I am reading Sutton & Bartos's Book "Introduction to reinforcement learning". In this book, the defined the optimal value function as:
$$v_*(s) = \max_{\pi} v_\pi(s),$$ for all $s \in \mathcal{S}$.
Do we take the max over all deterministic policies, or do we also look at stochastic policies (is there an example where a stochastic policy always performs better than a deterministic one?)
My intuition is that the value function of a stochastic policy is more or less a linear combination of the deterministic policies it tries to model, however, there are some self-references, so it is not mathematically true).
If we do look over all stochastic policies, shouldn't we take the supremum? Or do we know, that the supremum is achieved, and therefore it is truly a maximum?