<p>In this paper, we consider a multistage expected utility maximization problem where the decision maker’s utility function at each stage depends on historical path and the information on the true utility function is incomplete. To mitigate the adverse impact arising from ambiguity regarding 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 show that the multistage maximin problem is time consistent when the utility functions depend on the historical path and provide a counter example demonstrating that the time consistency is not retained when the utility functions are stagewise independent. With the time consistency, we show the maximin problem can be recursively solved by backward induction, where a one-stage maximin problem is solved at each stage starting from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>-ball approach where a ball of utility functions centered at a nominal utility function under <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\zeta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation>-metric is considered. To overcome the difficulty arising from solving the infinite-dimensional optimization problem in the computation of the worst-case expected utility value, we propose a piecewise linear approximation of the utility functions and derive error bounds for the approximation under moderate conditions. Finally, we use the stochastic dual dynamic programming method and the nested Benders’ decomposition method to solve the multistage historical-path-dependent preference robust problem and the scenario tree method to solve the stagewise-independent problem. We carry out comparative analysis on the efficiency of the computational schemes as well as out-of-sample performances of the historical-path-dependent and stagewise-independent models. The preliminary results show that the historical-path-dependent preference robust model displays overall superiority.</p>

Preference ambiguity and robustness in multistage decision making

期刊论文
发表日期：

2025年4月16日

查看全文

Jia Liu,
Zhiping Chen,
Huifu Xu

摘要

In this paper, we consider a multistage expected utility maximization problem where the decision maker’s utility function at each stage depends on historical path and the information on the true utility function is incomplete. To mitigate the adverse impact arising from ambiguity regarding 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 show that the multistage maximin problem is time consistent when the utility functions depend on the historical path and provide a counter example demonstrating that the time consistency is not retained when the utility functions are stagewise independent. With the time consistency, we show the maximin problem can be recursively solved by backward induction, where a one-stage maximin problem is solved at each stage starting from the last stage. Moreover, we propose two approaches to construct the ambiguity set: a pairwise comparison approach and a $\zeta $ $ζ$ -ball approach where a ball of utility functions centered at a nominal utility function under $\zeta $ $ζ$ -metric is considered. To overcome the difficulty arising from solving the infinite-dimensional optimization problem in the computation of the worst-case expected utility value, we propose a piecewise linear approximation of the utility functions and derive error bounds for the approximation under moderate conditions. Finally, we use the stochastic dual dynamic programming method and the nested Benders’ decomposition method to solve the multistage historical-path-dependent preference robust problem and the scenario tree method to solve the stagewise-independent problem. We carry out comparative analysis on the efficiency of the computational schemes as well as out-of-sample performances of the historical-path-dependent and stagewise-independent models. The preliminary results show that the historical-path-dependent preference robust model displays overall superiority.