Going, Going, ...., Gone!It may surprise you to learn how much insight into auctions mathematics has been able to provide in recent years...
Movie buffs will no doubt remember the tension and drama of the auction scenes in Alfred Hichcock's classic movie North by Northwest and in the James Bond thriller Octopussy. Auctions, whether in the movies or in real life, are filled with excitement - including the "ritual" sound of the auctioneer's gavel. Auctions have the tension of people anxious to get what they want - outbid their opponent but hopefully get a "bargain."
(Courtesy of West's of East Dean, England)
The World of Auctions
When someone has an item to sell, one approach to getting it sold is to find a "store," assign the item a price, and see if someone wants to buy it. However, for many items it is hard to know what price to assign. This is especially true for one-of-a-kind items, art works, photographs from the past, or the right to display your web page prominently when someone does a search with a particular keyword string (ad auctions).
The world is a complex place and when mathematics is used to model the world, it is both an art and a science to simplify the complex real world to obtain a model which is amenable to analysis. One wants the model to lead to mathematical questions that have been solved or for which new tools can be developed that lead to a solution, and which, when results are obtained,provide useful information about the original problem to be addressed. Furthermore, there are issues of fees for the auction house and who pays to have the painting shipped to the buyer if the purchase is successful, as well as what to do if the purchaser does not make good in paying for the purchase.
Types of Auctions
One very common thing for mathematicians to do when they study a "domain" of ideas is to develop a taxonomy or notation for different types of things in that domain. This is something that has been done with auctions, but because there is a large literature about auctions which does not have a mathematical origin and the systematic mathematical study of auctions is rather recent, there is no authoritative source for how the different types of auctions are named. In fact, one of the things that mathematicians do is to take existing ideas and to make them more general. This sometimes causes nomenclature issues. An example of an auction term that is not always used in a standardized way is "Dutch auction." I have seen this term used for an auction where there is a lot to be sold. A clock showing the current price is started from a value higher than what anyone would be willing to pay. The first bidder who says "stop the clock" gets the lot to be sold at the price shown on the clock. However, many people use the phrase Dutch auction to mean something rather different.
(Image Courtesy of Wiki)
Perhaps the biggest milestone in the theory of auctions was taken by the mathematical economist William Vickrey. Vickrey did his undergraduate studies in mathematics at Yale but his masters degree and doctorate degree were in economics from Columbia University.
(Courtesy of Wiki and Columbia University)
Dollar Auction of Martin Shubik
In dealing with policy issues or economics issues one hopes that people will be rational or use common sense. However, game theorists over a period of time have shown that there are games that challenge what common sense or being rational means. Examples of such games are chicken (who blinks first in a confrontation between two groups in Congress in negotiations over the debt ceiling) or prisoner's dilemma, which is sometimes used as a model for confrontation games. Auctions have their share of instances that challenge one's intuition and also raise questions about the discrepancy between theory and practice.
(Courtesy of Martin Shubik)
This fact, that people play this game in a way that seems "irrational" is a way to try to probe the psychology/behavior of real world human beings, rather than what theorists advise is wise or rational behavior for the dollar auction. There is empirical literature for the dollar auction game which parallels the large literature of empirical results for prisoner's dilemma. The set-up of prisoner's dilemma requires action by the players. In the dollar auction the players have the option not to participate.
Towards the Future
As is typical with situations where mathematical modeling comes into play, one usually starts by making very strong simplifying assumptions partly because the hope is this will lead to mathematical formulations that are easy enough to solve. If one is successful in getting insights from strong simplifying assumptions, one can try to tweak one's model to make it more realistic. This is in part the reason why most early theoretical studies of auctions involved the auctioning of a single item. However, for many reasons theory should not stop there. While it may make sense to be able to have an understanding of what is involved in auctioning a piece of the electromagnetic spectrum for cell phone traffic in a certain city, it may be more realistic to deal with the auctioning rights to electromagnetic spectrum in nearby cities also because the company that might buy the rights for one of the cities would also have an interest in the rights for the nearby city. It may be that company A would be reluctant to get the rights for one of the cities but not the other. However, if the rights to the spectra are sold separately, company B, knowing this, would run up the price for getting rights in one of the cities knowing that A will not be able to afford a "fair" price for getting the rights in the other city. Some of these issues can be addressed by having auctions where related items are sold together. These are the combinatorial auctions that were mentioned above. From a mathematical point of view one can model some of this using a "value" function v(S) where S is some item. One can inquire if v(S∪T) is equal to v(S) + v (T) or smaller (much smaller) or larger (much larger) than this quantity. One can also wonder about whether bidders, knowing the nature of the auction process, will negotiate prior to the auction (collude?) to better their own individual situations. Theorists are at work trying to understand which auction procedures encourage or discourage such behavior. As auctions are being used in a growing variety of situations, mathematics is responding with exciting new ways to meet a growing need for robust auction procedures.
The AMS encourages your comments, and hopes you will join the discussions. We review comments before they're posted, and those that are offensive, abusive, off-topic or promoting a commercial product, person or website will not be posted. Expressing disagreement is fine, but mutual respect is required.
Ausubel, L. and P. Cramton, Auctioning many divisible goods, J. of Euro. Econ. Assoc., 2 (2004) 480-493.
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