  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 adverse impact arising from ambiguity of the true utility, we propose
a maximin robust model where the optimal policy is based on the worst-case
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
zeta-ball approach where a ball of utility functions centered at a nominal
utility function under zeta-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 use the SDDP method and
the nested Benders' decomposition method to solve the multistage
state-dependent preference robust problem and the scenario tree method to solve
the state-independent problem, and carry out comparative analysis on the
efficiency of the computational schemes as well as out-of-sample performances
of the state-dependent and state-independent models.
