International Law

Shubik's Dollar Auction Game - Not Rational to Play?

In 1992, William Poundstone wrote a series of essays on game theory aimed at an introductory audience.  The essays are worth reading, as he is constantly developing upon the theme how these strategic problems occupy real world considerations.

One of his expositions was on Martin Shubik's Dollar Auction game,

Much like Thaler's winner's curse, this appears to be another auction not worth playing.

Here is the dollar auction game, as described by Poundstone:

"In his 1971 paper, Shubik describes the dollar auction as an "extremely simple, highly amusing and instructive parlor game." A dollar bill is auctioned with these two rules:

1. (As in any auction) the dollar bill goes to the highest bidder, who pays whatever the high bid was. Each new bid has to be higher than the current high bid, and the game ends when there is no new bid within a specified time limit.

2. (Unlike at Sotheby's!) the second-highest bidder also has to pay the amount of his last bid - and gets nothing in return. You really don't want to be the second-highest bidder.

Shubik wrote, "A large crowd is desirable. Furthermore, experience has indicated that the best time is during a party when spirits are high and the propensity to calculate does not settle in until at least two bids have been made."

Shubik's two rules swiftly lead to madness. "Do I hear 10 cents?" asks the auctioneer - "5 cents?"

Well, it's a dollar bill, and anyone can have it for a penny. So someone says 1 cent. The auctioneer accepts the bid. Now anyone can have the dollar bill for 2 cents. That's still better than the rate Chase Manhattan gives you, so someone says 2 cents. It would be crazy not to.

The second bid puts the first bidder in the uncomfortable position of being the second-highest bidder. Should the bidding stop now, he would be charged 1 cent for nothing. So this person has particular reason to make a new bid - "3 cents." And so on

Maybe you're way ahead of me. You might think that the bill will finally go for the full price of $1.00 - a sad comment on greed, that no one got a bargain. If so, you'd be way too optimistic.

Eventually someone does bid $1.00. That leaves someone else with a second-highest bid of 99 cents or less. If the bidding stops at $1.00, the underbidder is in the hole for as much as 99 cents. So this person has incentive to bid $1.01 for the dollar bill. Provided he wins, he would be out only a penny (for paying $1.01 for a dollar bill). That's better than losing 99 cents.

That leads the $1.00 bidder to top that bid. Shubik wrote,

"There is a pause and hesitation in the group as the bid goes through the one dollar barrier. From then on, there is a duel with bursts of speed until tension builds, bidding then slows and finally peters out."

No matter what the stage of the bidding, the second-highest bidder can improve his position by almost a dollar by barely topping the current high bid. Yet the predicament of the second-highest bidder gets worse and worse! This peculiar game leads to a bad case of buyer's remorse.

The highest bidder pays far more than a dollar for a dollar, and the second-highest bidder pays far more than a dollar for nothing.

Computer scientist Marvin Minsky learned of the game and popularized it at MIT.

Shubik reported: "Experience with the game has shown that it is possible to 'sell' a dollar bill for considerably more than a dollar. A total of payments between three and five dollars is not uncommon."

So, is this a game not worth playing? Possibly.

W. C. Fields said it best: "If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it."

Without at all diminishing my respect for W.C. Fields, I venture to suggest that there is a more reasonable way to play this game as opposed to quiting. What is it?

First, lets update the game to the 21st century and restrict it to two players. Replace the $1 with $20 and each bid must be a multiple of $1. Each person must bid at least once, or they can agree not to play at all.

What should they do? Suppose first bidder bids $1, and second bidder pays $2, what is the first bidder's reasonable response?

Right now, as a collective they are paying the auctioneer $3 to get $20, or netting $17 as a group.

So, the first player should demand that the second bidder pay him $9.50, or half of $17.00 plus the second bid, not to bid! (Alternatively, second bidder can offer the first bidder $9.50 not to bid again.)

If the second bidder accepts or is accepted, he will "win" the $20, paying $2 to the auctioneer, $9.50 to first bidder and so he nets $8.50. First bidder gets $9.50, pays $1 to auctioneer and nets $8.50, jointly sharing the $17.00 equally.

As I see it, $8.50 is better than nothing, giving lie to the claim that you cannot get something for nothing.

Full post as published by Misleading Advertising Law on March 19, 2009 (boomark / email).