Bellman Equation Calculator

Bellman Equation Calculator is useful calculator in the concept of dynamic programming and reinforcement learning. By using this calculator you will be capable of instant decision making in various fields from economics to Artificial intelligence.

Formula:

The formula is:

$V(s) = R(s) + \gamma \times V(s’)$

Where:

$V(s)$ stands for the value of the current state $s$ .
$R(s)$ stands for the reward received in the current state $s$ .
$\gamma$ (gamma) is the discount factor, which represents the importance of future rewards.
$V(s’)$ stands for the value of the next state $s’$ .

Variables

Variable	Meaning
$V(s)$	Value of the current state $s$
$R(s)$	Reward received in the current state $s$
$\gamma$	Discount factor (between 0 and 1)
$V(s’)$	Value of the next state $s’$

Solved Examples:

Example 1:

Given:

Reward in the current state $R(s)$ = 10
Discount factor $\gamma$ = 0.9
Value of the next state $V(s’)$ = 20

Calculation	Instructions
Step 1: $V(s) = R(s) + \gamma \times V(s’)$	Start with the formula.
Step 2: $V(s) = 10 + 0.9 \times 20$	Replace $R(s)$ , $\gamma$ , and $V(s’)$ with the given values.
Step 3: $V(s) = 10 + 18$	Multiply $\gamma$ by $V(s’)$ to get 18.
Step 4: $V(s) = 28$	Add the reward to the discounted value of the next state.

Answer: The value of the current state $V(s)$ is 28.

Example 2:

Given:

Reward in the current state $R(s)$ = 5
Discount factor $\gamma$ = 0.8
Value of the next state $V(s’)$ = 15

Calculation	Instructions
Step 1: $V(s) = R(s) + \gamma \times V(s’)$	Start with the formula.
Step 2: $V(s) = 5 + 0.8 \times 15$	Replace $R(s)$ , $\gamma$ , and $V(s’)$ with the given values.
Step 3: $V(s) = 5 + 12$	Multiply $\gamma$ by $V(s’)$ to get 12.
Step 4: $V(s) = 17$	Add the reward to the discounted value of the next state.

Answer: The value of the current state $V(s)$ is 17.

What is Bellman Equation Calculator?

The Bellman Equation is a the key concept in reinforcement learning and dynamic programming. It is used to calculate the value of a state in a Markov Decision Process (MDP), taking into account both the immediate reward and the future rewards that can be obtained from the next states. The Bellman Equation is central to algorithms that seek to find the optimal policy, which is a strategy that maximizes the cumulative reward over time.

The formula $V(s) = R(s) + \gamma \times V(s’)$ is used to recursively compute the value of a state $s$ , considering both the reward at that state and the discounted value of the subsequent state $s’$ . The discount factor $\gamma$ ensures that future rewards are given less weight than immediate rewards, reflecting the uncertainty of future events.

Conclusion

The Bellman Equation Calculator provides a powerful framework for analyzing decision-making processes in dynamic environments. By understanding the relationship between current and future rewards, practitioners can make informed decisions and optimize their strategies across different domains.