Leonard J. Schulman, Vijay V. Vazirani
We present a simple and natural non-pricing mechanism for allocating divisible goods among strategic agents having lexicographic preferences. Our mechanism has favorable properties of incentive compatibility (strategy-proofness), Pareto efficiency, envy-freeness, and time efficiency.
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Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods
针对二值估值下混合可分与不可分物品的公平分配问题，证明了不可分物品相同时问题为NP难，给出了可分物品相同时的多项式时间算法，为凸优化混合领域提供理论洞见。
When there is a dispute between players on how to divide multiple divisible assets, how should it be resolved? In this paper we introduce a multi-asset game model that enables cooperation between multiple agents who bargain on sharing K assets, when each player has a different value for each asset. It thus extends the sequential discrete Raiffa solution and the Talmud rule solution to multi-asset cases. keyword: resource allocation, game theory, Raiffa Bargaining Solution, Aumann Bankruptcy, non-transferable commodities
Envy-freeness up to any good (EFX) provides a strong and intuitive guarantee of fairness in the allocation of indivisible goods. But whether such allocations always exist or whether they can be efficiently computed remains an important open question. We study the existence and computation of EFX in conjunction with various other economic properties under lexicographic preferences--a well-studied preference model in artificial intelligence and economics. In sharp contrast to the known results for additive valuations, we not only prove the existence of EFX and Pareto optimal allocations, but in …
Envy-freeness and maximum Nash welfare for mixed divisible and indivisible goods
本文研究混合可分与不可分物品的公平分配，证明二值线性等估值下 MNW 分配满足更强的 EFXM，并扩展刻画、证明一般加性情形满足弱于 EFM 的公平性。
We consider the division of a finite number of homogeneous divisible items among three players. Under the assumption that each player assigns a positive value to every item, we characterize the optimal allocations and we develop two exact algorithms for its search. Both the characterization and the algorithm are based on the tight relationship two geometric objects of fair division: the Individual Pieces Set (IPS) and the Radon-Nykodim Set (RNS).
Maximin Share Allocations Beyond Weakly Lexicographic Valuations
本文研究不可分割商品公平分配中的最大最小份额（MMS）问题，证明了一类推广少值域和弱字典序估值的估值域存在精确MMS分配，并给出时间复杂度为O(nm log²(2+m))的算法。
We study fair allocation of indivisible goods and chores among agents with \emph{lexicographic} preferences -- a subclass of additive valuations. In sharp contrast to the goods-only setting, we show that an allocation satisfying \emph{envy-freeness up to any item} (EFX) could fail to exist for a mixture of \emph{objective} goods and chores. To our knowledge, this negative result provides the \emph{first} counterexample for EFX over (any subdomain of) additive valuations. To complement this non-existence result, we identify a class of instances with (possibly subjective) mixed items where an EF…
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…
We study the efficiency of the proportional allocation mechanism, that is widely used to allocate divisible resources. Each agent submits a bid for each divisible resource and receives a fraction proportional to her bids. We quantify the inefficiency of Nash equilibria by studying the Price of Anarchy (PoA) of the induced game under complete and incomplete information. When agents' valuations are concave, we show that the Bayesian Nash equilibria can be arbitrarily inefficient, in contrast to the well-known 4/3 bound for pure equilibria. Next, we upper bound the PoA over Bayesian equilibria by…
Fairly Allocating Goods and (Terrible) Chores
该研究针对词典偏好下不可分物品与杂务的混合公平分配问题，证明含极糟糕杂务实例中EFX分配存在性判定为NP完全，同时提出可保证EFX+PO分配存在且高效计算的偏好子类，并给出相关弱公平性分配的计算结论。
This paper is about allocation of an infinitely divisible good to several rational and strategic agents. The allocation is done by a social planner who has limited information because the agents' valuation functions are taken to be private information known only to the respective agents. We allow only a scalar signal, called a bid, from each agent to the social planner. Yang and Hajek [Jour. on Selected Areas in Comm., 2007] as well as Johari and Tsitsiklis [Jour. of Oper. Res., 2009] proposed a scalar strategy Vickrey-Clarke-Groves (SSVCG) mechanism with efficient Nash equilibria. We consider…
The Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation
研究可分割资源分配中MNW与PA机制的激励保证，证明MNW在蛋糕切割中激励比率为2，PA介于[e^(1/e),e]间，并给出无嫉妒机制的最优激励比率结果。
Balancing fairness and efficiency in resource allocation is a classical economic and computational problem. The price of fairness measures the worst-case loss of economic efficiency when using an inefficient but fair allocation rule; for indivisible goods in many settings, this price is unacceptably high. One such setting is kidney exchange, where needy patients swap willing but incompatible kidney donors. In this work, we close an open problem regarding the theoretical price of fairness in modern kidney exchanges. We then propose a general hybrid fairness rule that balances a strict lexicogra…
The study of matching theory has gained importance recently with applications in Kidney Exchange, House Allocation, School Choice etc. The general theme of these problems is to allocate goods in a fair manner amongst participating agents. The agents generally have a unit supply/demand of a good that they want to exchange with other agents. On the other hand, Bochet et al. study a more general version of the problem where they allow for agents to have arbitrary number of divisible goods to be rationed to other agents in the network. In this current work, our main focus is on non-bipartite netwo…
Fair Allocation of Divisible Goods under Non-Linear Valuations
研究非线性估值下可分物品的公平分配问题，针对最大最小份额保证设计算法，给出\(\frac{1}{2n - 1}\)-MMS分配，证明对MMS的近似比例几乎是紧的；还研究无嫉妒性，证明其检查存在性对\(n\)个主体和至少三种物品是NP难的，对单个物品给出多项式时间算法。
On the Fairness of Normalized p-Means for Allocating Goods and Chores
研究提出归一化p-均值目标，用于公平分配物品和杂务，实现公平和效率的分配方案。
We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for $n$ agents with piecewise …
Easier, but Not Easy: Nash Welfare under Lexicographic Valuations
本文针对字典序估值下的纳什福利最大化问题，提出了更优的近似算法与精确多项式框架，同时证明该问题仍存在显著计算难度，并非易事。
Optimal Allocations under Strongly Pigou-Dalton Criteria: Hidden Layer Structure & Efficient Combinatorial Approach
本文在强皮古-达尔文标准下研究最优社会福利分配问题，提出稳定分配算法并展示其高效性与稳定性。
In multi-type resource allocation (MTRA) problems, there are p $\ge$ 2 types of items, and n agents, who each demand one unit of items of each type, and have strict linear preferences over bundles consisting of one item of each type. For MTRAs with indivisible items, our first result is an impossibility theorem that is in direct contrast to the single type (p = 1) setting: No mechanism, the output of which is always decomposable into a probability distribution over discrete assignments (where no item is split between agents), can satisfy both sd-efficiency and sd-envy-freeness. To circumvent t…
The problem of fair division known as "cake cutting" has been the focus of multiple papers spanning several decades. The most prominent problem in this line of work has been to bound the query complexity of computing an envy-free outcome in the Robertson-Webb query model. However, the root of this problem's complexity is somewhat artificial: the agents' values are assumed to be additive across different pieces of the "cake" but infinitely complicated within each piece. This is unrealistic in most of the motivating examples, where the cake represents a finite collection of homogeneous goods. We…
A many-to-one job market: more about the core and the competitive salaries
本文研究带工资的多对一匹配市场，揭示其核与谈判集、核仁及极端配置结构的新性质，并用公理化和有向图刻画竞争性工资极值，提出可生成全部极端核配置的字典式程序。
Several resource allocation problems involve multiple types of resources, with a different agency being responsible for "locally" allocating the resources of each type, while a central planner wishes to provide a guarantee on the properties of the final allocation given agents' preferences. We study the relationship between properties of the local mechanisms, each responsible for assigning all of the resources of a designated type, and the properties of a sequential mechanism which is composed of these local mechanisms, one for each type, applied sequentially, under lexicographic preferences, …
A lone divider allocation algorithm with subjective divisibility
本文针对主观可分性分配模型，通过调整lone divider框架，证明存在3/5-MMS分配，拓展了该领域的相关研究成果。
Partial Allocations in Budget-Feasible Mechanism Design: Bridging Multiple Levels of Service and Divisible Agents
针对预算可行采购中的部分分配问题，在能完全负担每个智能体的假设下，设计了适用于多级服务和可分服务的确定性、真实且预算可行的近似机制，并将线性估值下的最佳近似比改进至2。
We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study the envy-free division of chores, and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete, even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up…
We consider upper and lower bounds for maxmin allocations of a completely divisible good in both competitive and cooperative strategic contexts. We then derive a subgradient algorithm to compute the exact value up to any fixed degree of precision.
Asymptotically Fair and Truthful Allocation of Public Goods
本文提出PPGA算法，通过引入差分隐私实现公共物品分配的渐近公平和 truthful，解决了传统方法在大规模场景下的局限性。
Almost Envy-Freeness under Weakly Lexicographic Preferences
本文研究弱字典序偏好下不可分物品公平分配问题，提出可定制算法同时保证EF1、EFX、MMS与帕累托最优，并引入偏好图和潜在嫉妒技术处理并列关系，同时分析了苦差事情形下的算法与公理差异。
In this report we construct two mechanisms that fully implement social welfare maximising allocation in Nash equilibria for the case of a single infinitely divisible good subject to multiple inequality constraints. The first mechanism achieves weak budget balance, while the second is an extension of the first, and achieves strong budget balance. One important application of this mechanism is unicast service on the Internet where a network operator wishes to allocate rates among strategic users in such a way that maximise overall user satisfaction while respecting capacity constraints on every …
Dominant resource fairness (DRF) is a popular mechanism for multi-resource allocation in cloud computing systems. In this paper, we consider a problem of multi-resource fair allocation with bounded number of tasks. Firstly, we propose the lexicographically max-min normalized share (LMMNS) fair allocation mechanism, which is a natural generalization of DRF, and design a non-trivial optimal algorithm to find a LMMNS fair allocation, whose running time is linear in the number of users. Secondly, we prove that LMMNS satisfies envy-freeness (EF) and group strategy-proofness (GSP), and analysis the …
We study fair allocation of indivisible goods among agents. Prior research focuses on additive agent preferences, which leads to an impossibility when seeking truthfulness, fairness, and efficiency. We show that when agents have binary additive preferences, a compelling rule -- maximum Nash welfare (MNW) -- provides all three guarantees. Specifically, we show that deterministic MNW with lexicographic tie-breaking is group strategyproof in addition to being envy-free up to one good and Pareto optimal. We also prove that fractional MNW -- known to be group strategyproof, envy-free, and Pareto op…
A Unified Framework for Uniform-Price Resource Allocation Mechanisms
提出一个统一框架，设计介于Kelly机制和第一价格拍卖之间的资源分配机制，在保持比例分配和统一定价简单性的同时，提高社会福利并保证收益。
We study the problem of fair and efficient allocation of a set of indivisible chores to agents with additive cost functions. We consider the popular fairness notion of envy-freeness up to one good (EF1) with the efficiency notion of Pareto-optimality (PO). While it is known that an EF1+PO allocation exists and can be computed in pseudo-polynomial time in the case of goods, the same problem is open for chores. Our first result is a strongly polynomial-time algorithm for computing an EF1+PO allocation for bivalued instances, where agents have (at most) two disutility values for the chores. To th…
The problem of fair division of indivisible goods has been receiving much attention recently. The prominent metric of envy-freeness can always be satisfied in the divisible goods setting (see for example \cite{BT95}), but often cannot be satisfied in the indivisible goods setting. This has led to many relaxations thereof being introduced. We study the existence of {\em maximin share (MMS)} allocations, which is one such relaxation. Previous work has shown that MMS allocations are guaranteed to exist for all instances with $n$ players and $m$ goods if $m \leq n+4$. We extend this guarantee to t…
Guaranteeing Envy-Freeness under Generalized Assignment Constraints
该研究针对广义分配约束下的物品公平分配问题，定义了可行无嫉妒（FEF）及任意物品级可行无嫉妒（FEFx）概念，证明了可分物品FEF分配的存在性与可计算性，以及不可分物品FEFx分配的存在性，并给出相应计算算法与近似方案。
Fair Division with Indivisible Goods, Chores, and Cake
本文研究不可分物品（含物品与杂务）和可分蛋糕的混合资源公平分配，提出EFM概念，并证明对任意数量具有可加效用的智能体EFM分配总存在。
Many network resource allocation problems can be viewed as allocating a divisible resource, where the allocations are constrained to lie in a polymatroid. We consider market-based mechanisms for such problems. Though the Vickrey-Clarke-Groves (VCG) mechanism can provide the efficient allocation with strong incentive properties (namely dominant strategy incentive compatibility), its well-known high communication requirements can prevent it from being used. There have been a number of approaches for reducing the communication costs of VCG by weakening its incentive properties. Here, instead we t…
Mixed Fair Division: A Survey
本文综述混合公平分配研究，梳理不可分割物品与苦差事、混合物品及带补贴不可分割物品三类场景的现状，并提出开放问题与未来方向。
Fair Allocation of Indivisible Goods with Variable Groups
本文研究可变群体下不可分物品的公平分配，证明任意群体规模下EF1结果总存在并给出高效算法，同时在路径约束和随机效用情形下刻画了无嫉妒结果存在与否的渐近条件。
Best-of-Both-Worlds Fair Allocation of Indivisible and Mixed Goods
本文研究不可分割及混合物品的事前与事后公平分配，给出两智能体多项式时间随机算法，并证明二值效用下多智能体可同时满足比例性、无嫉妒性、EFX/EFM 与分数 Pareto 最优。
According to the proportional allocation mechanism from the network optimization literature, users compete for a divisible resource -- such as bandwidth -- by submitting bids. The mechanism allocates to each user a fraction of the resource that is proportional to her bid and collects an amount equal to her bid as payment. Since users act as utility-maximizers, this naturally defines a proportional allocation game. Recently, Syrgkanis and Tardos (STOC 2013) quantified the inefficiency of equilibria in this game with respect to the social welfare and presented a lower bound of 26.8% on the price…
It is often beneficial for agents to pool their resources in order to better accommodate fluctuations in individual demand. Many multi-round resource allocation mechanisms operate in an online manner: in each round, the agents specify their demands for that round, and the mechanism determines a corresponding allocation. In this paper, we focus instead on the offline setting in which the agents specify their demand for each round at the outset. We formulate a specific resource allocation problem in this setting, and design and analyze an associated mechanism based on the solution concept of lex…
Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results
本综述聚焦不可分割资源的公平分配问题，梳理了相关公平准则、兼具公平与效率的算法、计算复杂度、公平代价及混合资源场景等研究进展，为该领域后续研究提供基础。
We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We s…
Envy-Free Allocation of Indivisible Goods under Leontief Preferences
本文研究 Leontief 偏好下不可分割物品的无嫉妒分配，证明至少两件物品时分配总存在，单物品时可在多项式时间判定，并分析福利最大化分配的 NP 难性与参数化复杂性。
In the allocation of resources to a set of agents, how do fairness guarantees impact the social welfare? A quantitative measure of this impact is the price of fairness, which measures the worst-case loss of social welfare due to fairness constraints. While initially studied for divisible goods, recent work on the price of fairness also studies the setting of indivisible goods. In this paper, we resolve the price of two well-studied fairness notions for the allocation of indivisible goods: envy-freeness up to one good (EF1), and approximate maximin share (MMS). For both EF1 and 1/2-MMS guarante…
We study the allocation of divisible goods to competing agents via a market mechanism, focusing on agents with Leontief utilities. The majority of the economics and mechanism design literature has focused on \emph{linear} prices, meaning that the cost of a good is proportional to the quantity purchased. Equilibria for linear prices are known to be exactly the maximum Nash welfare allocations. \emph{Price curves} allow the cost of a good to be any (increasing) function of the quantity purchased. We show that price curve equilibria are not limited to maximum Nash welfare allocations with two mai…
Best-of-Both-Worlds Guarantees with Fairer Endings
本文研究不可分物品公平分配中事前与事后保障的兼容性，提出在字典序偏好下实现事后EFX与PO及事前9/10-EF的多项式时间算法，并在单调和次可加估值下分别实现带慈善EFX与事前0.5-EF及带有界慈善EFX与事前0.5-比例性。
Optimal Subsidy Bounds for Goods and Chores: One Dollar Each Suffices
研究m个不可分割物品分给n个代理人的公平分配，在物品效用或负效用受限为1时，证明每人最多补贴一美元可保证无嫉妒分配且界限 tight，还能多项式时间计算，总补贴最多n - 1美元即可。
We study online auction settings in which agents arrive and depart dynamically in a random (secretary) order, and each agent's private type consists of the agent's arrival and departure times, value and budget. We consider multi-unit auctions with additive agents for the allocation of both divisible and indivisible items. For both settings, we devise truthful mechanisms that give a constant approximation with respect to the auctioneer's revenue, under a large market assumption. For divisible items, we devise in addition a truthful mechanism that gives a constant approximation with respect to t…
Proportional Fairness for Harmful Decisions
研究公共坏品分配问题，引入针对公共坏品的核心形式化定义，在特定条件下零尊重林达尔均衡满足定义且有公平保证、推广CEEI，无此条件时有不良行为及不可能性结果，还给出归约规则，为该领域研究奠定基础。
Existence and Computation of Fair Allocations under Constraints
本文研究了在预算约束下公平分配的问题，证明了可行公平分配的存在性，并探讨了其计算性和与帕累托最优性的兼容性。
The notion of \emph{envy-freeness} is a natural and intuitive fairness requirement in resource allocation. With indivisible goods, such fair allocations are unfortunately not guaranteed to exist. Classical works have avoided this issue by introducing an additional divisible resource, i.e., money, to subsidize envious agents. In this paper, we aim to design a truthful allocation mechanism of indivisible goods to achieve both fairness and efficiency criteria with a limited amount of subsidy. Following the work of Halpern and Shah, our central question is as follows: to what extent do we need to …
Allocating Mixed Goods with Customized Fairness and Indivisibility Ratio
研究混合可分与不可分物品的公平分配问题，提出基于不可分性比例的“至多分数”松弛及定制化非对称条件，并给出满足公平准则的紧上下界。
The Fairness of Maximum Nash Social Welfare Under Matroid Constraints and Beyond
本文研究拟阵及推广约束下物品公平分配，证明Max-NSW分配达到紧1/2-EF1并具帕累托最优，解决开放问题，并给出二值及p-可扩展情形下的近似保证。
Fair and Efficient Completion of Indivisible Goods
研究不可分割物品公平与高效完成问题的计算复杂性，发现受限偏好下Prop1和MMS有积极算法结果，但EF1和EF1+PO仍计算困难。
Discrete Budget Aggregation: Truthfulness and Proportionality
本文研究离散预算聚合中真实机制的存在性，证明分数选票下Gibbard-Satterthwaite定理成立，整数选票下通过移动幻影机制构造真实机制，并揭示强比例性与真实性不可兼得。
Symmetrically Fair Allocations of Indivisible Goods
该研究聚焦不可分割物品的对称公平分配问题，提出symEF1分配概念并证明其存在的图论充分条件，明确多类适用场景，还探究了指数级数量分配的存在条件并通过随机估值实验验证其出现概率。
Probing EFX via PMMS: (Non-)Existence Results in Discrete Fair Division
该研究针对不可分物品公平分配中的EFX与PMMS问题，构建实例证明二者的分离性，同时证明三类特殊估值下公平分配的存在性并给出多项式时间算法。
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