arXiv subjects

Nisarg Shah

Publications and source records attributed to Nisarg Shah.

At least 19 recordsLinked to original sources

arXiv2026

Constrained Fair Allocations via Partition Matroid Reductions

We study fair allocation of indivisible goods under additive valuations and matroid constraints. A challenging open question is whether a complete and feasible envy-free up to one good (EF1) allocation exists under every matroid that admits a complete and feasible allocation. The state-of-the-art result by Biswas and Barman [2018] positively resolves this question for partition matroids. Our first result positively resolves it for laminar matroids, which generalize partition matroids, when there are three agents. Our technique reduces this general existence question to finding an EF1 allocation satisfying a mild additional condition under a single finite-sized key laminar matroid, and we establish the required allocation by case analysis. We show that our technique somewhat extends to four agents, reducing the analogous problem to finding EF1 allocations under two finite-sized laminar matroids, although we are unable to establish their existence. We also use recent matroid decomposition results to establish EF1 existence under broader classes of matroids. Specifically, we show that EF1 allocations always exist under transversal matroids whenever a complete allocation is feasible, and obtain existence results for graphic matroids and gammoids under stronger assumptions.

cs.GT↗

arXiv2026

Fair Division Under Boolean Valuations: Beyond Normalization

We study fair division of indivisible items when agents have arbitrary two-level preferences: the value of each agent for any set of items is Boolean, which need not be monotone or additive. Notably, we do not impose the standard assumption of normalization, i.e., different agents may value the empty set at different Boolean levels. Since the preferences are nonmonotone, envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) each admit several variants, depending on which items are tested for removal and whether they are removed from the envious agent's bundle or the envied agent's bundle. This paper investigates the existence of these variants of EF1 and EFX, on their own and together with economic efficiency, incentive compatibility, feasibility constraints, and lottery-based randomization. Our results highlight that the existence landscape depends crucially on the number of normalized agents, who value the empty bundle at the lower Boolean level. The authors used significant assistance from GPT-5.6-Sol for deriving theoretical results, verified any AI-generated proofs for correctness, and expanded on the exposition and simplified arguments, with the aid of GPT-5.6-Sol and Claude Opus 5.

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arXiv2026

The Art of Calling the Winner by Asking Just Enough Questions: Competitive Preference Elicitation with Next-Best Queries

We study active elicitation of agent preferences for collectively choosing among $m$ alternatives using prominent voting rules. We focus on the next-best query model, in which an agent responds to a query by revealing their next favorite alternative, and measure the competitive ratio, which is the worst-case ratio between the number of queries made by the active elicitation algorithm and the minimum number of queries needed to reveal the winning alternative(s) in hindsight. We show that sublinear competitive ratios are achievable for many positional scoring rules, whereas every Condorcet-consistent rule has competitive ratio linear in $m$. For Borda count, we develop two complementary techniques: level-wise pruning, whose analysis extends to general concave scoring rules, and multi-scale score thresholding, which gives an $O(\sqrt m)$ worst-case guarantee for Borda. We also demonstrate strong empirical performance of level-wise pruning on real data.

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arXiv2026

Improving Randomized Metric Distortion to 2.1441

In metric social choice, voters rank candidates by their distances in an unknown metric space. A voting rule uses these rankings to select a candidate or a lottery over candidates, aiming to minimize the average distance to voters. Distortion measures the worst-case approximation ratio. While the best distortion of deterministic rules is $3$, prior work pins down the best distortion of randomized rules to $[2.1126,2.5]$. We improve the upper bound to $2.1441$, closing over $90\%$ of this gap. The proof introduces random-size stable lotteries, proves their existence, and derives the new bound through a potential argument. All the proofs were obtained using GPT-5.6-Sol with significant guidance from the author, who verified them and simplified the exposition.

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arXiv2026

Fair and Efficient Balanced Allocations for Additive Valuations

We study the existence of fair and efficient allocations of indivisible goods under the balancedness constraint, which requires that any two agents' bundles differ in size by at most one. Our main result establishes the existence of balanced allocations that satisfy envy-freeness up to one good (EF1) and fractional Pareto optimality (fPO) for arbitrary additive valuations. This generalizes a recent result of Kawase et al. (2026), which establishes existence only for personalized bivalued valuations or when there are at most two distinct valuation types. Our proof applies the Knaster-Kuratowski-Mazurkiewicz (KKM) lemma to a weighted-welfare duality framework and develops a novel price-interlacing lemma to overcome barriers encountered by prior work. We extend this technique to category constraints, also known as partition-matroid constraints. In this setting, we establish the existence of an fPO allocation satisfying a weaker, category-sensitive relaxation of EF1, under which envy can be eliminated by removing at most one good from each category. All proofs in this paper were obtained using GPT-5.6-Sol with guidance from the authors. The authors verified the proofs, expanded the exposition, and simplified the arguments with assistance from GPT-5.6-Sol and Claude Fable 5.

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arXiv2026

Proportional Fairness for Harmful Decisions

We study allocation of (divisible) public bads, where agents incur costs for alternatives and the goal is to pick a lottery over the alternatives. We show that the traditional definitions of the core, a central criterion of proportional representation for allocation of public goods, private goods, and private bads (chores), do not make sense for allocation of public bads. We introduce two formalizations of the core tailored to public bads. Under a structural condition which subsumes allocation of private bads, we show that zero-respecting Lindahl equilibria satisfy both formalizations, exhibit additional fairness guarantees, and strictly generalize competitive equilibria from equal incomes (CEEI) for allocation of private bads. Without this structural condition, we show that Lindahl equilibria exhibit undesirable behaviors, prove sharp impossibility results separating public bads from public goods, but show that a rule using a reduction to public goods recovers one of our formalizations of the core. Our results lay the groundwork for studying fair allocation of public bads, an overlooked yet fundamental problem, and highlight several structural and algorithmic directions that remain open.

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arXiv2026

Unifying Proportional Fairness in Centroid and Non-Centroid Clustering

Proportional fairness criteria inspired by democratic ideals of proportional representation have received growing attention in the clustering literature. Prior work has investigated them in two separate paradigms. Chen et al. [ICML 2019] study centroid clustering, in which each data point's loss is determined by its distance to a representative point (centroid) chosen in its cluster. Caragiannis et al. [NeurIPS 2024] study non-centroid clustering, in which each data point's loss is determined by its maximum distance to any other data point in its cluster. We generalize both paradigms to introduce semi-centroid clustering, in which each data point's loss is a combination of its centroid and non-centroid losses, and study two proportional fairness criteria -- the core and, its relaxation, fully justified representation (FJR). Our main result is a novel algorithm which achieves a constant approximation to the core, in polynomial time, even when the distance metrics used for centroid and non-centroid loss measurements are different. We also derive improved results for more restricted loss functions and the weaker FJR criterion, and establish lower bounds in each case.

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arXiv2025

Multi-Agent Risks from Advanced AI

The rapid development of advanced AI agents and the imminent deployment of many instances of these agents will give rise to multi-agent systems of unprecedented complexity. These systems pose novel and under-explored risks. In this report, we provide a structured taxonomy of these risks by identifying three key failure modes (miscoordination, conflict, and collusion) based on agents' incentives, as well as seven key risk factors (information asymmetries, network effects, selection pressures, destabilising dynamics, commitment problems, emergent agency, and multi-agent security) that can underpin them. We highlight several important instances of each risk, as well as promising directions to help mitigate them. By anchoring our analysis in a range of real-world examples and experimental evidence, we illustrate the distinct challenges posed by multi-agent systems and their implications for the safety, governance, and ethics of advanced AI.

cs.MA↗

arXiv2024

Constrained Fair and Efficient Allocations

Fairness and efficiency have become the pillars of modern fair division research, but prior work on achieving both simultaneously is largely limited to the unconstrained setting. We study fair and efficient allocations of indivisible goods under additive valuations and various types of allocation feasibility constraints, and demonstrate the unreasonable effectiveness of the maximum Nash welfare (MNW) solution in this previously uncharted territory. Our main result is that MNW allocations are 1/2-envy-free up to one good (EF1) and Pareto optimal under the broad family of (arbitrary) matroid constraints. We extend these guarantees to complete MNW allocations for base-orderable matroid constraints, and to a family of non-matroidal constraints (which includes balancedness) using a novel "alternate worlds" technique. We establish tightness of our results by providing counterexamples for the satisfiability of certain stronger desiderata, but show an improved result for the special case of goods with copies (Gafni et al. 2023). Finally, we also establish novel best-of-both-worlds guarantees for goods with copies and balancedness.

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arXiv2024

Fair Division with Market Values

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and with respect to the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

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arXiv2024

Proportional Fairness in Non-Centroid Clustering

We revisit the recently developed framework of proportionally fair clustering, where the goal is to provide group fairness guarantees that become stronger for groups of data points (agents) that are large and cohesive. Prior work applies this framework to centroid clustering, where the loss of an agent is its distance to the centroid assigned to its cluster. We expand the framework to non-centroid clustering, where the loss of an agent is a function of the other agents in its cluster, by adapting two proportional fairness criteria -- the core and its relaxation, fully justified representation (FJR) -- to this setting. We show that the core can be approximated only under structured loss functions, and even then, the best approximation we are able to establish, using an adaptation of the GreedyCapture algorithm developed for centroid clustering [Chen et al., 2019; Micha and Shah, 2020], is unappealing for a natural loss function. In contrast, we design a new (inefficient) algorithm, GreedyCohesiveClustering, which achieves the relaxation FJR exactly under arbitrary loss functions, and show that the efficient GreedyCapture algorithm achieves a constant approximation of FJR. We also design an efficient auditing algorithm, which estimates the FJR approximation of any given clustering solution up to a constant factor. Our experiments on real data suggest that traditional clustering algorithms are highly unfair, whereas GreedyCapture is considerably fairer and incurs only a modest loss in common clustering objectives.

cs.LG↗

arXiv2024

Temporal Fair Division

We study temporal fair division, whereby a set of agents are allocated a (possibly different) set of goods on each day for a period of days. We study this setting, as well as a number of its special cases formed by the restrictions to two agents, same goods on each day, identical preferences, or combinations thereof, and chart out the landscape of achieving two types of fairness guarantees simultaneously: fairness on each day (per day) and fairness over time (up to each day, or the weaker version, overall). In the most general setting, we prove that there always exists an allocation that is stochastically-dominant envy-free up to one good (SD-EF1) per day and proportional up to one good (PROP1) overall, and when all the agents have identical preferences, we show that SD-EF1 per day and SD-EF1 overall can be guaranteed. For the case of two agents, we prove that SD-EF1 per day and EF1 up to each day can be guaranteed using an envy balancing technique. We provide counterexamples for other combinations that establish our results as among the best guarantees possible, but also leaving open some tantalizing questions.

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arXiv2024

Group Fairness in Peer Review

Large conferences such as NeurIPS and AAAI serve as crossroads of various AI fields, since they attract submissions from a vast number of communities. However, in some cases, this has resulted in a poor reviewing experience for some communities, whose submissions get assigned to less qualified reviewers outside of their communities. An often-advocated solution is to break up any such large conference into smaller conferences, but this can lead to isolation of communities and harm interdisciplinary research. We tackle this challenge by introducing a notion of group fairness, called the core, which requires that every possible community (subset of researchers) to be treated in a way that prevents them from unilaterally benefiting by withdrawing from a large conference. We study a simple peer review model, prove that it always admits a reviewing assignment in the core, and design an efficient algorithm to find one such assignment. We use real data from CVPR and ICLR conferences to compare our algorithm to existing reviewing assignment algorithms on a number of metrics.

cs.GT↗

arXiv2024

Harm Ratio: A Novel and Versatile Fairness Criterion

Envy-freeness has become the cornerstone of fair division research. In settings where each individual is allocated a disjoint share of collective resources, it is a compelling fairness axiom which demands that no individual strictly prefer the allocation of another individual to their own. Unfortunately, in many real-life collective decision-making problems, the goal is to choose a (common) public outcome that is equally applicable to all individuals, and the notion of envy becomes vacuous. Consequently, this literature has avoided studying fairness criteria that focus on individuals feeling a sense of jealousy or resentment towards other individuals (rather than towards the system), missing out on a key aspect of fairness. In this work, we propose a novel fairness criterion, individual harm ratio, which is inspired by envy-freeness but applies to a broad range of collective decision-making settings. Theoretically, we identify minimal conditions under which this criterion and its groupwise extensions can be guaranteed, and study the computational complexity of related problems. Empirically, we conduct experiments with real data to show that our fairness criterion is powerful enough to differentiate between prominent decision-making algorithms for a range of tasks from voting and fair division to participatory budgeting and peer review.

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arXiv2024

What is Best for Students, Numerical Scores or Letter Grades?

We study letter grading schemes, which are routinely employed for evaluating student performance. Typically, a numerical score obtained via one or more evaluations is converted into a letter grade (e.g., A+, B-, etc.) by associating a disjoint interval of numerical scores to each letter grade. We propose the first model for studying the (de)motivational effects of such grading on the students and, consequently, on their performance in future evaluations. We use the model to compare uniform letter grading schemes, in which the range of scores is divided into equal-length parts that are mapped to the letter grades, to numerical scoring, in which the score is not converted to any letter grade (equivalently, every score is its own letter grade). Theoretically, we identify realistic conditions under which numerical scoring is better than any uniform letter grading scheme. Our experiments confirm that this holds under even weaker conditions, but also find cases where the converse occurs.

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arXiv2024

Optimized Distortion and Proportional Fairness in Voting

A voting rule decides on a probability distribution over a set of m alternatives, based on rankings of those alternatives provided by agents. We assume that agents have cardinal utility functions over the alternatives, but voting rules have access to only the rankings induced by these utilities. We evaluate how well voting rules do on measures of social welfare and of proportional fairness, computed based on the hidden utility functions. In particular, we study the distortion of voting rules, which is a worst-case measure. It is an approximation ratio comparing the utilitarian social welfare of the optimum outcome to the social welfare produced by the outcome selected by the voting rule, in the worst case over possible input profiles and utility functions that are consistent with the input. The previous literature has studied distortion with unit-sum utility functions (which are normalized to sum to 1), and left a small asymptotic gap in the best possible distortion. Using tools from the theory of fair multi-winner elections, we propose the first voting rule which achieves the optimal distortion $Θ(\sqrt{m})$ for unit-sum utilities. Our voting rule also achieves optimum $Θ(\sqrt{m})$ distortion for a larger class of utilities, including unit-range and approval (0/1) utilities. We then take a worst-case approach to a quantitative measure of the fairness of a voting rule, called proportional fairness. Informally, it measures whether the influence of cohesive groups of agents on the voting outcome is proportional to the group size. We show that there is a voting rule which, without knowledge of the utilities, can achieve a $Θ(\log m)$-approximation to proportional fairness, and thus also to Nash welfare and to the core, making it interesting for applications in participatory budgeting. For all three approximations, we show that $Θ(\log m)$ is the best possible.

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arXiv2023

Best of Both Distortion Worlds

We study the problem of designing voting rules that take as input the ordinal preferences of $n$ agents over a set of $m$ alternatives and output a single alternative, aiming to optimize the overall happiness of the agents. The input to the voting rule is each agent's ranking of the alternatives from most to least preferred, yet the agents have more refined (cardinal) preferences that capture the intensity with which they prefer one alternative over another. To quantify the extent to which voting rules can optimize over the cardinal preferences given access only to the ordinal ones, prior work has used the distortion measure, i.e., the worst-case approximation ratio between a voting rule's performance and the best performance achievable given the cardinal preferences. The work on the distortion of voting rules has been largely divided into two worlds: utilitarian distortion and metric distortion. In the former, the cardinal preferences of the agents correspond to general utilities and the goal is to maximize a normalized social welfare. In the latter, the agents' cardinal preferences correspond to costs given by distances in an underlying metric space and the goal is to minimize the (unnormalized) social cost. Several deterministic and randomized voting rules have been proposed and evaluated for each of these worlds separately, gradually improving the achievable distortion bounds, but none of the known voting rules perform well in both worlds simultaneously. In this work, we prove that one can achieve the best of both worlds by designing new voting rules, that simultaneously achieve near-optimal distortion guarantees in both distortion worlds. We also prove that this positive result does not generalize to the case where the voting rule is provided with the rankings of only the top-$t$ alternatives of each agent, for $t<m$.

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arXiv2022

Proportionally Fair Online Allocation of Public Goods with Predictions

We design online algorithms for the fair allocation of public goods to a set of $N$ agents over a sequence of $T$ rounds and focus on improving their performance using predictions. In the basic model, a public good arrives in each round, the algorithm learns every agent's value for the good, and must irrevocably decide the amount of investment in the good without exceeding a total budget of $B$ across all rounds. The algorithm can utilize (potentially inaccurate) predictions of each agent's total value for all the goods to arrive. We measure the performance of the algorithm using a proportional fairness objective, which informally demands that every group of agents be rewarded in proportion to its size and the cohesiveness of its preferences. In the special case of binary agent preferences and a unit budget, we show that $O(\log N)$ proportional fairness can be achieved without using any predictions, and that this is optimal even if perfectly accurate predictions were available. However, for general preferences and budget no algorithm can achieve better than $Θ(T/B)$ proportional fairness without predictions. We show that algorithms with (reasonably accurate) predictions can do much better, achieving $Θ(\log (T/B))$ proportional fairness. We also extend this result to a general model in which a batch of $L$ public goods arrive in each round and achieve $O(\log (\min(N,L) \cdot T/B))$ proportional fairness. Our exact bounds are parametrized as a function of the error in the predictions and the performance degrades gracefully with increasing errors.

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