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Because of the ample subjectivity that went into making a decision, the club wants to observe how the optimal decision would be affected if certain elements within the context of the decision were to change. For that, club analysts utilize sensitivity analysis, varying the following quantities between 0 and 200% of their original values:
The probability of construction being done before the next season commences.
The amount of extra refund extended by the club to buyers of unfulfilled tickets.
The tornado diagram on the left is the product of one-way sensitivity analysis. It is used to highlight the most influential variables over the specified range. The blue bars represent the numeric change in the expected value of the decision, in this case the expected season ticket revenue, as each variable is varied across its range. It is revealed that the extra refund amount is second to probability of construction in terms of importance over this range.
Shown above are the sensitivity graph and strategy region resultant of a one-way sensitivity of the probability of completing construction on time. The probability is varied between 0 and 60%. The graph on the left shows a plateau followed by an increasing linear function. This pattern can be understood in the context of the strategy region. The plateau applies for probabilities below 22% (roughly); over this interval, offering 40,000 tickets is the optimal choice. Beyond that, expected revenue increases linearly, and the preferred decision switches to 60,000 tickets. Again, at no point is offering 50,000 tickets the best option; however, for a probability of 21.978%, a value obtained using the Goal Seek feature in Excel, all three options yield the same expected revenue, and the club is indifferent between them.
The same analysis can be applied to the graphs related to the extra refund variable, which are shown below. For refund values below €155 (roughly), expected revenue decreases linearly, and the 60,000 tickets option is preferred. Beyond that, the optimal decision switches to 40,000 tickets. As discussed previously, the 50,000 ticket decision is never preferred over this range. Interestingly, graphs for the two preceding analyses are mirror images of each other.
Club analysts extend their analysis of those variables by conducting a two-way sensitivity analysis whereby the variables are varied simultaneously. The resulting strategy region is shown to the right, with the probability of completing construction on the y-axis and the value of extra refund on the x-axis. Highlighted in the figure is the current decision context, with 30% probability and €100 bonus refund. The figure reveals clear separation between the decision to offer 40,000 tickets and 60,000 tickets. Note the absence of the 50,000 tickets alternative. It can be inferred that a greater probability of construction completion favors the 60,000 tickets option. That is to be expected as the riskiness of this alternative is reduced. On the other hand, an increase in extra refund favors the decision to offer 40,000 tickets, a natural byproduct of the fact that with this decision, the club is guaranteed not to issue refunds, unlike with the other alternatives.
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