Post by patrick on May 4, 2017 0:42:58 GMT
Problem 3.2 asks us to consider the revenue target auction, which has k items to auction off and seeks to make revenue of at least R. It works by just asking if the top k bidders are willing to pay R/k each, and giving them to them if so, or otherwise asking if the top k-1 are willing to pay R/(k-1), and so on. Part (b) of this problem asks us to prove that this mechanism is DSIC, but I don't think that it is.
Consider the case k = 2, R = 1000, v_1 = 800, v_2 = 700, v_3 = 600. Then with truthful bidding, bidders 1 and 2 each get the item for a price of 500, and bidder three will get nothing. However, if bidder three lies and submits a bid of 900, they will get the item for a price of 500, for a profit of 100. Therefore truthful bidding is not a dominant strategy for this auction.
This seems like it will happen whenever v_(k+1) > R/k. (Assuming the convention that v_i > v_(i+1) for all i). The obvious fix is to set the price at max(R/|S|, b_(k+1)), but this destroys the property of (weak-) group-strategyproofness in problem 3.3. Is there something I'm missing here that makes it work out after all?
Consider the case k = 2, R = 1000, v_1 = 800, v_2 = 700, v_3 = 600. Then with truthful bidding, bidders 1 and 2 each get the item for a price of 500, and bidder three will get nothing. However, if bidder three lies and submits a bid of 900, they will get the item for a price of 500, for a profit of 100. Therefore truthful bidding is not a dominant strategy for this auction.
This seems like it will happen whenever v_(k+1) > R/k. (Assuming the convention that v_i > v_(i+1) for all i). The obvious fix is to set the price at max(R/|S|, b_(k+1)), but this destroys the property of (weak-) group-strategyproofness in problem 3.3. Is there something I'm missing here that makes it work out after all?
