The Barcelona Football Club is considering the signing of a player of international fame. The problem is that the player has a reputation for having a weak knee. The probability that the club assigns to the event that the player is injured during the season (state θ1) is 30%. The expected revenue is €18 million if there are no injuries (state θ2) and €3 million if there is an injury (state θ1). Assume that the club is risk neutral and wants to maximize the expected profit.
a) If the cost of the contract is €15 million, what is the club's optimal decision? What is the expected profit?
We now consider the possibility that the club, before making its decision, can have the player pass a medical examination. Doctors can issue a negative report (β1), suggesting that his knee is not strong enough to endure the season, or positive (β2), suggesting that his knee is fine. The probability that the medical report is negative when the knee is really bad, P(β1¦θ1), is 80%, the probability that the report is positive when the knee is really good, P(β2¦θ2), is also 80% (in other words, in each state there is a 20% chance that the doctors are wrong).
b) Represent the decision tree when the medical examination is done.
c) Compute the joint probabilities P(βj, θi), the probabilities of the signals P(βj), and the conditional probabilities P(θi¦βj).
d) What is the club's optimal strategy? What is the expected profit?
e) What is the value of the doctors' opinion?