# Value iteration

An alternative approach to policy iteration is provided by the value iteration algorithm. The main assumption is based on the empirical observation that the policy evaluation step converges rather quickly and it's reasonable to stop the process after a fixed number of steps (normally 1). In fact, policy iteration can be thought of as a game where the first player tries to find the correct values considering a stable policy, while the other player creates a new policy that is greedy with respect to the new values.

Clearly, the second step compromises the validity of the previous evaluation, forcing the first player to repeat the process. However, as the Bellman equation uses a single fixed point, the algorithm converges to a solution characterized by the fact that the policy doesn't change anymore and, consequently, the evaluation becomes stable. This process can be simplified by removing the policy improvement step and continuing the...