To find the expected value of the state $s$ using the Bellman equation, we first need to understand the components involved:

1. States and Rewards:

   • The agent is in state $s$.
   • From state $s$, the agent can transition to:
     • State $s'$ with probability 0.4, receiving a reward $R = 10$.
     • State $s''$ with probability 0.6, receiving a reward $R = 2$.

2. State Values:

   • The value of state $s'$ is given as $v(s') = 5$.
   • The value of state $s''$ is given as $v(s'') = 3$.

3. Discount Factor:

   • The discount factor is $\gamma = 0.5$.

Bellman Equation

The Bellman equation for the value of a state $s$ under a given policy is:

$v(s) = \sum_{s'} P(s' \mid s, a) \left[ R(s, a, s') + \gamma \, v(s') \right]$

In this problem, we are looking for the expected value of state $s$, which means we will sum over all possible outcomes (transitions) from state $s$.

Applying the Bellman Equation to the Problem

Given the transitions:

1. With probability 0.4:

   • Transition to $s'$ with reward $R = 10$ and $v(s') = 5$.

2. With probability 0.6:

   • Transition to $s''$ with reward $R = 2$ and $v(s'') = 3$.

The expected value of state $s$, $v(s)$, can be calculated as:

$v(s) = 0.4 \times \left(10 + 0.5 \times 5\right) + 0.6 \times \left(2 + 0.5 \times 3\right)$

Now, we compute the value step-by-step:

1. Calculate the contribution from the first transition to $s'$:

$0.4 \times (10 + 0.5 \times 5) = 0.4 \times (10 + 2.5) = 0.4 \times 12.5 = 5$

2. Calculate the contribution from the second transition to $s''$:

$0.6 \times (2 + 0.5 \times 3) = 0.6 \times (2 + 1.5) = 0.6 \times 3.5 = 2.1$

3. Sum the contributions to get the expected value $v(s)$:

$v(s) = 5 + 2.1 = 7.1$

Conclusion

The expected value of state $s$ is:

$\boxed{v(s) = 7.1}$

Would you like more details, or do you have any questions?

Related Questions to Explore Further:

1. How does the discount factor ($\gamma$) affect the value calculation in an MDP?
2. What is the difference between the Bellman expectation equation and the Bellman optimality equation?
3. How do policy changes affect the expected value of a state in an MDP?
4. What are the effects of different reward structures on the value function?
5. How can the Bellman equation be extended to handle continuous state and action spaces?

Tip:

When working with Markov Decision Processes, always ensure you understand the transitions, probabilities, and rewards for each action to accurately apply the Bellman equation.