Characterization of Efficient Influence Function for Off-Policy Evaluation Under Optimal Policies

Off-policy evaluation (OPE) provides a powerful framework for estimating the value of a counterfactual policy using observational data, without the need for additional experimentation. Despite recent progress in robust and efficient OPE across various settings, rigorous efficiency analysis of OPE under an estimated optimal policy remains limited. In this paper, we establish a concise characterization of the efficient influence function (EIF) for the value function under optimal policy within canonical Markov decision process models. Specifically, we provide the sufficient conditions for the existence of the EIF and characterize its expression. We also give the conditions under which the EIF does not exist.