History-Dependent Recursive Preferences in Markov Decision Processes

Abstract

In finite horizon dynamic programming with history-dependent preferences, the relevant state may be the entire realized history, even when the physical state is Markov. This paper develops a behavioral state-reduction theory for such Markov decision processes. Under behavioral axioms and a certainty-equivalent richness condition, the full-history problem admits a recursive representation composed of time and risk aggregators. We then derive a canonical preference-augmented (PA) state by quotienting histories that have the same current physical Markov state, are indifferent under every common continuation plan, and remain equivalent after every common one-step extension. This canonical PA state is minimal among reachable recursive factorizations of the underlying preferences. Under Markov feasibility and standard dynamic-programming regularity, a PA Bellman selector induces an optimal full-history policy. With additional rectangularity and exhaustiveness conditions, we reparameterize the preference memory into distinct belief and taste coordinates, and obtain a separated representation and Bellman recursion. We give a taxonomy of examples to illustrate the scope of our framework.