Fair Allocation of Indivisible Goods: Improvement

2021 ◽  
Author(s):  
Mohammad Ghodsi ◽  
Mohammad Taghi Hajiaghayi ◽  
Masoud Seddighin ◽  
Saeed Seddighin ◽  
Hadi Yami
Keyword(s):  
Assignment Problem ◽  
Competitive Equilibrium ◽  
Fundamental Problem ◽  
Impossibility Result ◽  
Utility Functions ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Approximation Result ◽  
Combinatorial Assignment

We study the problem of fair allocation for indivisible goods. We use the maximin share paradigm introduced by Budish [Budish E (2011) The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. J. Political Econom. 119(6):1061–1103.] as a measure of fairness. Kurokawa et al. [Kurokawa D, Procaccia AD, Wang J (2018) Fair enough: Guaranteeing approximate maximin shares. J. ACM 65(2):8.] were the first to investigate this fundamental problem in the additive setting. They showed that in delicately constructed examples, not everyone can obtain a utility of at least her maximin value. They mitigated this impossibility result with a beautiful observation: no matter how the utility functions are made, we always can allocate the items to the agents to guarantee each agent’s utility is at least 2/3 of her maximin value. They left open whether this bound can be improved. Our main contribution answers this question in the affirmative. We improve their approximation result to a 3/4 factor guarantee.

Fair Allocation of Indivisible Goods: Beyond Additive Valuations

2021 ◽  
pp. 103633
Author(s):  
Mohammad Ghodsi ◽  
MohammadTaghi HajiAghayi ◽  
Masoud Seddighin ◽  
Saeed Seddighin ◽  
Hadi Yami
Keyword(s):  
Fair Allocation ◽  
Indivisible Goods

scholarly journals The Complexity of Computing Maximin Share Allocations on Graphs

2020 ◽  
Vol 34 (02) ◽  
pp. 2006-2013
Author(s):  
Gianluigi Greco ◽  
Francesco Scarcello
Keyword(s):  
A Priori ◽  
Utility Functions ◽  
Indivisible Goods ◽  
Bounded Treewidth ◽  
Underlying Graph ◽  
Low Degree

Maximin share is a compelling notion of fairness proposed by Buddish as a relaxation of more traditional concepts for fair allocations of indivisible goods. In this paper we consider this notion within a setting where bundles of goods must induce connected subsets over an underlying graph. This setting received much attention in earlier literature, and our study answers a number of questions that were left open. First, we show that computing maximin share allocations is FΔ2P-complete, even when focusing on consistent scenarios, that is, where such allocations are a-priori guaranteed to exist. Moreover, the problem remains intractable if all agents have the same type, i.e., have the same utility functions, and if either the values returned by the utility functions are polynomially bounded, or the underlying graphs have a low degree of cyclicity (more precisely, have bounded treewidth). However, if these conditions hold all together, then computing maximin share allocations (or checking that none exists) becomes tractable. The result is established via machineries based on logspace alternating machines that use partial representations of connected bundles, which are interesting in their own.

scholarly journals Distributed fair allocation of indivisible goods

2017 ◽  
Vol 242 ◽  
pp. 1-22 ◽  
Author(s):  
Yann Chevaleyre ◽  
Ulle Endriss ◽  
Nicolas Maudet
Keyword(s):  
Fair Allocation ◽  
Indivisible Goods

scholarly journals A Combinatorial Assignment Problem

Nature ◽  
1961 ◽  
Vol 191 (4783) ◽  
pp. 100-100 ◽  
Author(s):  
C. RADHAKRISHNA RAO
Keyword(s):  
Assignment Problem ◽  
Combinatorial Assignment

scholarly journals Existence of General Competitive Equilibria: A Variational Approach

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
G. Anello ◽  
F. Rania
Keyword(s):  
Variational Inequalities ◽  
Finite Number ◽  
Variational Methods ◽  
Variational Approach ◽  
Existence Result ◽  
Competitive Equilibrium ◽  
Utility Functions ◽  
Locally Lipschitz ◽  
Competitive Equilibria

We study the existence of general competitive equilibria in economies with agents and goods in a finite number. We show that there exists a Walras competitive equilibrium in all ownership private economies such that, for all consumers, initial endowments do not contain free goods and utility functions are locally Lipschitz quasiconcave. The proof of the existence of competitive equilibria is based on variational methods by applying a theoretical existence result for Generalized Quasi Variational Inequalities.

Competitive equilibrium in two sided matching markets with general utility functions

2011 ◽  
Vol 10 (2) ◽  
pp. 34-36 ◽  
Author(s):  
Saeed Alaei ◽  
Kamal Jain ◽  
Azarakhsh Malekian
Keyword(s):  
Competitive Equilibrium ◽  
Utility Functions ◽  
Matching Markets ◽  
General Utility

scholarly journals Fair Division Under Cardinality Constraints

2018 ◽  
Author(s):  
Arpita Biswas ◽  
Siddharth Barman
Keyword(s):  
Resource Allocation ◽  
Fair Division ◽  
Division Problem ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Solution Concepts ◽  
Cardinality Constraints ◽  
The Given

We consider the problem of fairly allocating indivisible goods, among agents, under cardinality constraints and additive valuations. In this setting, we are given a partition of the entire set of goods---i.e., the goods are categorized---and a limit is specified on the number of goods that can be allocated from each category to any agent. The objective here is to find a fair allocation in which the subset of goods assigned to any agent satisfies the given cardinality constraints. This problem naturally captures a number of resource-allocation applications, and is a generalization of the well-studied unconstrained fair division problem.  The two central notions of fairness, in the context of fair division of indivisible goods, are envy freeness up to one good (EF1) and the (approximate) maximin share guarantee (MMS). We show that the existence and algorithmic guarantees established for these solution concepts in the unconstrained setting can essentially be achieved under cardinality constraints. Furthermore, focusing on the case wherein all the agents have the same additive valuation, we establish that EF1 allocations exist even under matroid constraints.

scholarly journals Maximin Share Allocations on Cycles

2020 ◽  
Vol 69 ◽  
pp. 613-655
Author(s):  
Miroslaw Truszczynski ◽  
Zbigniew Lonc
Keyword(s):  
Resource Allocation ◽  
Social Choice ◽  
Fundamental Problem ◽  
Fair Division ◽  
Connected Subgraph ◽  
Indivisible Goods ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Multi Agent ◽  
Share Value

The problem of fair division of indivisible goods is a fundamental problem of resource allocation in multi-agent systems, also studied extensively in social choice. Recently, the problem was generalized to the case when goods form a graph and the goal is to allocate goods to agents so that each agent’s bundle forms a connected subgraph. For the maximin share fairness criterion, researchers proved that if goods form a tree, an allocation offering each agent a bundle of at least her maximin share value always exists. Moreover, it can be found in polynomial time. In this paper we consider the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out to be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide in this case results on allocations guaranteeing each agent a certain fraction of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

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