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For each of the extended Shapley values due to Derks and Peters [3] and Peters and Zank [16], we first propose a corresponding definition of marginal contributions of a potential, and further demonstrate that these solutions can be generalized as the vector of corresponding marginal contributions of a potential. We further propose several equivalent relations and related axiomatizations to demonstrate that these two solutions are almost coincident in axiomatic approach except that the loss of amount is different in the axiom of equal loss.

This paper develops axiomatically a revealed preference theory of reference-dependent choice behavior. Instead of taking the reference for an agent as exogenously given in the description of a choice problem, we suitably relax the Weak Axiom of Revealed Preference to obtain, endogenously, the existence of reference alternatives as well as the structure of choice behavior conditional on those alternatives. We show how this model captures some well-known choice patterns such as the attraction effect.

Interval games are an extension of cooperative coalitional games, in which players are assumed to face payoff uncertainty. Characteristic functions thus assign a closed interval, instead of a real number. In this paper, we first examine the notion of solution mapping, a solution concept applied to interval games, by comparing it with the existing solution concept called the interval solution concept. Then, we define a Shapley mapping as a specific form of the solution mapping. Finally, it is shown that the Shapley mapping can be characterized by two different axiomatizations, both of which employ interval game versions of standard axioms used in the traditional cooperative game analysis such as efficiency, symmetry, null player property, additivity and separability.

Interval games are an extension of cooperative coalitional games, in which players are assumed to face payoff uncertainty. Characteristic functions thus assign a closed interval instead of a real number. This study revisits two interval game versions of Shapley values (i.e., the interval Shapley value and the interval Shapley-like value) and characterizes them using an axiomatic approach. For the interval Shapley value, we show that the existing axiomatization can be generalized to a wider subclass of interval games called size monotonic games. For the interval Shapley-like value, we show that a standard axiomatization using Young's strong monotonicity holds on the whole class of interval games.

This paper develops a formal axiomatic framework for Project Theory, introducing key concepts such as task dependencies, hierarchies, and project structures. The motivation behind this work is to achieve a global applicability by using universal mathematical language and logic, aiming to standardize the analysis of project management systems. We define projects in terms of well-founded relations and set-theoretic principles to rigorously describe their properties. Central to this theory are the Axiom of Compatibility, which ensures consistency across subprojects; the Axiom of Regularity, which prohibits infinite regress in project hierarchies and Mostowski’s Collapsing Theorem, which is applied to project structures to establish isomorphisms between transitive sets and projects. These theoretical results provide a robust foundation for modeling complex project management systems, enabling applications in areas such as operations research, workflow optimization, and software engineering.

Let$\\mathcal {P}(\\mathbf{N})$be the power set of N . We say that a function$\\mu ^\\ast : \\mathcal {P}(\\mathbf{N}) \\to \\mathbf{R}$is an upper density if, for all X , Y ⊆ N and h , k ∈ N + , the following hold: ( f1 )$\\mu ^\\ast (\\mathbf{N}) = 1$; ( f2 )$\\mu ^\\ast (X) \\le \\mu ^\\ast (Y)$if X ⊆ Y ; ( f3 )$\\mu ^\\ast (X \\cup Y) \\le \\mu ^\\ast (X) + \\mu ^\\ast (Y)$; ( f4 )$\\mu ^\\ast (k\\cdot X) = ({1}/{k}) \\mu ^\\ast (X)$, where k · X : = kx : x ∈ X ; and ( f5 )$\\mu ^\\ast (X + h) = \\mu ^\\ast (X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α -densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms ( f1 )–( f5 ), and we investigate various properties of upper densities (and related functions) under the assumption that ( f2 ) is replaced by the weaker condition that$\\mu ^\\ast (X)\\le 1$for every X ⊆ N . Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

Abstract The distinction between de re (of the thing) and de dicto (of what is said) readings of sentences has long been the topic of studies in logic and philosophy of language. The article proposes to apply these concepts to anonymity. It argues that, in the proposed setting, de dicto knowledge preserves anonymity, while de re knowledge does not. The article also considers a third, “overt,” form of knowledge. The main technical result is a sound and complete logical system that captures the interplay between a data traceability expression and the de re, de dicto, and overt knowledge modalities. The article also shows that the three knowledge modalities are not definable through each other even in the presence of the traceability expression.

Recently, we ( Synthese, 199 (5–6):13247–13281, 2021 ) proposed Kripke-like semantics for two quantum logics of interderminacy. These logics expand the vocabulary of standard Birkhoff-von Neumann propositional quantum logic with a pair of modal operators interpreted as “it is (in)determinate that”, allowing them to express in the object language statements such as “it is indeterminate that system S is spin-up in the x-direction”, as well as statements of any logical complexity involving ascriptions of (in)determinacy. We present an axiomatization of a logic closely related to one of these quantum logics of indeterminacy and prove that this axiomatization is sound and complete with respect to the Kripke-like semantics. We then prove that this logic is decidable.

When we say “I know why he was late”, we know not only the fact that he was late, but also an explanation of this fact. We propose a logical framework of “knowing why” inspired by the existing formal studies on why-questions, scientific explanation, and justification logic. We introduce the 𝒦y i operator into the language of epistemic logic to express “agent i knows why φ” and propose a Kripke-style semantics of such expressions in terms of knowing an explanation of φ. We obtain two sound and complete axiomatizations w.r.t. two different model classes depending on different assumptions about introspection. Finally we connect our logic with justification logic technically by providing an alternative semantics and an in-depth comparison on various design choices.

In a variety of systems, in particular in a financial system, entities hold liabilities on each other. The reimbursement abilities are intertwined, thereby potentially generating coordination failures and cascades of defaults calling for orderly resolution. With a single indebted firm, a bankruptcy law organizes such an orderly resolution. With cross-liabilities, a resolution rule should be defined at the system level to account for all those affected, directly or indirectly. This paper investigates such rules assuming their primary goal is to avoid defaults on creditors external to the system, say banks’ defaults on customers’ deposits. I define and characterize the constrained-proportional rule building on two approaches: the minimization of an inequality measure on the reimbursements (made and received) and the axiomatization through desirable properties.