Some people advocated the following modification of the auction rule. A bidder cannot bid for only one object, i.e., if at some point in time he withdraws from the bidding race for one object, he automatically withdraws the race for the other object. Every other aspect of the auction, including how prices increase over time, does not change. (a) What should a bidder do if his valuations for the two objects are 50 and 60; respectively? Explain your answer. (b) Does the auction lead to an efficient allocation? Explain your answer.
Let vij be bidder i's valuation for object j, where i in {1,2,3} and j in {1,2}. Bidder i knows its valuation vi; but other bidders only know that vi; is drawn uniformly from [0, 100]. If bidder i wins object 1 at
The auction proceeds as follows. The initial prices are zero for both objects. All bidders sit in front of their computers and observe the prices for both items in real-time. Initially, all bidders are invited to enter the bidding race for both items. At any moment in time, each bidder has the option to withdraw from the bidding race for either object or both. If a bidder withdraws from the bidding for one object, he can no longer get back to the bidding for that object, but he can stay in the bidding race for the other object if he hasn't withdrawn from it previously. The price for an object increases continuously over time as long as there are two or three bidders in the race for that object. The rate of price increase is 1; that is to say, the price increases to x if x units of time has passed, where x is a real number. The price for an object stops increasing if there is at most one bidder left for the bidding race for the object, and the price will stay at that level for the rest of the game; we say the price is locked in this case. (the price for the other object will continue to increase if more than one bidder stays in the bidding race for that object). The auction ends as soon as both prices stop increasing (i.e., at most one bidder remains in the bidding race for each object). An object is sold to the bidder (if any) who remains in the bidding race for that object at the locked price.
3. Some people advocated the following modification of the auction rule. A bidder cannot bid for only one object, i.e., if at some point in time he withdraws from the bidding race for one object, he automatically withdraws the race for the other object. Every other aspect of the auction, including how prices increase over time, does not change.
(a) What should a bidder do if his valuations for the two objects are 50 and 60; respectively? Explain your answer.
(b) Does the auction lead to an efficient allocation? Explain your answer.
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