# Distributionally Robust Markov Decision Processes and their Connection to Risk Measures

arXiv: https://arxiv.org/abs/2007.13103 · HTML: https://mlsanity.com/paper/2007.13103/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Jul 2020.

## Built on by (most cited first, 14 of 14)

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- [Soft Robust MDPs and Risk-Sensitive MDPs: Equivalence, Policy Gradient, and Sample Complexity](https://mlsanity.com/paper/2306.11626/) (2023)
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- [Q-Learning under Finite Model Uncertainty](https://mlsanity.com/paper/2407.04259/) (2024)
- [Distributionally Robust Deep Q-Learning](https://mlsanity.com/paper/2505.19058/) (2025)
- [Optimism as Risk-Seeking in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2509.24047/) (2025)
- [Decoupling Time and Risk: Risk-Sensitive Reinforcement Learning with General Discounting](https://mlsanity.com/paper/2602.04131/) (2026)
- [The Indoor-Training Effect: unexpected gains from distribution shifts in the transition function](https://mlsanity.com/paper/2401.15856/) (2024)
- [Risk Sensitivity in Markov Games and Multi-Agent Reinforcement Learning: A Systematic Review](https://mlsanity.com/paper/2406.06041/) (2024)
- [Markov Decision Processes of the Third Kind: Learning Distributions by Policy Gradient Descent](https://mlsanity.com/paper/2602.06567/) (2026)
- [Data-driven robust Markov decision processes on Borel spaces: performance guarantees via an axiomatic approach](https://mlsanity.com/paper/2603.08979/) (2026)
- [Robust Q-learning for mean-field control under Wasserstein uncertainty in common noise](https://mlsanity.com/paper/2606.20356/) (2026)

## Abstract (arXiv)

We consider robust Markov Decision Processes with Borel state and action spaces, unbounded cost and finite time horizon. Our formulation leads to a Stackelberg game against nature. Under integrability, continuity and compactness assumptions we derive a robust cost iteration for a fixed policy of the decision maker and a value iteration for the robust optimization problem. Moreover, we show the existence of deterministic optimal policies for both players. This is in contrast to classical zero-sum games. In case the state space is the real line we show under some convexity assumptions that the interchange of supremum and infimum is possible with the help of Sion's minimax Theorem. Further, we consider the problem with special ambiguity sets. In particular we are able to derive some cases where the robust optimization problem coincides with the minimization of a coherent risk measure. In the final section we discuss two applications: A robust LQ problem and a robust problem for managing regenerative energy.