Generalized Families of Fractional Stochastic Dominance

TL;DR

This paper introduces a flexible new framework for stochastic dominance by replacing a fixed parameter with a function, allowing for more nuanced distribution comparisons and decision-maker behavior modeling.

Contribution

It proposes multi-fractional and functional fractional stochastic dominance, expanding the utility classes and incorporating partial greediness for dynamic decision-making analysis.

Findings

Broader distribution ranking capabilities

Inclusion of local non-concavities in utility functions

Extension to almost stochastic dominance

Abstract

Introduced by M\"uller et al. in their seminal paper \cite{muller}, 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 $\gamma \in [0,1]$ in fractional SD with a function $\boldsymbol{\gamma}: \mathbb{R} \to [0,1]$. 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.