1. Strategy 1-Bid $264.9M - Bid your signal. What will happen? Give reasoning for your analysis Had all companies bid their signals, the losses would have been huge, ranging from 14.1 to 37.3 million dollars! Evidently, bidding one’s signal is a disastrous strategy. Strategy 1 not optimal because the extra profit values for the historic auctions are all negative, indicating that each winner paid more for the lease than it was worth to the company.

2. Strategy 1-Bid $264.9M

3. “Winner’s Extra Profit” is almost always negative. That is, the winning company will not get its needed fair return on the lease. Strategy 1 is not optimal because, the highest signal will almost always be well above the value of the lease and the winning company will have paid too much for the drilling rights. This is called the winner’s curse(we will learn how to calculate after simulation of Normal Errors).

4. Strategy 2-First plan-Company 1 will Bid=$264.9M-$25M=$239.9M Subtract Winner’s curse from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis 1)Equal probability of winning 2)Company 1 expected value is very small & other companies expected value is negative Strategy 2 is not optimal because the difference between the highest bid & second highest bid wasted(E(B))

5. If all companies bid 25 million dollars less than their signals, the expected value of the difference between the top two bids will still be 6.38 million dollars. Hence, the winning company will, on average, pay an unnecessary premium of 6.38 million. Strategy 2-First plan-Company 1 will Bid=$264.9M-$25M=$239.9M

6. Strategy 3-Second plan-Company 1 will Bid=$264.9M-$31.38M=$233.52M Subtract Winner’s curse & Winner’s blessing from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis Strategy 3 is not optimal because companies tend to improve its expected value & we need to incorporate other companies actions 1)Equal probability of winning 2) expected values very close

7. Strategy 4 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse and Winner’s blessing from their signals to obtain their bids.

8. Strategy 4

9. Strategy 4 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse and Winner’s blessing from their signals to obtain their bids. steps 1. Enter the sum of the WC & WB as the signal adjustment for all other companies cell 2. Change company 1 signal adjustment cell to get a set of expected values for company 1. 3. record results of expected value for company 1 good adjustment points(10 points ) Use 17,19,21,23,25,27,29,31,33,35 4. enter each good adj. ->hit F9(to recalculate)->manually record the expected values & create a table for company 1

10. on the project Probability, Mathematics, Tests, Homework, Computers Bidding on an Oil Lease for Company 1 must find the maximum expected value of adjustment(assuming that all other companies subtract both the curse and blessing) this best adjustment, a cb Strategy 4-Constructing f(a) function

11.  copy from the sheet Strategy in Auction Focus.xls to a new book  Let f(a) be the expected value for Company 1 for subtracting a million dollars from its signal, assuming that all other companies adjust their signals by both the curse and blessing.  Fit a 4 th degree polynomial trend line, which we will use as an approximate formula for the unknown function f.  Use solver to find the best adjustment f(a) function Strategy 4-

12. Strategy 4- Company 1 will Bid=$264.9M-$21.28M=$243.62M

13. Is Strategy 4 optimal ? No This is the real world of business, where we expect our competitors to be well-managed companies other 18 companies are sitting in their offices and boardrooms making the same calculations that we have just performed. Given the results, other companies will also elect to subtract less than 31.38million dollars from their signals. It is worth noting that the 21.28million dollar adjustment that is our appropriate response to a larger reduction by the other companies, is not itself a stable strategy. Since this is less than the winner’s curse of 25 million dollars, there would be a negative expected value for extra profit if all companies adjusted downward by 21.28million dollars.

14. Strategy 5 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse from their signals to obtain their bids.

15. Strategy 5- Company 1 will Bid=$264.9M-$30.81M=$234.09M

16. Strategy 5 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse from their signals to obtain their bids. steps 1. Enter the WC as the signal adjustment for all other companies cell 2. Change company 1 signal adjustment cell to get a set of expected values for company 1. 3. record results of expected value for company 1 good adjustment points(11 points ) Use 19,21,23,25,27,29,31,33,35,37,39 4. enter each good adj. ->hit F9(to recalculate)->manually record the expected values & create a table for company 1 change

17. on the project Probability, Mathematics, Tests, Homework, Computers Bidding on an Oil Lease Simulating, Focus  TI Auction Focus.xls (material continues) Class Project C for Company 1 must find the maximum expected value of adjustment(assuming that all other companies subtract both the curse and blessing) this best adjustment, a c Strategy 5- Constructing g(a) function

18.  USE the sheet Strategy in Auction Focus.xls.  Let g(a) be the expected value for Company 1 for subtracting a million dollars from its signal, assuming that all other companies adjust their signals by curse.  Fit a 4 th degree polynomial trend line, which we will use as an approximate formula for the unknown function g.  Use solver to find the best adjustment the use of Solver in Strategy shows that g(30.8) = 0. Hence, a c = 30.8 million dollars. Company 1’s best response to an adjustment of 25 million dollars by all other companies is to lower its signal by the considerably larger amount of $30.8M Strategy 5 -g(a) function

19. Strategy 5- Company 1 will Bid=$264.9M-$30.81M=$234.09M

20. Is Strategy 5 optimal? if we know what all other companies plan to do. Moreover, this same information is available to all of the bidders. Need a stable strategy??? If all companies made such a stable adjustment to their signals, then there would be no incentive for anyone to alter the strategy. A stable bidding strategy is also called a Nash equilibrium

21. on the project Probability, Mathematics, Tests, Homework, Computers Bidding on an Oil Lease Simulating, Focus (material continues) A signal adjustment of a s would be a stable strategy if Company 1’s best response to an adjustment of a s by all other companies, would be to also adjust by a s. If all companies made such a stable adjustment to their signals, then there would be no incentive for anyone to alter the strategy. A stable bidding strategy is also called a Nash equilibrium. The development of this game theoretic concept earned a share in the 1994 Nobel Prize in Economics for the mathematician and economist John F. Nash (1928 - ). It seems to be quite possible that there is a Nash equilibrium, or stable strategy, for our auctions. We have seen that Company 1 should respond with a smaller adjustment to a large adjustment by all other companies. Conversely, Company 1 should respond with a larger adjustment to a small adjustment by all other companies. We are looking for a universal intermediate strategy, a s.  TI Auction Focus.xls Class Project C 21.28 31.38 30.8 25 Strategy-4Strategy-5

22. Strategy 6 Finally, each team should experiment with Auction Equilibrium.xls to determine a stable Nash equilibrium bidding strategy for its auction scenario. This will lead to a modification of your team’s signal and a specific bid in the upcoming auction. Need to enter different values for the blue cell and experiment until the two cells are equal(this will take a LONG time. We will use 10 iterations and get the average $27.031M

23. Strategy 6

25. (2wc+wb)/2

26. Excel method for Strategy 6 How? (a)Use Auction Equilibrium.xls(. (b)FOLLOW THE INSTRUCTIONS IN THIS FILE! (c)Enter appropriate values in cells B10 through E10. (d)Enter a logical value in cell E39. Run the macro Optimize. (the first logical value to use- (2wc+wb)/2 (e)Enter another logical value in cell E39 and press the key F9.. record numbers in a table. (f)See table (g)Find the stable adj for strategy 6

27. Excel method for Strategy 6 Company 1 Optimal Adjustment, a max (use 4 decimals) All Other Companies Adjustment Subtracted From Signal New logical value 1 25.393328.19 (25.3933+28.19)/2= 26.7916 2.. 10 Avg of a max =final stable adj for strategy 6 first logical value to use for class project- (2wc+wb)/2=28.19

28. Strategy 6- Company 1 will Bid=$264.9M-$27.031M=$237.869M Using Auction Equilibrium.xls to determine 10 stable strategy values, and averaging the results, we find a signal adjustment of 27.031 million dollars. This provides our Nash equilibrium and is the final answer to our bidding strategy problem. If each company reduced its signal by $27,031,000 then there would be no expected gain for any one company, if it deviated from this plan. Specifically, we will reduce our signal of $264,900,000 by $27,031,000 and submit a bid of $237,869,000.

29. Finding the Nash Equilibrium for the exams-Example Nash equilibrium is when the two values of the columns are approximately equal to each other=24.322

30. Recall- Class Project-Goals  Determine what would be expected to happen if each company bid the same amount as its signal.  Determine the Company 1 bid under several uniform bidding strategies, and explore the expected values of these plans.  Find a stable uniform bidding strategy that could be followed by all companies, without any chance for improvement.

31. Recall-Project Assumptions Assumption 1. The same 18 companies will each bid on future similar leases only bidders for the tracts. Assumption 2. The geologists employed by companies equally expert on average, they can estimate the correct values of leases. each signal for the value of an undeveloped tract is an observation of a continuous random variable, S v,

32. Recall-Project Assumptions Assumption 3. Except for their means, the distributions of the S v ’s are all identical (The shape /The Spread) Assumption 4. All of the companies have the same profit margins

33. Strategies for bidding on an Oil Lease Strategy 1 -Bid your signal. What will happen? Give reasoning for your analysis Strategy 2(First Plan) -Subtract Winner’s curse from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis Strategy 3(Second Plan) -Subtract Winner’s curse and Winner’s blessing from your signal to obtain bid. Assume all other companies do the same process. What will happen? Give reasoning for your analysis

34. Strategies for bidding on an Oil Lease Strategy 4 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse and Winner’s blessing from their signals to obtain their bids. Strategy 5 -Find a optimal adjustment for company 1. Assume all other companies Subtract Winner’s curse from their signals to obtain their bids. Strategy 6 -Determine a stable Nash equilibrium bid. This stable strategy is such that any company will not have any incentive to deviate from

35. Error random variable, R We can assume that all of the individual error random variables represent a common error random variable, R. This gives error = signal  proven value.

36. Errors –R random variable is a Normal random varible

37. Simulation Need more than 20 auctions. Since we have to bid now, we cannot wait for more actual data to accumulate. The only practical solution is to use the small amount of historical data(380 errors) to train a computer to simulate thousands of similar auctions. Monte Carlo method of simulation to amplify the information that is contained in our original sample of data from 20 auctions

38. Important - Generate Errors 10000 leases class project has 19 companies->19 error columns Want Normal Errors with Mean 0 & Standard Deviation 13.53 using =NORMINV(RAND(),0,13.53)

39. Generate Normal Errors

40. Creating the fixed error matrix Copy all the generated and do a paste special(values only) in a new worksheet Once the values are copied YOU MUST DELETE ALL THE NORMINV FORMULAS IN THE ORIGINAL WORKSHEET IF YOU DON’T DO THIS THE FILE WILL BE TOO LARGE TO HANDLE

41. How to calculate Winner’s Curse- E(C) Let C be the continuous random variable which gives the largest number in a sample of 19(class project has 19 companies) observations of R. Assuming that each company bids its signal, E(C) will be the expected value of the winner’s curse.

42. Winner’s Curse-E(C) sample mean for the 10,000 observations of C is $25.00 M. If a company bids its signal and wins the auction, it can expect, on average, to fall $25M below its needed fair return on the lease.

43. How to calculate Winner’s Blessing- E(B) Let B be the continuous random variable which gives the difference between the largest and second largest errors in a random set of 19 observations of R E(B) will be the expected value of the winner’s blessing sample mean for the 10,000 observations of B is 6.38 million

44. E(C) & E(B)

45. Extra profit Extra profit is the amount by which the winning bid is below the fair value of a lease.

46. Expected Value of an adjustment (for all other companies-combined) The best measure of success is the expected value of an adjustment. Let X i be the continuous random variable giving the extra profit, in millions of dollars, for Company i. X i can be positive or negative, if Company i wins, but it will always be 0 if Company i does not win. Signal Adjustment uses the 10,000 simulated auctions to approximate E(X 1 ) and the average of E(X 2 ), E(X 3 ), , and E(X 19 ).

47. Extra Profit

49. Identifying the Random variables Let V be the continuous random variable that gives the fair profit value, in millions of dollars, for an oil lease that is similar to the 20 tracts in the data. The 20 proven values form a random sample for V. the proven leases have a sample mean of 229.8million dollars

50. Random variable V- fair profit value

51. Continuous random variable S v Recall from the project description that each signal is an observation of a continuous random variable S v, where v is the actual fair profit value of the given lease. We have assumed that E(S v ) = v, for every lease. To test the reasonableness of this assumption we have computed the sample mean for the 19 signals on each of the proven leases. For example historical lease number 1 –>Mean of the signals of 19 companies is $233.7M S 1, and proven value is $237.2M Even with the small sample sizes of 19, there is relatively good agreement between the sample means of the signals and the proven values of the leases.

52. Continuous random variable S v

53. Random variable, R v For each lease value, v, we define a new random variable, R v, that gives the error in a company’s signal. This is given by the signal minus the actual fair profit value of the lease, R v = S v  v. By project Assumption 2, the long term average of the signals for a fixed lease will be the actual value of the lease. Thus, for each value of v, the expected value of R v is assumed to be 0.

54. Assumption 3 According to project Assumption 3, the signals for each lease have similar distributions about their averages. This means that, for all values of v, the signal errors have the same distributions about 0. In terms of our random variables, we conclude that the R v ’s are all identical random variables.

55. Error random variable, R We can assume that all of the individual error random variables represent a common error random variable, R. This gives error = signal  proven value.

56. Distributions, Focus on the project Probability, Mathematics, Tests, Homework, Computers Bidding on an Oil Lease (material continues) Integration provides expressions for the expected values of V and R. Unfortunately, we have no way to know exact formulas for the p.d.f.’s, f V, for V. and f R for R. Thus, we cannot compute these means. The only thing that we can do is to replace the parameters E(V) and E(R) with their estimating statistics, the sample means. Hence, we assume that E(V) is 229.82 million dollars. Likewise, we have used project assumptions to conclude that E(R) = 0. Auction Focus.xls  TI Class Project C

57. on the project Probability, Mathematics, Tests, Homework, Computers Bidding on an Oil Lease Normal, Focus From the histogram, it appears that the distribution of V is reasonably close to being normal. From now on, we will assume that V is a normal random variable, with  V = 229.82 and  V = 38.73. With this assumption the p.d.f. for V has the following form. Auction Focus.xls  TI (material continues) Class Project C

58. on the project Probability, Mathematics, Tests, Homework, Computers Bidding on an Oil Lease Normal, Focus In the sheet Normal we have also computed values for the normal density, with a mean of 0 and a standard deviation of 13.53, at the midpoints of the bars in the histogram for our sample of R. We see that the distribution of R appears to be very close to that of a normal random variable. We will assume that R is a normal random variable, with  R = 0 and  R = 13.53. The p.d.f. for R has the following form. Auction Focus.xls  TI (material continues) Class Project C