Going, Going, ...., Gone!

 

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."

 

 

However, modern auctions are used for more than selling Picasso, Rembrandt and Goya paintings. They are used to sell government securities, leases to drill for gas or oil on public property, the right to use part of the electromagnetic spectrum for cell phone service, provide services for the Medicare health program run by the U.S. Government, or get rid of a piece of "junk" from your attic or closet by selling it at an on-line auction site.

While people are familiar with the role that mathematics plays in applications to engineering, chemistry, physics and biology, it may surprise you to learn how much insight into auctions mathematics has been able to provide in recent years. These insights are garnered in a variety of ways. First, there have been many theoretical developments which help people design auctions to better meet the goals that are present in conducting the auction. Second, there are lots of data for study about the way different styles of auctions behave because implementations of on-line auctions by companies like eBay allow buyers and sellers to exchange an amazing variety of goods using different kinds of auction rules. Third, there have been experiments carried out by economists and psychologists in "laboratory" settings to try to understand the nature of the way different people behave in auctions. Using data and theory, mathematicians have explored various problems that have resulted in both new mathematics and more responsive auction designs. Here I will sample some of what has been accomplished.

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).

Who are the people who study auctions? Like many situations where mathematics has been used outside of mathematics (mathematical modeling), the individuals who study auctions may be economists, psychologists, experts on accounting and business, computer scientists, and mathematicians. What mathematics is used in studying auctions? As is generally true when mathematical models are constructed, people who study, design and participate in auctions use many mathematical tools. In a general way auctions belong to the part of mathematics called game theory, a subject which deals primarily with situations where different "players" have different interests and goals and, thus, are in "competition" with each other. Players in these games may use complex strategies to achieve these goals - including "lying" about their true beliefs to get some advantage. Thus, if one looks at a paper on the theory of auctions one might see integral signs (calculus), matrices (linear algebra), graphs (both the kind from calculus and graph theory), partially ordered sets (discrete mathematics), ideas from complexity theory, mathematical economics and game theory. To read technical papers about auctions one would probably need to know terms like: core of game, Nash equilibrium, opportunity cost, price to clear the market, etc. Work on auctions not only draws on mathematical tools but on important ideas from economics such as concepts about price, supply and demand, and the way markets and firms operate.

Historically, auctions have been used in "exchanging" a wide variety of goods: art, flowers, fish, tobacco, treasury bills, and many others. While these exchanges have typically balanced the needs of buyers and sellers in a way that did not cause great notice, there is the story of the tulip "bubble" that occurred in the Netherlands in the 17th-century. There was a period when tulip bulbs, something that one can not eat, turn on like a TV set, have grace one's wall like a Rembrandt, changed hands for a sum of money which sometimes exceeded what a worker of the time (1637) could earn in a year. The "hysteria" which seems to have been responsible for this peculiar situation explains why psychologists and behavioral economists have taken an interest in auctions. There was no doubt some feature of the ordinary economics of supply and demand going on here - tulips were a desired commodity but the flower markets in 1637, as is true sometimes of commodity markets today, can have a very high volatility.

The idea behind an auction is the exchange of a good between the owner and a potential buyer of this good. One advantage of using mathematics to analyze auctions is that it is possible to see that there are different kinds of auctions and one can try to see which types are the same but look different, which ones are better for buyers and which ones are better for sellers. The most basic framework for an auction is that there is a single seller who is making available a single item for sale. The format under which the item is put up for sale can vary. There may be a physical place where the auction occurs and in order to participate one must be physically present. However, perhaps you can phone in bids or make bids in advance, and your wishes are carried out by an "agent" on site.

When mathematicians want to apply mathematics to a particular domain, they can use a mathematical model to get insight. It is often helpful to have a taxonomy of the situation at hand and to understand the components of the system investigated. When one holds an auction, who is involved? In simplified form, some of the components of a "real auction" that must be taken into account (modeled) are:

a. A seller (the person who owns the object and wishes to get someone to buy it)

b. Buyers (these are the people who might have an interest in acquiring the object)

c. Auctioneer (this is the person who typically represents a company hired by the buyer to conduct the auction and who provides the interface between the buyers and the seller. The auctioneer follows some predetermined procedure that is known to the seller and buyer. He/she is also trusted to carry out the duties of auctioneer "honestly" without favoring the buyers or the seller. Recently, the auctioneer's role may be carried out by a software system!

d. Sometimes there are additional parties who have an interest in an auction. For example, if a work of art from country X is a national treasure, the seller may be guided by only maximizing profit but if the buyer is from another country, some issues of national pride may be compromised. Sometimes what is being auctioned are mineral rights or parts of the electromagnetic spectrum. These items may be considered "common resources" but they might be sold since the public will benefit if use is made of the spectrum or resources; additional revenues accrue to the public.

Now that we have identified the "players" in an auction, we can consider some issues related to the goals of the auction. Sometimes the goal is to maximize the revenue of the seller. Sometimes the goal is fairness of access. Rather than give a contract for a government construction project say to a friend of the agency doing the work, one might want to have a broad field of applicants who might make bids to do the work. A different goal might be to maintain the privacy of the bidders. Some auctions require that one appear in public to make a bid; in other settings a procedure may be put in place to allow bona fide buyers to make their bids without disclosing their identities.


The following example helps give the flavor of some of the issues that come up in an auction environment.

Example 1:

Mr. X has contracted with an auction house to sell a painting that his grandmother had willed him. He has no way to display it at the current time and he needs cash. He takes his painting to a local auction house to sell on his behalf. While he is anxious to get cash for the painting, X has no idea of its value. He might take the painting to have it appraised in order to get some idea of its value; some auction houses will, for a fee, appraise the items they sell on your behalf. X may decide on the basis of the appraisal to set a secret "reserve" price for the minimum amount he is willing to sell it for. The auction house, especially when the painting is by a known artist, wants to be sure of the circumstances under which X obtained the painting. Perhaps grandma's husband was given the painting by the artist and it came to her as part of his estate. However, unknown to grandma the painting may have come to grandpa as stolen property, a fact he may or may not have been aware of.

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.

For simplicity we will pretend that there are only seven people who want to place bids in the auction. A₁, A₂, ...., A₇ (agents 1 to 7). Each of these "players" has a valuation for the painting as follows:
 

A₁: $510

A₂: $470

A₃: $450

A₄: $430

A₅: $400

A₆: $380

A₇: $340

Who do you think will get the painting and what is the price that will be paid? Note that above there are no identical bids but auction procedures must deal with what to do in the case of ties.

Unlike the example above, in some auction situations the buyer has the desire to purchase many of the items, and the seller might have many essentially identical items she wishes to sell. This might be true for a potential buyer at an auction who sells plants wholesale. The bidders in such an auction would have "private knowledge" of how many plants they believe they can sell with a markup based on the price that they pay. In other situations, a government might want to pay for an infrastructure project by selling securities. The idea is to float a bond issue to a collection of "buyers" who resell these securities to a broader group of people. The government involved wants to sell a large number of these bonds at a low interest rate.

Example 2:

In some situations where goods (commodities, securities) are auctioned there is a price each buyer is willing to pay but there may some bidders who are not interested in purchasing all of the goods for sale. In the situation below the buyers have the same amounts they are willing to bid (as in Example 1) on the full lot of 10 units, but some of the bidders are not interested in having 10 units. The results of the auction have to specify whether one can make a bid on less than the full number of units being sold.
 

What do you see happening in this auction if only bids for all 10 units can be accepted? Alternatively you should consider that an acceptance price can be set for amounts less than 10 units and that if more than 10 units are allocated at this price, there is a system of fairly distributing the promised sales of 10 units to bidders who collectively want more than 10 units. However, in this approach some bidders may not get their desired number of units. Can you see a price that "clears" the market in this way?

A key concern in auction models is to understand how people will bid in situations where they have limited amounts of information about the value of what they are buying, as well as the their knowledge of what other buyers know about the value of the items for sale. Thus, the structure of some auctions allows one to infer information about the value of what is being auctioned. Some have called attention to the phenomenon of the "winners curse."

If in a small bookstore in a rural state someone brought in an antique copy of Newton's Principia, the buyer and seller both might realize that this is a valuable piece of property and that perhaps the store owner cannot offer the seller the usual $1 for an "old worthless" book. In the age prior to the Internet, the same dealer and sophisticated seller might not know what a reasonable amount might be to sell a copy of the less well known Cauchy's Recherche sur les polyèdres - premier mémoire (bound as a separate "book") or Max Brückner's Vielecke und Vielfläche: Theorie und Geschichte (with the lithograph plates intact). Now, in the Internet age, one might be able to get some idea of how much someone might be willing to pay for such "rare" books and/or what amounts people did pay in recent sales. The importance of auctions as a "market" tool is that they organize the sale in a way that may increase the potential number of buyers. There is probably a tiny market for books by Cauchy and Brückner in a small rural town. There might be many more potential buyers in New York City or London, especially at a shop that specializes in the history of mathematics and science books. And in this Internet age there are many more potential buyers for such books via sales at an on-line auction site.

When I buy an item at an auction there are many uses to which I might put that item. For example, a mathematician book lover might buy a book to be part of his/her private library. Books might have different value depending on the name recognition of the author (Newton, Lagrange, Euler, Hilbert, etc.) or the buyer might know that the price being asked is much lower than the person can get for the item by reselling it to an interested person, a bookstore specializing in rare mathematics books, or at an auction of rare books being held by an international company specializing in auctions such as Sotheby's and Christie's.

Example 1 (with changes and more details)

Let us assume that none of the buyers know anything at all about the other buyers for the item to be sold, now thought of as a rare mathematics manuscript rather than a painting. A₃ does not realize that A₅ is a book dealer who owns a technical book store that specializes in both new and rare technical books. A₄ does not realize that A₇ represents a local auction house that twice a year sponsors rare book auctions. These circumstances are typical of the information (which can be hard to predict and to weigh) which is in the possession of particular individuals. If the stakes are high enough, there are times players will indulge in "questionable" behavior to acquire information. We see repugnant behavior in shows on TV all the time. Recently, people in the financial world have gone to jail for "insider trading." To know what a competitor might know about the geology of an oil field that drilling rights are being sold for, some company may be tempted to try to probe what their rivals know, sometimes even using industrial espionage.

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.

Here let me explore some of the commonly used types of auctions while at the same time fleshing out some of the other issues that enrich the modeling aspects of this environment.

Ascending open bid auction (English auction)

In this type of auction (the kind one typically sees in the movies), people are physically present (but sometimes can bid by phone or electronically) and make their bids in response to those of other bidders with bids being noted and "accepted" by the auctioneer. The bidder who makes the last bid gets the item at the final price that he/she offered.

Comment:

Often there is a secret reserve price which is specified by the seller and if that amount is not reached, then the item does not get sold. There may be an opening amount required for the bidding to go forward at the start, other than zero. The auctioneer may request new bids in specified amounts above the previous bid but bidders are often allowed to "jump" much higher than a previous bid. Note that in this type of auction at each stage the highest current bid is available to be known by the other bidders.

Descending open bid auction (Dutch auction)

The opening price is higher than what is thought anyone will be willing to pay. It is lowered using some "clock" which gets stopped someone bids. This person then gets that item (or lot) at the amount on the clock at the time of the bid. Such auctions have been used for tulips bulbs in the Netherlands and for items that might spoil, such as fish. This is the type of auction I alluded to earlier.
 

First price sealed bid auction

The buyer participants in this type of auction make one bid which is submitted to the seller (auctioneer) in a way that other bidders don't know the size of the bid which is made. The seller opens the "private" bids and the highest bidder gets the "object" at the price listed in the bid, with provisions for what to do in the case of a tie.

Comment: This type of auction is often used for government contracts and rights to conduct mining operations.

Second price sealed bid auction (Vickrey auction)

The buyers who participate make bids which are are sealed (private) to the seller (auctioneer). The auctioneer opens these bids and announces who will win based on the bid which is highest but where the price the highest bidder pays is not the amount of the highest bid but that of the second highest bidder. Again, there are provisions for ties.

Mathematics can give insights into the relationships between what happens when different auctions are employed. Though the way a descending open auction is described has a dynamic "flavor," in practice each bidder must decide on a price to call out to stop the auction assuming that no bidder has called out a bid already. Thus, the outcome of such an auction will be the same had the auction been conducted as a first price sealed bid auction. Put a bit differently, an auction participant choosing a particular bid gets the same payoff in both types of auctions.

Combinatorial auctions

An auction in which there are many items that are going to be auctioned at one time. Bidders can or must submit bids for sets of items. There are different types of combinatorial auctions depending on the way that the bids are accepted and how to distribute the items to the "winner(s)." For example, a bidder may specify bids for X and for Y but does not want to win unless given both of the items. The way the winners are decided is part of what the bidders know but the rules for deciding a winner can be very complicated. There is no guarantee that the person with the highest bid for some set of items will win the items.

Comment: This type of auction has prospered and been studied when it came to be realized that socially problematical results sometimes emerged from "sequential" auctions for different items which changed value when used in conjunction with each other.

In the table shown there are 4 bidders who have assigned values to being able to get the items they choose. The auctioneer must give each item to only one person. Line 3 means that A₃ is willing to pay 60 units to get items V and Z. Note that the bid indicated will only be honored if the person gets all the items that have an x indicated. Also, we assume (for similar kinds of examples) that one can't give a person some item for "free" to get a high sum. There are items people want and don't want.

 

Even though the highest bid was 70, A₁ does not get any items in one way to assign "winners." Giving A₃ items V and Z and A₄ items U and W disposes of all items. Furthermore, it yields a higher total revenue than giving A₁ items U and V and giving A₂ items W and Z.

As the kinds of objects that are being sold using auctions have become more varied and the evolution of on-line markets has progressed, there are many new ideas and approaches to auctions. Mathematicians (and others) have also found ways to think of phenomena that do not look like auctions from an auction point of view. For example, some scholars have studied what are called "all pay auctions." This is an auction where some "prize" is up for grabs but in attempting to get the prize lots of people pay, though the nature of the prize is "vague." Examples of situations where people think of what is happening as an all pay auction are political lobbying (where many people make contributions even if they don't get the prize) to affect a political outcome or contests where there is an entrance fee of some kind to participate but a prize goes only to one "winner."

Vickrey Auction

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.

 

Recall that a Vickrey auction is one with sealed bids where the item goes to the highest bidder but at the amount that the second highest bidder bid! Why would it make sense to do this? One answer is that it encourages the bidders to bid their true values! If one uses the information in Example 1 for a traditional sealed bid auction where the bidders bid their truthful values, the winner would be A₁, who would pay $ \$ 510 $ for the item (say a painting), since A₂ would bid only $ \$ 470$. However, A₂ might be tempted to bid less than what the item was worth to him for fear he/she was "overpaying." compared with what other people might think the item was worth. However, in the Vickrey auction each person's bid does not affect the price that the item changes hands for. The size of the highest bid determines who gets the item but the amount paid depends not on the high bidder's choice of bid but other peoples' valuations. Example 1 run as a Vickrey auction will give the item to A₁ but he/she will pay only $ \$ 470$. If A₂ bid $ \$ 500 $, A₂ will not get the item. If you bid $ \$ 500 $ when your true valuation is $ \$ 470 $ you run the risk in a traditional sealed auction of paying $ \$ 500 $ for an item you think is worth less.

Given the fact that Vickrey auctions encourage bids that are honest with respect to the true value that the bidder places on the object, why are there so few Vickrey auctions? Empirically, they are not used that often. One reason might be tradition - auctions by Sotheby's and Christie's predate by a long time Vickrey's theoretical observation. Another reason might be that in designing an auction the seller may be more interested in maximizing revenue over insuring that the bidders give forth accurate bids. If the excitement of an ascending bid auction encourages people to go beyond what they were originally planning to spend, this might seem a nice feature of this type of auction from the point of view of a seller and the organization running the auction. Some auction houses charge a percentage of the sale price for their role in selling the object. (The auction houses are "selling" their reputation for honesty, that the items being auctioned are authentic, not stolen, etc. but they do like to make money.)

Much work has been done since William Vickrey's original work. Vickrey's idea of an auction that elicits truthfulness has been extended to try to deal with auctions where there is more than one item being auctioned. Other examples where eliciting truthful behavior is a goal are school choice (parents ranking schools they would like their children to attend) or matching hospitals and residents as part of student's medical training. In this last case, the goal is meeting the needs of hospitals for "relatively cheap" and able medical practitioners while at the same time having the students be happy with their assignments. Good assignments in cases such as these are not only of importance to the individuals and hospitals involved but to society as a whole.

Vickrey's step forward falls within the area of what today would be called Mechanism Design. The idea is to try to design ways to carry out systems where the public has an interest in addition to that of the "players" involved in the system. For example, when corporations carry out elections for various purposes, when hospitals interact with residents who are seeking residencies to complete their medical training, when treasury notes or health care contracts are being auctioned, society has an interest in what happens above and beyond what happens to the players in the system. Mechanism design tries to make ways that the various systems work optimize the situations that the players find themselves in since society too has a stake. Consider for example the auctions that the Federal Government conducts in conjunction with the Medicare System. Recently, an expert on auctions, Peter Cramton, organized a campaign where he enlisted support from a large group of experts on economics, mathematics, computer science, and auctions for his position. The situation concerned how auctions are conducted involving private contractors that the government uses to deliver the services in its Medicare program to the public. The experts were arguing that these auctions were conducted in a manner that did not get the best "deal" for either the government or the users of the Medicare system, and in many cases, even the people delivering services. Examples such as Prisoner's dilemma and Braess's Paradox show that even when people make locally rational decisions in conflict situations or choose what routes to drive so they can get to where they want to go in as little time as possible, not only may their "rational" behavior harm them but it often harms every other person in society, too. Thus, having economic institutions where these kinds of unfortunate situations occur should be avoided as much as possible. While some people may resent "regulation," there are situations where regulation makes everyone better off!

Vickrey's ideas are now often mentioned in conjunction with what have come to be called VCG mechanisms. VCG stands for the Vickrey, Clarke, Groves. Clarke is Edward H. Clarke, an economist who was educated at Princeton and the University of Chicago, and who published his work in 1971. Groves is Theodore Groves, an economist educated at Harvard and Berkeley. The big idea, again, is to try to find an optimal design for a system which avoids "unpleasant" consequences resulting from rational behavior of individuals.

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.

A good example of this is an auction situation pioneered by Martin Shubik known as the dollar game.
 

A seller who also serves as the auctioneer is going to auction off something of clear value - say, a dollar bill. The smallest increment for a bid is a fixed amount, perhaps 1 cent (the idea is not to allow the game to go on forever by having the bids increment the previous bid by smaller and smaller amounts). The "rub" in this game is that the person who makes the next-to-highest bid must pay the seller/auctioneer as well as the highest bidder, who also get the dollar! There may be two or more players. To see the full effect of the "paradoxical" nature of this game it is better to have many potential bidders. While it is possible that the auctioneer/seller may lose money, very often this is not what happens. Often what happens is that several people bid above a dollar even though the winner will lose money and the next-to-last bidder will also lose money. To see why this might happen, imagine that the highest bid is 99 cents and that you have the next-highest bid, 98 cents. You don't want to lose 98 cents and so you will bid a dollar. Now the person who bid 99 cents stands to lose 99 cents so he or she will bid $1.01 (for now she will lose only one cent). The auction will continue well past bids of one dollar. Thus, people will bid over the value of the good and no one (except the auctioneer) will profit.

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.

One of the surprising developments of the application of mathematics to such topics as elections and voting was the discovery of some perhaps surprising lines of results:

a. Sometimes an appealing method of conducting a fairness procedure must be ruled out because the computational complexity of carrying out the work is such that a large version of the problem cannot be carried out in a reasonable amount of time.

b. Sometimes one can write down a small list of reasonable fairness conditions which no method can meet simultaneously.

c. Sometimes one can show that one cannot design a system so that "non-truthful" actions are invoked.

Similar "theorems" about manipulability, computational complexity, and the impossibility of achieving certain fairness goals have also been shown for auctions.

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.

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Shubik, M., The Theory of Money and Financial Institutions, Volume 1, MIT Press, Cambridge, 2004.

R. Thaler, The Winner's Curse: Paradoxes and Anomalies of Economic Life, Princeton U. Press, Princeton, 1964.

Vickrey, W., Counterspeculation, auctions, and competitive sealed tenders, J. of Finance, 16 (1987) 8-37.

Weininger, L., Escalation and cooperation in conflict situations: The dollar auction game revisited, J. of Conflict Resolution, 33 (1989) 231-254.

Wilson, R., Competitive bidding with disparate information, Management Sci., 15 (1969) 446-448.

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be found via the ACM Portal, which also provides bibliographic services.