The value you obtained, 5.8, for the state $s$ is correct given the assumptions you've made. Let's go through your reasoning step-by-step to understand why this value makes sense.

Given Information and Assumptions

1. Transitions and Rewards:

   • With probability 0.4, the agent transitions to state $s'$ with reward $R' = 10$ and $v(s') = 0$ (assumed).
   • With probability 0.6, the agent transitions to state $s''$ with reward $R'' = 3$ and $v(s'') = 0$ (assumed).

2. Discount Factor:

   • $\gamma = 0.5$.

3. Formula:

   • The Bellman equation for the expected value of state $s$ is: $v(s) = P(s' \mid s) \cdot (R' + \gamma \cdot v(s')) + P(s'' \mid s) \cdot (R'' + \gamma \cdot v(s''))$ Given: $v(s) = 0.4 \times (10 + 0.5 \times 0) + 0.6 \times (3 + 0.5 \times 0)$

Calculation

Let's break down the calculation step-by-step:

1. Substitute the values:

$v(s) = 0.4 \times (10 + 0.5 \times 0) + 0.6 \times (3 + 0.5 \times 0)$

Since $v(s') = 0$ and $v(s'') = 0$ by assumption:

$v(s) = 0.4 \times (10 + 0) + 0.6 \times (3 + 0)$

2. Simplify the expression:

$v(s) = 0.4 \times 10 + 0.6 \times 3$

$v(s) = 4 + 1.8$

3. Compute the final value:

$v(s) = 5.8$

Conclusion

The value of state $s$ is 5.8.

Why is this the answer?

• The value $5.8$ is correct based on the assumption that $v(s') = 0$ and $v(s'') = 0$.
• The calculation uses the probabilities of transitioning to each state and the associated immediate rewards, along with the discount factor and assumed future state values.
• Since both $v(s')$ and $v(s'')$ are assumed to be 0, the contribution from the future discounted values ($\gamma \cdot v(s')$ and $\gamma \cdot v(s'')$) is also 0.

Questions for Further Exploration

1. What happens to the value $v(s)$ if we change the assumed values of $v(s')$ and $v(s'')$?
2. How does changing the discount factor $\gamma$ affect the value $v(s)$?
3. What if the transition probabilities to $s'$ and $s''$ were different? How would that impact $v(s)$?
4. What role do immediate rewards play in determining the value of a state in an MDP?
5. Can you calculate the value of $s$ using different assumed values for $v(s')$ and $v(s'')$?

Tip

In reinforcement learning, the value of a state represents the expected return (cumulative future rewards) the agent can obtain starting from that state. Always check the assumptions (like values of $v(s')$ and $v(s'')$) to understand their impact on the calculation.