Business Mathematics II
Project 2: Auction Project
Final Report
Prepared for
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Math 115b, Section
University of Arizona
By
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Submitted on
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Table of Contents
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I.
Background
A. Company’s Signal for Next Auction:
B. Number of Bidders:
C. Excerpt of Historical Data
Leases
Signals
Number Proven Company 1 Company 2 Company 3 Company 4 Company 5
Value
Signals
Signals
Signals
Signals
Signals
1
2
3
4
5
237.2
196.8
154.9
262.6
229.4
II.
Objectives
III.
Assumptions
IV.
Effect of Bidding Signal
V.
Errors
A. Definitions of 𝑽, 𝑺𝒗 , and 𝑹
B. Computation
1. Formula for 𝑹
2. Excerpt of Results
Error 1
Error 2
Error 3
Error 4
Error 5
1
C. Analysis
1. Mean and Standard Deviation
a) Function
b) Results
Mean
Standard
Deviation
2. Histogram and Normal Curve (with source data)
VI.
Simulation
A. Function
B. Winner’s Curse and Blessing
1. Definitions of 𝑪, 𝑬(𝑪), 𝑩, and 𝑬(𝑩)
2
2. Excerpt of Simulated Values
Maximum
2nd Maximum
Difference
3. Estimated Expected Value of Winner’s Curse:
4. Estimated Expected Value of Winner’s Blessing:
5. Company 1 Bids
a) First Plan
b) Second Plan
VII. Strategies
A. Definitions of 𝑿𝒊 and 𝑬(𝑿𝒊 )
B. Adjustment of Other Company’s Signals by Curse and Blessing
1. Estimated Expected Value for Company 1 of Same Adjustment
Extra Profit of Winning Company
Company 1 Other Companies
Probability of winning
Mean extra profit, if company wins
Expected value of adjustment
2. Graph of Estimated Expected Values for Company 1 and Trend Line
(with source data)
3
3. Definition of and formula for 𝒇(𝒂)
4. Optimal Adjustment for Company 1
Increment
𝒉
0.00001
𝒂
Computation
𝒇(𝒂)
𝒇′ (𝒂)
5. Company 1 Bid
C. Adjustment of Other Company’s Signals by Curse Only
1. Estimated Expected Value for Company 1 of Same Adjustment
Extra Profit of Winning Company
Company 1 Other Companies
Probability of winning
Mean extra profit, if company wins
Expected value of adjustment
2. Graph of Estimated Expected Values for Company 1 and Trend Line
(with source data)
3. Definition of and formula for 𝒈(𝒂)
4. Optimal Adjustment for Company 1
Increment
𝒉
0.00001
𝒂
Computation
𝒈(𝒂)
𝒈′(𝒂)
5. Company 1 Bid
4
D. Nash Equilibrium
1. Definition of stable strategy
2. Equilibrium adjustment
Company 1
Adjustment
Added to Signal
10
13
16
19
22
25
28
31
34
37
40
Optimal
Expected Value of
Adjustment
Adjustment, 𝒂𝒎𝒂𝒙
All Other
Companies
Adjustment
Subtracted From
Signal
Simulation Data
Minimum
of All Errors
Maximum
of All Errors
3. Company 1 Bid
VIII. Further Analysis
IX.
Conclusion
5