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Consider an Markov Decision Process with 6 states s∈{0,1,2,3,4,5} and 2 actions a∈{C,M}, defined by the following transition probability functions

For states 1, 2, and 3:

T(s,M,s−1)=1
T(s,C,s+2)=0.7
T(s,C,s)=0.3

1 answer

To analyze the given Markov Decision Process (MDP) with 6 states and 2 actions, we need to clarify the transition probabilities for each state when the actions C (Continue) and M (Move) are taken.

Here's a breakdown of the transition probabilities for actions taken in states 1, 2, and 3 based on the information provided:

Transition Probabilities

  1. For States 1, 2, and 3:

    • Action M (Move):
      • From state 1: \( T(1, M, 0) = 1 \) (transition to state 0)
      • From state 2: \( T(2, M, 1) = 1 \) (transition to state 1)
      • From state 3: \( T(3, M, 2) = 1 \) (transition to state 2)
    • Action C (Continue):
      • From state 1:
        • \( T(1, C, 3) = 0.7 \) (transition to state 3)
        • \( T(1, C, 1) = 0.3 \) (stay in state 1)
      • From state 2:
        • \( T(2, C, 4) = 0.7 \) (transition to state 4)
        • \( T(2, C, 2) = 0.3 \) (stay in state 2)
      • From state 3:
        • \( T(3, C, 5) = 0.7 \) (transition to state 5)
        • \( T(3, C, 3) = 0.3 \) (stay in state 3)

  2. For States 0, 4, and 5:

    • The transition probabilities for states 0, 4, and 5 need to be defined as they haven't been provided. Let's assume:
      • From state 0, assuming it is an absorbing state:
        • \( T(0, a, 0) = 1 \) for both actions \( a \in {C, M} \)
      • For state 4:
        • Since we don't have specific values provided, we could set some hypothetical transitions to connect state 4 to states 3 or 5.
      • From state 5:
        • Similar to state 0, we could assume it is also absorbing.
        • \( T(5, a, 5) = 1 \) for both actions \( a \in {C, M} \)

Summary of Transitions

  • Action M leads to a deterministic move to the previous state (or state 0).
  • Action C provides a probabilistic nature to either progress two states up or stay in the current state, with a higher probability of progressing to the next state in the sequence.

Note:

The defined transitions will allow us to model the decision-making process in terms of expected rewards, value estimation, and optimal policy if we incorporate a reward structure and discount factor. If you have specific rewards for each state-action pair or other questions regarding value iteration, policy iteration, or specific scenarios in this MDP, please provide that information!