An All-Pay Auction

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 — 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.


Interesting. I wonder if there is some relation to the cost of the bid and the price to bid.

I think the rational strategy is to treat it like a contingent interest. How much would you pay for the chance to win the item at a fixed discount. This has an overlay of auction strategy. But if you view the cost to bid as the price of a contingent interest, and think you can calculate the odds you will win (the "contingency"), then it is rational to bid.

But I bet most people don't do this.


A non-swoopo comment :

"it's a game you want to avoid playing if you possibly can."

Actually, you don't have to avoid playing the game if there is an upper limit for the bids. If the prize is X (say, $20) and the upper limit is X+N (say, $20+$8=$28), then just bid the amount N ($8) and yell out that your next bid will X+N ($28) if somebody dares to go above your bid N. Nobody will since it's a credible threat, and you win X-N ($12). Having a crazy voice might help making the threat even more credible.

Victor Prometeo L. Frankenstein

I cannot understand why that way of making money is considered legal in the first place, but let's assume it is and You have all the means to bring into existence/being Your business.
At this point You have to choose the appropriate hardware and software to implement your wannabe "dollar auction sale". What Your choose will be considering that auction sales occur in "real time"?
In this last question I smell a rat (a fat one), so, I won't try to give a direct answer to it; instead, I'm going to deploy all my ignorance in what follow:
1) What kind of responsiveness You need to implement a real time auction sale? 10sec.? 1sec.? 0.5sec.? 0.05sec.? Most likely: the lesser it is, the better!
2) Are we sure that the 1sec. countdown is appropriate/suitable in relation to the number of users that are bidding up? I'm not sure of it! I think that the countdown should be determined (at least) by both the responsiveness and the number of users.
3) Has a person from Cape Horn the same chances to compete than a person that lives in UK, if, for example, both are connected to an overcrowded server that is in UK and both have the same internet speed? I might say no because of the number of routers that an internet packet has to pass through, but even if that was not a problem, You might want to ask Yourself why when You are watching what is called real time content you're going to need more than a 15sec. memory buffer.
The two point above (I recognise that the third is a little weak per se considering very high-bandwidth) lead me to think that if Your "dollar auction business" is not widespread enough to ensure that Your e-auction sale is carried out in a way to assure, to all the participants, the same chances to compete in the last few seconds, you are running an illicit business (You are breaking the law!).

At this point, my last (and unanswered) question: why keep illicit businesses going on and on?



@19: I agree that this isn't gambling and I'd furthermore say that basic method the site is running on is not a scam. The rules are clearly describing what happens and everyone with basic apprehension can easily reason that the site owners cash in disproportionally, while most clients will pay for nothing. The existence of this enterprise proves that computers have become so accessible that they can be operated by morons. Thank you Bill Gates!


Phil wrote: If all bidders were perfectly rational, what would be the best bidding strategy? ... It must be something like, "bid with probability 1 in (e times number of bidders)", or something weird like that.

I have a hazy memory of a columnist doing exactly this once. I think it was for the Scientific American Mathematical Games column. I don't think it was Martin Gardner; it might have been during the Douglas Hofstadter days.

If I recall correctly, the setup was that one person selected at random from all entries would win, and the amount of the prize would be $1000 divided by the number of entries. So if just one person entered, they would get $1000; if two, each had a 50-50 chance of $500; and so on. The column did the mathematical analysis of the situation, with a result along the lines you describe above; that is, enter with probability 1/N, for some large N. He even provided the value of N that everyone should use.

So, had everyone followed the suggestion, the total payout would be maximized. But he went on to write that he was sure he wouldn't end up paying the maximum, since most people would "cheat" and enter even if their random roll didn't come out right. The column a few months later verified this; the payout was under a dime, since many entries were received.

I wonder, though, if the same would be the case if the participants could coordinate their actions. What if, for example, in the original class, the first person were to loudly say "I am bidding 1 penny, which will be my only bid, and I will split the $20 evenly among us if I win." - would that work? An interesting experiment in its own right.



@25: Ken, I remember that ... I think it was the Hofstadter column in the early 80s. If I recall correctly, some entrants entered numbers of entries that were so huge, Hofstadter couldn't even compute them. And so not only couldn't he figure the winner, but the prize was so infinitesimal that it wouldn't have mattered.

Of course, it was free to enter that one. :)


"it's a game you want to avoid playing if you possibly can."

Easy money if you're quick, though. If I bid $19 first, then there's no incentive for anybody else to jump in with a higher round-dollar bid ($20, payoff = 0). I pay $19 to get $20 - a $1 profit.

Of course, you'd want all the other potential bidders to be rational...

Nigel Arlow

I use lowest unique bid sites all the time. It is part fun and part auction. I live in England and there are a proliferation of lowest unique bid auction sites.

I have won quite a few items on - and I can tell you that it isn't gambling. I use strategies to win the auctions and have been successful doing so.

Remember, with lowest unique bid you are told the status of your bid (if it is not unique, unique but not lowest or lowest unique bid) and if this changes in the game, they text me or mail me. SO it isn't a lottery. You can't dictate the outcome in a lottery, you just pick your numbers and take your chances

Its a bit of fun for me and in this economic climate, its refreshing to have some fun and win prizes.


If the first bidder bids $10, then it won't be worth anyone else's while bidding as the best they could do would be to get $20 for $21, so the bidder will get the $20 for $10 as their's is the only bid.


A similar website ( has been going for a long time in Germany and seems to be pretty successful. They work on selling packs of bids. I think some very interesting research could be done on the bidding psycology amongst users as well as on how varying the cost per bid changes the perceived prices as well as the overall result for the auctioneer.

Also, would be interesting to see how often users go back onto new auctions - I would assume either they get good at gaming the system or they get disgusted pretty quickly?

Can't help feeling this is a case of economics being used to exploit people looking for a good deal!


#22 Stephan and #27 Chris:
You are missing the fact that you don't get to bid a dollar amount. When you use one of your pre-purchased "bids" all it does is raise the price 15 cents


in Swoopo you can NOT place a "bid." All you can do is pay $0.75 to increase the current bid by $0.15. And after this bid increase the expiration of the auction is extended by 20 seconds.


@26: Phil, yes, that's right. The contest was slightly different than I remembered: Anyone could enter by mailing in a positive integer, and the largest number entered would win; but the prize was divided by the sum of all the numbers. As in the simpler version I (mis-)remembered, the maximum payout would be if exactly one person entered, sending the value 1; but there was strong individual incentive to cheat.

Bobby G

@ Kevin (#31),

Some people are presenting their solution to the dollar auction and others are hypothesizing about Swoopo.

As for the dollar auction, Chris (#27) has the strategy I came up with when thinking about it. If you're the first bidder, you should bid $19. If you're the second bidder, you could either bid $20 and be a nice person (assuming the first person bid $1) or you should not enter at all, since you will enter into that game where your incentives are to keep bidding, and you will be locked into a game of chicken with whoever you get into a bid-off into. In this game, the "winner" loses less.

If the dollar auction does not allow flat bid entries, only in $1 increments, then the solution would be to never play. It's an interesting scenario where if one person plays, he win bigs, but if two or more people play, two people lose big.

My thoughts on Swoopo? Given the way the game works, on average, users pay more money to Swoopo than the benefit they get from getting a few products cheap. Just like a casino. Not everyone wins and not everyone loses, but in the end there is more money going into the company than coming out. So it is like gambling... you could play a few times and hope you win something and then stop playing, to stay "up," or you should never play, as in the end you will lose more than you win.



Is it technically an all-pay auction if only the top two bidders pay?

I don't think so. A strange mixture of a Dutch and all-pay auction for sure, but if the third-highest bidder doesn't pay I can't see how it can reasonably be called an all-pay auction.

Ian McKay

The time added aspect is very interesting. This is how EBay should work. (No, not paying to bid, just the added time). That would make it more like a real auction. EBay is NOT a live auction like it claims to be.

On another note paying to bid should not make a bit of difference as to how much you are willing to pay. It obviously does though. This is a very interesting experiment in human psychology.


I am not sure whether Swoopo is a legal auction site... assuming it is, it is the most ingenious business model I have seen in a very long time. I wish I came up with that idea.


Most people paying for nothing is a great definition of gambling.


Big high and threaten to overbid?

Ingeniero Juan Carlos Garcìa

This is for real!
In mexico,if you want to sell something for the federal government ,you have to pay for the "bases de compra " ,-requirements for bidding-,that cost from 500 to 3000 US dollars.
The cost of the requirements depends of waht you want to sell,from uniforms for the local police,to a subway line full proyect.
And all is completely legal.
Only the company awarded with the contract gets back his money.