Dollar auction

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The Dollar auction is a non-zero sum sequential game designed by economist Martin Shubik^[1] to illustrate a paradox brought about by traditional rational choice theory in which players with perfect information in the game are compelled to make an ultimately irrational decision based completely on a sequence of rational choices made throughout the game.

Contents 1 Setup 2 Outcome 3 Refutations 4 Analysis 5 See also 6 References 7 Further reading

[edit] Setup

The setup involves an auctioneer who volunteers to auction off a dollar bill with the following rule: the dollar goes to the highest bidder, who pays the amount he bids. The second-highest bidder also must pay the highest amount that he bid, but gets nothing in return. Suppose that the game begins with one of the players bidding 1 cent, hoping to make a 99 cent profit. He will quickly be outbid by another player bidding 2 cents, as a 98 cent profit is still desirable. Similarly, another bidder may bid 3 cents, making a 97 cent profit. Alternatively, the first bidder may attempt to convert their loss of 1 cent into a gain of 97 cents by also bidding 3 cents. In this way, a series of bids is maintained. However, a problem becomes evident as soon as the bidding reaches 99 cents. Supposing that the other player had bid 98 cents, they now have the choice of losing the 98 cents or bidding a dollar even, which would make their profit zero. After that, the original player has a choice of either losing 99 cents or bidding $1.01, and only losing one cent. After this point the two players continue to bid the value up well beyond the dollar, and neither stands to profit.

[edit] Outcome

The game actually has no equilibrium, as two rational players in this game could theoretically lose all of their money to the auctioneer. Both players stand to lose money, but the winning bidder loses about 99 cents less than the losing bidder.

[edit] Refutations

People have proposed cooperative models by which the dollar auction can be profitable for the bidders and detrimental to the auctioneer. If the two parties bidding on the dollar agree to cooperate, they could bid two cents for the dollar together, and each profit 49 cents. However, this ignores the fact that the auction is a public auction. Though the end game only involves two bidders, before the bidding hits 98 cents it is still a "profitable" proposition for any player to enter the bidding.

To end the bidding war a bidder can bid 99 cents more than the previous bid, leaving no bid that offers a potentially higher profit (or smaller loss). (For example, Bidder 1 bids $x, Bidder 2 bids $x + $0.99. If Bidder 1 were to bid $x + $0.99 + $0.01, he would be bidding to pay $x + $0.99 + $0.01 for a prize of $1, or a total loss of $x-- the same as if he had not increased his previous bid.) As a special case of this, if the first bidder immediately bids $0.99, he will not be outbid by the other bidder, who has no potential to make a profit. The first bidder will earn $0.01 in profit and the second bidder will pay nothing and win nothing.

[edit] Analysis

The dollar auction is often used as a simple illustration of the irrational escalation of commitment. By the end of the game, though both players stand to lose money, they continue bidding the value up well beyond the point that the dollar difference between the winner's and loser's loss is negligible; they are fueled to bid further by their past investment.^[2]

The dollar auction is sometimes used as a refutation of Expected value calculations. The typical setup will show that at a bid price of 99 cents and a second bid price of 98 cents, the current bidder's application of probability calculations will determine to bid again based on his apparent choices of either:

• Losing 98 cents
• Gaining a dollar and paying one dollar.

However, it should be noted that this is an improper formation of an expected value calculation. The actual expected value of bidding again is not zero cents due to the unterminated nature of the game; the value of the bid is actually zero cents multiplied by the possibility of the other player giving up at that point, added to the value of losing two cents multiplied by the probability of the other player NOT giving up at that point, in an infinite series with unbounded loss.

[edit] See also

• War of attrition (game)
• All-pay auction

[edit] References

^  Martin Shubik: "The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation," The Journal of Conflict Resolution, 15, 1, 1971, 109-111.

^ Andrew M Colman: Game Theory and Its Applications in the Social and Biological Sciences (International Series in Social Psychology. London:Routledge Farmer. 1995. ISBN 0750623691.

[edit] Further reading

• William Poundstone, Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb, Anchor/Random House, 1993 (especially chapter 13: "The Dollar Auction")

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