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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.

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.

Outcome

The game has a unique mixed strategy Nash Equilibrium in which each player continues the game with some probability, determined by the minimum bid increment and size of the prize.

Refutations

People^[who?] 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.

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.

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.

@@ Line 5: / Line 5: @@
 ==Outcome==
+The game has a unique mixed strategy Nash Equilibrium in which each player continues the game with some probability, determined by the minimum bid increment and size of the prize.
-The game actually has no [[Nash equilibrium|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.
 ==Refutations==

v t e Topics in game theory
Definitions	Congestion game Cooperative game Determinacy Escalation of commitment Extensive-form game First-player and second-player win Game complexity Graphical game Hierarchy of beliefs Information set Normal-form game Preference Sequential game Simultaneous game Simultaneous action selection Solved game Succinct game
Equilibrium concepts	Bayes correlated equilibrium Bayesian Nash equilibrium Berge equilibrium Core Correlated equilibrium Epsilon-equilibrium Evolutionarily stable strategy Gibbs equilibrium Mertens-stable equilibrium Markov perfect equilibrium Nash equilibrium Pareto efficiency Perfect Bayesian equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Risk dominance Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfection Trembling hand
Strategies	Appeasement Backward induction Bid shading Collusion De-escalation Deterrence Escalation Forward induction Grim trigger Markov strategy Dominant strategies Pure strategy Mixed strategy Strategy-stealing argument Tit for tat
Classes of games	Bargaining problem Cheap talk Global game Intransitive game Mean-field game Mechanism design n-player game Perfect information Large Poisson game Potential game Repeated game Screening game Signaling game Stackelberg competition Strictly determined game Stochastic game Symmetric game Zero-sum game
Games	Go Chess Infinite chess Checkers Tic-tac-toe Prisoner's dilemma Gift-exchange game Optional prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede game Lewis signaling game Volunteer's dilemma Dollar auction Battle of the sexes Stag hunt Matching pennies Ultimatum game Rock paper scissors Pirate game Dictator game Public goods game Blotto game War of attrition El Farol Bar problem Fair division Fair cake-cutting Cournot game Deadlock Diner's dilemma Guess 2/3 of the average Kuhn poker Nash bargaining game Induction puzzles Trust game Princess and monster game Rendezvous problem
Theorems	Arrow's impossibility theorem Aumann's agreement theorem Folk theorem Minimax theorem Nash's theorem Negamax theorem Purification theorem Revelation principle Sprague–Grundy theorem Zermelo's theorem
Key figures	Albert W. Tucker Amos Tversky Antoine Augustin Cournot Ariel Rubinstein Claude Shannon Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin John Conway Jean Tirole Jean-François Mertens Jennifer Tour Chayes John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Axelrod Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey
Miscellaneous	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition Evolutionary game theory First-move advantage in chess Glossary of game theory List of game theorists List of games in game theory No-win situation Paradox of tolerance Solving chess Topological game Tragedy of the commons Tyranny of small decisions