# Why is the value function obtained from a greedy policy different from its original value function (i.e. $V_k \neq V_{\pi_k }$)?

Consider a vector of values $$V_k$$ and consider the related value $$V_{\pi_k}$$ obtained by coming the policy $$\pi_k$$ by acting greedily according to it. i.e.

$$\pi_k(i) := arg \min_{a \in A} \{ R(i,a) + \alpha \sum_{j \in S} P_{i,j}(a ) V_k \}$$

where $$P_{ij}( a ) = Pr[j \mid i, a]$$

my question is why is:

$$V_k \neq V_{\pi_k }$$

my intuition (which is obviously wrong) says that we are acting according to the value function $$V_k$$ so we should get the same value function out. Why is this reasoning incorrect?

Note that I am sure we could cook up a boring example that shows Im wrong. I don't want that or need that. That won't provide insights to WHY I am wrong, just that I am which I already know.

Your intuition is off because the policy $$\pi$$ used to form $$V_k$$ (or what I will denote as $$V^{\pi}$$) will actually be different than the policy $$\pi_k$$ if $$\pi$$ is not optimal. We can show they will be different in the event $$\pi$$ is not optimal by investigating the following. We can first state the following definitions
\begin{align} V^{\pi}(s) &= R(s, \pi(s)) + \gamma \mathbb{E}_{s' \sim P(s,\pi(s))} \left[V^{\pi}(s')\right] \\ Q^{\pi}(s,a) &= R(s,a) + \gamma \mathbb{E}_{s' \sim P(s,a)} \left[V^{\pi}(s')\right] \end{align}
If we make note of the above definitions, it is clear that $$V^{\pi}(s) = Q^{\pi}(s, \pi(s))$$. We can also use these definitions to restate $$\pi_k$$ in the following manner
\begin{align} \pi_k(s) &= \arg \min_{a \in A(s)} \left\lbrace R(s,a) + \gamma \mathbb{E}_{s' \sim P(s,a)} \left[V^{\pi}(s')\right] \right\rbrace \\ &= \arg \min_{a \in A(s)} Q^{\pi}(s,a) \\ &= \arg \min_{a \in A(s)} \left\lbrace Q^{\pi}(s,a) - Q^{\pi}(s, \pi(s))\right\rbrace \end{align}
If we look at the final expression for $$\pi_{k}(s)$$, it is clear that for a given state $$s$$, $$\pi_k(s)$$ will be a better action than $$\pi(s)$$ unless $$\pi(s)$$ is already an optimal action for the given state $$s$$. This implies that $$V^{\pi_k}(s) \leq V^{\pi}(s)$$ for all $$s \in S$$. The way one can view it is that the $$\arg \min$$ step to construct $$\pi_k(s)$$ is effectively saying "For each state $$s$$, choose the action $$a$$ that is most optimal relative to $$\pi(s)$$ to make as the action choice for $$\pi_k(s)$$".