1. You hold an oral, or English, auction among three bidders. You estimate that each bidder has a value of either $40 or $50 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
($)	($)	($)
$40	$40	$40	0 .125
$40	$40	$50	0.125
$40	$50	$40	0.125
$40	$50	$50	0.125
$50	$40	$40	0.125
$50	$40	$50	0.125
$50	$50	$40	0.125
$50	$50	$50	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
($)	($)
$50	$40	0 .125
$50	$40	0.125
$40	$40	0.125
$50	$50	0.125
$50	$40	0.125
$50	$50	0.125

With collusion, the expected price paid is....................

2. A reserve price is a minimum price set by the auctioneer. If no bidder is willing to pay the reserve price, the item is unsold at a profit of $0 for the auctioneer. If only one bidder values the item at or above the reserve price, that bidder pays the reserve price. An auctioneer faces two bidders, each with a value of either $18 or $48, with both values equally probable. Without a reserve price, the second highest bid will be the price paid by the winning bidder.

The following table lists the four possible combinations of bidder values. Each combination is equally likely to occur.

On the following table, indicate the price paid by the winning bidder with and without the stated reserve price.

Bidder 1 Value	Bidder 2 Value	Probability	Price Without Reserve	Price with $48 Reserve Price
($)	($)	($)
$18	$18	0 .25
$18	$48	0.25
$48	$18	0.25
$48	$48	0.25

Without a reserve price, the expected price is......................

With a reserve price of $48, the expected price is.................................

. Thus, the expected price is larger(with, without) the reserve price.

3.A reserve price is a minimum price set by the auctioneer. If no bidder is willing to pay the reserve price, the item is unsold at a profit of $0 for the auctioneer. If only one bidder values the item at or above the reserve price, that bidder pays the reserve price. An auctioneer faces two bidders, each with a value of either $204 or $272, with both values equally probable. Without a reserve price, the second highest bid will be the price paid by the winning bidder.

The following table lists the four possible combinations for bidder values. Each combination is equally likely to occur.

On the following table, indicate the price paid by the winning bidder with and without the stated reserve price.

Bidder 1 Value	Bidder 2 Value	Probability	Price Without Reserve	Price with $272 Reserve Price
($)	($)	($)
$204	$204	0 .25
$204	$272	0.25
$272	$204	0.25
$272	$272	0.25

Without a reserve price, the expected price is.......................... With a reserve price of $272, the expected price is................

. Thus, the expected price is larger................the reserve price.

4.

When a famous painting becomes available for saler it is often known which museum or collector will be the likely winner. Yet, the auctioneer actively woos representatives of other museums that have no chance of winning to attend anyway. Suppose a piece of art has recently become available for sale and will be auctioned off to the highest bidder, with the winner paying an amount equal to the second highest bid. Assume that most collectors know that Sean places a value of $3,000 on the art piece and that he values this art piece $75,000 2 more than any other collector. Suppose that if no one else shows up, Sean simplyr bids = $37,500 and wins the piece of art. The expected price paid by Sean, with no other bidders present, is . Suppose the owner of the artwork manages to recruit another bidder, Alex, to the auction. Alex is known to value the art piece at $60,000. The expected price paid by Sean, given the presence of the second bidder Alex, is