# When a tennis player serves he gets two chances to serve

When a tennis player serves he gets two chances to serve in bounds. If he fails to do so twice he loses the point. If he attempts to serve an ace he serves in bounds with probability If he serves a lob he serves in bounds with probability . If he serves an ace in bounds he wins the point with probability . With an inbounds lob he wins the point with probability If the cost is +1 for each point lost and -1 for each point won the problem is to determine the optimal serving strategy to minimize the (long-run) expected average cost per point. (a) Formulate this problem as a Markov decision process by identifying the states and decisions and then finding the Cik. (b) Identify all the (stationary deterministic) policies. For each one find the transition matrix and write an expression for the (longrun) expected average cost per point in terms of the unknown steady-state probabilities (0 1 . . . M). (c) Use your OR Courseware to find these steady-state probabilities for each policy. Then evaluate the expression obtained in part (b) to find the optimal policy by exhaustive enumeration.

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