Question

Following is the payoff table for the Pittsburgh Development Corporation (PDC) Condominium Project. Amounts are in millions of dollars.

	State of Nature
Decision Alternative	Strong Demand S1	Weak Demand S2
Small complex, d1	8	7
Medium complex, d2	14	5
Large complex, d3	20	-9

Suppose PDC is optimistic about the potential for the luxury high-rise condominium complex and that this optimism leads to an initial subjective probability assessment of 0.8 that demand will be strong (S₁) and a corresponding probability of 0.2 that demand will be weak (S₂). Assume the decision alternative to build the large condominium complex was found to be optimal using the expected value approach. Also, a sensitivity analysis was conducted for the payoffs associated with this decision alternative. It was found that the large complex remained optimal as long as the payoff for the strong demand was greater than or equal to $17.5 million and as long as the payoff for the weak demand was greater than or equal to -$19 million.

1. Consider the medium complex decision. How much could the payoff under strong demand increase and still keep decision alternative d₃ the optimal solution? If required, round your answer to one decimal place.
   
   The payoff for the medium complex under strong demand remains less than or equal to $ ??? million, the large complex remains the best decision.
   
2. Consider the small complex decision. How much could the payoff under strong demand increase and still keep decision alternative d₃ the optimal solution? If required, round your answer to one decimal place.
   
   The payoff for the small complex under strong demand remains less than or equal to $ ??? million, the large complex remains the best decision.

Answer 1

Probability that the demand will be strong is 0.8 (S1)

Probability that the demand will be weak is 0.2 (S2)

Decision Alternative	Strong Demand (S1) (million $)	Weak Demand (S2) (million $)
Small Complex (d1)	8	7
Medium Complex (d2)	14	5
Large Complex (d3)	20	-9

Finding the EMV for all the three decision:

Large Complex: 0.8*20 + 0.2*(-9) = 16 – 1.8 = 14.2

Medium Complex: 0.8*14 + 0.2*5 = 11.2 + 1 = 12.2

Small Complex: 0.8*8 + 0.2*7 = 6.4 + 1.4 = 7.8

As per the sensitivity report, Large complex is optimal if the payoff for strong demand is more than or equal to $17.5 million and payoff for weak demand is more than or equal to -$19 million.

Here the payoff is $20 million for strong demand and -$9 million for weak demand. So the Large complex(d3) is the optimal solution giving EMV $14.2 million.

For the Large complex decision to be optimal, the EMV for the large complex (d3) should be the maximum.

Question – a:

Increase in payoff for the medium complex under strong demand:

Let the payoff of medium complex strong demand be p2.

We need to compare the EMV for d3 and d2 (medium complex) as there is no change in the EMV of small complex (d1).

For large complex (d3) to be the best decision:

0.8*p2 + 0.2*5 <= 14.2

0.8*p2 +1 <= 14.2

0.8*p2 <= 13.2

p1 <= 13.2/0.8 = 16.5

Hence, Answer is: The pay off for the medium complex under strong demand remains less than or equal to $16.5 million, the large complex remains the best decision.

Question – b:

Let the payoff of small complex strong demand be p3

We need to compare the EMV for d3 and d1 (small complex) as there is no change in the EMV of medium complex (d2).

For large complex (d3) to be the best decision:

0.8*p3 + 0.2*7 <= 14.2

0.8*p3 + 1.4 <= 14.2

0.8*p3 <= 12.8

p3 <= 12.8/0.8 = 16

Answer is: The payoff for the small complex under strong demand remains less than or equal to $16 million, the large complex remains the best decision.

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