arXiv subjects

Xiaohui Bei

Publications and source records attributed to Xiaohui Bei.

At least 19 recordsLinked to original sources

arXiv2026

Non-Existence of PMMS Allocations and a $4/3$-PMMS Guarantee for Additive Chores

We study pairwise maximin share (PMMS) fairness for indivisible items with additive preferences. We give a polynomial-time reduction from chores to goods that preserves the existence of a PMMS allocation. Together with known nonexistence results for chores, this yields nonexistence for additive goods. In addition, we show that deciding if a given instance admits a PMMS allocation is NP-hard. We also give explicit instances whose PMMS factors are $226/227$ for goods and $1.102065$ for chores, certified by exact enumeration. Complementing these impossibility results, we prove that every additive-chore instance admits a $4/3$-PMMS allocation.

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arXiv2026

Individual Fairness in Budget Aggregation

We consider the problem of aggregating $n$ individual distributions over $m$ alternatives into a collective distribution, also known as budget aggregation. Existing fairness notions in this literature typically do not guarantee fairness to individual agents. To address this, we define two versions of individual fair share guarantees. We show that when agents' utilities are derived from $\ell_t$ metrics for any $t\geq 1$, both these guarantees can be satisfied along with Pareto efficiency, and the corresponding distributions can be computed in polynomial time. On the other hand, for $\ell_1$ utilities, we prove that Pareto efficiency, strategyproofness, and a very weak fairness notion called single-minded positive share are not always compatible for $n,m \ge 3$. For smaller parameters, we provide rules that satisfy these three axioms. We also establish similar impossibility results for $\ell_2$ utilities.

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arXiv2026

EconCSLib: A Lean Library for Computational Economics and AI-Assisted Research

Mathematical formalization uses interactive theorem provers to turn informal mathematical statements into machine-checkable artifacts. The success of mathlib, a large collaborative library for Lean, illustrates the potential of this approach. Recent progress in AI-assisted programming and theorem proving is also making large-scale formalization more practical. This paper presents EconCSLib, an early Lean 4 library for computational economics, as both infrastructure and a case study for AI-assisted formalization. The library aims to provide reusable definitions and theorems for game theory, mechanism design, social choice, and related areas. Beyond verified proofs of existing results, the library also aims to host machine-checked open problems and formalization of modern research papers. We discuss the design principles behind the library, the lessons learned from its development, and future directions for AI-assisted formalization in computational economics.

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arXiv2026

Fair Division by Contribution: A Shapley Value Perspective

In many resource allocation problems, agents' valuations are best interpreted not as subjective preferences, but as the value they generate from receiving resources. Such valuations capture productivity, effectiveness, or technology, and may differ significantly across agents. In these settings, classical fairness notions such as proportionality or envy-freeness fail to reflect agents' heterogeneous contributions to the collective outcome. Motivated by this perspective, we introduce \emph{Shapley Value Fairness (SVF)} for the allocation of divisible goods without monetary transfers. SVF interprets an agent's entitlement as her expected marginal contribution to optimal social welfare, and uses the Shapley value of the associated welfare maximization game as a normative fairness benchmark. We position SVF relative to existing fairness notions and show that it provides a natural bridge between fairness and efficiency in contribution-based environments. Since exact implementation of the Shapley value is generally infeasible without transfers, SVF naturally leads to the problem of finding allocations that approximate this benchmark as well as possible. We provide a systematic worst-case analysis of the achievable Shapley approximation ratio. For general concave valuations, we establish a tight $\Theta(\ln n)$ bound. For capped concave valuations with bounded demands, this bound improves to $\Theta(\ln D)$, where $D$ is the maximum aggregate demand for any item. For linear valuations, we further refine the bound to $\Theta(\min\{k, \ln \gamma, \ln n\})$ in terms of the number of agent types $k$ and the value fluctuation ratio $\gamma$, and show that all bounds are asymptotically tight. Regarding per-instance guarantees, we show that a near-optimal approximation allocation can be computed efficiently (with high probability) via sampling for general concave valuations.

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arXiv2026

Approximate Maximin Share with Subjective Divisibility: Beating the 1/2 Barrier

Maximin share (MMS) stands out as a central notion in fair resource allocation. It is known that exact MMS fairness is not always attainable, especially when agents differ along two dimensions: their valuations and their perceptions of the divisibility of resources. The former case with heterogeneous valuations has been widely studied in the literature. The latter, referred to as subjective divisibility by Bei et al., [Games Econ. Behav. 2025], remains much less explored. We study MMS approximation under subjective divisibility. First, we prove that even in the unary valuation setting, where all items have equal value, the optimal approximation ratio is 2/3. This result is somewhat surprising since in the objective setting, even when agents have heterogeneous valuations, the best possible approximation ratio is at least 7/9 [Huang and Zhou, 2025]. We then address the general case with both valuation heterogeneity and subjective divisibility. Previous work shows the existence of a 1/2-approximate MMS allocation. In this paper, we develop new algorithmic techniques that overcome the difficulties posed by subjective divisibility, and improve the approximation guarantee to 5/9. Finally, we complement this result with small-agent cases. For up to four agents, we give polynomial-time algorithms that compute 2/3-approximate MMS fair allocations. These bounds are tight. Our results deepen the understanding of MMS fairness under heterogeneous valuations and subjective divisibility, and provide a new perspective for this emerging model.

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arXiv2025

Nash Social Welfare with Submodular Valuations: Approximation Algorithms and Integrality Gaps

We study the problem of allocating items to agents with submodular valuations with the goal of maximizing the weighted Nash social welfare (NSW). The best-known results for unweighted and weighted objectives are the $(4+ε)$ approximation given by Garg, Husic, Li, Végh, and Vondrák~[STOC 2023] and the $(233+ε)$ approximation given by Feng, Hu, Li, and Zhang~[STOC 2025], respectively. In this work, we present a $(3.56+ε)$-approximation algorithm for weighted NSW maximization with submodular valuations, simultaneously improving the previous approximation ratios of both the weighted and unweighted NSW problems. Our algorithm solves the configuration LP of Feng, Hu, Li, and Zhang~[STOC 2025] via a stronger separation oracle that loses an $e/(e-1)$ factor only on small items, and then rounds the solution via a new bipartite multigraph construction. Some key technical ingredients of our analysis include a greedy proxy function, additive within each configuration, that preserves the LP value while lower-bounding the rounded solution, together with refined concentration bounds and a series of mathematical programs analyzed partly by computer assistance. On the hardness side, we prove that the configuration LP for weighted NSW with submodular valuations has an integrality gap of at least $(2^{\ln 2}-ε) \approx 1.617 - ε$, which is larger than the current best-known $e/(e-1)-ε\approx 1.582-ε$ hardness~[SODA 2020]. For additive valuations, we show an integrality gap of $(e^{1/e}-ε)$, which proves the tightness of the approximation ratio in~[ICALP 2024] for algorithms based on the configuration LP. For unweighted NSW with additive valuations, we show an integrality gap of $(2^{1/4}-ε) \approx 1.189-ε$, again larger than the current best-known $\sqrt{8/7} \approx 1.069$-hardness~[Math. Oper. Res. 2024].

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arXiv2025

The Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation

We study the problem of allocating divisible resources among $n$ agents, hopefully in a fair and efficient manner. With the presence of strategic agents, additional incentive guarantees are also necessary, and the problem of designing fair and efficient mechanisms becomes much less tractable. While there are flourishing positive results against strategic agents for homogeneous divisible items, very few of them are known to hold in cake cutting. We show that the Maximum Nash Welfare (MNW) mechanism, which provides desirable fairness and efficiency guarantees and achieves an incentive ratio of $2$ for homogeneous divisible items, also has an incentive ratio of $2$ in cake cutting. Remarkably, this result holds even without the free disposal assumption, which is hard to get rid of in the design of truthful cake cutting mechanisms. Moreover, we show that, for cake cutting, the Partial Allocation (PA) mechanism proposed by Cole et al. (EC'13), which is truthful and $1/e$-MNW for homogeneous divisible items, has an incentive ratio between $[e^{1 / e}, e]$ and when randomization is allowed, can be turned to be truthful in expectation. Given two alternatives for a trade-off between incentive ratio and Nash welfare provided by the MNW and PA mechanisms, we establish an interpolation between them for both cake cutting and homogeneous divisible items. Finally, we study the optimal incentive ratio achievable by envy-free cake cutting mechanisms. We first give an envy-free mechanism for two agents with an incentive ratio of $4 / 3$. Then, we show that any envy-free cake cutting mechanism with the connected pieces constraint has an incentive ratio of $Θ(n)$.

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arXiv2024

Fair Division with Subjective Divisibility

The classic fair division problems assume the resources to be allocated are either divisible or indivisible, or contain a mixture of both, but the agents always have a predetermined and uncontroversial agreement on the (in)divisibility of the resources. In this paper, we propose and study a new model for fair division in which agents have their own subjective divisibility over the goods to be allocated. That is, some agents may find a good to be indivisible and get utilities only if they receive the whole good, while others may consider the same good to be divisible and thus can extract utilities according to the fraction of the good they receive. We investigate fairness properties that can be achieved when agents have subjective divisibility. First, we consider the maximin share (MMS) guarantee and show that the worst-case MMS approximation guarantee is at most $2/3$ for $n \geq 2$ agents and this ratio is tight in the two- and three-agent cases. This is in contrast to the classic fair division settings involving two or three agents. We also give an algorithm that produces a $1/2$-MMS allocation for an arbitrary number of agents. Second, we study a hierarchy of envy-freeness relaxations, including EF1M, EFM and EFXM, ordered by increasing strength. While EF1M is compatible with non-wastefulness (an economic efficiency notion), this is not the case for EFM, even for two agents. Nevertheless, an EFXM and non-wasteful allocation always exists for two agents if at most one good is discarded.

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arXiv2024

Approval-Based Voting with Mixed Goods

We consider a voting scenario in which the resource to be voted upon may consist of both indivisible and divisible goods. This setting generalizes both the well-studied model of multiwinner voting and the recently introduced model of cake sharing. Under approval votes, we propose two variants of the extended justified representation (EJR) notion from multiwinner voting, a stronger one called EJR for mixed goods (EJR-M) and a weaker one called EJR up to 1 (EJR-1). We extend three multiwinner voting rules to our setting -- GreedyEJR, the method of equal shares (MES), and proportional approval voting (PAV) -- and show that while all three generalizations satisfy EJR-1, only the first one provides EJR-M. In addition, we derive tight bounds on the proportionality degree implied by EJR-M and EJR-1, and investigate the proportionality degree of our proposed rules.

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arXiv2024

Welfare Loss in Connected Resource Allocation

We study the allocation of indivisible items that form an undirected graph and investigate the worst-case welfare loss when requiring that each agent must receive a connected subgraph. Our focus is on both egalitarian and utilitarian welfare. Specifically, we introduce the concept of egalitarian (resp., utilitarian) price of connectivity, which captures the worst-case ratio between the optimal egalitarian (resp., utilitarian) welfare among all allocations and that among connected allocations. We provide tight or asymptotically tight bounds on the price of connectivity for several large classes of graphs in the case of two agents -- including graphs with vertex connectivity $1$ or $2$ and complete bipartite graphs -- as well as for paths, stars, and cycles in the general case where the number of agents can be arbitrary.

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arXiv2023

Auction Design for Bidders with Ex Post ROI Constraints

Motivated by practical constraints in online advertising, we investigate single-parameter auction design for bidders with constraints on their Return On Investment (ROI) -- a targeted minimum ratio between the obtained value and the payment. We focus on ex post ROI constraints, which require the ROI condition to be satisfied for every realized value profile. With ROI-constrained bidders, we first provide a full characterization of the allocation and payment rules of dominant-strategy incentive compatible (DSIC) auctions. In particular, we show that given any monotone allocation rule, the corresponding DSIC payment should be the Myerson payment with a rebate for each bidder to meet their ROI constraints. Furthermore, we also determine the optimal auction structure when the item is sold to a single bidder under a mild regularity condition. This structure entails a randomized allocation scheme and a first-price payment rule, which differs from the deterministic Myerson auction and previous works on ex ante ROI constraints.

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arXiv2022

Bidder Subset Selection Problem in Auction Design

Motivated by practical concerns in the online advertising industry, we study a bidder subset selection problem in single-item auctions. In this problem, a large pool of candidate bidders have independent values sampled from known prior distributions. The seller needs to pick a subset of bidders and run a given auction format on the selected subset to maximize her expected revenue. We propose two frameworks for the subset restrictions: (i) capacity constraint on the set of selected bidders; and (ii) incurred costs for the bidders invited to the auction. For the second-price auction with anonymous reserve (SPA-AR), we give constant approximation polynomial time algorithms in both frameworks (in the latter framework under mild assumptions about the market). Our results are in stark contrast to the previous work of Mehta, Nadav, Psomas, Rubinstein [NeurIPS 2020], who showed hardness of approximation for the SPA without a reserve price. We also give complimentary approximation results for other well-studied auction formats such as anonymous posted pricing and sequential posted pricing. On a technical level, we find that the revenue of SPA-AR as a set function $f(S)$ of its bidders $S$ is fractionally-subadditive but not submodular. Our bidder selection problem with invitation costs is a natural question about (approximately) answering a demand oracle for $f(\cdot)$ under a given vector of costs, a common computational assumption in the literature on combinatorial auctions.

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arXiv2022

Fair and Efficient Multi-Resource Allocation for Cloud Computing

We study the problem of allocating multiple types of resources to agents with Leontief preferences. The classic Dominant Resource Fairness (DRF) mechanism satisfies several desired fairness and incentive properties, but is known to have poor performance in terms of social welfare approximation ratio. In this work, we propose a new approximation ratio measure, called \emph{\fratio}, which is defined as the worst-case ratio between the optimal social welfare (resp. utilization) among all \emph{fair} allocations and that by the mechanism, allowing us to break the lower bound barrier under the classic approximation ratio. We then generalize DRF and present several new mechanisms with two and multiple types of resources that satisfy the same set of properties as DRF but with better social welfare and utilization guarantees under the new benchmark. We also demonstrate the effectiveness of these mechanisms through experiments on both synthetic and real-world datasets.

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arXiv2022

The Price of Connectivity in Fair Division

We study the allocation of indivisible goods that form an undirected graph and quantify the loss of fairness when we impose a constraint that each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We introduce the price of connectivity to capture the largest gap between the graph-specific and the unconstrained maximin share, and derive bounds on this quantity which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. For instance, with two agents we show that for biconnected graphs it is possible to obtain at least $3/4$ of the maximin share with connected allocations, while for the remaining graphs the guarantee is at most $1/2$. In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents, and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretic tools and concepts to fair division problems.

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arXiv2022

Truthful Cake Sharing

The classic cake cutting problem concerns the fair allocation of a heterogeneous resource among interested agents. In this paper, we study a public goods variant of the problem, where instead of competing with one another for the cake, the agents all share the same subset of the cake which must be chosen subject to a length constraint. We focus on the design of truthful and fair mechanisms in the presence of strategic agents who have piecewise uniform utilities over the cake. On the one hand, we show that the leximin solution is truthful and moreover maximizes an egalitarian welfare measure among all truthful and position oblivious mechanisms. On the other hand, we demonstrate that the maximum Nash welfare solution is truthful for two agents but not in general. Our results assume that mechanisms can block each agent from accessing parts that the agent does not claim to desire; we provide an impossibility result when blocking is not allowed.

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arXiv2021

Least Square Calibration for Peer Review

Peer review systems such as conference paper review often suffer from the issue of miscalibration. Previous works on peer review calibration usually only use the ordinal information or assume simplistic reviewer scoring functions such as linear functions. In practice, applications like academic conferences often rely on manual methods, such as open discussions, to mitigate miscalibration. It remains an important question to develop algorithms that can handle different types of miscalibrations based on available prior knowledge. In this paper, we propose a flexible framework, namely least square calibration (LSC), for selecting top candidates from peer ratings. Our framework provably performs perfect calibration from noiseless linear scoring functions under mild assumptions, yet also provides competitive calibration results when the scoring function is from broader classes beyond linear functions and with arbitrary noise. On our synthetic dataset, we empirically demonstrate that our algorithm consistently outperforms the baseline which select top papers based on the highest average ratings.

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arXiv2021

Maximin Fairness with Mixed Divisible and Indivisible Goods

We study fair resource allocation when the resources contain a mixture of divisible and indivisible goods, focusing on the well-studied fairness notion of maximin share fairness (MMS). With only indivisible goods, a full MMS allocation may not exist, but a constant multiplicative approximate allocation always does. We analyze how the MMS approximation guarantee would be affected when the resources to be allocated also contain divisible goods. In particular, we show that the worst-case MMS approximation guarantee with mixed goods is no worse than that with only indivisible goods. However, there exist problem instances to which adding some divisible resources would strictly decrease the MMS approximation ratio of the instance. On the algorithmic front, we propose a constructive algorithm that will always produce an $α$-MMS allocation for any number of agents, where $α$ takes values between $1/2$ and $1$ and is a monotone increasing function determined by how agents value the divisible goods relative to their MMS values.

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arXiv2021

The Price of Fairness for Indivisible Goods

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We also introduce the concept of strong price of fairness, which captures the efficiency loss in the worst fair allocation as opposed to that in the best fair allocation as in the price of fairness. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions, for both the price of fairness and the strong price of fairness.

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