Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance

Version 1 : Received: 28 February 2023 / Approved: 28 February 2023 / Online: 28 February 2023 (14:22:40 CET)

A peer-reviewed article of this Preprint also exists.

Yin, H.H.F.; Wang, X.; Mak, H.W.L.; Au Yong, C.S.; Chan, I.Y.Y. Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance. Axioms 2023, 12, 564. Yin, H.H.F.; Wang, X.; Mak, H.W.L.; Au Yong, C.S.; Chan, I.Y.Y. Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance. Axioms 2023, 12, 564.

Abstract

A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In early days, Fishburn investigated SD from the view of distributions and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD may not be generalized directly since some properties of fractional calculus do not have a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be generalized directly from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle them.

Keywords

fractional-order stochastic dominance; discrete stochastic dominance; discrete utility; fractional sum

Subject

Computer Science and Mathematics, Discrete Mathematics and Combinatorics

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