**1.** You are given the following payoff table for a decision analysis problem.

Alternative State of Nature

S1 S2 S3

A1 4000 0 0

A2 0 2000 0

A3 3000 0 1000

Prior probability 0.2 0.5 0.3

**Answer the following questions.**

**a)** Which alternative should be chosen under the maximin payoff criterion?

**b)** Which alternative should be chosen under the maximum likelihood criterion?

**c)** Which alternative should be chosen under Bayes’ decision rule?

**d)** Using Bayes’ decision rule, do sensitivity analysis graphically with respect to the prior probability of states S1 and S2 (without changing the prior probability of states S3) to determine the crossover point(s) where the decision shifts from one alternative to the other. Then use algebra to calculate this crossover point(s)

**e)** Find EVPI.

**f)** You are given the opportunity to spend $1000 to obtain more information about which state of nature is likely to occur. Given your answer to part e), might it be worthwhile to spend this money?

**2.**You are given the opportunity to guess whether a coin is fair or two-headed, where the prior probabilities are 0.5 for each of these possibilities. If you are correct, you win $5; otherwise, you lose $5. You are also given the option of seeing a demonstration flip of coin before making your guess.

**a)** Develop a payoff table with identifying the alternative actions, states of nature.

**b)** Calculate the posterior probability if the demonstration flip is a tail. Do the same if the flip is a head.

Hint: Draw a decision tree.

**c)** Now suppose that you must pay to see the demonstration flip. What is the most that you should be willing to pay?

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