Decision Making Under Cumulative Prospect Theory: An Alternating Direction Method of Multipliers

Abstract

This paper proposes a novel numerical method for solving the problem of decision making under cumulative prospect theory (CPT), where the goal is to maximize utility subject to practical constraints, assuming only finite realizations of the associated distribution are available. Existing methods for CPT optimization rely on particular assumptions that may not hold in practice. To overcome this limitation, we present the first numerical method with a theoretical guarantee for solving CPT optimization using an alternating direction method of multipliers (ADMM). One of its subproblems involves optimization with the CPT utility subject to a chain constraint, which presents a significant challenge. To address this, we develop two methods for solving this subproblem. The first method uses dynamic programming, whereas the second method is a modified version of the pooling-adjacent-violators algorithm that incorporates the CPT utility function. Moreover, we prove the theoretical convergence of our proposed ADMM method and the two subproblem-solving methods. Finally, we conduct numerical experiments to validate our proposed approach and demonstrate how CPT’s parameters influence investor behavior, using real-world data. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms: Continuous. Funding: This research was supported by the National Natural Science Foundation of China [Grants 12171100, 71971083, and 72171138], the Natural Science Foundation of Shanghai [Grant 22ZR1405100], the Major Program of the National Natural Science Foundation of China [Grants 72394360, 72394364], the Program for Innovative Research Team of Shanghai University of Finance and Economics [Grant 2020110930], Fundamental Research Funds for the Central Universities, and the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science, Ministry of Education, East China Normal University. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0243 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0243 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .