87 research outputs found
Current Open Questions in Complete Mixability
Complete and joint mixability has raised considerable interest in recent few
years, in both the theory of distributions with given margins, and applications
in discrete optimization and quantitative risk management. We list various open
questions in the theory of complete and joint mixability, which are
mathematically concrete, and yet accessible to a broad range of researchers
without specific background knowledge. In addition to the discussions on open
questions, some results contained in this paper are new
Quasi-convexity in mixtures for generalized rank-dependent functions
Quasi-convexity in probabilistic mixtures is a common and useful property in
decision analysis. We study a general class of non-monotone mappings, called
the generalized rank-dependent functions, which include the preference models
of expected utilities, dual utilities, and rank-dependent utilities as special
cases, as well as signed Choquet integrals used in risk management. As one of
our main results, quasi-convex (in mixtures) signed Choquet integrals precisely
include two parts: those that are convex (in mixtures) and the class of scaled
quantile-spread mixtures, and this result leads to a full characterization of
quasi-convexity for generalized rank-dependent functions. Seven equivalent
conditions for quasi-convexity in mixtures are obtained for dual utilities and
signed Choquet integrals. We also illustrate a conflict between convexity in
mixtures and convexity in risk pooling among constant-additive mappings
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