Strategy%201-Bid%20$264.9M%20-Bid%20your%20signal.%20What%20will%20happen?%20Give%20reasoning%20for%20your%20analysis

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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 ones 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.

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Strategy 1-Bid 264.9M
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Strategy 1-Bid 264.9M

• Winners 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 winners curse(we
  will learn how to calculate after simulation of
  Normal Errors).

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Strategy 2-First plan-Company 1 will
Bid264.9M-25M239.9M Subtract Winners
curse from your signal to obtain bid. Assume all
other companies do the same process. What will
happen? Give reasoning for your analysis
Strategy 2 is not optimal because the difference
between the highest bid second highest bid
wasted(E(B))
1)Equal probability of winning 2)Company 1
expected value is very small other companies
expected value is negative
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Strategy 2-First plan-Company 1 will
Bid264.9M-25M239.9M

• 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.

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Strategy 3-Second plan-Company 1 will
Bid264.9M-31.38M233.52M Subtract Winners
curse Winners 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
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• Strategy 4
• -Find a optimal adjustment for company 1. Assume
  all other companies Subtract Winners curse and
  Winners blessing from their signals to obtain
  their bids.

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Strategy 4
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Strategy 4-Find a optimal adjustment for company
1. Assume all other companies Subtract Winners
curse and Winners 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. -gthit F9(to
  recalculate)-gtmanually record the expected values
  create a table for company 1

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Strategy 4-Constructing f(a) function
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, acb
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f(a) function
Strategy 4-

• 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 4th degree polynomial trend line, which we
  will use as an approximate formula for the
  unknown function f.
• Use solver to find the best adjustment

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Strategy 4- Company 1 will Bid264.9M-21.28M2
43.62M
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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 winners curse of 25 million dollars,
  there would be a negative expected value for
  extra profit if all companies adjusted downward
  by 21.28million dollars.

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Strategy 5

• -Find a optimal adjustment for company 1. Assume
  all other companies Subtract Winners curse from
  their signals to obtain their bids.

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Strategy 5- Company 1 will Bid264.9M-30.81M2
34.09M
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Strategy 5-Find a optimal adjustment for company
1. Assume all other companies Subtract Winners
curse from their signals to obtain their bids.
change

• 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. -gthit F9(to
  recalculate)-gtmanually record the expected values
  create a table for company 1

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Simulating, Focus

Strategy 5- Constructing g(a) function
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, ac
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Strategy 5 -g(a) function

• 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 4th 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, ac 30.8 million dollars.
• Company 1s 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

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Strategy 5- Company 1 will Bid264.9M-30.81M2
34.09M
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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

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Simulating, Focus
A signal adjustment of as would be a stable
strategy if Company 1s best response to an
adjustment of as by all other companies, would be
to also adjust by as. 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, as.
Strategy-4
Strategy-5
21.28
30.8
31.38
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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 teams 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
27.031M
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Strategy 6
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Strategy 6
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Strategy 6
(2wcwb)/2
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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.
• Enter a logical value in cell E39. Run the macro
  Optimize.
• (the first logical value to use- (2wcwb)/2
• Enter another logical value in cell E39 and press
  the key F9.. record numbers in a table.
• See table
• Find the stable adj for strategy 6

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Excel method for Strategy 6
Company 1 Optimal Adjustment, amax (use 4 decimals) All Other Companies Adjustment Subtracted From Signal New logical value
1 25.3933 28.19 (25.393328.19)/2 26.7916
2 26.7916
.
.
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Avg of amaxfinal stable adj for strategy 6
first logical value to use for class project-
(2wcwb)/228.19
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Strategy 6- Company 1 will Bid264.9M-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.
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Finding the Nash Equilibrium for the exams-Example
Nash equilibrium is when the two values of the
columns are approximately equal to each
other24.322
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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.

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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, Sv,

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Recall-Project Assumptions

• Assumption 3. Except for their means, the
  distributions of the Svs are all identical
• (The shape /The Spread)
• Assumption 4.
• All of the companies have the same profit
  margins

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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 Winners 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 Winners curse and Winners blessing
  from your signal to obtain bid. Assume all other
  companies do the same process. What will happen?
  Give reasoning for your analysis

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Strategies for bidding on an Oil Lease

• Strategy 4
• -Find a optimal adjustment for company 1. Assume
  all other companies Subtract Winners curse and
  Winners blessing from their signals to obtain
  their bids.
• Strategy 5
• -Find a optimal adjustment for company 1. Assume
  all other companies Subtract Winners 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

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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.

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Errors R random variable is a Normal random
varible
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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

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Important - Generate Errors

• 10000 leases
• class project has 19 companies-gt19 error columns
• Want Normal Errors with Mean 0 Standard
  Deviation 13.53
• using NORMINV(RAND(),0,13.53)

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Generate Normal Errors
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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 DONT DO THIS THE FILE WILL BE TOO LARGE
  TO HANDLE

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How to calculate Winners 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 winners curse.

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Winners 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.

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How to calculate Winners 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 winners
  blessing
• sample mean for the 10,000 observations of B is
  6.38 million

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E(C) E(B)
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Extra profit

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

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Expected Value of an adjustment (for all other
companies-combined)

• The best measure of success is the expected value
  of an adjustment. Let Xi be the continuous
  random variable giving the extra profit, in
  millions of dollars, for Company i. Xi 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(X1) and the average of
  E(X2), E(X3), ?, and E(X19).

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Extra Profit
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Extra Profit
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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

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Random variable V- fair profit value
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Continuous random variable Sv

• Recall from the project description that each
  signal is an observation of a continuous random
  variable Sv, where v is the actual fair profit
  value of the given lease. We have assumed that
  E(Sv) 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 gtMean of
  the signals of 19 companies is 233.7M S1, 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.

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Continuous random variable Sv
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Random variable, Rv

• For each lease value, v, we define a new random
  variable, Rv, that gives the error in a companys
  signal. This is given by the signal minus the
  actual fair profit value of the lease, Rv Sv ?
  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 Rv is assumed to be 0.

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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 Rvs are all
  identical random variables.

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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.

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Distributions, Focus
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Normal, Focus
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Normal, Focus
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