Dollar auction
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The dollar auction is a nonzero sum sequential game designed by economist Martin Shubik to illustrate a paradox brought about by traditional rational choice theory in which players are compelled to make an ultimately irrational decision based completely on a sequence of apparently rational choices made throughout the game.^{[1]}
Contents
Play
The setup involves an auctioneer who volunteers to auction off a dollar bill with the following rule: the bill goes to the winner; however, the secondhighest bidder also loses the amount that they bid. The winner can get a dollar for a mere 5 cents, but only if no one else enters into the bidding war. The secondhighest bidder is the biggest loser by paying the top amount they bid without getting anything back. The game begins with one of the players bidding 5 cents (the minimum), hoping to make a 95cent profit. They can be outbid by another player bidding 10 cents, as a 90cent profit is still desirable. Similarly, another bidder may bid 15 cents, making an 85cent profit. Meanwhile, the second bidder may attempt to convert their loss of 10 cents into a gain of 80 cents by bidding 20 cents, and so on. Every player has a choice of either paying for nothing or bidding 5 cents more on the dollar. Any bid beyond the value of a dollar is a loss for all bidders alike. A series of rational bids will reach and ultimately surpass one dollar as the bidders seek to minimize their losses. If the first bidder bids 95 cents, and the second bidder bids one dollar (for no net gain or loss), the first bidder stands to lose ninety five cents unless he bids $1.05, in which case they rationally bid more than the value of the item for sale (the dollar) in order to reduce their losses to only five cents. Bidding continues with the second highest bidder always losing more than the highest bidder and therefore always trying to become the high bidder. Only the auctioneer gets to profit in the end.^{[1]}
Analysis
The game is analogous to the war of attrition and penny auction and has a symmetric mixed strategy equilibrium (there are also asymmetric pure equilibria). Suppose we start with two players; player 1 moves in odd periods, while player 2 moves in even periods. When a player is behind, they are indifferent between raising and dropping out. If the opponent drops out with probability raising gives the player an expected payoff of On the other hand, dropping out yields zero, so indifference requires that the opponent drops out with 5% probability each period. Notice that this probability is independent of how high the bids currently are; this follows from the fact that past bids are a sunk cost. The maximum bid follows a binomial distribution with mean 0.5 (recall the bidders make zero profits each, on average).
See also
 Allpay auction
 Bidding fee auction
 Escalation of commitment
 Tragedy of the commons
 War of attrition (game)
Notes
 ^ ^{a} ^{b} Shubik: 1971. Page 109
References
 Shubik, Martin (1971). "The Dollar Auction Game: A Paradox in Noncooperative Behavior and Escalation" (PDF file, direct download 274 KB). Journal of Conflict Resolution. 15 (1): 109–111. doi:10.1177/002200277101500111.
 Poundstone, William (1993). "The Dollar Auction". Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb. New York: Oxford University Press. ISBN 019286162X.