and perfection with CLB-$1$ coinciding with PROP and CLB-$n$ being strictly weaker than perfection.\n\nThis paper gives an algorithm that computes a CHB-$n$ allocation in $O(n^4)$ queries and shows that no algorithm can compute a CLB-$2$ allocation with a bounded number of queries. Then, the paper considers an approximation version of CLB and provides an algorithm for finding a $\\delta$-CLB-$n$ allocation using $O(\\frac{n^6 \\ln (1 / \\delta)}{\\delta \\ln(n)})$ queries. Both algorithms exploit the observation that finding a complete, proportional, and nearly perfect allocation suffices, and invokes the algorithm for finding nearly perfect allocations given by prior work.\n\nSTRENGTHS AND WEAKNESSES:\nThe study of cake-cutting has been initiated decades ago, and it still remains an important and interesting topic in algorithmic game theory. This paper contributes to this line of literature by proposing new fairness notions that interpolate between PROP and EF/perfection and providing a variety of intersting results. Personally, I find both notions quite natural in hindsight as they come from certain conditions implied by EF and perfection, and they nicely bridge between PROP and EF/perfection. On the other hand, it is less obvious that these notions can be interpreted as fairness even though they imply PROP, one of the most well-studied fairness concepts. This drawback makes it hard to judge whether such notions alone would find applications in either theory or practice.\n\nThe following algorithmic and hardness results in the paper are particularly good to know: (1) CHB-$n$ allocations, which is intuitively quite close to EF, can be computed in polynomially many queries, and (2) CLB-$2$ allocations, a seemingly mild relaxation of PROP toward perfection, inherits the hardness of computing perfect allocations. I believe these results will be of great interest to the fair division community as they seem to serve as an important step toward bridging the dichotomy between PROP and EF/perfection and closing the gap between upper and lower bounds for the query complexity of EF.\n\nMoreover, the paper is in general well-written, and I really enjoy reading it, especially for the technical overview in the introduction. The proofs are correct to the extent that I checked. Overall, I believe that this paper exceeds the bar of NeurIPS and should be accepted.\n\nHere are some detailed comments:\n- Line 99-101: it is unclear why such a step forces us to deal with the weighted case.\n- Line 126: It is unclear what \"uniform\" means here.\n- Line 143: \"$\\delta$-CLB-$n$\" -> \"$\\delta$-CLB-$k$\".\n- Line 144: \"such that $k$\" -> \"such that if $k$\".\n- Line 161-164: The statement here seems obvious and redundant, since finding EF allocations has already been a strict subproblem of finding CHB-$n$ allocations.\n- Line 13 in Algorithm 1: $S$ is a set of pieces instead of a piece, although it only contains one element at that time.\n- In the proof of Theorem 1, it is not argued that Line 6 of Algorithm 1 can always be implemented, i.e., such a mark on $R$ is well-defined. Although this should follow easily from the same argument for the last remaining agent, it should be added for completeness. Same for the proof of Theorem 2.\n- Line 325: \"runtime\" -> \"query complexity\".\n- Line 333-334: This sentence seems to be grammatically problematic.\n- Line 344: Should $I_{2,left}$ be $I'$?\n\nThe following papers are also relevant and are recommended to be cited:\n\nA recent paper which also proposes a hierarchical type of notion for fair division, although it is for the strategic setting:\n- It's Not All Black and White: Degree of Truthfulness for Risk-Avoiding Agents. Eden Hartman, Erel Segal-Halevi, Biaoshuai Tao.\n\nPapers on computing approximate EF allocations:\n- Envy-Free Cake-Cutting for Four Agents. Alexandros Hollender, Aviad Rubinstein.\n- Hardness of Approximate Sperner and Applications to Envy-Free Cake Cutting. Ruiquan Gao, Mohammad Roghani, Aviad Rubinstein, Amin Saberi.\n- Algorithmic Solutions for Envy-Free Cake Cutting. Xiaotie Deng, Qi Qi, Amin Saberi.\n\nPapers on cake cutting with strategic agents:\n- ​The Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation. Xiaohui Bei, Biaoshuai Tao, Jiajun Wu, Mingwei Yang.\n- On Existence of Truthful Fair Cake Cutting Mechanisms. Xiaolin Bu, Jiaxin Song, Biaoshuai Tao.\n\nQUESTIONS:\n- Line 272: Should $O(n^2)$ be $O(n^4)$? This is because there are $n$ iterations, and in each iteration, each agent takes $O(n^2)$ queries to mark. Or perhaps one can use the same trick as in the proof of Theorem 2 to reduce the last $O(n^2)$ cost to $O(1)$?\n\n- In the statement of Theorem 2, how are $d$ and $\\epsilon'$ defined? Can they be computed efficiently given the instance parameters?\n\n- Line 355: Should $O(n^2)$ be $O(n^3)$? This is because there are $n$ iterations, and in each iteration, each agent needs to query $O(n)$ times.\n\n- For the CHB-$k$ and CLB-$k$, do they imply any approximate version of EF? I'd especially like to know how far (whatever it means) CHB-$n$ is from EF in order to evaluate the significance of the $O(n^4)$ query complexity for CHB-$n$.\n\n- Is there an intuitive description (ideally a description without using any math) of CHB-$k$ or CLB-$k$ for $k>1$?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces two new fairness notions, including Complement Harmonically Bounded-$k$ (CHB-$k$) and Complement Linearly Bounded-$k$ (CLB-$k$), to the cake-cutting model. There are several contributions in this paper. \n\n(1) They study the relationship between the above two notions and the existing fairness concepts, including EF and PROP. \n\n(2) For CHB-$n$, they design an algorithm to compute such an allocation in $O(n^4)$ queries with RW model and provide a lower bound of $\\Omega(n^2)$ queries for CHB-2 allocations. \n\n(3) For CLB-$k$, they construct an instance where CLB-2 allocations cannot be computed in bounded queries. Then, they focus on the approximate CLB-$n$ allocation and propose an algorithm to compute it in bounded queries.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n(1) This paper provides a new perspective of fairness besides EF and PROP, which fills the gap in the cake cutting model.\n\n(2) They smartly adapt the current technique to develop new algorithms to compute CHB-$n$ and the approximate CLB-$n$ allocations.\n\n(3) The whole paper is well written, and it is easy for us to read.\n\nWeakness:\n\n(1) In the relationship graph, we can easily find the position of CHB and CLB, which gives us a mathematical view of them. It may be a little bit hard to understand the practical motivation or meanings of these two notions, compared to the classic notions, including EF and PROP. For example, for PROP, each agent has a value threshold, and then compares the value of her bundle with it, which is easy for us to understand.\n\n(2)  For the CHB-$n$ allocations, the algorithm design is nice, but there is still a gap between the lower and upper bounds of queries. For the approximate CLB-$n$ allocations, it seems that the lower bound of queries is missing.\n\nQUESTIONS:\nQ1: Could you please give me some practical examples or non-mathematical explanation of the proposed notions?\n\n\nQ2: For the computation of approximate CLB-$n$ allocations, I am a little curious about the lower bound of queries. So, could you please provide some insights for it?\n\nLIMITATIONS:\nN/A", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of fair cake-cutting. There are some well-studied notions of fairness in this problem including proportionality and envy-freeness. This paper introduces two hierarchical fairness notions to bridge the gaps in the number of queries  (in the RW model) needed to achieve different notions. These notions are called Complement Harmonically Bounded-k (CHB-$k$) and Complement Linearly Bounded-$k$ (CLB-$k$). An allocation is CHB-$k$ if for any subset of size $s \\le k$ of agents and agent $i$ in this subset, value of $i$ for the union of the pieces received by the agents not in this set is at most $\\frac{n-s}{n-s+1}$. This bound changes to $\\frac{n-s}{n}$for CLB-$k$. Both CLB-1 and CHB-1 are equal to proportionality and CLB-$n$ coincides with super envy-freeness.\n\nThey show upper and lower bounds on the number of queries required for these notions, and also study approximate versions of them. Figure 1 show a great overview of the main results.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\nThe paper is well-written and easy to follow.\n\nThe idea of introducing such fairness notions is novel and interesting and the results advance our understanding of fair cake cutting in the Robertson-Webb model.\n\nThe algorithms are mostly intuitive and proofs are solid.\n\nThe paper provides a nice overview of the related literature.\n\n\n## Weaknesses\n\nI pretty much like the contribution of this paper. My only concern is whether this is the right venue for this work. The paper is rather technical and due to lack of space most of the interesting part had to move to the appendix. In general I would suggest WINE/EC/SODA for this paper.\n\nI think after the definition of these notions, lower-bounds are the most interesting part of this work. I was a bit disappointed that they were all in the appendix.\n\nIn the definitions of CLB and CHB, $S$ is defined as the set that includes $I$ but then n-$S$ is used in the bounds. I don't see the reason for this as you can instead define $S$ as a set that does not include $i$  and use $|S|$ instead of $n-|S|$ everywhere.\n\nQUESTIONS:\nDo you have some intuition on how relaxed versions of these notions look in the indivisible setting? Let's say we want to talk about CLB-$k$ up to one item/any item.\n\nLIMITATIONS:\n-", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-04"}