A seller decides to sell an object by means of an English auction without a reservation price. There are two bidders. The seller believes that for each of the two bidders there is a probability of that the bidder's value for the object is $500 and a probability of that the bidder's value is $200. The seller believes that these probabilities are independent between bidders. If the bidders bid rationally, what is the seller's expected revenue from the auction?

Question

A seller decides to sell an object by means of an English auction without a reservation price. There are two bidders. The seller believes that for each of the two bidders there is a probability of  that the bidder's value for the object is $500 and a probability of  that the bidder's value is $200. The seller believes that these probabilities are independent between bidders. If the bidders bid rationally, what is the seller's expected revenue from the auction?

Quizplus · Accepted Answer

Since there is no reserve price, the bidder with the highest value will win the auction at a price equal to the second-highest value. For each bidder, there is a 50% chance that their value for the object is $500 and a 50% chance that their value is $200. Therefore, we can construct the following table:

|    Bidder A   |    Bidder B   |  Winning Bid |
|:-------------:|:-------------:|:------------:|
| $500 | $500 | $500 |
| $500 | $200 | $500 |
| $200 | $500 | $500 |
| $200 | $200 | $200 |

The only case in which the seller earns less than $275 is if both bidders value the object at $200, in which case the auction will end at a price of $200. The probability of this occurring is (0.5)(0.5) = 0.25.

The expected revenue for the seller is therefore:

($500)(0.5)(0.5) + ($500)(0.5)(0.5) + ($500)(0.5)(0.5) + ($200)(0.5)(0.5) = $275

So the answer is C.

A Seller Decides to Sell an Object by Means of an English