Consider an undiscounted MDP having three states, (1, 2, 3), with rewards $-1$, $-2$, $0$, respectively. State 3 is a terminal state. In states 1 and 2 there are two possible actions: $a$ and $b$. The transition model is as follows:
- In state 1, action $a$ moves the agent to state 2 with probability 0.8 and makes the agent stay put with probability 0.2.
- In state 2, action $a$ moves the agent to state 1 with probability 0.8 and makes the agent stay put with probability 0.2.
- In either state 1 or state 2, action $b$ moves the agent to state 3 with probability 0.1 and makes the agent stay put with probability 0.9.
Answer the following questions:
1. What can be determined qualitatively about the optimal policy in states 1 and 2?
2. Apply policy iteration, showing each step in full, to determine the optimal policy and the values of states 1 and 2. Assume that the initial policy has action $b$ in both states.
3. What happens to policy iteration if the initial policy has action $a$ in both states? Does discounting help? Does the optimal policy depend on the discount factor?