Randomization of spectral risk measures and distributional robustness

In this paper we consider a situation where the risk preferences of a decision maker can be described by a spectral risk measure (SRM) but there is no single SRM that can be used to represent the decision maker’s preferences consistently, due to some kind of randomness in these preferences. Consequently, we propose to randomize the SRM by introducing a random parameter into the risk spectrum. The randomized SRM (RSRM) allows us to describe the decision maker’s random preferences in different states with different SRMs. When the distribution of the random parameter is known (ie, the randomness of the decision maker’s preference can be described by a probability distribution), we introduce a new risk measure: the mean value of the RSRM. In the case when the distribution is unknown, we propose a distributionally robust formulation of the RSRM. The RSRM paradigm provides a new framework for interpreting the well-known Kusuoka representation of law-invariant coherent risk measures and addressing the inconsistency issues arising from observation/ measurement errors or erroneous responses in a preference elicitation process. We discuss in detail computational schemes for solving the optimization problems based on the RSRM and the distributionally robust RSRM. The preliminary numerical tests show that the proposed models and computational schemes work very well. Finally, we discuss how to use step-like approximation and sample-average approximation to deal with the case when the randomized risk spectrums are not step-like and/or the random parameter is continuously distributed, and we derive error bounds to justify the approximations.