February 03, 2006

The Dollar Game

A friend related to me the following game: (Game as in Game Theory, I'm afraid)

A certain person is auctioning off a dollar. (Or a pound, or whatever) The deal is this – the highest bidder will win the prize, but every bidder – including losers – will be forced to pay whatever price they bid.

So what happens in the game? Notionally, two players can make a profit by agreeing to share the prize and refusing to escalate the situation. But such a deal is an inherently unstable one – each player has a lot to gain by screwing over the other. Furthermore, additional players can jump in and demand their share, blackmailing the cooperators by threatening to take all the prize for himself.

At each stage in the game, then, the player always profits by raising his offer to be above that of his rival. And unlike with the various pricing games we learn about in economics, this game doesn't just stop with zero profit. Even when the players are bidding above the prize, they still have an incentive to bid up a little more in order to try and claw back $1 worth of their losses. Extrapolating, we get to the conclusion that the players all end up paying infinite amounts of money for $1.

The only way to win the game is not to play.

Which is pretty cool. I wonder if this can be used to describe some real world situations…

- 5 comments by 1 or more people Not publicly viewable

  1. The lottery falls into this category (mathematical arguments cast aside for one moment). My numbers never come up, so the only way I can "win" (relatively) is by not playing.

    03 Feb 2006, 14:22

  2. The radio says that you can get U2 concert tickets for free if you ring a certain number within the next minute. However to win the tickets you need to stay on the phone longer then anyone else.

    24 Sep 2006, 18:40

  3. Moritz


    you haven’t taken into account that each player’s financial resources are limited. As always in real life there is no real infinity—which is sad from a mathematician’s point of view.

    05 Nov 2006, 10:37

  4. Fang

    It doesn’t really make a difference. Even if he had no money left, it is still better, at each stage, to increase his bid – simply as during each turn, increasing the bid is the move that minimises his loss. As the bids increase, eventually he gets into a situation where he either loses all the money he has, or loses everything and gains the prize on offer minus one pound. So, of course, he has to raise his bid. From then on, it’s either lose money he doesn’t have, or try to end up with some slight amount of cash remaining. Even when eventually even with the prize money, he doesn’t have enough case, he still has to keep bidding up, because whatever situation he is in, he is worse off by not bidding.

    The difference that saves the player in real life is that he is not a rational player. After some point, people stop considering such decisions as choices to be judged, and go on an instinctual level of ‘Let’s get out of here!’. So, funnily enough, irrational action is the preferred action here, though of course a rational person wouldn’t have got into the game in the first place.

    06 Nov 2006, 01:11

  5. Moritz

    No, there is a fundamental difference. At some point you have no more money. Period. Nothing in your pocket, house sold, bank account empty, nobody gives you any more money because they know you’ll spend it on some stupid game ;-)
    If you could always spend one more pound, you’d have inifinetly much money (and you could solve many problems by donating where apropriate).

    So if there is a lot to win, and you know that you havethe most money by far, you can safely play. (Of course nobody else would play if they acted rational…)

    Just my 2 p’s ;-)

    06 Nov 2006, 10:54

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