You work for Dell Computer Corp. and are bidding to supply computers to Loyola University. The cost to Dell to produce and supply the computers is $3,250,000. If you do not win the contract, your costs are zero, your revenues are zero, and thus your profits are also zero. Note that this is a low-bidder wins auction because Loyola wants to buy their computers as cheaply as possible. You dont know what the opposition will bid. But you believe that a bid of $5,000,000 is certain to lose; it is way too high and one of your competitors can beat that price and win the contract. You also believe that a bid of $3,000,000 is certain to win the contract; but then your costs would be larger than your revenues and youd lose money. Finally, you believe that a bid of $4,000,000 (the midpoint between these to values) gives you a 50/50 chance of winning. Thus, you believe wins are a random variable drawn from a symmetric triangular(a,b) distribution with a=3,000,000 and b=5,000,000.

Now lets focus on the effect of the variance (or standard deviation) of the distribution of bids. 1

(a) Presume that bids come from a triangular distribution where a = 2, 000, 000 and b = 6, 000, 000. i. What is the mean and standard deviation of bids? ii. What is the optimal bid? iii. Graph your bids vs expected profits, with your bids on the x-axis and the expected profits on the y-axis. (b) Now presume that bids come from a triangular distribution where a = 1, 000, 000 and b = 7, 000, 000. i. What is the mean and standard deviation of bids? ii. What is the optimal bid? iii. Graph your bids vs expected profits. (c) Why is the optimal bid related to variance in this way?