Question

Alden Construction is bidding against Forbes Construction for a project. Alden believes that Forbes’s bid is a random variable B with the following mass function: P(B $6,000) .40, P(B $8,000) .30, P(B $11,000) .30. It will cost Alden $6,000 to complete the project. Use each of the decision criteria of this section to determine Alden’s bid. Assume that in case of a tie, Alden wins the bidding. (Hint: Let p Alden’s bid. For p 6,000, 6,000 p 8,000, 8,000 p 11,000, and p 11,000, determine Alden’s profit in terms of Alden’s bid and Forbes’s bid.)

Answer 1

Let p be the Alden's bid,

For p 6000

Forbes’s bid	Probability	Alden wins the bid	Payoff
$6,000	0.4	Yes	p - 6000
$8,000	0.3	Yes	p - 6000
$11,000	0.3	Yes	p - 6000

Expected payoff = 0.4 * (p - 6000) + 0.3 * (p - 6000) + 0.3 * (p - 6000) = (p - 6000)

Since p 6000, expected payoff is negative or 0 for p = 6000.

For 6000 < p 8000

Forbes’s bid	Probability	Alden wins the bid	Payoff
$6,000	0.4	No	0
$8,000	0.3	Yes	p - 6000
$11,000	0.3	Yes	p - 6000

Expected payoff = 0.4 * 0 + 0.3 * (p - 6000) + 0.3 * (p - 6000) = 0.6 (p - 6000)

For p = 8000,  Expected payoff = 0.6 * (8000 - 6000) = $1200

For 8000 < p 11000

Forbes’s bid	Probability	Alden wins the bid	Payoff
$6,000	0.4	No	0
$8,000	0.3	No	0
$11,000	0.3	Yes	p-6000

Expected payoff = 0.4 * 0 + 0.3 * 0 + 0.3 * (p-6000) = 0.3(p - 6000)

For p = 11000,  Expected payoff = 0.3 * (11000 - 6000) = $1500

For p > 11000

Forbes’s bid	Probability	Alden wins the bid	Payoff
$6,000	0.4	No	0
$8,000	0.3	No	0
$11,000	0.3	No	0

Expected payoff = 0.4 * 0 + 0.3 * 0 + 0.3 * 0 = 0

So, the maximum expected payoff if for p = 11,000.

Thus, Alden’s bid should be $11,000