Balkinization: An All-Pay Auction

An All-Pay Auction

Ian Ayres

Martin Shubik invented a famous game-theory exercise, sometimes called “the dollar auction,” where a teacher auctions off a $20 bill to the highest bidder. Bids have to be in round dollar amounts, but the twist is that both the highest and the second-highest bidder have to pay. When uninitiated students start to play this game, someone rushes to bid $3 or $4 dollars for the prospect of winning $20, and then other students respond by bidding up the price.

But then something amazing happens as the auction price starts approaching $20. The remaining bidders realize that they could end up having to pay a lot of money and not win the auction. Imagine that you had bid $19, and another bidder upped the ante by bidding $20. What would you do? Is it better to bid $21 for a $20 prize or to remain silent and pay $19 for nothing?

What starts off as a feel-good exercise to take advantage of a generous professorial offer suddenly becomes a sickening war of attrition, where the last two bidders pay more than what the prize is worth. These games routinely end with the winning bid being 50 percent higher than the value of the prize. Since both the highest and second-highest bidders pay, this means that the professor rakes in about three times the amount being auctioned.

This is an example of what auction theorists call an “all-pay” auction, and it’s a game you want to avoid playing if you possibly can.

But Barry Nalebuff pointed me toward a scary website — called swoopo.com — that seems to be exploiting the low-price allure of all-pay auctions. And it seems to be working.

Swoopo auctions off desirable (gotta have) electronic items (Wii’s, smartphones) for really low prices and with really short fuses — often less than a minute before the auction expires. It’s kind of seductive to watch these fast-paced auctions — because if someone ups the high bid, 15 seconds of extra time is added to the auction length. I found myself waiting to see if a TomTom GPS device would really end up selling for $18.

But there is an important hitch: you have to pay Swoopo $1 every time you bid. This creates an analogous all-pay effect. Swoopo may only sell a Wii for $30, but it might collect an extra $1,000 from bids. This website is a great experiment to see whether sunk costs matter. I’m thinking that someone who has already invested $5 in bidding costs is more likely to keep bidding to “protect” his or her sunk investments.

Of course, there is also the concern that you might end up competing against a Swoopo-bot that outbids you just before the time is about to expire. This is a game that I don’t want to start playing.

Posted 3:31 PM by Ian Ayres [link]

Comments:

But isn't this just a form of community center style raffle, underneath it all? The player puts down a small sum in the hope of winning a desirable item against a limited group of people, while the auctioner/raffler collects more than the value of the item. If so, why should be worried about people going for a $5 flutter on a highly discounted Wii? I realize the setup of Swoopo drives people to keep bidding to try to secure their item, but to some extent that's a feature of a raffle too.

# posted by

Raffi : 3:56 PM

Douglas Hofstadter, when he was doing the "Metamagical Themas" successor to Martin Gardner's famous Mathematical Games" feature in Scientific American, talked a bit about "Prisoner's Dilemma" type non-zero-sum games and other such game situations concerning competition versus co-operation.

He once held what he called the "Luring Lottery" (IIRC). He had to convince Scientific American to run it. It might have been treated as a bona finde lottery [or auction, and they were concerned they might be on the hook for big money, as the professor might have been in your example had someone bid $1, and no one else bid. The gist of the game was that the prize was $1 million ... divided by the number of entries received. To make it easy, he said that the entrants need only put the number of entries they wanted to make (essentially, to "bid") on a single post card. Their chances of winning would be the number of their entries divided by the total number.

Scientific American really needn't have worried about getting stiffed for the full $ million.

As Hofstadter expected, not everyone tried to figure out the rational profit-maximising number of entries (one can determine the estimated number of entries assuming you know the approximate number of participants and also assuming that everyone does the same math and acts to maximise their profit as well; everyone should then make the same number of entries). Even if they had done so, the most likely eventual prize would have been far less than $1M. But there were spoilers; people that wanted to "win" (or just show their 'brilliance') even if it meant no money. These people made up the largest numbers they could think up (or formulate) and put that on the cards they sent in. In a talk Hofstadter gave later, I asked about the result, and he said that he could have waved a gold coin in the direction of the winning entrant, and enough gold atoms would sublimate and waft over to the winner to satisfy the prize payoff. Many complicated formulae tried for very large numbers, but one of Hofstadter's favourites was a card that said "I enter 9!!!!!!!! [...filling the card with exclamation marks (i.e factorial operators)] times" Simple yet elegant, this produces a very large number; try just the first couple iterations if you don't believe me.

There's always spoilsports (or possibly, "poisoners" colluding with the auction/lottery runner). And then there's that don't know enough to figure out the best strategy.

But there is a rational winning strategy for such; in Hofstadter's case, a finite number of entries (in his case, less than 1, actually; the entrants should roll a die with many faces and simply send in 1 entry if the marked die face shows up; the "best" number of entries is far less than 1).

In the case of the $20 auction, if the students collude and agree that only one person will bid, and the profit be shared by all (or assigned via subsequent lottery), or alternatively, that a person winning a secondary lottery should be the one and only bidder of $1, then profit is maximised for all (at the expense of the professor). But to do that; they have to co-operate ... and act rationally. Poor information, poor analytical skills, or just human orneriness will help the professor.

Cheers,

# posted by

Arne Langsetmo : 5:54 PM

FWIW, one web link about Hofstadter's "Luring Lottery" and some of the considerations involved in such games.

Cheers,

# posted by

Arne Langsetmo : 7:04 PM

More here.

Cheers,

# posted by

Arne Langsetmo : 7:11 PM

Mark Kleiman was already there on December 13.

# posted by

James Wimberley : 7:04 AM

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