Calculate the missing value in the Bellman equation from reward, discount factor, and next state value to get the updated state value.
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Bellman Equation Formula
The calculator uses the simplified one-step Bellman equation. It assumes one immediate reward and one next-state value.
V(s) = R + \gamma V(s')
To solve for a missing input, the same equation is rearranged:
R = V(s) - \gamma V(s')
\gamma = \frac{V(s) - R}{V(s')}V(s') = \frac{V(s) - R}{\gamma}- V(s) is the updated value of the current state.
- R is the immediate reward received after taking the action.
- γ is the discount factor, usually between 0 and 1.
- V(s’) is the value of the next state.
The calculator lets you leave exactly one field empty. If you leave Updated Value empty, it calculates V(s). If you leave Reward empty, it solves for R. If you leave Discount Factor empty, it solves for γ. If you leave Value of Next State empty, it solves for V(s’).
Discount Factor and Bellman Value Interpretation
| Discount factor γ | Meaning | Effect on V(s) |
|---|---|---|
| 0 | Only the immediate reward matters. | V(s) = R |
| 0.1 to 0.4 | Future value has a small influence. | The updated value stays close to the reward. |
| 0.5 to 0.8 | Immediate and future values both matter. | The next-state value noticeably changes V(s). |
| 0.9 to 1 | Future value is weighted heavily. | V(s) is strongly affected by V(s’). |
| Input left blank | What is calculated | Required nonzero condition |
|---|---|---|
| Reward R | R = V(s) – γV(s’) | None |
| Discount factor γ | γ = (V(s) – R) / V(s’) | V(s’) cannot be 0 |
| Next-state value V(s’) | V(s’) = (V(s) – R) / γ | γ cannot be 0 |
| Updated value V(s) | V(s) = R + γV(s’) | None |
Example Problems
Example 1: Calculate the updated value
Suppose the reward is 5, the discount factor is 0.9, and the next-state value is 20.
V(s) = 5 + 0.9(20)
V(s) = 5 + 18 = 23
The updated value is 23.
Example 2: Calculate the reward
Suppose the updated value is 14, the discount factor is 0.8, and the next-state value is 10.
R = 14 - 0.8(10)
R = 14 - 8 = 6
The reward is 6.
FAQ
What does the Bellman equation calculate?
The Bellman equation calculates the value of a state by combining the immediate reward with the discounted value of the next state. In this simplified one-step version, it answers: what is this state worth if you receive reward R now and then move to a next state worth V(s’)?
Why is the discount factor usually between 0 and 1?
The discount factor controls how much future value counts. A value near 0 makes the calculation focus mostly on the immediate reward. A value near 1 makes the future state value almost as important as the immediate reward. In many reinforcement learning problems, γ is kept between 0 and 1 to prevent future rewards from growing without bound.
Can the next-state value or reward be negative?
Yes. A negative reward can represent a penalty or cost. A negative next-state value can represent a bad future state. The same formula still applies, but negative values will reduce the updated state value depending on the discount factor.
