shill bidding in which a buyer uses multiple identities in the auction in order to maximize its profit. It does not necessarily maximize seller revenues; seller revenues may even be zero in VCG auctions. If the purpose of holding the auction is to maximize profit for the seller rather than just allocate resources among buyers, then VCG may be a poor choice. The seller's revenues are non-monotonic with regard to the sets of bidders and offers. The non-monotonicity of seller's revenues with respect to bids (without introducing the VCG opportunity-cost mechanism described at the bottom of this article) can be shown by the following example. Consider three bidders A, B, and C, and two homogeneous items bid upon, Y and Z. A wants both items and bids $2 for the package of Y and Z. B and C both bid $2 each for a single item (bid $2 for Y or Z), as they really want one item but don't care if they have the second. Now, Y and Z are allocated to B and C, but the price is $0, as can be found by removing either B or C respectively. If C bid $0 instead of $2, then the seller would make $2 instead of $0. Because the seller's revenue can go up when bids are either increased or decreased, the seller's revenues are non-monotonic with respect to bids. Proof of dominance of truthful bidding The dominant strategy in a Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item. Let be bidder i's value for the item. Let be bidder i's bid for the item. The payoff for bidder i is The strategy of overbidding is dominated by bidding truthfully. Assume that bidder i bids . If then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case. If then the bidder would lose the item either way so the strategies have equal payoffs in this case. If then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be zero. Thus the strategy of bidding higher than