Two promising drugs—X14Alpha and X14Beta—are ready for testing to combat rare, but dangerous virus X14. If one of the drugs proves to have high effectiveness, 500 lives could be saved in the coming winter, but only 200 lives would be saved if the a drug proved to have just moderate effectiveness. Also, the tests may prove inconclusive, which would mean no drugs could be released.

Unfortunately, only 10 patients are available to serve as test subjects. Researchers can test X14Alpha on all 10, or X14Beta on all 10, or split the subjects 5 and 5 between the drugs, with the most successful being considered for release. The following table shows the predicted probabilities of different kinds of test results depending on which subjects are tested.

(a) Taking states i = 1, 2, 3 for the high, moderate, and inconclusive outcomes, identify all the elements of definition 9.44 to formulate this drug testing challenge as a MDP maximizing the expected number of saved lives.

(b) Sketch a digraph depicting your model of part

(a) over nodes for 3 states and arcs for possible transitions. Clearly label all transition arcs with the decision to which they are attached, the associated probability, and the implied reward.

(c) Form the functional equations for all states i.

(d) Solve the equations of

(c) to compute an the optimal testing action.

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Test Design High Probability of Proven Effectiveness Moderate Inconclusive All X14Alpha 0.30 0.40 0.30 All X14Beta 0.40 0.20 0.40 Split 0.15 0.60 0.25