policy-indexed state-value function

$v_{\pi }(s)={\mathbb{E}}_{\pi }[G_t|S_t=s]$ The policy-indexed state-value function $v_{\pi }(s)$ is the expected total future reward $G_t$ (excludes $R_t$) starting at state $s$ if the agent follows the policy $\pi$ at $s$ til the end in an MDP. This reflects whether or not transitioning to state $s$ is rewarding in the long-term.