Abstract
It is known that the quasi-arithmetic means can be characterized in various ways, with an essential role of a symmetry property. In the expected utility theory, the quasi-arithmetic mean is called the certainty equivalent and it is applied, e.g., in a utility-based insurance contracts pricing. In this paper, we introduce and study the quasi-arithmetic type mean in a more general setting, namely with the expected value being replaced by the generalized Choquet integral. We show that a functional that is defined in this way is a mean. Furthermore, we characterize the equality, positive homogeneity, and translativity in this class of means.
Highlights
The notion of quasi-arithmetic mean, playing an important role in several branches of mathematics and its applications, was introduced in the book by Hardy, Littlewood, and Pólya [1]
If I ⊆ R is an interval and u : I → R is a strictly monotone continuous function, the quasi-arithmetic mean Mu : n∈N I n → I, which is generated by u, is given by
Various axiomatic characterizations of the quasi-arithmetic mean have been independently established by de Finetti [2], Kolmogorov [3], and Nagumo [4]
Summary
The notion of quasi-arithmetic mean, playing an important role in several branches of mathematics and its applications, was introduced in the book by Hardy, Littlewood, and Pólya [1]. If X : S → R is an F −measurable essentially bounded function on a given probability space (S, F , P), the quasi-arithmetic mean of X, generated by a strictly monotone continuous function u : R → R, is defined as follows. In the expected utility theory developed by von Neumann and Morgenstern [9], the quasi-arithmetic mean, as given by (1), is known as the certainty equivalent. It establishes a symmetry in preferences between a risk that is represented by X and a deterministic payoff M(u,P) ( X ), in a Symmetry 2020, 12, 2104; doi:10.3390/sym12122104 www.mdpi.com/journal/symmetry. In the whole paper (S, F ) stands for a measurable space
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
