ou hold an oral, or English, auction among three bidders. You estimate that each bidder has a value of either $48 or $60 for the item, and you attach probabilities to each value of 50%. The winning bidder must pay a price equal to the second highest bid. The following table lists the eight possible combinations for bidder values. Each combination is equally likely to occur.
ou hold an oral, or English, auction among three bidders. You estimate that each bidder has a value of either $48 or $60 for the item, and you attach probabilities to each value of 50%. The winning bidder must pay a price equal to the second highest bid. The following table lists the eight possible combinations for bidder values. Each combination is equally likely to occur.
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter11: Data Analysis And Probability
Section11.8: Probabilities Of Disjoint And Overlapping Events
Problem 2C
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Question
You hold an oral, or English, auction among three bidders. You estimate that each bidder has a value of either $48 or $60 for the item, and you attach probabilities to each value of 50%. The winning bidder must pay a price equal to the second highest bid.
The following table lists the eight possible combinations for bidder values. Each combination is equally likely to occur.
On the following table, indicate the price paid by the winning bidder.
|
Bidder 1 Value
|
Bidder 2 Value
|
Bidder 3 Value
|
Probability
|
Price
|
|---|---|---|---|---|
|
($)
|
($)
|
($)
|
||
| $48 | $48 | $48 | 0.125 | |
| $48 | $48 | $60 | 0.125 | |
| $48 | $60 | $48 | 0.125 | |
| $48 | $60 | $60 | 0.125 | |
| $60 | $48 | $48 | 0.125 | |
| $60 | $48 | $60 | 0.125 | |
| $60 | $60 | $48 | 0.125 | |
| $60 | $60 | $60 | 0.125 |
The expected price paid is ______
.
Suppose that bidders 1 and 2 collude and would be willing to bid up to a maximum of their values, but the two bidders would not be willing to bid against each other. The probabilities of the combinations of bidders are still all equal to 0.125. Continue to assume that the winning bidder must pay a price equal to the second highest bid.
On the following table, indicate the price paid by the winning bidder.
|
Maximum of Bidder 1 and 2
|
Bidder 3 Value
|
Probability
|
Price
|
|---|---|---|---|
|
($)
|
($)
|
||
| $48 | $48 | 0.125 | |
| $48 | $60 | 0.125 | |
| $60 | $48 | 0.125 | |
| $60 | $60 | 0.125 | |
| $60 | $48 | 0.125 | |
| $60 | $60 | 0.125 | |
| $60 | $48 | 0.125 | |
| $60 | $60 | 0.125 |
With collusion, the expected price paid is ______
.
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