We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell optimality. Many existing approaches to computing average-optimal policies solve for discount-optimal policies with a discount factor close to $1$, but they only work under strong or hard-to-verify assumptions on the MDP structure such as unichain or ergodicity. We are the first to highlight the shortcomings of the classical definition of Blackwell optimality, which does not lead to simple algorithms for computing Blackwell-optimal policies and overlooks the pathological behaviors of optimal policies as regards the discount factors. To resolve this issue, in this paper, we show that when the discount factor is larger than the Blackwell discount factor $\gamma_{\sf bw}$, all discount-optimal policies become Blackwell-and average-optimal, and we derive a general upper bound on $\gamma_{\sf bw}$. Our upper bound on $\gamma_{\sf bw}$, parametrized by the bit-size of the rewards and transition probabilities of the MDP instance, provides the first reduction from average and Blackwell optimality to discounted optimality, without any assumptions, along with new polynomial-time algorithms. Our work brings new ideas from polynomials and algebraic numbers to the analysis of MDPs. Our results also apply to robust MDPs, enabling the first algorithms to compute robust Blackwell-optimal policies.

Large reasoning models (LRMs) often consume excessive tokens, inflating computational cost and latency. We challenge the assumption that longer responses improve accuracy. By penalizing reasoning tokens using a discounted reinforcement learning setup (interpretable as a small token cost) and analyzing Blackwell optimality in restricted policy classes, we encourage concise yet accurate reasoning. Experiments confirm our theoretical results that this approach shortens chains of thought while preserving accuracy.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell opti-mality.

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell optimality. Many existing approaches to computing average-optimal policies solve for discount-optimal policies with a discount factor close to 1, but they only work under strong or hard-to-verify assumptions on the MDP structure such as unichain or ergodicity. We are the first to highlight the shortcomings of the classical definition of Blackwell optimality, which does not lead to simple algorithms for computing Blackwell-optimal policies and overlooks the pathological behaviors of optimal policies as regards the discount factors. To resolve this issue, in this paper, we show that when the discount factor is larger than the Blackwell discount factor \gamma_{\sf bw}, all discount-optimal policies become Blackwell- and average-optimal, and we derive a general upper bound on \gamma_{\sf bw} .

We introduce the Blackwell discount factor for Markov Decision Processes (MDPs). Classical objectives for MDPs include discounted, average, and Blackwell optimality. Many existing approaches to computing average-optimal policies solve for discounted optimal policies with a discount factor close to $1$, but they only work under strong or hard-to-verify assumptions such as ergodicity or weakly communicating MDPs. In this paper, we show that when the discount factor is larger than the Blackwell discount factor $\gamma_{\mathrm{bw}}$, all discounted optimal policies become Blackwell- and average-optimal, and we derive a general upper bound on $\gamma_{\mathrm{bw}}$. The upper bound on $\gamma_{\mathrm{bw}}$ provides the first reduction from average and Blackwell optimality to discounted optimality, without any assumptions, and new polynomial-time algorithms for average- and Blackwell-optimal policies. Our work brings new ideas from the study of polynomials and algebraic numbers to the analysis of MDPs. Our results also apply to robust MDPs, enabling the first algorithms to compute robust Blackwell-optimal policies.