550 research outputs found
Generalized probabilities in statistical theories
In this review article we present different formal frameworks for the
description of generalized probabilities in statistical theories. We discuss
the particular cases of probabilities appearing in classical and quantum
mechanics, possible generalizations of the approaches of A. N. Kolmogorov and
R. T. Cox to non-commutative models, and the approach to generalized
probabilities based on convex sets
The lattice of embedded subsets
In cooperative game theory, games in partition function form are real-valued function on the set of so-called embedded coalitions, that is, pairs where is a subset (coalition) of the set of players, and is a partition of containing . Despite the fact that many studies have been devoted to such games, surprisingly nobody clearly defined a structure (i.e., an order) on embedded coalitions, resulting in scattered and divergent works, lacking unification and proper analysis. The aim of the paper is to fill this gap, thus to study the structure of embedded coalitions (called here embedded subsets), and the properties of games in partition function form.Partition; Embedded subset; Game; Valuation; k-monotonicity
Entropy of capacities on lattices and set systems
We propose a definition for the entropy of capacities defined on lattices.
Classical capacities are monotone set functions and can be seen as a
generalization of probability measures. Capacities on lattices address the
general case where the family of subsets is not necessarily the Boolean lattice
of all subsets. Our definition encompasses the classical definition of Shannon
for probability measures, as well as the entropy of Marichal defined for
classical capacities. Some properties and examples are given
Measure and integral with purely ordinal scales
We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at on the real interval . Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and
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