Preference robust state-dependent distortion risk measure on act space and its application in optimal decision making

Decision-making under uncertainty involves three fundamental components: acts, states of nature, and consequences, as first introduced by Savage (The foundations of statistics, Wiley, New York, 1954). An act is uniquely determined by a number of random consequences that are associated with different states of nature. If the consequences are identical across all states of nature, then the act is state-independent. Prior research on distortion risk measures (DRMs) has primarily focused on state-independent acts. In this paper, we extend the research to state-dependent acts by introducing a state-dependent DRM (SDRM) under the Anscombe–Aumann’s framework (Anscombe and Aumannin in Ann Math Stat 34(1):199–205, 1963). The proposed SDRM is the weighted average of DRMs at each state, where the weights are determined by the decision maker’s (DM’s) subjective probabilities of the states. In situations where there is incomplete information about the DM’s true distortion function and/or the true subjective probabilities of the states, we introduce a preference robust SDRM (PRSDRM) for acts. The PRSDRM is based on the worst-case state-dependent distortion function and the worst-case subjective probabilities over a dependent joint ambiguity set constructed with partially available information. To compute the PRSDRM numerically, we show that when the distortion functions are concave, it can be formulated as a biconvex program and further as a convex program by changing some variables. As a motivation and application, we use the PRSDRM for decision-making problems and propose an alternating iterative algorithm for solving it. Finally, we conduct numerical experiments to assess the performance of our proposed model and computational scheme.