Math Trivia: Auctions

Here are two common mechanisms for auctions:

1) Everyone privately bids, and whoever bids the highest wins and pays what they bid.

2) Everyone privately bids, and whoever bids the highest wins and pays the bid of whoever bids the second highest.

In the second mechanism, the optimal strategy for each bidder is to bid however much they actually value what they are bidding for. In the first mechanism, the optimal strategy is more complicated, and players should bid less than their true value.

Lets assume that there is a fixed number of bidders. The value each bidder assigns to the object being auctioned is generated independently at random from a fixed continuous probability distribution that everyone knows. If everyone follows an optimal strategy, then the amount that the object sells for will follow some probability distribution.

Not surprisingly, the distribution that the auction house sells the object for is dependent on which auction mechanism they use. and the two different mechanism actually do produce different distributions.

Which mechanism do you think will produce a better expected value for the auction house. In the first, people will bid less, but in the second, the highest bid is ignored. Amazingly, in spite of producing different distributions, these two auction mechanisms produce the same expected value for the auction house!

This is actually part of a more general revenue equivalence theorem, which shows that the auction house expects the same revenue for any of a large class of auction mechanisms. This class even includes the auction where all players have to pay whatever they bid, even though only only the highest bidder wins.