A professor wants to sell a 1948 edition of Paul Samuelson’s economics textbook

Question

A professor wants to sell a 1948 edition of Paul Samuelson’s economics textbook at an auction. Ten people are interested in buying this book. The valuations of the 10 potential buyers are shown in the following table.

(a) If the professor holds a second-price, sealed-bid auction, who will win the auction and how much will this person pay for the book? Please explain your answer. [2 points]Each bidder knows his or her own valuation of owning the book. A bidder’s payoff is equal to his/her valuation minus what he/she pays for the book, if he/she wins the auction; the payoff is zero otherwise.

(b) Suppose the professor uses a second-price Dutch auction. All potential buyers sit in a large room; they don’t see other people’s actions but they all can see a big screen on the wall. When the auction starts, the screen shows an initial price of $150. In every three seconds, the price is lowered by $1. Each potential buyer may push a button during the auction. The price on the screen stops changing if two buyers have pushed the button. The person who first pushed the button wins the auction, and he/she pays the price that was stopped on the screen. Show that a bidder has a weakly dominant strategy to decide when to push the button in this auction. Who will win the auction and how much will this person pay for the book? Please explain your answer carefully. [3 points]

(c) Suppose that the bidders’ valuations are common knowledge among themselves. That is, each bidder knows the valuations of all other bidders and each bidder knows this fact. If the professor holds a first-price, sealed-bid auction, who will win the auction and how much will he or she pay for the book? Please explain your answer. [2 points]

(d) Now suppose the professor knows that the buyers’ valuations are 45, 53, 97, 61, 26, 79, 84, 70, 66, and 58, but he does not know exactly which buyer has which valuation. The buyers know their own valuations but not one another’s valuations. Suppose the professor runs the following first-price, sealed-bid auction with a reserve price. The professor first announces a reserve price P. Then potential buyers simultaneously and independently choose their bids. Any bids below the reserve price will be automatically disqualified and therefore will not win. The highest bidder wins the auction and has to pay his/her bid for the book. What is the optimal reserve price P for the professor to announce? Who will win the auction? How much will the winner pay for the book? [3 points]

Explanation / Answer

(a)In a second-price sealed bid auction, each bidder submits a sealed bid to the seller. The high bidder wins and pays the amount which is quoted in second-highest bid.

There here Carl with bid amount $97 will win the bid and have to pay second highest bid amount that is $84.

In a sealed-bid second-price auction the best strategy is the bid for true price because the bidder can get maximum payoff when the bidding price is equal to the true price and no deviation from this bid can improve their payoff, regardless of what strategy the other bidders are using.

(b)In a Dutch auction initially offered at a very high price here it $150, well in excess of the amount the seller expects to receive. Bids are not sealed it is an open offer. The price is lowered in decrements as here by $1 until a bidder accepts the current price. That bidder wins the auction and pays that price for the item.

The bidder has a weakly dominant strategy to decide when to push the button in this auction and bidding the true value is a weakly dominant strategy. By this strategy again Carl with bid amount $97 will win the bid but this time he have to pay his own bid amount that is $97.

(c)In the first-price sealed bid auction, each bidder writes down their bid and places it in a sealed envelope. The auctioneer gathers up all the bids, opens the envelopes and awards the item to the highest bidder that is Carl with highest bid amount $97. He has to pay the highest bid amount $ 97 which is his own bid.

(d)In the first-price, a sealed-bid auction with a reserve price. The optimal reserve price P can be taken as mean of all the bids

P = (45+53+97+61+26+79+84+70+60+58)/10 = 633 /10 = $63.3

In that case there will be only 4 successful bidders. As the bid is first-price sealed-bid auction with a reserve price. The next process will be same as first-price sealed-bid auction and Carl with highest bid amount $97 will be winner of the bid. He has to pay the highest bid amount $ 97 which is his own bid.

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