Fair Division with a Secretive Agent

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 the (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 that finds such a fair rent division under 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 epsilon-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively. This work also addresses fairness in terms of proportionality and maximin shares. Our key result here is an efficient algorithm that, even with a secretive agent, finds a 1/19-approximate maximin fair allocation (of indivisible goods) under submodular valuations of the non-secretive agents. One of the main technical contributions of this paper is the development of novel connections between different fair-division paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm.Comment: 27 page

On the complexity of Pareto-optimal and envy-free lotteries

We study the classic problem of dividing a collection of indivisible resources in a fair and efficient manner among a set of agents having varied preferences. Pareto optimality is a standard notion of economic efficiency, which states that it should be impossible to find an allocation that improves some agent's utility without reducing any other's. On the other hand, a fundamental notion of fairness in resource allocation settings is that of envy-freeness, which renders an allocation to be fair if every agent (weakly) prefers her own bundle over that of any other agent's bundle. Unfortunately, an envy-free allocation may not exist if we wish to divide a collection of indivisible items. Introducing randomness is a typical way of circumventing the non-existence of solutions, and therefore, allocation lotteries, i.e., distributions over allocations have been explored while relaxing the notion of fairness to ex-ante envy freeness. We consider a general fair division setting with $n$ agents and a family of admissible $n$-partitions of an underlying set of items. Every agent is endowed with partition-based utilities, which specify her cardinal utility for each bundle of items in every admissible partition. In such fair division instances, Cole and Tao (2021) have proved that an ex-ante envy-free and Pareto-optimal allocation lottery is always guaranteed to exist. We strengthen their result while examining the computational complexity of the above total problem and establish its membership in the complexity class PPAD. Furthermore, for instances with a constant number of agents, we develop a polynomial-time algorithm to find an ex-ante envy-free and Pareto-optimal allocation lottery. On the negative side, we prove that maximizing social welfare over ex-ante envy-free and Pareto-optimal allocation lotteries is NP-hard.Comment: 22 page

Diprosopus tetraophthalmos: a rare congenital anomaly

Diprosopus is an extremely rare form of conjoined twins which is found in newborns where there is partial or complete duplication of face. The etiology and pathophysiology remain unknown and no genetic mutations have definitively associated with the condition so far. This article described a case of an infant born at 33.4 weeks gestation with multiple congenital anomalies including diprosopus tetraophthalmos and discussed the possible hypothesis and implications of prenatal diagnosis

A Rare Case of Parotid Tuberculosis in a Pediatric Patient: A Case Report

Even in countries like India where the general dictum is to consider Tuberculosis as one of the differential diagnosis to any condition that is still under evaluation. Parotid Tuberculosis is a very rare entity. There are fewer than 200 cases reported in literature worldwide of Parotid gland Tuberculosis, a form of extrapulmonary Tuberculosis. We present an interesting case of Parotid Tuberculosis which presented as unilateral swelling in the parotid region of a four-year-old male child with an underlying abscess

Existence and Computation of Epistemic EFX Allocations

We consider the problem of allocating indivisible goods among $n$ agents in a fair manner. For this problem, one of the best notions of fairness is envy-freeness up to any good (EFX). However, it is not known if EFX allocations always exist. Hence, several relaxations of EFX allocations have been studied. We propose another relaxation of EFX, called epistemic EFX (EEFX). An allocation is EEFX iff for every agent $i$, it is possible to shuffle the goods of the other agents such that agent $i$ does not envy any other agent up to any good. We show that EEFX allocations always exist for additive valuations, and give a polynomial-time algorithm for computing them. We also show how EEFX is related to some previously-known notions of fairness.Comment: Edit: include additional authors and some minor change

A Comparative study between conventional method and the Bethesda system for reporting thyroid cytopathology (TBSRTC)

Background: FNAC is widely used for the diagnosis of thyroid lesions. TBSRTC was introduced to streamline the reporting of thyroid aspirates. Objectives was to study the spectrum of various thyroid lesions in and around Meerut and comparison of conventional method and TBSRTC.Methods: A total of 464 patients were evaluated both by conventional system and TBSRTC.Results: As per conventional system of reporting, maximum number of cases were diagnosed as benign (87%), followed by malignant (6.4%), follicular neoplasia (2.5%), equivocal (2.1%) and inadequate (1.7%). As per TBSRTC, maximum number of cases were reported as category 2, 86.2% followed by category 6, 6%, category 1, 2.5% and cat 3,2.5%, category 4, 1.5% and category 5, 1%. Among benign category, maximum number of cases were of colloid goiter/multinodular goiter (MNG) (65%) followed by Hashimoto`s thyroiditis (15%), Grave`s disease (6%). Among the malignant category, maximum number of cases were of papillary carcinoma thyroid (PCT) (5.3%) followed by medullary carcinoma thyroid (MCT) (0.43%), non-Hodgkin`s lymphoma NHL (0.21%), metastatic carcinoma (0.21%), and anaplastic carcinoma (0.21%) of each.Conclusions: In present study it was found that there is an excellent agreement between the conventional system and TBSRTC. Results of both systems of reporting are comparable. In some cases, TBSRTC is almost equivalent to the conventional system in the pattern of reporting (viz. benign and malignant cat) while in some cases, TBSRTC is better in reporting (viz. FN and equivocal cat). The sensitivity of TBSRTC (93%) is slightly more as compared to the conventional system (90%) while its specificity (86%) slightly less as compared to conventional system (88%). Thus, as compared to the conventional system, TBSRTC may be viewed as a better screening test for thyroid lesions, though at the cost of specificity

Clinico-pathological correlation of ovarian tumors and tumor like lesions with role of CA125 and HE4 as biomarkers for discrimination of benign and malignant ovarian tumors

Background: Ovarian tumors and tumor like lesions of ovary frequently form pelvic masses and are associated with hormonal manifestations. Clinically or surgically they can mimic malignancy but pathologically they could be benign tumors or tumor like lesions.Methods: The aim of present study is to do clinico-histopathological correlation of ovarian tumors and tumor like lesions of ovary and also evaluate the role of serum CA125, HE4 and calculate risk of ovarian malignancy algorithm (ROMA), for differentiation of benign and malignant ovarian tumors.233 cases of ovarian tumors and tumor like lesions were studied. Tumors were classified according to WHO classification. Clinical and histological findings were compiled on proforma and subjected to analysis.Results: In present study, out of total 233 cases, 41.2% were ovarian tumors and 58.8% tumor like lesions of ovary. Among tumor like lesions, follicular cyst was commonest lesion while among ovarian tumors, benign serous surface epithelial tumor was commonest. In patients with ovarian tumors, blood samples were collected, before and after the treatment for analysis of CA125, HE4 and ROMA.Conclusions: Serum values of CA125and HE4 as well as ROMA were highly elevated in women with malignant epithelial tumors as compared to women with benign lesions. Also, all the parameters i.e. HE4, CA125 and ROMA showed significant difference before and after surgery. Hence measuring serum HE4 and CA125 along with ROMA calculation may provide higher accuracy for detecting malignant epithelial ovarian tumor