optimal policy

An MDP must have an optimal policy ${\pi }_{\ast }$ that is either as good or better than all other policies (in terms of the rewards the agent receives). All optimal policies achieve the optimal state-value function and optimal action-value function (the latter of which can be used to produce the optimal policy). There is always a deterministic optimal policy for any MDP.

$${\pi }_{\ast }(a|s)=\{\begin{array}{ll}1& \text{if }a=\underset{a\in \mathcal{A}}{\mathrm{argmax}}{q}_{\pi }(s,a)\\ 0& \text{otherwise}\end{array}$$

A policy $\pi$ is more optimal than ${\pi }^{\prime }$ ($\pi \ge {\pi }^{\prime }$) if $v_{\pi }(s)\ge v_{{\pi }^{\prime }}(s),\forall s$.