Simulating Normal Random Variables Simulation can provide a great deal of information about the behavior of a random variable

Simulating Normal Random Variables Two types of simulations (1) Generating fixed values - Uses Random Number Generation (2) Generating changeable values - Uses NORMINV function

Simulating Normal Random Variables Fixed Values Random Number Generation is found under Tools/Data Analysis Values will never change Useful if you need to show how your specific results are tabulated

Simulating Normal Random Variables Sample: Number of columns Number of rows Type of distribution Mean Standard Deviation (Leave blank) Cell where data is placed

Simulating Normal Random Variables Ex. Generate a fixed sample of data containing 25 values that has a normal distribution with a mean of 13 and a standard deviation of 4.6

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Simulating Normal Random Variables Ex. Generate a fixed sample of data containing 30 values in cells A1:F5 that has a normal distribution with a mean of 81 and a standard deviation of 21

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Simulating Normal Random Variables Ex. Suppose X is a normal random variable whose mean is 140 and standard deviation 18. Find a 90% confidence interval for sample mean for samples of size 6 of the random variable X.

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Simulating Normal Random Variables Ex. Generate a fixed sample of data containing 1000 samples of size 6 each, where the values in the sample represent final exam scores and have a normal distribution with a mean of 140 and a standard deviation of 18.

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Simulating Normal Random Variables Ex. For the 1000 samples created in the previous example, find the sample mean for each. Then use the DCOUNT function to determine how many of the 1000 sample mean values lie within the 90% confidence interval. Soln: Approximately 90% of the values should lie within the 90% confidence interval.

Simulating Normal Random Variables Caution! Occasionally the values Random Number Generation gives might yield poor values These values are more than 6 standard deviations from the mean Find the max and min to be certain the values are within the appropriate range, replacing any values outside of the range

Simulating Normal Random Variables Changeable Values NORMINV function Values will change by pressing F9 or by setting calculation in automatic mode Useful if you want several samples to average (eliminates a small number of poor values)

Simulating Normal Random Variables NORMINV will be used to run simulations for the project Sample data looks similar to Random Number Generation sample data Difference is that values can change by pressing F9

Simulating Normal Random Variables Sample: - Any number between 0 and 1 (Use RAND( ) for random values) - Mean - Standard Deviation

Simulating Normal Random Variables Ex. Generate a changeable sample of data containing 30 values in cells A1:F5 that has a normal distribution with a mean of 81 and a standard deviation of 21

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Simulating Normal Random Variables Ex. Generate a changeable sample of data containing 1000 samples of size 6 each, where the values in the sample represent final exam scores and have a normal distribution with a mean of 140 and a standard deviation of 18.

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Simulating Normal Random Variables Ex. For the 1000 samples created in the previous example, find the sample mean for each. Then use the DCOUNT function to determine how many of the 1000 sample mean values lie within the 90% confidence interval.

Simulating Normal Random Variables Project: Several options for bid: (1) Bid our signal (2) Develop several strategies (3) Develop stable bidding strategy

Simulating Normal Random Variables Project: Bidding the geologist’s signal is a bad idea Leads to an average loss of approx. $21.7 million What should be done?

Simulating Normal Random Variables Project: Create a simulation of auctions including all companies (use NORMINV) on a new worksheet Find maximum error for each auction (represented by the variable C) Find average of maximum values: E(C)

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Simulating Normal Random Variables Project: The average of the maximum values is called the “Winner’s Curse” Defined as the average amount the winner of the auction would overbid by bidding their signal

Simulating Normal Random Variables Project: “First Plan” for a reasonable profit is to subtract E(C) from the geologist’s signal This ensures that the winning company would break even on average

Simulating Normal Random Variables Project: This plan would not be reasonable, since the goal of a company is to make a fair and reasonable profit To find a better strategy, find the gap between 1 st place company and 2 nd place company

Simulating Normal Random Variables Project: The monetary gap between 1 st and 2 nd is a wasted addition to the amount bid The 1 st place company must only outbid the 2 nd place company by $0.01

Simulating Normal Random Variables Project: Find the value of the 2 nd place company for each of the sample auctions (use LARGE function) Once the 2 nd place values are found, find the difference between 1 st and 2 nd (represented by variable B)

Simulating Normal Random Variables Project: For differences, find the average: E(B) The average difference between 1 st and 2 nd place is called the “Winner’s Blessing” Find 10 samples of E(C) and E(B) and average them

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Simulating Normal Random Variables Project: “Second Plan” ensures the winner of making a profit on average Average profit is equal to E(B) Strategy is not stable

Simulating Normal Random Variables Project: Remember that every company uses the same techniques to arrive at curse and blessing values One company could decide to deviate from subtracting both the “Winner’s Curse” and the “Winner’s Blessing”

Simulating Normal Random Variables Project: Assume E(C) = $22.5 million and E(B) = $5.8 million A company could decide to subtract $22.5 million and subtract $3.0 million (trying to outbid a company with a starting value higher than theirs)

Simulating Normal Random Variables Project: Ex: Company A: Signal $100 million (1 st place) Company B: Signal $99 million (2 nd place)

Simulating Normal Random Variables Project: Ex: Company A subtracts ($22.5 M and $5.8 M) leaving a bid of $71.7 million. Company B subtracts ($22.5 M and $3.0 M) deviating from a plan to subtract curse and blessing fully. This leaves a bid of $73.5 million.

Simulating Normal Random Variables Project: This process could continue endlessly Find a stable bidding strategy A stable bidding strategy means that any deviation from the suggested bid would not be beneficial over a large number of trials

Simulating Normal Random Variables Project: Stable bidding strategy has 2 steps (1) Find a trend line to predict a fair and reasonable adjustment amount if other companies subtract both E(C) and E(B). ($ $5.8 in millions) (2) Use the adjustment value in the Excel file Auction Equilibrium.xls to find a stable adjustment

Simulating Normal Random Variables Project: Create a new worksheet that calculates the errors for the simulated auctions minus 28.3 (error – (curse + blessing)) for all other companies For your company (company 1), take the errors for the simulated auctions minus several different values (start around an adjustment of 13 and continue until an adjustment of around 31

Simulating Normal Random Variables Project: Find the maximum value (representing a winning company) A negative maximum means that the winning company’s bid was below the proven value (making positive profit) A positive maximum means that the winning company’s bid was above the proven value (company incurs a loss)

Simulating Normal Random Variables Project: Determine if company 1 (our company) wins each auction by using an IF statement Determine how much profit the company would make by using IF statements

Simulating Normal Random Variables Project: Determine the probability of company 1 winning using COUNTIF, average extra profit, and the expected value of the adjustment Determine the probability of every other company winning using COUNTIF, average extra profit, and the expected value of the adjustment

Simulating Normal Random Variables Project: Run the simulation several times to ensure accuracy Average the results from several simulations Record the results for adjustment (13, 15, 17, …, 31) and the expected values received

Simulating Normal Random Variables Project: Plot the points on a graph (x-axis will be adjustment values, y-axis will be expected value) Fit a polynomial trend line of order 4 through the points Estimate the maximum point (want a reasonable x-value) This tells us how much we might want to subtract from estimate

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Simulating Normal Random Variables Project: The peak is around x = 21 with y = 0.53

Simulating Normal Random Variables Project: The peak is around x = 21 with y = 0.53 This means that a company should subtract approximately $21 million from their signal and receive an average of $0.53 million profit per auction

Simulating Normal Random Variables Project: This strategy is still unstable (we assumed every other company would subtract $28.3 million That means that there would now be a different strategy to compute (every company subtracts $21 million, what should we do?)

Simulating Normal Random Variables Project: This process would continue until we reached a point where the adjustment we should make is equal to the adjustment all other companies should make To achieve this, we use Auction Equilibrium.xls

Simulating Normal Random Variables Project: Need 4 pieces of information: # of companies, standard deviation, winner’s curse, and winner’s blessing Found in cells B10:E10

Simulating Normal Random Variables Project: In cell E39, type the adjustment all other companies will make (initially this is curse + blessing) Run the macro so that a value for Company 1’s adjustment appears in the yellow box in cell D39

Simulating Normal Random Variables Project: Average the values in cells D39 and E39 Replace cell E39 value with the average that was just computed Press F9 to recompute values

Simulating Normal Random Variables Project: Continue finding the average and replacing the E39 value until the D39 and E39 values are the same (our adjustment is equal to other company’s adjustment) The optimal adjustment is called a “Nash Equilibrium” value

Simulating Normal Random Variables Project: “Nash Equilibrium” is named after a mathematician named John Nash The optimal situation for one person (company) may not be most beneficial to the whole group (all companies)