If you know the rules of the world perfectly, you don't need to guess. Dynamic Programming allows an agent to calculate the perfect strategy through recursive logic.
1The Bellman Equations
The fundamental insight of Dynamic Programming (DP) is that the value of a state can be defined recursively. The Bellman Equation tells us that the value of being in state $s$ is the immediate reward we expect, plus the discounted value of where we might end up next. By turning this equation into an 'Update Rule', we can iteratively refine our estimates of how 'good' every position in our world truly is.
2Solving the MDP
There are two primary ways to find the optimal solution: Policy Iteration and Value Iteration. Policy Iteration alternates between evaluating the current strategy and improving it. Value Iteration is faster—it effectively combines these steps by directly updating the state values to the maximum possible expected return at each step. Both are guaranteed to converge to the optimal solution for finite MDPs with known transitions.
3Computational Limits
While DP is mathematically perfect, it suffers from the Curse of Dimensionality. Because it requires a 'Sweep' over every single state in the environment, it becomes impossibly slow for complex worlds like Chess, Go, or robotics. This is why we move toward Approximation and Model-Free methods in later stages—but the core principles of DP remain the target they all aim to hit.
