“The Dollar Auction game: a paradox in noncooperative behavior and escalation” was published in the Journal of Conflict Resolution in March of 1971. Martin Shubik is the author. Here’s how it works: The auctioneer says, “I’m going to give a dollar to the highest bidder, just like a regular auction. The only difference is that both the highest bidder (like any regular auction) and the second highest bidder (unlike a regular auction) have to pay the auctioneer the value of their bid.”
Mr. Smith offers an initial bid of a nickel. This bid certainly seems reasonable. Who wouldn’t want to get a dollar for five cents? And, according to the rules of the dollar auction, if no one else bids, Mr. Smith will indeed get a dollar for a nickel. Not a bad deal in these tough economic times.
Ms. Jones, however, bids a dime, reasoning as follows: I can get a dollar for ten cents. That’s a good deal. I’ll have to pay a dime and Mr. Smith will have to pay a nickel. But I’ll get a dollar for my dime and he’ll get nothing for his nickel.
(For the sake of simplicity, I will limit the explanation to just Mr. Smith and Ms. Jones. The argument doesn’t change with more people.)
Mr. Smith is now in a tricky spot: If he remains quiet, he loses a nickel. Ms. Jones will get the dollar, paying only a dime and he will get nothing for his nickel. Any rational person in Mr. Smith’s place will bid 15 cents. Mr. Smith will still get 85 cents profit. And he certainly doesn’t want to get nothing in return for his nickel. So Mr. Smith bids 15 cents.
But Ms. Jones now speaks up. She bids 20 cents. If she doesn’t bid again, she will lose her dime. By bidding 20 cents, she will earn 80 cents if Mr. Smith doesn’t bid again.
But of course, Mr. Smith does bid again. He has to. Otherwise he’ll lose the amount of his previous bids. Similarly, Ms. Jones has to keep bidding.
Consider their reasoning as the value of the bids approaches a dollar: Ms. Jones has bid 95 cents. Mr. Smith bids a dollar. That’s right. He bids a dollar to win a dollar. What choice does he have? If he doesn’t bid, he loses his 90 cents and Ms. Jones wins the dollar and gets a profit of five cents.
So what will Ms. Jones do? She will do what any rational person would do. She can’t lose her 95 cents and get nothing in return. She must keep bidding. She must bid $1.05. She has to bid more than a dollar to win a dollar.
Mr. Smith has to bid more than $1.05. Otherwise he’ll lose his dollar and get nothing in return. He bids $1.10. Surely any reasonable person would agree it’s better to pay $1.10 for a dollar than to lose $1.00 and get nothing. Losing a dime isn’t as bad as losing a dollar.
Except of course that now Ms. Jones has to bid $1.15 by the same reasoning. She bids $1.15 to win the dollar.
Both Mr. Smith and Ms. Jones must now continue to bid amounts well in excess of the dollar. There is no point at which it makes sense to stop bidding. When Mr. Smith bids $4.00 to preserve his $3.90, Ms. Jones will bid $4.15 so she doesn’t have to pay $4.05 and get nothing.
I’d like to write more about the dollar auction, but I’m too busy thinking about a completely different situation: These folks I know are getting a divorce. Coincidentally enough, Ms. Jones, the mom, kept her maiden name when she married Mr. Smith. One recent evening, Mr. Smith said he didn’t feel like doing the dishes. Ms. Jones said she didn’t love Mr. Smith. Mr. Smith said he had never loved Ms. Jones. Ms. Jones threw some of Mr. Smith’s clothes out into the street. Mr. Smith froze the assets in Ms. Jones’s bank account. Ms. Jones had an affair. Mr. Smith hired an attorney. Ms. Jones hired a forensic accountant. Mr. Smith accused Ms. Jones of giving their adolescent children drugs and alcohol. Ms. Jones accused Mr. Smith of being inappropriate with their daughter. Mr. Smith took Ms. Jones’s name off the emergency contact card for their children at school. Ms. Jones refused to allow Mr. Smith to see the children last weekend when it was his turn for visitation. Mr. Smith told the family mediator that Ms. Jones is unfit to be a mother. Ms. Jones told the judge in family court that Mr. Smith is actually an alien from another galaxy.
Just kidding. Of course, I’m exaggerating. Ms. Jones would never go before a judge in family court and say that her husband, the father of her children, was an alien from another galaxy. What she said was that her husband was an alien from another solar system.
In any case, extra credit for any gentle reader who can solve the following problem: Given the rules of the dollar auction as described above, what is the only way to win? Answer in the newsletter next week.
