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A significant part of Real Madrid's season ticket decision is the uncertainty regarding construction progress. In setting probabilities for completion, club higher-ups took a pessimistic stance in an attempt to minimize the repercussions resultant of a potential delay both financially and on a PR level. The club's president, Florentino Perez, is a shrewd and connected businessman. At a networking event organized by the city of Madrid, Mr. Perez was introduced to a gentleman in the field of construction with vast experience in project management. Upon hearing of the club's dilemma, the consultant offered his services to Mr. Perez, claiming his expertise and sound judgement could aid the club in making this decision.
Mr. Perez is faced with a new decision; whether or not to hire the consultant. Once again, he asks for input from the club's analysts and tasks them with evaluating the benefit of hiring the consultant.
The analysts start by adding a new branch to the tree shown to the left. In this case, the club only makes a decision regarding ticket offering after receiving input from the consultant. An implicit assumption in this branch is that the analyst's prediction are always accurate. In other words, if the probability of construction being completed before the start of next season is 30%, then the consultant will predict that construction will be completed before the start of next season with the same probability of 30%. If the consultant predicts on-time completion, then the club can offer more than the current capacity of 40,000 tickets and not have to pay any refunds. If, on the other hand, he predicts delay and the club still decides to offer more than 40,000 tickets, then there will definitely be unfulfilled tickets that the club will be forced to refund. As shown in the figure, the new branch has an expected monetary value (EMV) of €14,365,528.48. The original tree has an EMV of €13,609,230.42. The expected value of the consultant's prediction is calculated as €14,365,528.48-€13,609,230.42= €756,298.06.
To that extent that the preceding analysis assumed the consultant is always right, the added expected value calculated above represents the expected value of perfect information (EVPI). What if the consultant's prediction were prone to a fault, as most things in life are? In that case, analysts are interested in the expected value of imperfect information (EVII).
They start by approaching Mr. Perez for more insight into the background of the consultant. Using his many connections, the president acquires extensive information about the consultant and his track record in construction projects, which he relays to the analysts.
The information is summarized in the figure to the left. The leftmost column represents the consultant's assessment of completion, whereas the top row represents the actual outcome that transpired. Historically, the consultant has a 90% chance of correctly predicting a construction project would be completed on time and an 81% chance of correctly predicting a delayed project.
The analysts use this information to construct the following conditional probabilities. Events 'complete' and 'delayed' define actual outcomes for construction projects. Events 'C' and 'D' represent consultant predictions for on-time completion and delay, respectively.
P(C|Complete)= 0.9
P(C|Delayed)= 0.19
P(D|Complete)= 0.1
P(D|Delayed)= 0.81
Next, the analysts adjust the decision tree for this decision by adding an additional branch shown to the left. The new branch is similar to the EVPI branch, albeit with a few modifications. Again, the club decides how many tickets to offer only after being informed of the consultant's prediction of construction progress, but unlike the previous case, the actual outcomes of construction progress might differ from the consultant's prediction. In other words, the probabilities in the right-most branches can be written as P(Complete|C), P(Complete|D), P(Delayed|C), and P(Delayed|D). In assuming perfect information, those probabilities were implicity set to 1. In this case, they are unknown to the analysts and should be calculated. In estimating those probabilities, the analysts utilize the concept of Bayesian revision, whereby the conditional probabilities introduced earlier are flipped using Bayes' theorem. The process is carried out using PrecisionTree and involves transforming the tree in the leftmost figure below to the one on the right.
From that, the analysts infer the following probabilities:
P(Complete|C)= 66.9975%
P(Delayed|C)= 33.0025%
P(Complete|D)= 5.0251%
P(Delay|D)= 94.9749%
P(C), the probability that the consultant predicts completion, is equal to 40.3%
P(D), the probability that the consultant predicts delay, is equal to 59.7%
The obtained probabilities are subsequently input into the new branch to calculate its Expected Monetary Value (EMV) as €14,106,766.5. The original tree has an EMV of €13,609,230.42. EVII is calculated as €14,106,766.5-€13,609,230.42= €497,536.08.
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