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.

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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