Dynamic Programming in Ordered Vector Space

New approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered vector spaces. The integrated algebraic and order structure in such spaces leads to sharper fixed-point results. These fixed-point results can then be exploited to obtain optimality properties. We illustrate our results through applications ranging from firm management to data valuation. These applications include features from the recent literature on dynamic programming, including risk-sensitive preferences, nonlinear discounting, and state-dependent discounting. In all cases, we establish existence of optimal policies, characterize them in terms of Bellman optimality relationships, and prove convergence of major algorithms.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2025.1971.