scholarly journals Fair Division with a Secretive Agent

2019 ◽  
Vol 33 ◽  
pp. 1732-1739
Author(s):  
Eshwar Ram Arunachaleswaran ◽  
Siddharth Barman ◽  
Nidhi Rathi
Keyword(s):  
Partial Information ◽  
Fair Division ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Indivisible Goods ◽  
Cake Cutting ◽  
Division Problems ◽  
Divisible Goods ◽  
Arbitrary Selection ◽  
Selection Of

We study classic fair-division problems in a partial information setting. This paper respectively addresses fair division of rent, cake, and indivisible goods among agents with cardinal preferences. We will show that, for all of these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n − 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.For the rent-division setting we prove that well-behaved utilities of n − 1 agents suffice to find a rent division among n rooms such that, for every possible room selection of the secretive agent, there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (fairness). We complement this existential result by developing a polynomial-time algorithm for the case of quasilinear utilities. In this partial information setting, we also develop efficient algorithms to compute allocations that are envy-free up to one good (EF1) and ε-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively.One of the main technical contributions of this paper is the development of novel connections between different fairdivision paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm.

Task-Level Learning from Demonstration and Generation of Action Examples for Hierarchical Control Structure

2014 ◽  
Vol 565 ◽  
pp. 194-197
Author(s):  
Anna Gorbenko
Keyword(s):  
Polynomial Time ◽  
Control Structure ◽  
Hierarchical Control ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Robot Learning ◽  
Experimental Results ◽  
Learning From Demonstration ◽  
Selection Of ◽  
Task Level

We consider the problem of the task-level robot learning from demonstration. In particular, we consider a model that uses the hierarchical control structure. For this model, we propose the problem of selection of action examples. We present a polynomial time algorithm for solution of this problem. Also, we consider some experimental results for task-level learning from demonstration.

scholarly journals Maximin-Aware Allocations of Indivisible Goods

2019 ◽  
Author(s):  
Hau Chan ◽  
Jing Chen ◽  
Bo Li ◽  
Xiaowei Wu
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Indivisible Goods ◽  
Know How

We study envy-free allocations of indivisible goods to agents in settings where each agent is unaware of the goods allocated to other agents. In particular, we propose the maximin aware (MMA) fairness measure, which guarantees that every agent, given the bundle allocated to her, is aware that she does not envy at least one other agent, even if she does not know how the other goods are distributed among other agents. We also introduce two of its relaxations, and discuss their egalitarian guarantee and existence. Finally, we present a polynomial-time algorithm, which computes an allocation that approximately satisfies MMA or its relaxations. Interestingly, the returned allocation is also 1/2-approximate EFX when all agents have sub- additive valuations, which improves the algorithm in [Plaut and Roughgarden, 2018].

scholarly journals Fair Division of Mixed Divisible and Indivisible Goods

2020 ◽  
Vol 34 (02) ◽  
pp. 1814-1821
Author(s):  
Xiaohui Bei ◽  
Zihao Li ◽  
Jinyan Liu ◽  
Shengxin Liu ◽  
Xinhang Lu
Keyword(s):  
Piecewise Linear ◽  
Fair Division ◽  
Efficient Algorithms ◽  
Indivisible Goods ◽  
Direct Generalization ◽  
Two Agents ◽  
Divisible Goods

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 linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for ϵ-envy-freeness for mixed goods (ϵ-EFM), and present an algorithm that finds an ϵ-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and 1/ϵ.

scholarly journals Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter
Keyword(s):  
Control Problem ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Lower And Upper Bounds ◽  
Proportional Allocation ◽  
Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

scholarly journals On the Max-Min Fair Stochastic Allocation of Indivisible Goods

2020 ◽  
Vol 34 (02) ◽  
pp. 2070-2078
Author(s):  
Yasushi Kawase ◽  
Hanna Sumita
Keyword(s):  
Approximate Solution ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Indivisible Goods ◽  
Additive Utility ◽  
Ex Ante ◽  
Stochastic Setting

We study the problem of fairly allocating a set of indivisible goods to risk-neutral agents in a stochastic setting. We propose an (approximation) algorithm to find a stochastic allocation that maximizes the minimum utility among the agents. The algorithm runs by repeatedly finding an (approximate) allocation to maximize the total virtual utility of the agents. This implies that the problem is solvable in polynomial time when the utilities are gross-substitutes (which is a subclass of submodular). When the utilities are submodular, we can find a (1 − 1/e)-approximate solution for the problem and this is best possible unless P=NP. We also extend the problem where a stochastic allocation must satisfy the (ex ante) envy-freeness. Under this condition, we demonstrate that the problem is NP-hard even when every agent has an additive utility with a matroid constraint (which is a subclass of gross-substitutes). Furthermore, we propose a polynomial-time algorithm for the setting with a restriction that the matroid constraint is common to all agents.

scholarly journals Fairness Towards Groups of Agents in the Allocation of Indivisible Items

2019 ◽  
Author(s):  
Nawal Benabbou ◽  
Mithun Chakraborty ◽  
Edith Elkind ◽  
Yair Zick
Keyword(s):  
Social Welfare ◽  
Polynomial Time ◽  
Utility Maximization ◽  
Marginal Utility ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Indivisible Goods ◽  
Allocation Of Indivisible Items

In this paper, we study the problem of matching a set of items to a set of agents partitioned into types so as to balance fairness towards the types against overall utility/efficiency. We extend multiple desirable properties of indivisible goods allocation to our model and investigate the possibility and hardness of achieving combinations of these properties, e.g. we prove that maximizing utilitarian social welfare under constraints of typewise envy-freeness up to one item (TEF1) is computationally intractable. We also define a new concept of waste for this setting, show experimentally that augmenting an existing algorithm with a marginal utility maximization heuristic can produce a TEF1 solution with reduced waste, and also provide a polynomial-time algorithm for computing a non-wasteful TEF1 allocation for binary agent-item utilities.

scholarly journals Comparing Approximate Relaxations of Envy-Freeness

2018 ◽  
Author(s):  
Georgios Amanatidis ◽  
Georgios Birmpas ◽  
Vangelis Markakis
Keyword(s):  
Fair Division ◽  
Complete Picture ◽  
Indivisible Goods ◽  
Worst Case ◽  
Approximation Quality ◽  
Division Problems

In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have also been considered. In this work, we investigate further the connections  between the four notions mentioned above and their approximate versions. We establish several tight or almost tight results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions. 

Fair Division

2009 ◽  
Author(s):  
Steven J. Brams
Keyword(s):  
Fair Division ◽  
Indivisible Goods ◽  
Review Of The Literature ◽  
Trade Offs ◽  
Divisible Goods ◽  
Heterogeneous Good

This article provides a review of the literature on fair division, which has flourished in recent years. It focuses on three different literatures in the field: the allocation of several indivisible goods, the division of a single heterogeneous good, and the division, in whole or in part, of several divisible goods. The article discusses problems that arise in allocating indivisible goods, and highlights the trade-offs that must be made when not all of the criteria of fairness can be satisfied at the same time. It also describes and provides the procedures for dividing divisible goods fairly, which is based on different criteria of fairness.

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