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in second priced auctions, the highest bidder wins the auction and pays the price of the second highest bidder. What happens if there's only one bidder? That bidder would have won the auction, but does the winner pay his/her own bid? Or does the winner pay zero, imagining a second bidder with zero value bid?

Thanks

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The purpose of a 2nd price auction (Vickrey Auction) is to encourage bidders to bid their true maximum willing payment for a good rather than the lowest price they expect will win the auction. The reason for this is because they pay the value of the unknown second bid they don't have to worry about bidding to high because the highest bid will always pay the exact lowest they would have had to pay in order to obtain the product and don't have to worry about overbidding.

For a auction to be called an auction there must be multiple bidders. If I know that I am the only bidder I would always just bid $.01 and get everything for free. To prevent this and other types of bidder collusion all auction houses have a minimum starting bid for a product usually determined by the seller. If no one is willing to pay more than the minimum then the product simply wont be sold. Individual rule for vickrey auctions differ depending on the venue but I imagine if only one bid is made then the price would default to the minimum bid set before the auction.

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While I wholeheartedly agree with TheSaint321's answer, here is a different take on the question. We could think of a Becker DeGroot Marschak mechanism as a 1 player 2nd price auction. That is, say we want the player to truthfully reveal her value, $v$, of an object. We know the value is between 0 and 10. We could her to report her value $\hat v$, then randomly draw a number, $x$ from a continuous distribution between 0 and 10. If $x > \hat v$, she gets gets nothing. If her $\hat v \geq x$, she gets the object and pays a cost $x$.

It is pretty easy to show that this is strategically equivalent to a 2nd price auction with 2 players where the 2nd player has a valuation according to the induced distribution. For this reason, it is incentive compatible for the (single, real) player to report her true value. Note however, that is is not an efficient auction, since the player with the highest bid (trivially the only player) does not always get the object.

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