(A simple bandit model). A decision maker observes a discrcte-time system which moves between states (s,, s2, sj, s4) according to the following transition probability matrix:
At each point of time, the decision maker may leave the system and receive a reward of R = 20 units, or alternatively remain in the system and receive a reward of r(s,) units if the system occupies state s,. If the decision maker decides to remain in the system, its state at the next decision epoch is determined by P. Assume a discount rate of 0.9 and that ds,) = i.
a. Formulate this model as a Markov decision process.
b. Use policy iteration to find a stationary policy which maximizes the expected total discounted reward
c. Find the smallest value of R so that it is optimal to leave the system in state 2.
d. Show for arbitrary P and r(.) that there exists a value R, in each state such that it is optimal to leave the system in state s, only if R 2 R,.
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