Class : More Formulas for Derivative and Their Applications

Introduction to Auction Project

1. What is an auction?

2. What types of auctions are there?

3. What sorts of things are sold by auction? What does the government sell by auction?

4. What is a first-price, sealed-bid auction?

This project is about an auction for the lease to extract oil and gas from a tract of sea bed under the Gulf of Mexico. You represent an oil company bidding on such a lease. To help your company decide how much to bid, you hire a geologist who gives you a signal of the value of the lease, which is an estimate of the expected profit after all the oil is extracted. Each company against whom you are bidding (each team in the class has a different number of competitors) has their own geologist and their own signal, but of course you don’t know what their signals are. You also have a set of historical signals by the same geologists for 20 tracts of known value, where the oil has already been extracted. Notice that the signals are distributed around the proven value: Some are too high, some are too low. To prevent collusion among the bidders, the government sets a reserve price. If the highest bid is less than that price, the auction is closed and a lease for those tracts is not sold.

5. What is a bidding ring?

6. What do you observe from the result of the auction? Who wins? How much do they make?

Probability in the Project: “Each signal for the value of an undeveloped lease is a value of a random variable Sv whose mean is the true value of the lease, v. Except for having different means, the distributions of the Sv ‘s are all identical.” We are going to figure out what this means.

We need to recall from Math 115a: What is a random variable? What is the mean of a random variable? What is the distribution of a random variable?

Example: The random variable X is the number of heads on two tosses of a fair coin. The distribution of X shows the values, x, that X takes and the probabilities of each. (Note that capital X is the random variable and lower case x is a number.) 7. List all the possible values X can take.

8. Find the probability that X = 0.

9. Find the probability that X takes each of its other possible values.

10. Fill in the values in the following table: x

1 P(X = x) P(X ≤ x) The function P(X = x) is called the probability mass function, p.m.f. of X. The function P(X ≤ x) is called the cumulative distribution function, c.d.f. of X. 11. What does P(X = 1) mean in words? (No x’s, just words!)

12. What does P(X ≤ 1) mean in words?

The random variable X in #5-10 is called a Binomial random variable with n =2 (because the coin is tossed twice) and p = 0.5 (because the chance of getting heads is 0.5 each time). It is Binomial because each toss of the coin is independent and has the same probability of giving a head. In Excel, you can find the Binomial probabilities using =BINOMDIST(x, n, p, TRUE or FALSE) To get the pmf, put FALSE in the last slot; to get the cdf, put TRUE in the last slot.

13. Fill in the table below for the pmf and cdf of the number of heads in three tosses of a fair coin. Use Excel. x P(X = x) P(X ≤ x)

14. What is the probability that you get two or more heads in three tosses of a fair coin? (Use #11.)

(Example from Text) A saleswoman knows that one in five contacts lead to a sale. You will find the probability that she makes exactly 7 sales if she sees 28 contacts. Let X be the number of sales made. 15. Is X a Binomial random variable? Why? If so, what are n and p?

16. Find the probability of exactly 7 sales using Excel.

17. Find the probability of 7 or more sales. (Use Excel)

18. Find the probability of between 5 and 10 sales, inclusive. (Use Excel)

19. By common sense, what do you think is the average number of sales from 28 customers?

20. Look at your answer to #17: How can you get your answer from the values of n and p that you are using? Use this to guess the general formula for the mean of a Binomial random variable.

Back to the Project

21. Each signal for the value of an undeveloped lease is a value of a random variable Sv whose mean is the true value of the lease, v. What does this mean?

2 22. Think about the historical data. Suppose you subtract the proven value of each signal, giving a new random variable, the errors in the distributions. What can you say about the distribution of the errors?