# Bellman Equations

In this gridworld example, once the agent selects an action,

• it always moves in the chosen direction (contrasting general MDPs where the agent doesn’t always have complete control over what the next state will be), and
• the reward can be predicted with complete certainty (contrasting general MDPs where the reward is a random draw from a probability distribution).

In this simple example, we saw that the value of any state can be calculated as the sum of the immediate reward and the (discounted) value of the next state.

## There are 3 more Bellman Equations!

In this video, you learned about one Bellman equation, but there are 3 more, for a total of 4 Bellman equations.

All of the Bellman equations attest to the fact that value functions satisfy recursive relationships.

For instance, the Bellman Expectation Equation (for $v_\{pi}$​) shows that it is possible to relate the value of a state to the values of all of its possible successor states.

After finishing this lesson, you are encouraged to read about the remaining three Bellman equations in sections 3.5 and 3.6 of the textbook. The Bellman equations are incredibly useful to the theory of MDPs.

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