Math Problem Statement

In a simple MDP, an agent is in a state s, and the actions it can take can lead to the following outcomes: • With probability 0.4, the agent transitions to state , with reward , and 5 IIT KHARAGPUR AI4ICPS I HUB FOUNDATION Hands-on Approach to AI, Cohort-2, July – October 2024 Assignment 7: Reinforcement Learning ϵ s′ R = 10 v(s′) = • With probability 0.6, the agent transitions to state , with reward , and 3. The discount factor is 0.5. Using Bellman equation, find the expected value of state . s′′ R = 2 v(s′) = γ s

Solution

Based on the image you provided, it seems that you're dealing with a Markov Decision Process (MDP) problem where you need to calculate the expected value of state ss using the Bellman equation. Let's break down the problem step by step.

Problem Details:

  • Transition Probabilities:
    • With probability 0.40.4, the agent transitions to state ss' with a reward R=10R = 10.
    • With probability 0.60.6, the agent transitions to state ss'' with a reward R=2R = 2.
  • Discount Factor (γ\gamma): 0.50.5.
  • Values of v(s)v(s') and v(s)v(s''): These values are not explicitly given in the problem. Therefore, let's assume them as v(s)=xv(s') = x and v(s)=yv(s'') = y for now.

Bellman Equation:

The Bellman equation for the state-value function V(s)V(s) is given by: [ v(s) = \mathbb{E}[G_t \mid S_t = s] = \mathbb{E}[R_{t+1} + \gamma v(S_{t+1}) \mid S_t = s] ] In this context, the expected value of v(s)v(s) can be calculated as: v(s)=0.4×(Rs+γv(s))+0.6×(Rs+γv(s))v(s) = 0.4 \times (R_{s'} + \gamma v(s')) + 0.6 \times (R_{s''} + \gamma v(s'')) Substituting the values: v(s)=0.4×(10+0.5×v(s))+0.6×(2+0.5×v(s))v(s) = 0.4 \times (10 + 0.5 \times v(s')) + 0.6 \times (2 + 0.5 \times v(s''))

Step-by-Step Calculation:

  1. Substitute v(s)v(s') and v(s)v(s''):

    • If you have specific values for v(s)v(s') and v(s)v(s''), substitute them here.

    But if these values are not given, let's assume:

    • v(s)=xv(s') = x
    • v(s)=yv(s'') = y
  2. Expression: v(s)=0.4×(10+0.5x)+0.6×(2+0.5y)v(s) = 0.4 \times (10 + 0.5x) + 0.6 \times (2 + 0.5y)

  3. Simplify the Expression: v(s)=0.4×10+0.4×0.5x+0.6×2+0.6×0.5yv(s) = 0.4 \times 10 + 0.4 \times 0.5x + 0.6 \times 2 + 0.6 \times 0.5y v(s)=4+0.2x+1.2+0.3yv(s) = 4 + 0.2x + 1.2 + 0.3y v(s)=5.2+0.2x+0.3yv(s) = 5.2 + 0.2x + 0.3y

Conclusion:

  • The final expression for v(s)v(s) is: v(s)=5.2+0.2x+0.3y\boldsymbol{v(s) = 5.2 + 0.2x + 0.3y}
  • To get a numerical value for v(s)v(s), you would need the specific values for v(s)v(s') and v(s)v(s''). If those are not provided, this is the most simplified expression based on the given data.

Would you like more details or have any further questions?

Related Questions:

  1. What is the interpretation of the discount factor (γ\gamma) in reinforcement learning?
  2. How does the Bellman equation help in solving MDP problems?
  3. What assumptions are made when calculating expected value in MDPs?
  4. Can you explain how the transition probabilities affect the state-value function?
  5. How would changing the reward values impact the value of v(s)v(s)?

Tip:

When working with MDPs, always ensure that you have the correct and complete values for all states and rewards to accurately compute the expected state value.

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Math Problem Analysis

Mathematical Concepts

Markov Decision Process (MDP)
Bellman Equation
Expected Value Calculation

Formulas

Bellman equation for state-value function

Theorems

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Suitable Grade Level

Advanced undergraduate level