Whenever I teach the Vickrey auction in my undergraduate classes I give this question:

We have seen that when a single object is being auctioned, the Vickrey  (or second-price) auction ensures that bidders have a dominant strategy to bid their true willingness to pay. Suppose there are k>1 identical objects for sale.  What auction rule would extend the Vickrey logic and make truthful bidding a dominant strategy?

Invariably the majority of students give the intuitive, but wrong answer.  They suggest that the highest bidder should pay the second-highest bid, the second-highest bidder should pay the third-highest bid, and so on.

Did you know that Google made the same mistake?  Google’s system for auctioning sponsored ads for keyword searches is, at its core, the auction format that my undergraduates propose (plus some bells and whistles that account for the higher value of being listed closer to the top and Google’s assessment of the “quality” of the ads.)  And indeed Google’s marketing literature proudly claims that it “uses Nobel Prize-winning economic theory.”  (That would be Vickrey’s Nobel.)

But here’s the remarkable thing.  Although my undergraduates and Google got it wrong, in a seemingly miraculous coincidence, when you look very closely at their homebrewed auction, you find that it is not very different at all from the (multi-object) Vickrey mechanism.  (In case you are wondering, the correct answer is that all of the k highest bidders should pay the same price: the k+1st highest bid.)

In a famous paper, Edelman, Ostrovsky and Schwarz (and contempraneously Hal Varian) studied the auction they named The Generalized Second Price Auction (GSPA) and showed that it has an equilibrium in which bidders, bidding optimally, effectively undo Google’s mistaken rule and restore the proper Vickrey pricing schedule.  It’s not a dominant strategy, but it is something pretty close:  if everyone bids this way no bidder is going to regret his bid after the auction is over. (An ex post equilibrium.)

Interestingly this wasn’t the case with the old style auctions that were in use prior to the GSPA.  Those auctions were based on a first-price model in which the winners paid their own bids.  In such a system you always regret your bid ex post because you either bid too much (anything more than your opponents’ bid plus a penny is too much) or too little.  Indeed, advertisers used software agents to modify their standing bids at high-frequencies in order to minimize these mistakes.  In practice this meant that auction outcomes were highly volatile.

So the Google auction was a happy accident.  On the other hand, an auction theorist might say that this was not an accident at all.  The real miracle would have been to come up with an auction that didn’t somehow reduce to the Vickrey mechanism.  Because the revenue equivalence theorem says that the exact rules of the auction matter only insofar as they determine who the winners are.  Google could use any mechanism and as long as its guaranteed that the bidders with the highest values will win, that can be accomplished in an ex post equilibrium with the bidders paying exactly what they would have paid in the Vickrey mechanism.

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