You hold an oral, or English, auction among three bidders. You estimate that each bidder has a value of either $12 or $15 for the item, and you attach probabilities to each value of 50%. The winning bidder must pay a price equal to the second highest bid. The following table lists the eight possible combinations for bidder values. Each combination is equally likely to occur.
***On the following table, indicate the price paid by the winning bidder.***
Bidder 1 Value Bidder 2 Value Bidder 3 Value
Probability Price
($) ($) ($)
$12 $12 $12 0.125
$12 $12 $15 0.125
$12 $15 $12 0.125
$12 $15 $15 0.125
$15 $12 $12 0.125
$15 $12 $15 0.125
$15 $15 $12 0.125
$15 $15 $15 0.125
The expected price paid is ?
Suppose that bidders 1 and 2 collude and would be willing to bid up to a maximum of their values, but the two bidders would not be willing to bid against each other. The probabilities of the combinations of bidders are still all equal to 0.125. Continue to assume that the winning bidder must pay a price equal to the second highest bid.
On the following table, indicate the price paid by the winning bidder.
Maximum of Bidder 1 and 2 Bidder 3 Value
Probability Price
($) ($)
$12 $12 0.125
$12 $15 0.125
$15 $12 0.125
$15 $15 0.125
$15 $12 0.125
$15 $15 0.125
$15 $12 0.125
$15 $15 0.125