EXAMPLE 6 The Vickrey Auction

To see why this theorem is true, let’s assume that a lamp is being auctioned off and that it is worth $100 to a bidder named Bob. This means that Bob would prefer winning the lamp and paying less than $100 to losing the auction, but that he would rather lose the auction than wind up paying more than $100 for the lamp.

Let denote the highest of the bids other than Bob’s bid of $100. Either , in which case Bob wins the auction and pays dollars, or , in which case Bob loses the auction. We’ll consider these two cases separately and show that no bid for Bob can ever do better than his (sincere!) bid of $100.

  • Case 1: Bob wins the auction (so ). Any bid by Bob greater than yields the same outcome for Bob as does his bid of $100. He wins the auction and gets the lamp for dollars. So these bids are no better for Bob than his bid of $100. On the other hand, any bid less than is strictly worse for Bob than that achieved with his bid of $100 because he would lose the auction instead of getting the lamp for less than he actually thought it was worth.

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  • Case 2: Bob loses the auction (so ). Any bid by Bob less than yields the same outcome for Bob as does his bid of $100. He loses the auction. So these bids are no better for Bob than his bid of $100. On the other hand, any bid greater than is strictly worse for Bob than that achieved with his bid of $100 because he would win the auction and pay more for the lamp than he actually thought it was worth, instead of just losing the auction.