ps3 - Networks Fall 2011 David Easley and Eva Tardos...

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Networks: Fall 2011 Homework 3 David Easley and Eva Tardos Due before class on September 26, 2011 As noted on the course home page, homework solutions must be submitted by upload to the CMS site, at https://cms.csuglab.cornell.edu/ . This means that you should write these up in an electronic format (Word files, PDF files, and most other formats can be uploaded to CMS). Homework will be due at the start of class on the due date, and the CMS site will stop accepting homework uploads after this point. We cannot accept late homework except for University- approved excuses (which include illness, a family emergency, or travel as part of a University sports team or other University activity). Reading: The questions below are primarily based on the material in Chapters 9 and 10 of the book. (1) In this problem we will examine a second-price, sealed-bid auction for a single item. Assume that there are three bidders who have private values v i which are independently and uniformly distributed on the interval [0 , 1]. You are bidder 1. You know that bidders 2 and 3 bid according to the bidding rules: b 2 ( v 2 ) = α 2 v 2 and b 3 ( v 3 ) = α 3 v 3 where both α 2 and α 3 are greater than 1 and less than 2. You know your value v 1 , but not the values of bidders 2 and 3. How much should you bid? Provide an explanation for your answer; a formal proof is not necessary. (2) In this problem we will examine a second-price, sealed-bid auction for a single item. We’ll consider a case in which true values for the item may differ across bidders, and it requires extensive research by a bidder to determine her own true value for an item — maybe this is because the bidder needs to determine her ability to extract value from the item after purchasing it (and this ability may differ from bidder to bidder). There are three bidders. Bidders 1 and 2 have values v 1 and v 2 , each of which is a random number independently and uniformly distributed on the interval [0 , 1]. Through having per- formed the requisite level of research, bidders 1 and 2 know their own values for the item, v 1 and v 2 , respectively, but they do not know each other’s value for the item. Bidder 3 has not performed enough research to know his own true value for the item. He does know that he and bidder 1 are extremely similar, and therefore that his true value v 3 is exactly equal to the true value v 1 of bidder 1. The problem is that bidder 3 does not know this value v 1 (nor does he know v 2 ). He does know that bidders 1 and 2 know their own true values. (a) How should bidder 2 bid in this auction? How should bidder 1 bid?
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