Question

A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of $5 and the other has a buyer value of $7 and, with probability 1/2, one has a buyer value of $3 and the other has a buyer value of $10. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must set before the auction starts. To maximize his expected profits, he should set the reserve price at:_______

Answer 1

$6

Step-by-step explanation:

bidder 1:

50% chance paying $5

50% chance paying $7

expected value = ($5 x 50%) + ($7 x 50%) = $6

bidder 2:

50% chance paying $3

50% chance paying $10

expected value = ($3 x 50%) + ($10 x 50%) = $6.50

According to Myerson's optimal reserve price theory, the reserve price does not depend on the number of bidders, instead it depends on the distribution of the buyers' valuation. In this case, both buyers' valuations are very similar, $6 and $6.50, so one of them should be the reserve price. Since this is an open bid and the reserve price will probably serve as reference point, the reserve price should be $6 so that both bidders can participate and hopefully the price will exceed $7 (the highest second bid).

Answer 2

Reserve price = $6.25

Approximately $7

Step-by-step explanation:

Reserve price is the lowest set price that is acceptable by a seller at an auction.

With probability 1/2:

Buyer values = $5 and $7

Buyer values= $3 and $10

Since the seller knows there are two bidders for the object he is selling and he wants to maximize his expected profits. He also sells it in an English auction which he must set the reserve price before the commencement of the auction. His reserve price should be:

`((5*1)/(4)+(7*1)/(4)) + ((3*1)/(4)+(10*1)/(4))`

= (1.25 + 1.75) + (0.75 + 2.5)

= 3 + 3.25

$6.25

The maximize his expected profit the buyer should set his reserve price at approximately $7