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The decision tree for this model is shown to the left. The structural flow of the tree mimics that of the influence diagram. The club is weighing three options for ticket offering: 40,000, 50,000, or 60,000 tickets. The first offering ensures no refunds will be issued because all offered seats are currently available. The other two options are dependent upon the progress of construction at the start of next season. In that context, based on input from the construction company, club management predicts a pessimistic 70% chance of construction being delayed beyond September.
To model the uncertainty in ticket demand, club analysts assess subjective probabilities for different demand levels. They believe the club will not sell less than 20,000 tickets. Further, they argue, the club has one of the most passionate fanbases in the world. With that in mind, they predict a relatively high chance that demand will exceed the offered number of tickets, a presumption supported by historical records. The Bernabeu's maximum capacity of 80,000 is used as an upper bound for demand.
Next, analysts define an empirical cumulative density function (CDF) for the assessed probabilities. Utilizing values from the CDF, they use the Extended Pearson-Tukey (EP-T) method to approximate values for demand, which are translated to three distinct branches at each demand node in the tree. In essence, this allows them to model continuous demand using a discrete approximation, allowing for decision making using the tree.
In making a decision, the tree is linked to a mathematical model carried out using Excel. The club sells three types of season tickets: standard, premium, and VIP, at a cost of €200, €300, and €1,500 each, respectively. Of all tickets offered, 65% are standard, 25% are premium, and the rest are VIP. In the event of refunds, the club will reimburse the price of a ticket with a €100 bonus for good measure to appease disgruntled fans. Ticket prices, refund information, the number of offered tickets, and ticket demand are used as inputs to the model.
The following calculations are made to determine total revenue from season ticket sales. The minimum function is used in revenue calculations because the club cannot sell more tickets than it offers.
Standard Revenue= €200*65%*minimum(demand, offered)
Premium Revenue= €300*25%*minimum(demand, offered)
VIP Revenue= €1,500*10%*minimum(demand, offered)
# of Standard Refunds= 65%*# of Total Refunds
# of Premium Refunds= 25%*# of Total Refunds
# of VIP Refunds= 10%*# of Total Refunds
Standard Refunds= €200*# of Standard Refunds+€100
Premium Refunds= €300*# of Premium Refunds+€100
VIP Refunds= €1,500*# of VIP Refunds+€100
Total Refunds= Standard Refunds+Premium Refunds+VIP Refunds
Total Revenue= Standard Revenue+Premium Revenue+VIP Revenue-Total Refunds
The number of total refunds is determined as follows:
When the number of offered tickets is more than the current availability of 40,000 tickets:
If demand is less than 40,000, no refunds are necessary.
If demand is more than 40,000 and less than tickets offered, the number of refunds is calculated as demand-40,000.
If demand is more than 40,000 and more than tickets offered, the number of refunds is tickets offered-40,000.
When the number of offered tickets is less than or equal the current availability of 40,000 tickets, no refunds are necessary. The only applicable scenario in the context of this decision is the first option of offering 40,000 tickets.
Calculations are then carried out and linked to the tree accordingly. The optimal decision is identified as the third option of offering 60,000 tickets, with an expected revenue from ticket sales of €13,609,230.42.
A risk profile and a cumulative version of it for the club's decision are shown above. The risk profile (left) shows possible revenue outcomes based on each option and their respective probabilities. The decision to offer 40,000 tickets has two different expected outcomes, while that for 50,000 tickets has three. In offering 60,000 tickets, the club faces five potential outcomes for revenue ranging from €8,875,000 to €21,300,000. The scope of that range, coupled with the highest standard deviation amongst all three options, is a testament to the riskiness of this option. The cumulative risk profile offers significant insight. At no point in the graph does the decision to offer 50,000 tickets lie to the right of the other two options, implying that at no point is it the optimum choice under current assumptions. This is attributable to the all-or-nothing situation with the unavailable seats. If seats were returned to use gradually rather than in a bulk of 20,000, the decision might very well change. As it stands, the potential reduction in refunds offered by the 50,000 seats decision is not enough to overshadow the certainty associated with the 40,000 seats decision or the added revenue of the risky 60,000 seats option. This is unsurprising given that demand based on EP-T approximation is much more likely to be above 50,000 tickets than below.
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