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

TL;DR

This paper introduces a new numerical approach using ADMM for decision making under cumulative prospect theory, providing theoretical guarantees and practical algorithms for optimization with real-world data.

Contribution

It presents the first ADMM-based numerical method with convergence guarantees for CPT optimization, including novel subproblem solutions using dynamic programming and modified PAV algorithms.

Findings

Validated the approach with numerical experiments

Demonstrated influence of CPT parameters on investor behavior

Provided theoretical convergence proofs

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, while 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.