Dollar auction

The dollar auction is a non-zero sum sequential game designed by economist Martin Shubik 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 apparently rational choices made throughout the game.^[1]

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 second-highest bidder also loses the amount that they bid. The winner can get a dollar for a mere five cents, but only if no one else enters into the bidding war. The second-highest bidder is the biggest loser by paying the top amount he or she bid without getting anything back. The game begins with one of the players bidding five cents (the minimum), hoping to make a ninety-five-cent profit. He can be outbid by another player bidding ten cents, as a ninety-cent profit is still desirable. Similarly, another bidder may bid fifteen cents, making an eighty-five-cent profit. Meanwhile, the second bidder may attempt to convert his loss of ten cents into a gain of eighty cents by bidding twenty cents, and so on. Every player has a choice of either paying for nothing or bidding five 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 ninety five 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 he rationally bids more than the value of the item for sale (the dollar) in order to reduce his 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 seemingly rational decisions during the game are in fact clearly irrational once one realizes that they are nothing more than a greedy algorithm, which is of course not guaranteed to give a globally optimal solution. In this case there is actually a globally optimal solution, which is to not play the game at all unless one is certain that there are no other players.

Or if you're the first player to bid you can bid on the dollar at 95 cents and providing everyone else is rational and won't bid at $1 to spite you then you can win 5 cents. Any player wishing to play this strategy of spite must also realise that the first player must bet more and so start the greedy algorythm to minimise their losses and so the bet is not a value of zero bid.

Notes

1 2 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 0-19-286162-X.

Topics in game theory

Definitions	Normal-form game Extensive-form game Escalation of commitment Graphical game Cooperative game Succinct game Information set Hierarchy of beliefs Preference

Equilibrium concepts	Nash equilibrium Subgame perfection Mertens-stable equilibrium Bayesian Nash equilibrium Perfect Bayesian equilibrium Trembling hand Proper equilibrium Epsilon-equilibrium Correlated equilibrium Sequential equilibrium Quasi-perfect equilibrium Evolutionarily stable strategy Risk dominance Core Shapley value Pareto efficiency Gibbs equilibrium Quantal response equilibrium Self-confirming equilibrium Strong Nash equilibrium Markov perfect equilibrium

Strategies	Dominant strategies Pure strategy Mixed strategy Tit for tat Grim trigger Collusion Backward induction Forward induction Markov strategy

Classes of games	Symmetric game Perfect information Simultaneous game Sequential game Repeated game Signaling game Screening game Cheap talk Zero-sum game Mechanism design Bargaining problem Stochastic game n-player game Large Poisson game Nontransitive game Global games Strictly determined game Potential game

Games	Prisoner's dilemma Traveler's dilemma Coordination game Chicken Centipede 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 games 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 Prisoners and hats puzzle Trust game Princess and monster game Monty Hall problem Rendezvous problem

Theorems	Minimax theorem Nash's theorem Purification theorem Folk theorem Revelation principle Arrow's impossibility theorem

Key figures	Albert W. Tucker Amos Tversky Ariel Rubinstein Daniel Kahneman David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Hervé Moulin Jean Tirole Jean-François Mertens John Harsanyi John Maynard Smith John Nash John von Neumann Kenneth Arrow Kenneth Binmore Leonid Hurwicz Lloyd Shapley Melvin Dresher Merrill M. Flood Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert B. Wilson Roger Myerson Samuel Bowles Thomas Schelling William Vickrey

See also	All-pay auction Alpha–beta pruning Bertrand paradox Bounded rationality Combinatorial game theory Confrontation analysis Coopetition List of game theorists List of games in game theory No-win situation Topological game Tragedy of the commons Tyranny of small decisions

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Dollar auction

Play

Analysis

See also

Notes

References