Weak Monotonicity Research Articles

Weak differential monotonicity and axiomatization of convex combinations of solutions to cooperative games

Weak differential monotonicity and axiomatization of convex combinations of solutions to cooperative games

Weak monotonicity property of Korevaar-Schoen norms on nested fractals

In this paper, we study the weak monotonicity property of Korevaar-Schoen (p-energy) norms on nested fractals for 1<p<∞. Such property has many important applications on fractals and other metric measure spaces, such as constructing p-energies (when p=2 this is basically a Dirichlet form), generalizing the classical Sobolev type inequalities and the celebrated Bourgain-Brezis-Mironescu convergence.

Generalization of weighted-egalitarian Shapley values

In this study, we provide a new class of solutions that generalize the weighted-egalitarian Shapley values by permitting heterogeneity of players in both the Shapley value and equal division value. To derive the proposed value, we present axioms of weak monotonicity, weak differential null player out, efficiency, null game, ratio invariance for null players, and weak superweak differential marginality.

Finite distortion curves: Continuity, differentiability and Lusin’s (N) property

We define finite distortion \omega -curves and we show that for some forms \omega and when the distortion function is sufficiently exponentially integrable, the map is continuous, differentiable almost everywhere and has Lusin’s (N) property. This is achieved through some higher integrability results about finite distortion \omega -curves. It is also shown that this result is sharp both for continuity and for Lusin’s (N) property. We also show that if we assume weak monotonicity for the coordinates of a finite distortion \omega -curve, we obtain continuity.

The critical variational setting for stochastic evolution equations

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn–Hilliard equation, tamed Navier–Stokes equations, and Allen–Cahn equation.

Large deviation principles for SDEs under locally weak monotonicity conditions

Large deviation principles for SDEs under locally weak monotonicity conditions

Weak Monotonicity With Trend Analysis for Unsupervised Feature Evaluation.

Performance in an engineering system tends to degrade over time due to a variety of wearing or ageing processes. In supervisory controlled processes there are typically many signals being monitored that may help to characterize performance degradation. It is preferred to select the least amount of information to obtain high quality of predictive analysis from a large amount of collected data, in which labeling the data is not always feasible. To this end a novel unsupervised feature selection method, robust with respect to significant measurement disturbances, is proposed using the notion of ``weak monotonicity'' (WM). The robustness of this notion makes it very attractive to identify the common trend in the presence of measurement noises and population variation from the collected data. Based on WM, a novel suitability indicator is proposed to evaluate the performance of each feature. This new indicator is then used to select the key features that contribute to the WM of a family of processes when noises and variations among processes exist. In order to evaluate the performance of the proposed framework of the WM and suitability, a comparative study with other nine state-of-the-arts unsupervised feature evaluation and selection methods is carried out on well-known benchmark datasets. The results show a promising performance of the proposed framework on unsupervised feature evaluation in the presence of measurement noises and population variations.

Causal effects of mental health on food security

Although mental health conditions are known to be associated with socioeconomic hardships, their causal effects remain largely unexplored. Using a sample of low-income families in the National Health Interview Survey (NHIS), we assess causal effects of serious mental illness (SMI) and related mental health conditions on family food security. We apply partial identification methods to account for fundamental endogeneity and measurement identification problems in a unified framework. To implement these methods, we combine a proxy measure of SMI in the NHIS with an estimate of the true rate of SMI from the Substance Abuse and Mental Health Services Administration. We also develop an innovative approach to approximate true prevalence rates when only self-reported prevalence rates are available. Applying relatively weak monotonicity assumptions on latent food security outcomes, we find that alleviating SMI would improve the food security rate by at least 9.5 percentage points, or 15 %.JEL codes: C21, I10, I38

Gas of Particles Obeying the Monotone Statistics.

The present note is devoted to the detailed investigation of a concrete model satisfying the block-monotone statistics introduced in a previous paper (joint, with collaborators) of the author. The model under consideration indeed describes the free gas of massless particles in a one-dimensional environment. This investigation can have consequences in two fundamental respects. The first one concerns the applicability of the (block-)monotone statistics to concrete physical models, yet completely unknown. Since the formula for the degeneracy of the energy-levels of the one-particle Hamiltonian of a free particle is very involved, the second aspect might be related to the, highly nontrivial, investigation of the expected thermodynamics of the free gas of particles obeying the block-monotone statistics in arbitrary spatial dimensions. A final section contains a comparison between the various (block, strict, and weak) monotone schemes with the Boltzmann statistics, which describes the gas of classical particles. It is seen that the block-monotone statistics, which takes into account the degeneracy of the energy-levels, seems the unique one having realistic physical applications.

A framework for generalized monotonicity of fusion functions

The relaxation of the property of monotonicity is a trend in the theory of aggregation and fusion functions and several generalized forms of monotonicity have been introduced, most of which are based on the notion of directional monotonicity. In this paper, we propose a general framework for generalized monotonicity that encompasses the different forms of monotonicity that we can find in the literature. Additionally, we introduce various new forms of monotonicity that are not based on directional monotonicity. Specifically, we introduce dilative monotonicity, which requires that the function increases when the inputs have increased by a common factor, and a general form of monotonicity that is dependent on a function T and a subset of the domain Z. This two new generalized monotonicities are the basis to propose a set of different forms of monotonicity. We study the particularities of each of the new proposals and their links to the previous relaxed forms of monotonicity. We conclude that the introduction of dilative monotonicity complements the conditions of weak monotonicity for fusion functions and that (T,Z)-monotonicity yields a condition that is slightly stronger than weak monotonicity. Finally, we present an application of the introduced notions of monotonicity in sentiment analysis.

An Inertial Subgradient Extragradient Method for Approximating Solutions to Equilibrium Problems in Hadamard Manifolds

In this work, we are concerned with the iterative approximation of solutions to equilibrium problems in the framework of Hadamard manifolds. We introduce a subgradient extragradient type method with a self-adaptive step size. The use of a step size which is allowed to increase per iteration is to avoid the dependence of our method on the Lipschitz constant of the underlying operator as has been the case in recent articles in this direction. In general, operators satisfying weak monotonicity conditions seem to be more applicable in practice. By using inertial and viscosity techniques, we establish a convergence result for solving a pseudomonotone equilibrium problem under some appropriate conditions. As applications, we use our method to solve some theoretical optimization problems. Finally, we present some numerical illustrations in order to demonstrate the quantitative efficacy and superiority of our proposed method over a previous method present in the literature.

𝕃2-solutions of multidimensional generalized BSDEs with weak monotonicity and general growth generators in a general filtration

Abstract In this paper, we study multidimensional generalized backward stochastic differential equations (GBSDEs), in a general filtration supporting a Brownian motion and an independent Poisson random measure, whose generators are weakly monotone and satisfy a general growth condition with respect to the state variable y. We show that such GBSDEs admit a unique 𝕃 2 {\mathbb{L}^{2}} -solution. The main tools and techniques used in the proofs are the a-priori-estimation, the convolution approach, the iteration, the truncation, and the Bihari inequality.

Feature Identification Framework for Back Injury Risk in Repetitive Work With Application in Sheep Shearing.

Lower back injuries are a serious global problem. Most of these injuries occur over time with repeated sub-acute stresses. Neuromuscular control dysfunction could predict injury, however injuries are almost never observed alongside this data. No labels are available to identify important features that may be predictive of injury. While there are many individual differences in injury development, the population trend is that each individual's injury tolerance decreases over time with exposure, indicating a monotonic process. This paper proposes a framework for identifying key features of injury using an unsupervised technique that exploits knowledge of injury aetiology by analysing which features contribute to the popular trend using weak monotonicity from data segmented by task repetitions. The feature selection also evaluates feature redundancy. The efficacy of the framework is demonstrated through data from on-site sheep shearers over one day using 17 wearable inertial measurement units and 16 surface electromyography (sEMG) sensors. Consistent with literature, the results demonstrate sEMG features derived from the erector spinae and multifidus muscles are the most important indicators for lower back injury. To evaluate the performance of the proposed population-trend based unsupervised feature selection technique, the self-reported fatigue information is treated as some 'ground truth' information so that this proposed technique can compare with 5 existing unsupervised feature selection techniques. The proposed technique is shown to be the most consistent with the self-reported fatigue information, demonstrating the effectiveness of the proposed method.

On the Thermodynamics of Particles Obeying Monotone Statistics

The aim of the present paper is to provide a preliminary investigation of the thermodynamics of particles obeying monotone statistics. To render the potential physical applications realistic, we propose a modified scheme called block-monotone, based on a partial order arising from the natural one on the spectrum of a positive Hamiltonian with compact resolvent. The block-monotone scheme is never comparable with the weak monotone one and is reduced to the usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian are non-degenerate. Through a detailed analysis of a model based on the quantum harmonic oscillator, we can see that: (a) the computation of the grand-partition function does not require the Gibbs correction factor n! (connected with the indistinguishability of particles) in the various terms of its expansion with respect to the activity; and (b) the decimation of terms contributing to the grand-partition function leads to a kind of "exclusion principle" analogous to the Pauli exclusion principle enjoined by Fermi particles, which is more relevant in the high-density regime and becomes negligible in the low-density regime, as expected.

Learning and not using? The effect of degree attainment on illicit drug use among at-risk youth

ABSTRACT This paper examines the causal effect of earning a GED or vocational degree on future illicit drug use, employing random assignment into the United States’ most comprehensive education and vocational training program for at-risk youth – Job Corps – as a source of exogenous variability in degree attainment. Nonparametric bounds under relatively weak monotonicity assumptions are constructed to allow the random assignment to violate the exclusion restriction when used as an instrument. We also use a fixed effect model and propensity score weighting to supplement the results. The results from different methods suggest that degree attainment may have the most significant effect in reducing the illicit drug use of blacks, while the results for whites and Hispanics are less conclusive.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR FRACTIONAL BSDES WITH WEAK MONOTONICITY COEFFICIENTS

In this paper, we deal with the fractional backward stochastic differential equations (F-BSDEs in short) with Hurst parameter $H\in (\frac{1}{2},1)$ when the driver $g$ is weak monotone. Via an approximation theory, we derive the existence and uniqueness of solutions to F-BSDEs. The comparison theorem is also established.

Higher-Order Methods for Convex-Concave Min-Max Optimization and Monotone Variational Inequalities

We provide improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness. In min-max settings where the $p^{th}$-order derivatives are Lipschitz continuous, we give an algorithm HigherOrderMirrorProx that achieves an iteration complexity of $O(1/T^{\frac{p+1}{2}})$ when given access to an oracle for finding a fixed point of a $p^{th}$-order equation. We give analogous rates for the weak monotone variational inequality problem. For $p>2$, our results improve upon the iteration complexity of the first-order Mirror Prox method of Nemirovski [2004] and the second-order method of Monteiro and Svaiter [2012]. We further instantiate our entire algorithm in the unconstrained $p=2$ case.

Mean Field Games with Common Noises and Conditional Distribution Dependent FBSDEs

In this paper, we consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. We assume that the cost function satisfies a convexity and a weak monotonicity property. We use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE). We prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by Hu and Peng \cite{YH2}. We apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.

Continuity postulates and solvability axioms in economic theory and in mathematical psychology: a consolidation of the theory of individual choice

This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of continuity (full, separate, Wold, weak Wold, Archimedean, mixture) with those of solvability (restricted, unrestricted) under the completeness and transitivity of a binary relation. Theorem 2 uses the primitive notion of a separately continuous function to answer the question when an analogous property on a relation is fully continuous. Theorem 3 provides a portmanteau theorem on the equivalence between restricted solvability and various notions of continuity under weak monotonicity. Finally, Theorem 4 presents a variant of Theorem 3 that follows Theorem 1 in dispensing with the dimensionality requirement and in providing partial equivalences between solvability and continuity notions. These theorems are motivated for their potential use in representation theorems.

Logical differencing in dyadic network formation models with nontransferable utilities

This paper considers a semiparametric model of dyadic network formation under nontransferable utilities (NTU). Such dyadic links arise frequently in real-world social interactions that require bilateral consent but by their nature induce additive non-separability. In our model we show how unobserved individual heterogeneity in the network formation model can be canceled out without requiring additive separability. The approach uses a new method we call logical differencing. The key idea is to construct an observable event involving the intersection of two mutually exclusive restrictions—derived based on weak multivariate monotonicity—on the fixed effects. Based on this identification strategy we provide consistent estimators of the network formation model under NTU. Finite-sample performance of our method is analyzed in a simulation study, and an empirical illustration using the risk-sharing network data from Nyakatoke demonstrates that our proposed method is able to obtain economically intuitive estimates.