Question

Problem 1. An “antique” table is for sale in a sealed bid auction. It will go to the high bidder at the price the high bidder bids. You don’t know if it is a fake or not, but you do know that 20% of all antiques that look like this one are fakes. You are not able to have an appraiser examine it. If it is a fake, you will know this after you buy it and it will be worthless to you. If it is real, it will be worth $1,000 to you.

You know that the only other possible bidder is an expert antique dealer has also looked at this table and you know that this dealer can always tell a fake from a real antique. You know that if the antique dealer finds that it is a fake, she will bid zero for it. If she finds that it is real, she will bid $500 for it if she has a similar table in stock, and will $800 for it if she has another table like it in stock. Suppose that you believe that it is equally likely that the dealer has another table like it in stock.

A) What do you believe is the probability that the antique dealer will bid $500 for the table? What do you believe is the probability that she will bid $800 for the table?

B) If you bid $300 for the table and you are the high bidder, what is your probability that the table is a genuine antique? What is your expected profit (or loss) if you bid $300?

C) If you bid $501 for the table, what is the probability that you will be the high bidder. What is the probability that the table is genuine if you bid $501 and are the high bidder? What is your expected profit (or loss) if you bid $501?

D) If you bid $801 for the table, what is the probability that you will be the high bidder? What is your expected value for the table if you are the high bidder? What is your expected profit (or loss) if you bid $801?

E) Suppose you could choose to bid any amount between 0 and $1000 for the table. What bid would maximize your expected payoff? Explain your answer.

Answer 1

Probability that the antique is fake = 0.2
=> Probability that antique is genuine = 0.8

Probability (expert bids 0) = 0.2
Probability (expert bids $500) = P(antique is genuine & similar table in stock) = P(genuine) * P(similar) = 0.8*1/2 = 0.4
Probability (expert bids $800) = P(genuine & not similar) = P(genuine)*P(not similar) = 0.8*1/2 = 0.4

A) P(Antique dealer bids $500) = 0.4 (as calculated above)
P(Antique dealer bids $800) = 0.4 (as calculated above)

B) Our bid = $300
P(genuine | $300 is high bid) = P(genuine & $300 is high bid) / P($300 is high bid)

But we know that, if antique is genuine, antique dealer bids either $500 or $800. Thus $300 cannot be the high bid. Thus P(genuine & $300 is high bid) = 0.
Hence antique is definitely fake.

Expected profit/loss = (1000-300)*P(genuine) + (-300)*P(fake) = 0 + -300 = $ -300 (loss)

C) If we bid $501, to be the high bidder, the antique dealer must not bid $800. In all other cases, we are the highest bidder.
Thus P($501 high bid) = P(antique dealer does not bid $800) = 1 - P(dealer bids $800) = 1-0.4 = 0.6

P(genuine | $501 is high bid) = P(genuine & $501 is high bid) / P($501 is high bid) = 0.8*1/2 / 0.6 = 2/3
This result can be obtained by Bayes Theorem too
P(genuine | 501 high) = P(501 high | genuine) * P(genuine)/ [ P(501 high | genuine)*P(genuine) + P(501 high | fake)*P(fake) ]
= 0.5*0.8 / (0.5*0.8 + 1*0.2) = 2/3

E(profit) = (1000-501)*0.4 + 0*0.4 + -501*0.2 = $ 99.4 (profit)

D) P($801 is high) = 1
E(profit) = (1000-801)*0.8 + -801*0.2 = $ -1 (loss)
Expected profit is always the same as $801 is high bid with probability 1

E)