On the Manipulability of Maximum Vertex-Weighted Bipartite bb-matching Mechanisms

In this paper, we study the Maximum Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs in a new game-theoretical environment. In contrast to other game-theoretical settings, we consider the case in which the value of the tasks is public and common to every agent so that the private information of every agent consists of edges connecting them to the set of tasks. In this framework, we study three mechanisms. Two of these mechanisms, namely \MB and \MD, are optimal but not truthful, while the third one, \MG, is truthful but sub-optimal. Albeit these mechanisms are induced by known algorithms, we show \MB and \MD are the best possible mechanisms in terms of Price of Anarchy and Price of Stability, while \MG is the best truthful mechanism in terms of approximated ratio. Furthermore, we characterize the Nash Equilibria of \MB and \MD and retrieve sets of conditions under which \MB acts as a truthful mechanism, which highlights the differences between \MB and \MD. Finally, we extend our results to the case in which agents' capacity is part of their private information.Comment: 10 pages, 0 figure

Edge Manipulations for the Maximum Vertex-Weighted Bipartite b-matching

In this paper, we explore the Mechanism Design aspects of the Maximum Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs $(A\cup T, E)$. The set $A$ comprises agents, while $T$ represents tasks. The set $E$ is the private information of either agents or tasks. In this framework, we investigate three mechanisms - \MB, \MD, and \MG - that, given an MVbM problem as input, return a $b$-matching. We examine scenarios in which either agents or tasks are strategic and report their adjacent edges to one of the three mechanisms. In both cases, we assume that the strategic entities are bounded by their statements: they can hide edges, but they cannot report edges that do not exist. First, we consider the case in which agents can manipulate. In this framework, \MB and \MD are optimal but not truthful. By characterizing the Nash Equilibria induced by \MB and \MD, we reveal that both mechanisms have a Price of Anarchy ($PoA$) and Price of Stability ($PoS$) of $2$. These efficiency guarantees are tight; no deterministic mechanism can achieve a lower $PoA$ or $PoS$. In contrast, the third mechanism, \MG, is not optimal, but truthful and its approximation ratio is $2$. We demonstrate that this ratio is optimal; no deterministic and truthful mechanism can outperform it. We then shift our focus to scenarios where tasks can exhibit strategic behaviour. In this case, \MB, \MD, and \MG all maintain truthfulness, making \MB and \MD truthful and optimal mechanisms. In conclusion, we investigate the manipulability of \MB and \MD through experiments on randomly generated graphs. We observe that (1) \MB is less prone to be manipulated by the first agent than \MD (2) \MB is more manipulable on instances in which the total capacity of the agents is equal to the number of tasks (3) randomizing the agents' order reduces the agents' ability to manipulate \MB.Comment: 37 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:2307.1230

A Bilevel Formalism for the Peer-Reviewing Problem

Due to the large number of submissions that more and more conferences experience, finding an automatized way to well distribute the submitted papers among reviewers has become necessary. We model the peer-reviewing matching problem as a {\it bilevel programming (BP)} formulation. Our model consists of a lower-level problem describing the reviewers' perspective and an upper-level problem describing the editors'. Every reviewer is interested in minimizing their overall effort, while the editors are interested in finding an allocation that maximizes the quality of the reviews and follows the reviewers' preferences the most. To the best of our knowledge, the proposed model is the first one that formulates the peer-reviewing matching problem by considering two objective functions, one to describe the reviewers' viewpoint and the other to describe the editors' viewpoint. We demonstrate that both the upper-level and lower-level problems are feasible and that our BP model admits a solution under mild assumptions. After studying the properties of the solutions, we propose a heuristic to solve our model and compare its performance with the relevant state-of-the-art methods. Extensive numerical results show that our approach can find fairer solutions with competitive quality and less effort from the reviewers.Comment: 14 pages, 7 figure

The Equivalence of Fourier-based and Wasserstein Metrics on Imaging Problems

We investigate properties of some extensions of a class of Fourier-based probability metrics, originally introduced to study convergence to equilibrium for the solution to the spatially homogeneous Boltzmann equation. At difference with the original one, the new Fourier-based metrics are well-defined also for probability distributions with different centers of mass, and for discrete probability measures supported over a regular grid. Among other properties, it is shown that, in the discrete setting, these new Fourier-based metrics are equivalent either to the Euclidean-Wasserstein distance $W_2$, or to the Kantorovich-Wasserstein distance $W_1$, with explicit constants of equivalence. Numerical results then show that in benchmark problems of image processing, Fourier metrics provide a better runtime with respect to Wasserstein ones.Comment: 18 pages, 2 figures, 1 tabl

Naples Prognostic Score Predicts Tumor Regression Grade in Resectable Gastric Cancer Treated with Preoperative Chemotherapy

: Despite recent progresses, locally advanced gastric cancer remains a daunting challenge to embrace. Perioperative chemotherapy and D2-gastrectomy depict multimodal treatment of gastric cancer in Europe, shows better results than curative surgery alone in terms of downstaging, micrometastases elimination, and improved long-term survival. Unfortunately, preoperative chemotherapy is useless in about 50% of cases of non-responder patients, in which no effect is registered. Tumor regression grade (TRG) is directly related to chemotherapy effectiveness, but its understanding is achieved only after surgical operation; accordingly, preoperative chemotherapy is given indiscriminately. Conversely, Naples Prognostic Score (NPS), related to patient immune-nutritional status and easily obtained before taking any therapeutic decision, appeared an independent prognostic variable of TRG. NPS was calculated in 59 consecutive surgically treated gastric cancer patients after neoadjuvant FLOT4-based chemotherapy. 42.2% of positive responses were observed: all normal NPS and half mild/moderate NPS showed significant responses to chemotherapy with TRG 1-3; while only 20% of the worst NPS showed some related benefits. Evaluation of NPS in gastric cancer patients undergoing multimodal treatment may be useful both in selecting patients who will benefit from preoperative chemotherapy and for changing immune-nutritional conditions in order to improve patient's reaction against the tumor

Assessment of the DNA Mismatch Repair System Is Crucial in Colorectal Cancers Necessitating Adjuvant Treatment: A Propensity Score-Matched and Win Ratio Analysis

A deficient DNA mismatch repair (MMR) system is identified in a non-negligible part of sporadic colorectal cancers (CRCs), and its prognostic value remains controversial. High tumor mutational burden, along with a poor response to conventional chemotherapy and excellent results from immunotherapy, are the main features of this subset. The aim of this study was to evaluate the predictive value of DNA MMR system status for its best treatment. Four hundred and three CRC patients, operated on from 2014 to 2021 and not treated with immunotherapy, entered this study. Immunohistochemistry and polymerase chain reaction, as appropriate, were used to unequivocally group specimens into microsatellite stable (MSS) and instable (MSI) tumors. The win-ratio approach was utilized to compare composite outcomes. MSI tumors accounted for 12.9% of all series. The right tumor location represented the most important factor related to MSI. The status of the DNA MMR system did not appear to correlate with outcome in early-stage CRCs not requiring adjuvant treatment; in advanced stages undergoing conventional chemotherapy, MSI tumors showed significantly poorer overall and disease-free survival rates and the highest win ratio instead. The determination of DNA MMR status is crucial to recommending correct management. There is clear evidence that instable CRCs needing adjuvant therapy should undergo appropriate treatments

The Fourier Discrepancy Function

In this paper, we introduce the p-Fourier Discrepancy Functions, a new family of metrics for comparing discrete probability measures, inspired by the χr-metrics. Unlike the χr-metrics, the p-Fourier Discrepancies are well-defined for any pair of measures. We prove that the p-Fourier Discrepancies are convex, twice differentiable, and that their gradient has an explicit formula. Moreover, we study the lower and upper tight bounds for the p-Fourier Discrepancies in terms of the Total Variation distance