A Multiple Unit Auction Algorithm: Some Theory and a Web ...

Keywords: multiple unit auctions; World Wide Web; e-commerce; intelligent agents; bots

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The paper discusses an electronic multiple unit discriminative auction and its implementation on the world wide web. The algorithm is designed to reduce the level of price discrimination and provide a useful and easy-to-use electronic auction mechanism. The decision of the bid price is suggested to the bidder, making the auction less cumbersome than traditional multiple unit auctions. We discuss the special cases of auction owner specified reservation prices for different quantity levels and bidder acceptance of partial quantities or not. Implications of the four resulting cases are explored and discussed. Images from the web-site, providing a prototype implementation of the auction algorithm, are displayed.

A Multiple Unit Auction Algorithm: Some Theory and a Web Implementation JEFFREY TEICH, HANNELE WALLENIUS, JYRKI WALLENIUS AND ALEXANDER ZAITSEV

INTRODUCTION

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Copyright & 1999 Electronic Markets Volume 9 (3): 199–205. www.electronicmarkets.org

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RESEARCH

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Jeffrey Teich is an Associate Professor at the New Mexico State University. His research interests and publications are in the areas of negotiation modeling and decision support. In addition, he has served as Visiting Professor at the Helsinki School of Economics teaching in its international programs. Hannele Wallenius is a Senior Lecturer at the Department of Industrial Engineering and Management, the Helsinki University of Technology. Her research interests and publications are in the areas of public sector operations research and negotiation modeling. Jyrki Wallenius is Professor of Management Science and Director of the International Center at the Helsinki School of Economics. He is also the Director of the Interactive Telecommunications Program at the Helsinki School of Economics, an intensive training program in cutting edge telecommunications technologies, applications and multimedia. His academic interests and published research lie in the areas of decision making and negotiation modeling. Alexander Zaitsev is a student at Moscow State University with an interest in Web-Based electronic commerce and multiple criteria decision making.

We develop an algorithm for multiple unit auctions of homogeneous goods. We assume bidders desire multiple units of the good and thus their bids consist of the desired quantity and per unit price. In the terminology of auction literature, we discuss a single auction with one seller and many bidders. Our algorithm is ascending price, sealed in the sense that a bidder does not see the bids of the others, yet open in the sense that we recommend a bid price which would activate a bid. Bids are accepted sequentially, and when outbid a bidder is informed and invited to resubmit. The advantages of this auction mechanism over the traditional Yankee auction is that bidders will not see each others bids, and bidders do not have to decide what amount to bid to make their bid active, simplifying the process, and they have the option to autobid at reduced quantities. The advantage to the seller is that they can set varying levels of reservation prices for different quantities leading to price discrimination. But, since the results are not presented to the bidders, ‘winner’s curse’ situation does not arise. There is extensive literature which

discusses and tests different types of auctions using multiple performance measures, including revenue equivalence, the extent of price discrimination, and the efficiency of auctions (Feldman and Mehra 1993a, 1993b; Kagel and Roth 1995; Smith, 1991). Single-issue (price) auctions are discussed by, among others, Ashenfelter (1989); Engelbrecht-Wiggans (1988); Hausch (1986); McAfee and McMillan (1987) and Milgrom (1989). Among the few who discuss multiunit (price and quantity) auctions are McCabe et al. (1990), Cox et al. (1984) and Tenorio (1993). For an interesting combinatorial (grouping of assets) auction see Rothkopf et al. (1995). Web-based electronic auctions have recently become very popular. For a review, see Wurman et al. (1998), Wellman and Wurman (1998), Klein (1997), and Teich et al. (1998). As discussed by Schwartz (1998), hundreds of different types of electronic auctions exist. Examples are Ebay, Onsale, AuctionBot, Kasbah, Bid4it, Priceline, and Ubid (see ‘References’ for URLs). Design features include whether or not sellers specify reservation prices; whether there is automatic bidding (for example AuctionBot, Kasbah, Bid4it); whether bidders rate each other via blacklists.

e : l: l - 1: A:

A=k: SA: I:

minimum increment in bid price index associated with lowest active bid price; if tied, it is associated with the latest entered bid index associated with semi-active bid price if one exists set of active bids (possibly empty), corresponding to indexes i ˆ l, . . ., n when bids are ordered from lowest to highest the set of active bids reduced by k lowest priced active bids set of one semi-active bid (possibly empty), corresponding to i ˆ l - 1 set of inactive bids (possibly empty), corresponding to i ˆ 1,. . ., l - 2 (or 1, . . ., l - 1 if SA empty)

The steps of the basic algorithm are as follows: Step 1. Auction owner specifies D, RP, e , and the closing time of the auction. Step 2. Potential bidder presents a bid query i ˆ n ‡ 1 by specifying a desired quantity q i . Step 3. Bidder requests a price for q i :

THE ALGORITHM

In the algorithm, the bid required to enter the ‘action’ is posted while keeping the actual bids sealed. The revenue to the seller increases with each new bid. The algorithm encourages active bidding by allowing some degree of price discrimination, but less than in a typical discriminative auction. ‘Winners curse’ effect can thus be reduced. An active bid is one, which would be accepted at that price and quantity, if the auction were to close at that point in time. An inactive bid is one, which has expired because it has been outbid. A semi-active bid is one in which the bidder will only receive a partial quantity if the auction closed at that point in time. Our basic algorithm assumes that bidders will accept partial quantities, if their bid is semi-active. Next we will define the notation. Let:

Si :

n: n ‡ 1: D: SP: RP: qsr :

( pi , q i ), be the ith bid, where p i ˆ per unit price for bid i, and q i ˆ quantity desired in bid i, (i ˆ 1, . . ., n) number of existing bids, and index associated with highest bid price entering bid number of units of good for sale Suggested Price to make a bid active Reservation Price quantity for the semi-active bid

Case 1: Suggested P Price is the Reservation Price. n SP ˆ RP if q n‡1 ‡ iˆl q i < D; Case 2: Suggested Price is calculated. P ‡ i2 A q i < D; if not, b) a) SP ˆ p l - 1 ‡ e if q n‡1 P < D; if not, c) SP ˆ p l ‡ e if q n‡1 ‡ P i2 A=1 q i SP ˆ p l‡1 ‡ e if q n‡1 ‡ i2P D; or generically, d) A=2 q i < SP ˆ p l‡ k- 1 ‡ e if q n‡1 ‡ i2 A= k q i < D, where k ˆ 0, 1, 2, etc. until the above condition is met. Step 4. Bidders submit their bid S i ˆ (SP, q i ) or ( pi , q i ) where p i > SP ; update n ˆ n ‡ 1; or bidders drop out. Step 5. Bidders whose status changes from A to SA or I or from SA to I or whose qsr changes are informed and requested to make a decision. Reorder bids from low to high. Return to step 2. Repeat until auction closes. The revenue, increasing at every iteration, is calculated as follows: Total revenue ˆ

X i2 A

pi q i ‡ q sr p s ,

where s refers to the semi-active bid. The residual quantity for the semi-active bid s is: q sr ˆ D -

X i2 A

iff

X i2 A

qi ‡ qs .

D, otherwiseq sr ˆ 0:

See the flowchart illustrating the steps of the algorithm:

A Multiple Unit Auction Algorithm

Basic Algorithm: Partial Quantity Acceptable, One Reservation Price

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Other design features include rules regarding the closing, the type of merchandise (new, used) and the quantities offered (one or multiple), and whether we have a regular or reverse/procurement auction (see Priceline). Multiple unit auctions are complex by nature. Sellers may be willing to accept lower prices for higher quantities of the good and therefore have different reservation prices for the various quantities. As an option, the seller could specify these various levels and associated reservation prices. Additionally, buyers may only be willing to accept the full lot, in other words only the total quantity and not a partial quantity in their bid. This results in four pure cases and one mixed case, the simplest being partial quantities accepted with only one reservation price. In the following section we begin with the simplest case and extend it to the other cases, illustrated with examples. We also describe a prototype Java-based web-site implementing the basic auction algorithm.

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Note: In step 3 case 1, the suggested price is simply the reservation price because all bids do not exceed the supply. In case 2, the suggested price is an e amount above the last not active (in- or semi-active) bid price (2a) or an e above the lowest active bid price (2b), or an e above the second lowest active bid price (2c), etc. (2d), until the demand does not exceed the supply of the good. In our system we do not explicitly identify a reservation price to bidders ahead of time, as such. The bidders do not know if they are outbidding previous bidders, or if they are meeting a reservation price for a specific quantity. In both cases, all they know is that the requested bid price will be active if they make it. As an example, assume a seller has 100 units of a homogeneous good to auction, with a reservation price of $10 per unit, and bid increment (e ) is $1. At time 1, Bidder 1 enters the auction, specifies a quantity of 40 and requests a price from the auction mechanism. Since there are no other bids, the reservation price is suggested to the bidder (case 1). He/she makes his/her bid (10, 40) and it becomes active in status. At time 2, Bidder 2 enters, specifies a quantity of 30 and requests a price. Again the reservation price of $10 is suggested (case 1), because the supply has not yet been depleted. Bidder 2 makes his/her bid (10, 30) and it becomes active in status. At time 3, bidder 3 enters the auction, and specifies a quantity of 50. At this point the supply is depleted and a new price must be calculated. The auction mechanism calculates a price at the bid increment ($1) above the latest price, and a price of $11 is suggested to the bidder (case 2, rule b). He/she then makes his/her bid (11, 50). Bidder two then is outbid and thus becomes semi-active with a quantity of 10 units, because he/she was the last one to bid at the price of $10. Bidder one remains active. At time 4, bidder two has three options. He/she can withdraw from the auction completely, stay semi-active in status, or re-bid. Assume he/she decides to re-bid at the same quantity of 30 units,

and requests a price. The price $11 is suggested by the mechanism because at that price the quantities of bidders 2 and 3 would be met by the supply (case 2, rule b). Bidder 2 then makes his/her bid (11, 30). Bidder 3 remains active in status and bidder 1 becomes semi-active with a quantity of 20 units. At time 5, bidder 1 has, again, three options, i.e. withdraw, remain semi-active or re-bid. Assume he/she decides to re-bid and requests a price. The mechanism then returns a price of $12 because at $11 the demand is greater than the supply. Therefore the new price must be calculated $1 above the most recent price suggested ($11) (case 2, rule b). If he/she makes this bid (12, 40), he/she will become active in status, bidder 3 will remain active, and bidder 2 will become semi-active with a quantity of 10. The process repeats until the auction closes. See Table 1 for the sequence of bids. In our algorithm, the price discrimination is reduced to an e difference if the bidders accept the suggested bid. If, however, they bid above the suggested bid price, then, the price discrimination level could be higher. Why would a bidder be willing to pay above the suggested price level? This could happen if the bidder wants to decrease the probability of being outbid. If an automatic bidding mechanism is used, then the bidder could specify the top price to bid at this quantity, and the mechanism would automatically re-bid on his/her behalf up to that point. Beyond that point, he/she could specify a reduced quantity up to another level and so on.

Partial Quantity Acceptable, Many Reservation Prices Effectively, reservation price levels only play a role if bids are low. Once the bids go above the highest reservation level, the auction algorithm behaves the same as presented in the previous section. As an example, again assume the auction owner has 100 units of good available. He/she specifies reservation levels as follows: RP(for bids between 1–30 units) ˆ $10; RP(31–40) ˆ 9:50; RP(41 and above) ˆ 9:25. These reservation levels then function basically as a quantity discount provided to individual large quantity bidders. Bidder 1 specifies a quantity of 40, and the suggested price is 9.5, the reservation level for that quantity. Bidder 2 specifies a quantity of 30 and the suggested price is at the reservation level of 10. Bidder 3 Table 1. Example 1: Basic Algorithm

time

bidder/bid #

q

p

Status of bid\ in time

1 2 3 4 5

1/1 2/1 3/1 2/2 1/2

40 30 50 30 40

10 10 11 11 12

A\1 A\2 A\3 A\4 A\5

SA\4 I\5 SA\3 I\4 SA\5

In Table 3, we illustrate what might happen if partial quantities are not accepted by any/some of the bidders and the aberrant behavior that can result in the algorithm. In Example 3, a bidder requires 99 units and bids $10. A new bidder requiring 2 units queries the auction and receives a suggested price of 11. Using the basic algorithm the first bid becomes inactive and the revenue to the auction owner drops substantially from 990 to 22. If, however, the total revenue is restricted to increasing at every bid, we have Example 4 (Table 4). Bidder two would be informed of a suggested price of 446, which also seems unreasonable. An auction owner must decide which rules to follow. If the revenue must increase at each iteration, suggested prices will be unusual in certain situations as in Example 4 (Table 4). Even though an auction owner may object to the decreased revenue in Example 3, expected revenue may be higher in the long run without such a restriction. Bidders will then be forced out, and if they want to re-enter, price must be bid up. If total revenue Table 2. Example 2: Algorithm with Many Reservation Prices time

bidder/bid #

q

p

Status\in time

1 2 3 4 5

1/1 2/1 3/1 1/2 3/2

40 30 50 40 50

9.5 10 10 11 11

A\1 A\2 A\3 A\4 A\5

SA\3 I\4 SA\5 SA\4 I\5 SA\5

1. Revenue must increase at every iteration (Example 4, Table 4). This rule gives the most power to a large Full lot-bidder. 2. Ties in bids are broken by making the most recent bidder inactive in status. Revenue can drop, as in Example 3. This rule gives the least power to a large Full lot-bidder, and is the same as our basic algorithm described in section 2 (Example 5, Table 5).

Partial Quantity Not Acceptable to Everyone, One Reservation Price: Mixed Case This mixed case is probably the most commonly occurring case in reality, and the most difficult to deal with. We offer three possible auction rules to the auction owner: 1. As Rule 1 above. 2. Ties in bids are broken by making the most recent bidder semi-active or inactive in status. Revenue can drop, as in Example 3. This rule gives the least power to a large Full lot-bidder, and is the same as our basic algorithm described in section 2 (Example 6, Table 6). 3. Ties in bids are broken by making the partials accepting bidder semi-active or inactive without regard to time of bid (Example 7, Table 7). Revenue can drop, as in Example 3. This rule gives medium power to a Full lotbidder. In rule number one above, a large bidder has power to control the auction. A small quantity bidder may, however, decide to ‘pool’ with existing or new bidders in an attempt to outbid the large bidder. In example four, the suggested price to bidder two is 446. This will be an unreasonable Table 5. Example 5: full lots only – rule 2

Table 3. Example 3: extreme case 1 – full lots only

time

bidder/bid # q

p

Status\in time Revenue

time

bidder/bid # q

p

Status\in time Revenue

1 2

1/1 2/1

10 11

A\1 A\2

1 2 3

1/1 2/1 3/1

10 10 11

A\1 A\2 A\3

99F 2F

I\2

990 22

40F 30F 50F

I3

400 700 950

F ˆ Full lots only

F ˆ Full lots only

Table 6. Example 6: mixed case – rule 2 Table 4. Example 4: extreme case 2 – full lots only time

bidder/bid # q

p

Status\in time Revenue

1 2

1/1 2/1

10 446

A\1 A\2

F ˆ Full lots only

99F 2F

I\2

990 992

time

bidder/bid # q

p

Status\in time Revenue

1 2 3

1/1 2/1 3/1

10 10 11

A\1 A\2 A\3

40P 30P 50F

F ˆ Full lots only, P ˆ Partials accepted

SA\3

400 700 1050

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Partial Quantity Not Acceptable to Anyone, One Reservation Price

must increase at each iteration, large bidders will have extreme power to control the auction and restrict the entry of new bidders. We offer two possible auction rules to the auction owner:

A Multiple Unit Auction Algorithm

specifies a quantity of 50, but because he/she must outbid bidder 1, the suggested price is 10. The algorithm mechanism then performs the same for the bids that follow since they now are above the reservation prices (See Table 2).

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Table 7. Example 7: mixed case – rule 3 time

bidder/bid # q

p

Status\in time Revenue

1 2 3

1/1 2/1 3/1

10 10 11

A\1 A\2 A\3

40P 30F 50F

SA\3

400 700 1050

F ˆ Full lots only, P ˆ Partials accepted

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suggested price to the bidder. He could be offered the possibility of submitting an inactive bid of 11, which may become active through pooling with other bidders. The main difference here, with pooling, is that inactive bids can become active if they are pooled. In the previous cases inactive bids always remain inactive. The pooled bid would be considered a full lot bid once it is formed. The bidders would indicate their willingness to pool with other bidders at a suggested pool price. This option is only required in rule one, and not in rules two and three because the large full lot bidders are easily outbid and thus inactivated.

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The final variation on this theme is the possibility that partial quantities are not acceptable for some bidders, and the auction owner wishes to set reservation prices for various quantities of the good, again serving as quantity discounts. There were no new features different from those discussed in Partials Not Acceptable, One Reservation Price case, once the bids go above the highest reservation levels. An auction owner would need to select one of the above auction mechanisms. The decision which to select is neither easy, trivial, nor obvious. In the following section we have selected the general case of partial quantity acceptable, with many reservation prices, for the webbased prototype implementation. With this option, any bidder who will not accept partial quantities must check the auction status right before closing to ensure he/she is not semi-active in status. If he/she is semi-active with a partial quantity, he/she may re-bid or drop out. This should not be so troubling because a maximum of one bidder will be semi-active in any auction.

The Java application server is the main communication center. It connects directly to the SQL server and processes all client connections and requests. The auction algorithms are implemented here. Each client accesses the system through a web browser and then runs it in a separate window. Clients connect and communicate with the application server using Java Remote Method Invocation (RMI) technology. RMI is a Java analogue of CORBA. It makes real distributed computing possible. Users have a fast online access to information and processing procedures located on the application server. For a discussion concerning the advantages which Java/CORBA-based applications offer, see Fan et al. (1998). Upon entering the site a user is provided options and menus with links to pages of the site. If one selects ‘run project’, the Java-program loads. The first screen that pops after the loading is the main screen (Figure 1). The main screen is presented to auction owners and bidders alike. A new user must first register and log in. The main screen consists of menu option buttons across the top which allow a user to register, enter sessions, create sessions, make bids etc. In the window in the upper left, the auctions that the user has either entered or created, are listed. The user can highlight the specific auction by pointing and clicking on it, and the bidding activity for that auction is reported in the window in the upper right. All bids are reported to the auction owner, but a bidder only sees his/her own bids listed. The bottom window is a messaging system where system messages are received which tells users when new auctions are created, and reports changes in bid status (from active to semi-active to inactive) to bidders. An auction owner can start a new auction by selecting the ‘new session’ button. At this point a new window appears as shown in Figure 2. The auction owner defines the product, enters the product description and the definition of the auction which in this paper is the quantity/price auction. Then in a new window (Figure 3) the auction owner specifies the quantity to be auctioned, the reservation price, the bid increment, and the closing

WEB-SITE DESCRIPTION Next we describe our Java-coded auction system implementing our basic auction algorithm. Netscape 4.06 or above or Internet Explorer with plug-ins is required to run the site (http://kvstu001.hkkk.fi/NSS/). The system uses web-based distributed computing as the platform. It consists of three main parts: database SQL server, Java application server, and Java-clients. The SQL server stores the information concerning the auctions, users, bids, etc.

Figure 1. Main Screen

Figure 3. New Session: Auction Definition

date and time of the auction. A system message is then sent to all users that a new auction is open. A bidder logs on, enters the session and selects the button ‘new bid’ (Figure 1). A new bid window appears, as shown in Figure 4, and the bidder specifies the quantity desired. He/she can then enter the bid price or select ‘request price’ button, and the system returns a suggested price. The bidder now has several options: 1. cancel and withdraw; 2. hit ‘ok’ and simply accept the suggested price; 3. enter a bid price which is higher than the suggested price; or 4. accept the suggested price or change the price as in ‘3’ above and select ‘autobid to’ and insert a higher bid price in the associated box. If the bidder selects the ‘autobid’ option, the system will automatically outbid other bidders up to the specified

A Multiple Unit Auction Algorithm

Figure 2. New Session Screen

price. The system then shows the ID of the bid, the status of the bid (A ˆ Active, I ˆ Inactive, S ˆ Semi-active), Sell or Buy bid (S/B) in the ‘positions’ section of the main screen window as shown in Figure 1. Each bidder is only informed of his/her own bid status, suggested prices for the new bids, but not the bidding activity of other bidders. Once the status of a bid changes to ‘semi-active’ or ‘inactive’, a system message is sent to the bidder, with information about the semi-active quantity. The auction owner receives complete information regarding the bids as shown in Figure 5. He/she sees the bids arrive, the status of the bids, the user names of the bidders, and the prices and quantities. Users may also send messages to one another using the real-time message facility. Once the auction closes, the accepted bids are reported to all users in the ‘Deals’ window as shown in Figure 6. The issues of concurrency in a real-time web-based auction environment are important. They have been dealt with in three ways. First, the auction does not close exactly on the time specified, but only some minutes after the last bid has been received (after the closing time). Second, we have incorporated an autobid feature where bidders in different time zones can still bid higher if they have been out bid. As an option, they may also autobid at three

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Figure 4. New Bid Window

Figure 5. History of Auction

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Figure 6. Deals of the Auction

different levels of quantity and price. Third, we accept the first bid received as active in status. If a bidder requests a price, accepts that price and makes the bid, but because of web congestion another bidder comes in ahead and makes the same bid, then the bid is simply registered as inactive and the bidder will be informed immediately and he/she may make another bid.

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We have described a multiple unit auction algorithm and its Java-coded prototype Web-based implementation. The algorithm is for auctioning homogeneous goods. The bids are sealed, but the bidders are provided information about the status of their bids and required price to make their bid active. We have discussed issues involving reservation prices and the possible constraint of full lots. Novelty of the algorithm is the computer calculated suggested bid price, which is complicated by the reservation price restrictions as well as partial/full lots. Our future work involves testing the performance of the algorithm against existing multiple unit auction mechanisms, such as the English clock auction and the sealed bid auction.

References URLs AuctionBot: http://auction.eecs.umich.edu/ Bid4it: http://www.bid4it.com Ebay: http://www.ebay.com/ Jango: http:// www.jango.com Kasbah: http://kasbah.media.mit.edu Onsale: http://www.onsale.com/ Priceline: http://www.priceline.com Ubid: http://www.ubid.com/

Ashenfelter, Orley (1989) ‘How Auctions Work for Wine and Art’, Journal of Economic Perspectives 3: 23–36. Cox, J.C., Smith, V.L. and Walker, J.M. (1984) ‘Theory and Behavior of Multiple Unit Discriminative Auctions’, The Journal of Finance 34: 983–1010. Engelbrecht-Wiggans, R. (1988) ‘Revenue Equivalence in Multi-Object Auctions’, Economics Letters 26: 15–19. Fan, M., Stallaert, J. and Whinston, A. (1998) ‘Creating Electronic Markets’, Dr. Dobb’s Journal 23 (11), 52–7. Feldman, R.A. and Mehra, R. (1993a) ‘Auctions: Theory and Applications’, International Monetary Fund Staff Papers 40: 485–504. Feldman, R.A. and Mehra, R. (1993b) ‘Auctions: A Sampling of Techniques’, Finance and Development, September: 32– 5. Hausch, D.B. (1986) ‘Multi-Object Auctions: Sequential vs. Simultaneous Sales’, Management Science 32: 1599–610. Kagel, J. and Roth, A. (eds.) (1995) Handbook of Experimental Economics, Princeton, NJ: Princeton University Press. Klein, S. (1997) ‘EM-Electronic Auctions’, EM-Electronic Markets 7(4). McAfee, R.P. and McMillan, J. (1987) ‘Auctions and Bidding’, Journal of Economic Literature 25: 699–738. McCabe, K.A., Rassenti, S. and Smith, V.L. (1990) ‘Auction Institutional Design: Theory and Behavior of Simultaneous Multiple-unit Generalizations of the Dutch and English Auctions’, The American Economic Review 80: 1276– 283. McCabe, K.A., Rassenti, S. and Smith, V.L. (1991) ‘Smart Computer-Assisted Markets’, Science 254: 534 –8. Milgrom, P. (1989) ‘Auctions and Bidding: A Primer’, Journal of Economic Perspectives 3: 3–22. Rothkopf, M.H., Pekec, A. and Harstad, R.M. (1995) ‘Computationally Manageable Combinational Auctions’, DIMACS Technical Report 95-09, Forthcoming in Management Science, 44: 1131–47. Schwartz, E.I. (1998) ‘At On-line Auctions, Good, and Raw, Deals’, New York Times, 5 March 1998. Smith, V.L. (1991) Papers in Experimental Economics, New York: Cambridge University Press. Teich, J., Wallenius, H. and Wallenius, J. (1998) ‘Multiple Issue Auction and Market Algorithms for the World Wide Web’, Decision Support Systems, 26: 49–66. Tenorio, R. (1993) ‘Revenue Equivalence and Bidding Behavior in a Multi-Unit Auction Market: An Empirical Analysis’, The Review of Economics and Statistics May: 302– 14. Wellman, M.P. and Wurman, P.R. (1998) ‘Real Time Issues for Internet Auctions’, First IEEE Workshop on Dependable and Real-Time E-Commerce Systems (DARE98), Denver, Co., USA, June 1998. Wurman, P.R., Walsh, W.W. and Wellman, M.P. (1998) ‘Flexible Double Auctions for Electronic Commerce: Theory and Implementation’, University of Michigan, Artificial Intelligence Laboratory, Decision Support Systems, 24: 17–27.