  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.

Optimization and Control