Multistage Utility Preference Robust Optimization

Abstract

In this paper, we consider a multistage expected utility maximization problem where the decision maker's utility function at each stage depends on historical data and the information on the true utility function is incomplete. To mitigate the risk arising from ambiguity of the true utility, we propose a maximin robust model where the optimal policy is based on the worst sequence of utility functions from an ambiguity set constructed with partially available information about the decision maker's preferences. We then show that the multistage maximin problem is time consistent when the utility functions are state-dependent and demonstrate with a counter example that the time consistency may not be retained when the utility functions are state-independent. With the time consistency, we show the maximin problem can be solved by a recursive formula whereby a one-stage maximin problem is solved at each stage beginning from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a ζ-ball approach where a ball of utility functions centered at a nominal utility function under ζ-metric is considered. To overcome the difficulty arising from solving the infinite dimensional optimization problem in computation of the worst-case expected utility value, we propose piecewise linear approximation of the utility functions and derive error bound for the approximation under moderate conditions. Finally we develop a scenario tree-based computational scheme for solving the multistage preference robust optimization model and report some preliminary numerical results.

Preference Robust Modified Optimized Certainty Equivalent

“…In practice, decision maker's utility preferences are often elicited through questionnaires. For example, a customer's utility preference may be elicited via the customer's willingness to pay at certain price points [43,32]. From computational point of view, piecewise linear utility function may bring significant convenience to calculation of OCE, see Nouiehed et al [36].…”

“…Hu and Stepanyan [25] propose a so-called reference-based almost stochastic dominance method for constructing a set of utility functions near a reference utility which satisfies certain stochastic dominance relationship and use the set to characterize the decision maker's preference. Over the past few years, the research on PRO has received increasing attentions in the communities of stochastic/robust optimization and risk management, see for instances [22,21,14,52,31,32] and references therein.…”