You hold an auction on eBay and expect that exactly two bidders show up at any t

Question

You hold an auction on eBay and expect that exactly two bidders show up at any time. You estimate that each bidder has a value of either $5 or $8 for the item, and you attach probabilities to each value of 50% (there are equal number of each type in the population). Your own value for the item is zero. You can set a reserve price, a price below which you will not accept bids for the item. What reserve price should you set, and what is your expected revenue from auctioning the item with that optimal reserve price?

Explanation / Answer

Two-Bidder Auction

Bidder 1

Bidder 2

Probability

Winning bid with $8 Reserve

$5

$5

.25

no one

$5

$8

.25

$8

$8

$5

.25

$8

$8

$8

.25

$8

With a reserve price of $8, 75% of the time, we expect at least one high-value bidder, so the winning price is $8. The other 25% of the time, the item will not be sold. The expected revenue and surplus or profit is .75($8)=$6.

Three-Bidder Auction

In a three bidder auction, the expected revenue is (.5)*5 + (.5)*8=6.50. A reserve price of 8 leads to expected revenue of (.125)*0+(0.875)*8=7.  

Bidder 1

Bidder 2

Probability

Winning bid with $8 Reserve

$5

$5

.25

no one

$5

$8

.25

$8

$8

$5

.25

$8

$8

$8

.25

$8

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