The dollar auction is a non-
zero sum
Zero-sum game is a mathematical representation in game theory and economic theory of a situation which involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is e ...
sequential game
In game theory, a sequential game is a game where one player chooses their action before the others choose theirs. The other players must have information on the first player's choice so that the difference in time has no strategic effect. Sequen ...
explored by
economist
An economist is a professional and practitioner in the social sciences, social science discipline of economics.
The individual may also study, develop, and apply theories and concepts from economics and write about economic policy. Within this ...
Martin Shubik
Martin Shubik (1926-2018) was an American mathematical economist who specialized in game theory, defense analysis, and the theory of money and financial institutions. The latter was his main research interest and he coined the term "mathematical ...
to illustrate a
paradox
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
brought about by traditional
rational choice theory
Rational choice theory refers to a set of guidelines that help understand economic and social behaviour. The theory originated in the eighteenth century and can be traced back to political economist and philosopher, Adam Smith. The theory postula ...
in which players are compelled to make an ultimately
irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
decision based completely on a sequence of apparently
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
choices made throughout the game.
[ Shubik: 1971. Page 109]
Play
The setup involves an auctioneer who volunteers to
auction
An auction is usually a process of buying and selling goods or services by offering them up for bids, taking bids, and then selling the item to the highest bidder or buying the item from the lowest bidder. Some exceptions to this definition ex ...
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, making them the biggest loser in the auction. The winner can get a dollar for a mere 5 cents (the minimum bid), but only if no one else enters into the bidding war.
However, entering the auction with a low bid may result in a problematic outcome. For instance, a player might begin by bidding 5 cents, hoping to make a 95-cent profit. They can be outbid by another player bidding 10 cents, as a 90-cent profit is still desirable. Similarly, another bidder may bid 15 cents, making an 85-cent 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.
A series of short-term 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 95 cents unless they bid $1.05, in which case bidding more than the value of the dollar would reduce their losses to only 5 cents.
Bidding continues with the second highest bidder always losing more than the highest bidder and therefore always trying to become the high bidder. Once the highest bid reaches one dollar, only the auctioneer will profit in the end.
Analysis
The game is a type of
bidding fee auction A bidding fee auction, also called a penny auction, is a type of all-pay auction in which all participants must pay a non-refundable fee to place each small incremental bid. The auction is extended each time a new bid is placed, typically by 10 to 2 ...
which is a discrete version of the
war of attrition
The War of Attrition ( ar, حرب الاستنزاف, Ḥarb al-Istinzāf; he, מלחמת ההתשה, Milhemet haHatashah) involved fighting between Israel and Egypt, Jordan, the Palestine Liberation Organisation (PLO) and their allies from ...
. Like these games, the dollar auction 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
In economics and business decision-making, a sunk cost (also known as retrospective cost) is a cost that has already been incurred and cannot be recovered. Sunk costs are contrasted with '' prospective costs'', which are future costs that may be ...
.
The maximum bid follows a
binomial distribution with mean 0.5 (recall the bidders make zero profits each, on average).
See also
*
All-pay auction
In economics and game theory, an all-pay auction is an auction in which every bidder must pay regardless of whether they win the prize, which is awarded to the highest bidder as in a conventional auction. As shown by Riley and Samuelson (1981), equ ...
*
Bidding fee auction A bidding fee auction, also called a penny auction, is a type of all-pay auction in which all participants must pay a non-refundable fee to place each small incremental bid. The auction is extended each time a new bid is placed, typically by 10 to 2 ...
*
Escalation of commitment
Escalation of commitment is a human behavior pattern in which an individual or group facing increasingly negative outcomes from a decision, action, or investment nevertheless continue the behavior instead of altering course. The actor maintains ...
*
Tragedy of the commons
Tragedy (from the grc-gre, τραγῳδία, ''tragōidia'', ''tragōidia'') is a genre of drama based on human suffering and, mainly, the terrible or sorrowful events that befall a main character. Traditionally, the intention of tragedy ...
*
War of attrition (game)
In game theory, the ''war of attrition'' is a dynamic timing game in which players choose a time to stop, and fundamentally trade off the strategic gains from outlasting other players and the real costs expended with the passage of time. Its precis ...
Notes
References
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{{game theory
Non-cooperative games
Paradoxes in economics
Social science experiments
Types of auction