Generalized families of fractional stochastic dominance

Abstract

Abstract Introduced by Müller et al. in their seminal paper (2017), fractional stochastic dominance (SD) offers a nuanced approach to ordering distributions. In this paper we propose a fundamentally new framework by replacing the fixed parameter gamma element of left bracket 0 comma 1 right bracket γ ∈ [ 0 , 1 ] $\gamma \in [0, 1]$ in fractional SD with a function bold gamma colon double struck upper R right arrow left bracket 0 comma 1 right bracket 𝛄 : R → [ 0 , 1 ] $\unicode{x1D6C4}\colon \mathbb{R} \to [0, 1]$ . This yields two novel families, multi-fractional stochastic dominance (MFSD) and functional fractional stochastic dominance (FFSD). They enable the ranking of a broader range of distributions and incorporate a more informative utility class, including those with local non-concavities whose steepness varies depending on the location. Furthermore, our framework introduces the concept of partial greediness, which dynamically captures how behaviour of decision makers adapts to changes in wealth. We also extend this framework to encompass almost stochastic dominance. We provide the mathematical foundations of our generalized framework and study how it offers a novel tool for ordering distributions across various settings.