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. 

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.

scholarly journals Fair Division Through Information Withholding

2020 ◽  
Vol 34 (02) ◽  
pp. 2014-2021
Author(s):  
Hadi Hosseini ◽  
Sujoy Sikdar ◽  
Rohit Vaish ◽  
Hejun Wang ◽  
Lirong Xia
Keyword(s):  
Real World ◽  
Fair Division ◽  
Optimal Number ◽  
The Other ◽  
Indivisible Goods ◽  
Worst Case ◽  
Optimal Amount ◽  
Preference Data ◽  
Amount Of Information ◽  
Information Withholding

Envy-freeness up to one good (EF1) is a well-studied fairness notion for indivisible goods that addresses pairwise envy by the removal of at most one good. In the worst case, each pair of agents might require the (hypothetical) removal of a different good, resulting in a weak aggregate guarantee. We study allocations that are nearly envy-free in aggregate, and define a novel fairness notion based on information withholding. Under this notion, an agent can withhold (or hide) some of the goods in its bundle and reveal the remaining goods to the other agents. We observe that in practice, envy-freeness can be achieved by withholding only a small number of goods overall. We show that finding allocations that withhold an optimal number of goods is computationally hard even for highly restricted classes of valuations. In contrast to the worst-case results, our experiments on synthetic and real-world preference data show that existing algorithms for finding EF1 allocations withhold a close-to-optimal amount of information.

scholarly journals Picking Sequences and Monotonicity in Weighted Fair Division

2021 ◽  
Author(s):  
Mithun Chakraborty ◽  
Ulrike Schmidt-Kraepelin ◽  
Warut Suksompong
Keyword(s):  
Political Parties ◽  
Fair Division ◽  
Complete Picture ◽  
Coalition Government ◽  
Population Monotonicity ◽  
Resource Population ◽  
Monotonicity Properties ◽  
Divisor Methods ◽  
Division Problems ◽  
Apportionment Methods

We study the problem of fairly allocating indivisible items to agents with different entitlements, which captures, for example, the distribution of ministries among political parties in a coalition government. Our focus is on picking sequences derived from common apportionment methods, including five traditional divisor methods and the quota method. We paint a complete picture of these methods in relation to known envy-freeness and proportionality relaxations for indivisible items as well as monotonicity properties with respect to the resource, population, and weights. In addition, we provide characterizations of picking sequences satisfying each of the fairness notions, and show that the well-studied maximum Nash welfare solution fails resource- and population-monotonicity even in the unweighted setting. Our results serve as an argument in favor of using picking sequences in weighted fair division problems.

scholarly journals A Novel Dissipativity-Based Control for Inexact Nonlinearity Cancellation Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Giacomo Innocenti ◽  
Paolo Paoletti
Keyword(s):  
Control Strategy ◽  
Feedback Loop ◽  
Controller Design ◽  
Natural Control ◽  
Standard Methods ◽  
Approximation Quality ◽  
Linear Controller ◽  
Benchmark System ◽  
Global Linearization

When dealing with linear systems feedback interconnected with memoryless nonlinearities, a natural control strategy is making the overall dynamics linear at first and then designing a linear controller for the remaining linear dynamics. By canceling the original nonlinearity via a first feedback loop, global linearization can be achieved. However, when the controller is not capable of exactly canceling the nonlinearity, such control strategy may provide unsatisfactory performance or even induce instability. Here, the interplay between accuracy of nonlinearity approximation, quality of state estimation, and robustness of linear controller is investigated and explicit conditions for stability are derived. An alternative controller design based on such conditions is proposed and its effectiveness is compared with standard methods on a benchmark system.

Grid Quality Measures for PEBI Grids

2021 ◽  
Author(s):  
Ilya Mishev ◽  
Ruslan Rin
Keyword(s):  
Flow Behavior ◽  
Quality Measures ◽  
Approximation Properties ◽  
Approximation Quality ◽  
Flux Approximation ◽  
The Matrix ◽  
Eventual Loss ◽  
Dynamic Measures ◽  
The Impact

Abstract Combining the Perpendicular Bisector (PEBI) grids with the Two Point Flux Approximation (TPFA) scheme demonstrates a potential to accurately model on unstructured grids, conforming to the geological and engineering features of real grids. However, with the increased complexity and resolution of the grids, the PEBI conditions will inevitably be violated in some cells and the approximation properties will be compromised. The objective is to develop accurate and practical grid quality measures that quantify such errors. We critically evaluated the existing grid quality measures and found them lacking predictive power in several areas. The available k-orthogonality measures predict error for flow along the strata, although TPFA provides an accurate approximation. The false-positive results are not only misleading but can overwhelm further analysis. We developed the so-called "truncation error" grid measure which is probably the most accurate measure for flow through a plane face and accurately measures the error along the strata. We also quantified the error due to the face curvature. Curved faces are bound to exist in any real grid. The impact of the quality of the 2-D Delaunay triangulation on TPFA approximation properties is usually not taken into account. We investigate the impact of the size of the smallest angles that can cause considerable increase of the condition number of the matrix and an eventual loss of accuracy, demonstrated with simple examples. Based on the analysis, we provide recommendations. We also show how the size of the largest angles impacts the approximation quality of TPFA. Furthermore, we discuss the impact of the change of the permeability on the TPFA approximation. Finally, we present simple tools that reservoir engineers can use to incorporate the above-mentioned grid quality measures into a workflow. The grid quality measures discussed up to now are static. We also sketch the further extension to dynamic measures, that is, how the static measures can be used to detect change in the flow behavior, potentially leading to increased error. We investigate a comprehensive set of methods, several of them new, to measure the static grid quality of TPFA on PEBI grids and possible extension to dynamic measures. All measures can be easily implemented in production reservoir simulators and examined using the suggested tools in a workflow.

Application of Variational Geometry to the Analysis of Mechanical Tolerances

1989 ◽  
Author(s):  
P. M. Martino ◽  
G. A. Gabriele
Keyword(s):  
Mechanical Design ◽  
Worst Case Analysis ◽  
Prototype System ◽  
Tolerance Design ◽  
Worst Case ◽  
Geometric Tolerances ◽  
Solid Models ◽  
The Cost ◽  
Selection Of

Abstract The proper selection of tolerances is an important part of mechanical design that can have a significant impact on the cost and quality of the final product. Yet, despite their importance, current techniques for tolerance design are rather primitive and often based on experience and trial and error. Better tolerance design methods have been proposed but are seldom used because of the difficulty in formulating the necessary design equations for practical problems. In this paper we propose a technique for the automatic formulation of the design equations, or design functions, which is based on the use of solid models and variational geometry. A prototype system has been developed which can model conventional and statistical tolernaces, and a limited set of geometric tolerances. The prototype system is limited to the modeling of single parts, but can perform both a worst case analysis and a statistical analysis. Results on several simple parts with known characteristics are presented which demonstrate the accuracy of the system and the types of analysis it can perform. The paper concludes with a discussion of extensions to the prototype system to a broader range of geometry and the handling of assemblies.

Effective Intelligent Data Mining Using Dempster-Shafer Theory

2008 ◽  
pp. 2943-2963
Author(s):  
Malcolm J. Beynon
Keyword(s):  
Data Mining ◽  
Missing Values ◽  
Uncertain Reasoning ◽  
Worst Case ◽  
Data Mining Techniques ◽  
Imperfect Data ◽  
Dempster Shafer Theory ◽  
External Management ◽  
Shafer Theory

The efficacy of data mining lies in its ability to identify relationships amongst data. This chapter investigates that constraining this efficacy is the quality of the data analysed, including whether the data is imprecise or in the worst case incomplete. Through the description of Dempster-Shafer theory (DST), a general methodology based on uncertain reasoning, it argues that traditional data mining techniques are not structured to handle such imperfect data, instead requiring the external management of missing values, and so forth. One DST based technique is classification and ranking belief simplex (CaRBS), which allows intelligent data mining through the acceptance of missing values in the data analysed, considering them a factor of ignorance, and not requiring their external management. Results presented here, using CaRBS and a number of simplex plots, show the effect of managing and not managing of imperfect data.

Differential privacy preserving clustering using Daubechies-2 wavelet transform

2015 ◽  
Vol 13 (04) ◽  
pp. 1550028 ◽  
Author(s):  
Mohammad Reza Ebrahimi Dishabi ◽  
Mohammad Abdollahi Azgomi
Keyword(s):  
Wavelet Transform ◽  
Clustering Analysis ◽  
Differential Privacy ◽  
Original Data ◽  
Privacy Preserving ◽  
The Other ◽  
High Dimensionality ◽  
Worst Case ◽  
Strong Notion

Most of the existing privacy preserving clustering (PPC) algorithms do not consider the worst case privacy guarantees and are based on heuristic notions. In addition, these algorithms do not run efficiently in the case of high dimensionality of data. In this paper, to alleviate these challenges, we propose a new PPC algorithm, which is based on Daubechies-2 wavelet transform (D2WT) and preserves the differential privacy notion. Differential privacy is the strong notion of privacy, which provides the worst case privacy guarantees. On the other hand, most of the existing differential-based PPC algorithms generate data with poor utility. If we apply differential privacy properties over the original raw data, the resulting data will offer lower quality of clustering (QOC) during the clustering analysis. Therefore, we use D2WT for the preprocessing of the original data before adding noise to the data. By applying D2WT to the original data, the resulting data not only contains lower dimension compared to the original data, but also can provide differential privacy guarantee with high QOC due to less noise addition. The proposed algorithm has been implemented and experimented over some well-known datasets. We also compare the proposed algorithm with some recently introduced algorithms based on utility and privacy degrees.

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