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(β_j, θ_i), the probabilities of the signals P(β_j), and the conditional probabilities P(θ_i¦β_j).

d) What is the club's optimal strategy? What is the expected profit?

e) What is the value of the doctors' opinion?