Randomization of Spectral Risk Measure and Distributional Robustness

TL;DR

This paper introduces a randomized spectral risk measure (RSRM) framework to model decision maker preferences more flexibly, addressing inconsistencies and measurement errors, with computational methods for optimization under known or unknown preference distributions.

Contribution

It proposes a novel RSRM approach that generalizes spectral risk measures, including distributional robustness, and offers computational schemes for optimization problems.

Findings

RSRM captures diverse risk preferences effectively.

Distributionally robust RSRM enhances modeling under uncertainty.

Framework aligns with Kusuoka's representation of risk measures.

Abstract

In this paper, we consider a situation where a decision maker's (DM's) risk preference can be described by a spectral risk measure (SRM) but there is not a single SRM which can be used to represent the DM's preferences consistently. Consequently we propose to randomize the SRM by introducing a random parameter in the risk spectrum. The randomized SRM (RSRM) allows one to describe the DM's preferences at different states with different SRMs. When the distribution of the random parameter is known, i.e., the randomness of the DM's preference can be described by a probability distribution, we introduce a new risk measure which is the mean value of the RSRM. In the case when the distribution is unknown, we propose a distributionally robust formulation of RSRM. The RSRM paradigm provides a new framework for interpreting the well-known Kusuoka's representation of law invariant coherent risk measures and addressing inconsistency issues arising from observation/measurement errors or erroneous responses in preference elicitation process. We discuss in detail computational schemes for solving optimization problems based on the RSRM and the distributionally robust RSRM.