One of the bidders, let's say it is Alice, is a friend of the auctioneer. If Alice is not the high bidder then Alice loses, just like all the other low bidders. So far, nothing special. Now for the twist. If Alice is the high bidder, the auctioneer tells Alice this fact and lets her revise her bid.
Alice was not aware that this would happen and neither were any of the other bidders when they placed their bids. Moreover, the auctioneer is not without any ethics-he doesn't tell Alice what the next highest bid is.
Obviously Alice has no incentive to raise her bid since that would just lead her to pay more money. The question is should she lower her bid and if so, by how much. Recognize that if Alice lowers her bid too much so that she is no longer the high bidder then she no longer wins the auction and there is nothing her friend can do.
Although this behavior would be highly unethical and hence you might not want to take advantage of this option for that reason, I would like to know how you would use this information casting aside ethics for a moment.
One last simplification: just as in the first problem, you should assume that the value of the good is independent across the bidders so there is no issue of winner's curse. In figuring out how much to adjust the bid, if it helps, you can assume that there are only two bidders, Alice and Bob. And just like above, Alice thinks Bob's valuation is drawn from a uniform distribution between 0 and 100 and Bob, when formulating his bid, thinks that Alice's distribution is drawn from a uniform distribution between 0 and 100.
Write down your answer and, more importantly, your reasoning. All the points in this problem come from the reasoning part.