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N-discount optimality was introduced as a hierarchical form of policy- andvalue-function optimality, with Blackwell optimality lying at the top level ofthe hierarchy Veinott (1969); Blackwell (1962). We formalize notions of myopicdiscount factors, value functions and policies in terms of Blackwell optimalityin MDPs, and we provide a novel concept of regret, called Blackwell regret,which measures the regret compared to a Blackwell optimal policy. Our mainanalysis focuses on long horizon MDPs with sparse rewards. We show thatselecting the discount factor under which zero Blackwell regret can be achievedbecomes arbitrarily hard. Moreover, even with oracle knowledge of such adiscount factor that can realize a Blackwell regret-free value function, an$\epsilon$-Blackwell optimal value function may not even be gain optimal.Difficulties associated with this class of problems is discussed, and thenotion of a policy gap is defined as the difference in expected return betweena given policy and any other policy that differs at that state; we provecertain properties related to this gap. Finally, we provide experimentalresults that further support our theoretical results.

Issues concerning realizability of Blackwell optimal policies in reinforcement learning