# Why is the better policy defined with respect to all the states values being greater?

In Sutton & Barto (Section 3.6 - Optimal Policies and Optimal Value Functions), they say that :

Value functions define a partial ordering over policies. A policy $$\pi$$ is defined to be better than or equal to a policy $$\pi^{'}$$ if its expected return is greater than or equal to that of $$\pi^{'}$$ for all states. In other words, $$\pi \ge \pi^{'}$$ if and only if $$v_\pi(s) \ge v_\pi^{'}(s)$$ for all $$s \in \mathbb{S}$$.

My question is, why is the better policy defined with respect to all the states values being correspondingly greater instead of a combined metric of all state values of a policy?

If there is a policy that gives me the highest rewards on 99 out of 100 states, but gives me a lower reward on the last state compared to a second policy (which performs poorly on the other 99 states), would this first policy not be considered an optimal policy according to the definition above?

No, the answer of @foreverska is wrong, otherwise they would have said “better givena specific $$\mu(s)$$”. The reason is simply that given 2 policies, where one performs better than the other only in a subset of states, then you can create a strictly better third policy combining the two, where you consider the first one in the states where it outperforms the second one, and the other way around