442 research outputs found

    The Good, the Bad and the Submodular: Fairly Allocating Mixed Manna Under Order-Neutral Submodular Preferences

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    We study the problem of fairly allocating indivisible goods (positively valued items) and chores (negatively valued items) among agents with decreasing marginal utilities over items. Our focus is on instances where all the agents have simple preferences; specifically, we assume the marginal value of an item can be either βˆ’1-1, 00 or some positive integer cc. Under this assumption, we present an efficient algorithm to compute leximin allocations for a broad class of valuation functions we call order-neutral submodular valuations. Order-neutral submodular valuations strictly contain the well-studied class of additive valuations but are a strict subset of the class of submodular valuations. We show that these leximin allocations are Lorenz dominating and approximately proportional. We also show that, under further restriction to additive valuations, these leximin allocations are approximately envy-free and guarantee each agent their maxmin share. We complement this algorithmic result with a lower bound showing that the problem of computing leximin allocations is NP-hard when cc is a rational number

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Fair Allocation of goods and chores -- Tutorial and Survey of Recent Results

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    Fair resource allocation is an important problem in many real-world scenarios, where resources such as goods and chores must be allocated among agents. In this survey, we delve into the intricacies of fair allocation, focusing specifically on the challenges associated with indivisible resources. We define fairness and efficiency within this context and thoroughly survey existential results, algorithms, and approximations that satisfy various fairness criteria, including envyfreeness, proportionality, MMS, and their relaxations. Additionally, we discuss algorithms that achieve fairness and efficiency, such as Pareto Optimality and Utilitarian Welfare. We also study the computational complexity of these algorithms, the likelihood of finding fair allocations, and the price of fairness for each fairness notion. We also cover mixed instances of indivisible and divisible items and investigate different valuation and allocation settings. By summarizing the state-of-the-art research, this survey provides valuable insights into fair resource allocation of indivisible goods and chores, highlighting computational complexities, fairness guarantees, and trade-offs between fairness and efficiency. It serves as a foundation for future advancements in this vital field

    Distorted optimal transport

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    Classic optimal transport theory is built on minimizing the expected cost between two given distributions. We propose the framework of distorted optimal transport by minimizing a distorted expected cost. This new formulation is motivated by concrete problems in decision theory, robust optimization, and risk management, and it has many distinct features compared to the classic theory. We choose simple cost functions and study different distortion functions and their implications on the optimal transport plan. We show that on the real line, the comonotonic coupling is optimal for the distorted optimal transport problem when the distortion function is convex and the cost function is submodular and monotone. Some forms of duality and uniqueness results are provided. For inverse-S-shaped distortion functions and linear cost, we obtain the unique form of optimal coupling for all marginal distributions, which turns out to have an interesting ``first comonotonic, then counter-monotonic" dependence structure; for S-shaped distortion functions a similar structure is obtained. Our results highlight several challenges and features in distorted optimal transport, offering a new mathematical bridge between the fields of probability, decision theory, and risk management

    A Tight Competitive Ratio for Online Submodular Welfare Maximization

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    In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given nn bidders each equipped with a general (not necessarily monotone) submodular utility and mm items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of \nicefrac{1}{4}. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of 3βˆ’22β‰ˆ0.1715733-2\sqrt{2}\approx 0.171573 to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of β‰ˆ0.27493\approx 0.27493, improving the previously known \nicefrac{1}{4} guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest

    Weighted Fair Division with Matroid-Rank Valuations: Monotonicity and Strategyproofness

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    We study the problem of fairly allocating indivisible goods to agents with weights corresponding to their entitlements. Previous work has shown that, when agents have binary additive valuations, the maximum weighted Nash welfare rule is resource-, population-, and weight-monotone, satisfies group-strategyproofness, and can be implemented in polynomial time. We generalize these results to the class of weighted additive welfarist rules with concave functions and agents with matroid-rank (also known as binary submodular) valuations.Comment: Appears in the 16th International Symposium on Algorithmic Game Theory (SAGT), 202

    Cut Sparsification and Succinct Representation of Submodular Hypergraphs

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    In cut sparsification, all cuts of a hypergraph H=(V,E,w)H=(V,E,w) are approximated within 1Β±Ο΅1\pm\epsilon factor by a small hypergraph Hβ€²H'. This widely applied method was generalized recently to a setting where the cost of cutting each e∈Ee\in E is provided by a splitting function, ge:2eβ†’R+g_e: 2^e\to\mathbb{R}_+. This generalization is called a submodular hypergraph when the functions {ge}e∈E\{g_e\}_{e\in E} are submodular, and it arises in machine learning, combinatorial optimization, and algorithmic game theory. Previous work focused on the setting where Hβ€²H' is a reweighted sub-hypergraph of HH, and measured size by the number of hyperedges in Hβ€²H'. We study such sparsification, and also a more general notion of representing HH succinctly, where size is measured in bits. In the sparsification setting, where size is the number of hyperedges, we present three results: (i) all submodular hypergraphs admit sparsifiers of size polynomial in n=∣V∣n=|V|; (ii) monotone-submodular hypergraphs admit sparsifiers of size O(Ο΅βˆ’2n3)O(\epsilon^{-2} n^3); and (iii) we propose a new parameter, called spread, to obtain even smaller sparsifiers in some cases. In the succinct-representation setting, we show that a natural family of splitting functions admits a succinct representation of much smaller size than via reweighted subgraphs (almost by factor nn). This large gap is surprising because for graphs, the most succinct representation is attained by reweighted subgraphs. Along the way, we introduce the notion of deformation, where geg_e is decomposed into a sum of functions of small description, and we provide upper and lower bounds for deformation of common splitting functions

    Stateful Posted Pricing with Vanishing Regret via Dynamic Deterministic Markov Decision Processes

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    Incentive Ratios for Fairly Allocating Indivisible Goods: Simple Mechanisms Prevail

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    We study the problem of fairly allocating indivisible goods among strategic agents. Amanatidis et al. show that truthfulness is incompatible with any meaningful fairness notions. Thus we adopt the notion of incentive ratio, which is defined as the ratio between the largest possible utility that an agent can gain by manipulation and his utility in honest behavior under a given mechanism. We select four of the most fundamental mechanisms in the literature on discrete fair division, which are Round-Robin, a cut-and-choose mechanism of Plaut and Roughgarden, Maximum-Nash-Welfare and Envy-Graph Procedure, and obtain extensive results regarding the incentive ratios of them and their variants. For Round-Robin, we establish the incentive ratio of 22 for additive and subadditive cancelable valuations, the unbounded incentive ratio for cancelable valuations, and the incentive ratios of nn and ⌈m/nβŒ‰\lceil m / n \rceil for submodular and XOS valuations, respectively. Moreover, the incentive ratio is unbounded for a variant that provides the 1/n1/n-approximate maximum social welfare guarantee. For the algorithm of Plaut and Roughgarden, the incentive ratio is either unbounded or 33 with lexicographic tie-breaking and is 22 with welfare maximizing tie-breaking. This separation exhibits the essential role of tie-breaking rules in the design of mechanisms with low incentive ratios. For Maximum-Nash-Welfare, the incentive ratio is unbounded. Furthermore, the unboundedness can be bypassed by restricting agents to have a strictly positive value for each good. For Envy-Graph Procedure, both of the two possible ways of implementation lead to an unbounded incentive ratio. Finally, we complement our results with a proof that the incentive ratio of every mechanism satisfying envy-freeness up to one good is at least 1.0741.074, and thus is larger than 11 by a constant

    Nonbossy Mechanisms: Mechanism Design Robust to Secondary Goals

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    We study mechanism design when agents may have hidden secondary goals which will manifest as non-trivial preferences among outcomes for which their primary utility is the same. We show that in such cases, a mechanism is robust against strategic manipulation if and only if it is not only incentive-compatible, but also nonbossy -- a well-studied property in the context of matching and allocation mechanisms. We give complete characterizations of incentive-compatible and nonbossy mechanisms in various settings, including auctions with single-parameter agents and public decision settings where all agents share a common outcome. In particular, we show that in the single-item setting, a mechanism is incentive-compatible, individually rational, and nonbossy if and only if it is a sequential posted-price mechanism. In contrast, we show that in more general single-parameter environments, there exist mechanisms satisfying our characterization that significantly outperform sequential posted-price mechanisms in terms of revenue or efficiency (sometimes by an exponential factor)
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