The Shubik-Bazerman Auction

Today I will introduce a baffling piece of economics, psychology, or game theory, which has been called the dollar auction, though I consider that a bit of a misleading name. I believe Martin Shubik was the first to introduce it as a thought experiment, and Max Bazerman is the only person I have found who carried it out in practice. So I will dub it the Shubik-Bazerman auction for the time being.

A $20 bill is up for auction according to the following rules.

  • The bidding starts at $1 and goes in exactly $1 increments, and it is not allowed to bid twice in a row.
  • The two highest bidders pay the full price of their bid, but only the highest bidder wins the $20.
  • There is no communication allowed other than bidding.

A handful of people open bids for the $20. “Whatever, it’s just a bid, it couldn’t hurt, and I can drop out any time I want.” By the time the bidding reaches about $10 all but two contenders drop out — and the top two realize that they will either win the $20 or shell out cash to the auctioneer. Around $16 they seem to realize what is happening: the auctioneer is going to make a profit, and one of them is going to make almost nothing while the other pays almost $20.

Finally, bidder A bids $20 for the $20 bill. The immediate options available to bidder B are: drop out and lose $19, or bid $21 for the possibility of only losing $1. The hope of mitigating losses causes bidder B to bid $21 for a $20 bill. Bidder B is confronted by a similar dilemma, and bids $22. This continues for some time. Seven times (of 180) the bidding has cleared $100.

This phenomenon fascinates me, and I have tried in vain to formalize it. Say you are already above $20 and are considering making a bid. Your (unconscious?) thought process may be something like: he will surely drop out before he bids 10 more times, because this is ridiculous, at which point I will lose less than I would if I dropped out now. After all, 20 bids is a long time and it’s hard to see paying that much more for a measly $20 bill. But this thought process is going through both parties’ heads (perhaps unknowingly), so 20 more rounds continue without one dropping out. And the idea is continuous: neither really has a hard cut-off in mind, but each is playing as if the other has a hard cut-off in mind (and assuming said cut-off is reasonable, then they are both playing optimally).

Assuming you accidentally engaged in such an auction (say for $1), how could you ask mathematics what to do? What is the best strategy assuming you are playing against your own strategy (this strategy must work whether you were the first or second bidder)? A simple solution is to stop immediately and share the money with the other bidder, but let’s define that away and assume that the winner keeps all the money. Are the given conditions a guaranteed loss? If so, isn’t it fascinating that one can “give away” $20 and, under so-called “rational” conditions, not expect to lose any money?

What are readers’ ideas about how to analyze this? Do any interesting analogies come to mind?

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18 thoughts on “The Shubik-Bazerman Auction

  1. This seems recursive. Given an infinite supply of money, the second-highest bidder should /always/ bid higher. In this purely rational scenario the auction would in fact never terminate.

    We see these “real” outcomes when the money supply or number of bids is bounded.

  2. Psychological research regarding attachment and risk tolerance / aversion might provide insights that game theory washes away under assumptions.
    Attachment: People will run into the road for a $20 bill that fell out of their wallet, but not for a $20 bill that is already out there.
    Risk tolerance: People will go to greater lengths to avoid risk than to obtain reward and the bidding scales with that. At low bids the risk of being #2 is spread among many people, the losses are minimal, and the reward is medium. At higher the losses become larger, increasing people’s motivation to continue. Auctions also bring out the competitive side in people, and are self-selecting for those who are highly competitive.

  3. Your own google ads give a good clue where to look for analogies… There are lots of sites like that appear to making a bundle by setting up almost exactly this kind of auction. In those cases it’s pay-per-bid, which is in some ways even worse, but I think it exploits the same pathological psychology. An interesting phenomenon.

  4. Let’s assume I have already pushed the bidding up to $49. If the other person bids $50, I will lose my $49, so it makes more sense to bid $51 and only lose $31. But that only makes sense if I consider that bid in isolation.

    If I consider the fact the bidding has already gone past the value of the prize, and that nothing prevents the bidding from getting to $100 — in which case I will lose $80 even if I “win” — it makes more sense to stay at $49 and cut my losses.

    I have no idea how to formulate that mathematically.

  5. Two shopping sites are built on this concept: and

    It feels a little shady to build a business on this concept.

  6. The best strategy it would seem to me, is simply not to take part, or failing that, to drop out as soon as possible.

    The second best strategy would be to make everybody believe that you will carry on bidding *absolutely no matter what*. They will quickly realize that even though they can make you lose a lot of money, they cannot do that without themselves losing a lot of money. The only rational thing to do then would be to drop out fairly soon (below the $20 mark) at which point you make a profit.

  7. The first thing that came to my mind was was whether or not someone would choose to actually play this game because they may feel that just $20 is not worth bidding for. Perhaps if it was $100 up for bidding, an individual would be more likely to participate because of the perceived increased utility that would result of “winning” a bidding war for a larger sum of money.

    This was a good read, it made me think back to a game theory class I took last year (which I usually slept through).

  8. Analogies:
    If a business needs great investment, but it may be profitable for only one – the winner – participant. For example: building a railroad to a small town

  9. There is a problem with Colm’s strategy. In psych/economics games, people will spend $20 in order to force an opponent to loose $10 when the object of the game is to be the person with the most money at the end.
    The mechanics aren’t the same, but it tells you that people will abandon a winning strategy to stick it to the other guy.

  10. Drax :
    Really DUMB. The only intelligent bet is $1. Any outbidding is unintelligent and greedy.

    I disagree entirely. It is completely and utterly rational self-interest that drives this game. First, it is entirely rational to pay up to $19 for the $20, because that’s a 5.25% (or so) return on your risk. At $20 the rationale is to not lose, bringing the return on your risk to 0% and beyond that, it is loss mitigation. All things most functional humans, who tend to be less risk-averse, will do every time. Few plan on losing in this game on the outset, believing that everyone else will quit once the return hits 0%, but finding that the pride factor of losing *only* $30 while your competitor loses $49 (for example) turns into the main driver of the competition which then becomes a valuation of the loser’s pride (since pride is has its own purely subjective value).

    Basically, this experiment finds the self-imposed value of the pride of the second-place bidder, as a percentage of immediately spendable assets. Absolutely no greed, and absolutely no dumbness–except, perhaps, in agreeing to play the game in the first place.

    Makes me think it would make a great casino game…for the casino.

  11. I say that you should open the bidding (with your first bid) at $20. This way, there would be no gains or losses and it would discourage anyone else from out-bidding, unless they are just trying to screw you over.

  12. J.J. unfortunately, that is forbidden by rule #1.

    That’s very compassionate of you. Now that I’ve seen it, I’d just not bid and have a good laugh at the others. :-P

  13. I would say if i play only one time this auction makes sense to go until 10. Expected payoff versus risk. If i am planning to enter auction of this time for many times it is a different game. I need to come up with a startegy that on the long run keep me earning money.

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