We propose a new solution concept to address the problem of sharing a surplus among the agents generating it. The problem is formulated in the preferences-endowments space. The solution is defined recursively, incorporating notions of consistency and fairness and relying on properties satisfied by the Shapley value for Transferable Utility (TU) games. We show a solution exists, and call it an O...

The subject of this paper is Modal μ-Calculus, an extension of Modal Logic with operators for least and greatest fixpoints of monotone functions on sets. This logic, introduced by Kozen in [17], is a powerful formalism capable of expressing inductive as well as coinductive concepts and beyond (e.g. safety, liveness, fairness, termination, etc.) and is widely used in the area of verification of ...

This paper proves some extensions of Brenier's [1] theorem that an integrable vectorvalued function u, satisfying a non-degeneracy condition, admits a unique polar factorisation u = u s. Here u is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show tha...

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of m...

Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue...

We study multiwinner voting problems when there is an additional requirement that the selected committee should be fair with respect to attributes such as gender, ethnicity, or political parties. Every setting of an attribute gives rise to a group, and the goal is to ensure that each group is neither over nor under represented in the selected committee. Prior work has largely focused on designi...

In this paper, we discuss a method to nd a critical casebase to classify boolean concepts. By a critical casebase we mean that any subset of the casebase cannot represent a concept correctly. This notion is important for reduction of not only the size of casebase but also classi cation speed. In this paper, we consider set-inclusion based similarity which is originated from a legal reasoning sy...

Consumers with inequity aversion experience some psychological disutility when buying products at unfair prices. Empirical evidence and behavioral research suggest that consumers may perceive a firm’s price as unfair when its profit margin is too high relative to consumers’ surplus. We develop a game-theoretic model to investigate the effects of the consumer’s inequity aversion on a firm’s pric...

This paper studies minimax aspects of nonparametric classification. We first study minimax estimation of the conditional probability of a class label, given the feature variable. This function, say f , is assumed to be in a general nonparametric class. We show the minimax rate of convergence under square L2 loss is determined by the massiveness of the class as measured by metric entropy. The se...

Multipartite entanglement is an indispensable resource in quantum communication and computation; however, it a challenging task to faithfully quantify this global property of multipartite systems. In work we study the concurrence fill, which admits geometric interpretation measure genuine tripartite for three-qubit system [Xie Eberly, Phys. Rev. Lett. 127, 040403 (2021)]. First, use well-known ...