Why would an auction make the winner pay only the second-highest bid?

Because it makes honesty the best policy. In a Vickrey auction, also called a sealed-bid second-price auction, bidders submit written bids without seeing each other's. The highest bidder wins, but the price paid is the second-highest bid. That strange rule gives bidders an incentive to bid their true value, and it is strategically similar to the familiar English auction, where the auctioneer raises the price until no one is willing to go higher.

Why is lying pointless? Suppose an item is worth 100 to you. If you overbid, the only case where that changes the result is when the second-highest bid lies above your value, so you win but pay more than the item is worth to you: a negative payoff, where the truthful bid would have given zero. If you underbid, the only case where that changes the result is when you lose an item you could have won at a price below your value, giving up a positive payoff. In every other case the bid does not change what you pay, because the price is set by somebody else's bid. So bidding your true value is a dominant strategy: it is best whatever the others do.

The idea has a longer history than its name. William Vickrey, a Columbia University professor, described it academically in 1961, but stamp collectors had used it since 1893, and in 1797 Goethe sold a manuscript with a sealed-bid, second-price auction. Vickrey shared the 1996 Nobel Memorial Prize in Economic Sciences, though he never personally received it: it was announced three days before his death.

It is not a universal answer. Vickrey auctions are much studied in the economic literature but uncommon in practice.