Electronic Negotiation through Internet-based Auctions

CITM Working Paper 96-WP-1019

December 1996

Carrie Beam, Arie Segev, and J. George ShanthikumarFisher Center for Information Technology & Management

Walter A. Haas School of BusinessUniversity of California, Berkeley

Berkeley, CA 94720

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Abstract

Electronic commerce is changing the way business is conducted. We show how electronicnegotiation is not yet a full part of the electronic commerce toolset, and outline two majordifficulties with automating the negotiation process. We provide a literature review of differentexisting approaches to the electronic negotiation problem, and show how none adequatelyaddresses the problem. We next introduce the case study of Onsale, Inc., a fully functionalInternet-based auction house, and show how this auction solves some of the stumbling blocks toelectronic negotiation. The Internet-based auction changes the parameters of the traditionalauction, and allows the seller many more degrees of freedom. We analyze the new problem ofhow many items to optimally offer in each auction, formulating the optimization problem as adynamic program and showing a solution using existing data taken from Onsale. This paperconcludes by identifying areas for further research in this new domain of electronic auctions usedto conduct semi-automatic negotiation over the Internet.

AcknowledgmentsThis paper reflects work done for my qualifying examination in the Department of IndustrialEngineering and Operations Research at the University of California, Berkeley. I would like toespecially thank Professors Shmuel Oren, Stuart Dreyfus, and Hal Varian for their help andsuggestions during the course of this research; without their insight, this paper would not havebeen possible. Additionally, I would like to express appreciation to the Fisher Center’s Jim Ware,Amir Hartman and Malu Roldan for their valuable comments on an earlier draft of this paper.And finally, many thanks to my co-authors, Professors Arie Segev and George Shanthikumar, fortheir help, past, present, and future.

--Carrie Beam

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Table of Contents

1. Introduction............................................................................................................................12. Background: Electronic Commerce and Electronic Catalogs................................................... 13. Electronic Negotiations........................................................................................................... 2

3.1 Difficulties Facing Electronic Negotiations........................................................................ 33.2 Current Approaches: DSS/NSS and DAI.......................................................................... 4

4. Workable Negotiation Using Current Technology................................................................... 64.1 Onsale, Inc.: A Case Study...............................................................................................74.2 Auctions: A Brief Overview..............................................................................................94.3 Onsale in Terms of Auction Theory................................................................................. 11

5. Onsale as an Optimization Problem........................................................................................ 125.1 Background and Assumptions.......................................................................................... 125.2 Variables......................................................................................................................... 135.3 Formulation..................................................................................................................... 14

6. Solution and Data Analysis.................................................................................................... 156.1 Data Collection Technique............................................................................................... 156.2 Parameter Estimation....................................................................................................... 17

6.2.1 Estimation of s, h, N, and T...................................................................................... 176.2.2 Estimation of λ ......................................................................................................... 186.2.3 Estimation of a and b................................................................................................ 19

6.3 Solution Based on Model and Parameters........................................................................ 206.4 Properties of the Optimal Solution................................................................................... 22

7. Information System Requirements......................................................................................... 248. Price Formation and Bidder Strategy..................................................................................... 249. Summary and Conclusions..................................................................................................... 2610. Mathematical Appendix....................................................................................................... 28

10.1 Order Statistics Derivation............................................................................................. 2810.2 Lemma 1....................................................................................................................... 2910.3 Solution to the Larger Integral....................................................................................... 31

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List of Figures

Figure 1: Possible Changes in Commerce Process Which Use Negotiation.................................. 3Figure 2: Portion of Onsale Auction Screen................................................................................ 8Figure 3: Bid Arrival Process Over Time.................................................................................. 16Figure 4: Summary of Parameters Estimated for Item 2050...................................................... 17Figure 5: Distribution of Maximum Bids for Item 2050............................................................ 20Figure 6: Winning Bids from Five Auctions, Item 2050............................................................ 21Figure 7: Effect of Holding Cost on Optimal Allocation of Items.............................................. 23

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1. Introduction

The Internet and the World Wide Web have brought many changes to the business landscape, and

will bring many more. This paper analyzes the effect of the Internet and Web on price negotiation in the

supply chain. While there has been little success in implementing full-scale automated negotiations, a

similar, simpler problem, that of the electronic auction, has been implemented quite successfully. We

look at the case study of Onsale, Inc., as one in which an electronic catalog has successfully run auctions

over the Internet. We show ways the electronic commerce technology has changed the rules of the

traditional auction for both the buyer and the seller, using results from auction theory as applicable. We

next identify new optimization problems and present a solution to one problem. We conclude with

further research opportunities in this area.

2. Background: Electronic Commerce and Electronic Catalogs

Electronic commerce refers to the buying and selling of products and services, and the transfer of

funds, through public or private digital networks. It includes all types of inter-company and intra-

company business interactions, transactions and functions such as marketing, advertising, sales, support,

recruiting, research & development, administration, and corporate communication. Technologies include

e-mail, EDI, electronic funds transfer, the World Wide Web, video conferencing, electronic forums and

bulletin boards, distributed databases, and electronic catalogs. EDI is widely used in such diverse settings

as automobile manufacturing by General Motors and food processing by Frito-Lay, and Internet-based

Financial EDI is used in the banking industry by BankAmerica [24], [25], [27]. Government requests for

proposals (RFP) are often sent out electronically, and in 1995 the government began requiring some

businesses to file income tax returns using EFT. Consortia such as CommerceNet study electronic

commerce, and have much additional information [4].

An electronic catalog, as defined by Segev, Wan, and Beam [26] is a World Wide Web site

displayed to the Internet community with the intent to sell a commercial good or service. Early electronic

catalogs resembled their paper counterparts, often presenting pages of ‘flat’ product displays with little

cross-indexing or additional information. Current electronic catalogs provide capabilities far beyond sales

and marketing such as customer support, order processing, customer feedback, and search processes.

Closer to the cutting edge are the “smart” and “virtual” electronic catalogs outlined by Keller [10], which

bring together heterogeneous product databases into a searchable, annotated listing of product

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announcements. These catalogs use considerable processing power to present the user with timely,

usable information. Electronic catalogs are an increasingly important part of the automated supply chain;

hence, the ability to negotiate electronically is becoming increasingly important.

3. Electronic Negotiations

Negotiation, as defined in this paper, is the process in which two or more parties multilaterally

bargain goods or information for mutual intended gain. The ability to offer and counteroffer is a key

capability here. There are some sales in which the calculation takes place “behind the scenes;” the seller

offers a good at a fixed price, and the buyer decides to take it or leave it. This interchange, however, is

not considered negotiation in this context; the rest of this paper will focus on automated negotiation in

electronic catalogs using the auction mechanism. Fully automated negotiation requires all parties

involved to be software agents; semi-automated negotiation involves a human negotiating with a software

agent. The terms ‘electronic negotiation’ and ‘automated negotiation’ in this paper will refer to either

fully automated negotiation or semi-automated negotiation processes; ‘manual negotiation’ will refer to

processes in which all parties are human.

If automated negotiation could be implemented, the changes in the business landscape are

potentially immense. Figure 1 outlines briefly some possible changes.

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3.1 Difficulties Facing Electronic Negotiations

Why does current electronic commerce technology not support automated negotiation? The

answer, briefly, is that negotiation is difficult, and automated negotiation from an electronic catalog is

even more so.

To give only two of many examples highlighting this difficulty, the following issues are outlined

here: the need for an ontology, and the possibility of agents inferring each others’ strategies.

An ontology is a way of categorizing objects such that they are semantically meaningful to a

software agent. An ontology is required to ensure the agents are referring to exactly the same good.

Figure 1: Possible Changes in Commerce Process Which Use NegotiationProcess Who

negotiatesCurrent constraints Possible changes brought by

electronic commerceAuctions Auctioneer

and multiplebuyers

For open auctions,requires all partiesparticipate at thesame time; limit onnumber of items sold

Parties may participate from differentplaces at different times; otherparameters may be adjusted also.

PricingPolicies

Buyer andseller

Customer identityfrequently unknown;difficult to measurewillingness to pay

Customer identity can be knownduring negotiation; it is possible tosegment the market into 1-customerunits, offering each a custom-tailoredprice.

Purchasing Purchasingagent andseller

Can only negotiatewith one other partyat any given time

Possible simultaneous concurrentnegotiations; multi-agent deals easierto manage.

Inventorymanagement

Warehousemanager,customers,suppliers

Not always aware ofreal-time inventorysituation

Can negotiate depending oninventory situation; allow moreflexible response to market.

Retailing Retailer andcustomer

Retailer can onlycoarsely segmentmarket with pricingand promotions;does not always havepersonal customerinformation

Retailer can finely segment marketinto very large number of different“deals;” customers self-select.Retailer can also gather and leverageindividual customer informationbefore presenting sale options.

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With a compact disc, it is relatively easy; but specifying an automobile, or a food product, or a delivery

schedule can be very difficult. Moreover, with many give-and-take negotiations, attributes such as

delivery time, delivery quantity, and batch quality, and financing terms are up for debate; it is crucial that

an agent be able to evaluate the tradeoffs and implications of all the variables.

The second difficulty outlined here is that of negotiation strategy. If one agent’s negotiation

strategy is known to the other agent, the first agent may be at a significant disadvantage. Suppose the

buyer knows that the seller’s strategy is to accept all offers above a certain (unknown) threshold value.

The buyer can begin at $0.00, and repeatedly offer the seller a penny more each time, until the seller’s

threshold value is reached,, at which point the (worst possible, for the seller) deal is made. This is but

one example of mechanism design; Varian [29] outlines many more issues with economic mechanism

design for computerized agents, including some ways to ensure against losses due to strategy inference.

Despite the potentially large payoffs, because of these and other difficulties, electronic agents and

electronic catalogs are deafeningly silent with respect to negotiation functionality.

3.2 Current Approaches: DSS/NSS and DAI

While the problem of completely automating negotiation is a difficult, complex one, there has

been much progress made over the past few years with respect to the fields of Decision Support Systems

(DSS), Negotiation Support Systems (NSS), and Distributed Artificial Intelligence (DAI). Electronic

negotiation is a complex subject; for a more comprehensive overview than this section has space to

provide, see Beam and Segev [1]. This section will give a brief overview of some current literature and

current results in these fields.

Decision Support Systems (DSS) are a class of systems, often computer software, built with the

purpose of helping human decision-makers make better decisions. A Negotiation Support System (NSS)

is a DSS which is specially geared towards negotiation situations. Jelassi and Foroughi [9] present an

overview of design issues and existing software. They place a fair emphasis on human factors issues such

as behavioral characteristics, cognitive differences, and negotiation theories, issues which have also been

brought up by Raiffa [18] and Fisher and Ury [7]. Nunamaker et. al. [15] provide another literature

review, and present results of laboratory and field experiments using negotiation support systems to solve

cooperative negotiation problems. They find that electronic systems can be quite useful under certain

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conditions. Foroughi [8] presents a survey of current usage of NSS in business negotiations, and Perkins

et. al. [17] investigate practical implications of NSS for purchasing managers.

While DSS and NSS are quite powerful tools, and can often support negotiations which are more

productive than would be possible without them, they are far from being able to support automated

negotiation on their own. NSS require near-constant human input, and both the initial problem setup and

all final decisions are left to the human negotiators.

The field of Distributed Artificial Intelligence (DAI) has also approached the negotiation problem.

Rosenschein and Zlotkin [19] provide a somewhat technical overview of problems encountered when

negotiating between automated agents, and outline some problem domains in which the negotiation

problem may be less difficult to solve. Sandholm [22] and Sandholm and Lesser [23] present automated

agent negotiation in the domains of vehicle routing, package delivery, and coalition formation. Chavez

and Maes [3] created Kasbah, a marketplace for negotiating the purchase and sale of goods using

intelligent software agents. 1 While the agents are capable of simultaneous multiple negotiations, the

agents receive their complete strategies through a World Wide Web form from the users, who specify

strategy and retain final control at all times. Another approach is that of genetic programming, which

uses techniques akin to Darwinian evolution to select winning negotiation strategies from a large

population of initial possibilities; see Oliver [16], and Dworman, Kimbrough and Laing [5],[6] for more

details. Rust [21] writes of the Santa Fe Double Auction, a programming contest in which 30 computer

programs played both the role of buyer and seller in a series of double auction tournaments, vying for a

cash price of $10,000. The runaway winner was a very simple program, which used the strategy of

waiting in the background until the bid/ask spread was within 10%, and then jump in and “steal the

deal.”2

This section shows clearly that automated negotiation, while the subject of much interesting

research, has not reached a point where it is capable of negotiating on behalf of an electronic catalog.

The issues of ontology and strategy continue to persist, and solutions such as the vehicle routing, the

Kasbah marketplace, genetic programming, and those which surfaced in the Santa Fe double auction are

not robust enough to competently negotiate on behalf of a seller through an electronic catalog interface.

1 The case study had users buying and selling playing cards in an attempt to form a winning poker hand.2 When asked about his extraordinary success, contest winner and economics graduate student Todd Kaplan allegedly said,“Other entrants were worried about proving a theoretical result; I just really needed $10,000.” Anecdote thanks to HalVarian, conversation, 9/12/96.

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4. Workable Negotiation Using Current Technology

To summarize the previous sections, electronic commerce and electronic catalogs are becoming a

major force in the business landscape. While electronic catalogs do provide some basic functionality,

including searching, order fulfillment, and customer information, very few currently have the ability to

negotiate, even along the single dimension of price, with either an electronic or a human customer. The

inability to negotiate represents a large business need; however, electronic negotiation is a difficult,

complex process, and success with it is limited at best.

Despite these difficulties, however, there is one subset of negotiation processes which is admirably

well-suited to the current terrain of electronic commerce: the auction.

To borrow a definition from McAfee and McMillan, an auction is “a market institution with an

explicit set of rules determining resource allocation and prices on the basis of bids from the market

participants.” [12] This paper looks at auctions over the Internet as a first step towards more

sophisticated negotiation and bargaining between intelligent agents. Auctions have a number of

characteristics which fit well with intelligent agents, including:

• Auctions restrict the bargaining to a single dimension, price, which is relatively easy for a

software agent to manipulate.

• The ontology issue is somewhat resolved. The item for sale is displayed, and the bidders may

usually inspect it to gather its specifications. Moreover, items sold by auction are usually sold

“as is,” eliminating the need to intelligently trade off different levels of different variables.

• The strategy issue can be resolved from the seller’s standpoint. Recall that the seller whose

strategy was to sell above a threshold value was at a distinct disadvantage if a buyer inferred

that strategy. However, a seller who intends to sell an item by auction, to the highest bidder,

is not at a disadvantage if the bidders divine the seller’s strategy; in fact, the seller may want

to make this strategy publicly known. (The buyers’ strategies may or may not be as neatly

resolved.)

Auctions are, in fact, so well suited to electronic commerce and Internet activity that several are

up and running on the Internet today.

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The original English auction is constrained by time and space. Bidders must participate

simultaneously, and must usually gather in the same place (although telephone participation may be

possible). The auctions proceed quickly; each item usually sells in less than five minutes. Auctions are

held relatively infrequently, and several items are usually sold (either simultaneously or sequentially) in

the same auction proceeding.

Electronic commerce tools have dramatically decreased the cost of bidder participation: to

participate requires only an Internet connection and the time it takes to evaluate the current auction. The

ease of participation and the wide reach of the Internet have dramatically raised the number of potential

bidders. The auction may take place more frequently without the penalty of decreasing the number of

bidders.

As a result, the seller has many more degrees of freedom. One investigated here is the seller’s

new ability to decide how many items to auction off and when to auction them. Given 30 items at the

beginning of the month, the seller may choose to auction all 30 on a single day, all 30 over 30 days, or a

single item per day each day of the month. This raises new optimization problems: given holding costs,

bidder distribution, and bidder strategies, what is the optimal number of items to offer and what is the

optimal number of time periods to do it in? This is the problem which is addressed in this paper.

The case study covered in this paper is that of Onsale, Inc., and electronic auction house which

also functions as a virtual catalog. The next section outlines Onsale as a case study.

4.1 Onsale, Inc.: A Case Study

“We’ve created a new channel. It’s part Las Vegas, part QVC.”

This quote by company founder and Silicon Valley entrepreneur Jerry Kaplan captures the online

atmosphere of Onsale, at http://www.onsale.com. After his pen computer company, Go Corp., failed in

1994, Kaplan went on to found Onsale in June 1995 in Mountain View, CA. Onsale sells refurbished or

“end-of-life”3 computer and high-technology goods by auction over the Internet. By July 1996, Onsale

was averaging $700,000 in gross receipts per week, netting 13-20% margins on sales, and sustaining a

company growth rate of 15-20% per month [11].

The auctions are real, live, and run over the Internet. They close every Monday, Wednesday, and

Friday, and customers from all over the United States (and the occasional foreign country) place bids for

3 Kaplan considers any piece of hardware below a Pentium 133 “end of life!” [11]

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the merchandise displayed and described in the catalog pages. Each auction usually offers a small number

(1-10) of identical items, and a customer may bid for one or more of the items. Bids are accepted 24

hours a day, 7 days a week, and are updated on the screen several times per day. Many auctions specify a

minimum bid (reserve price) and a minimum bid increment. The bids placed by the customers require a

credit card number, and are considered legally binding offers to buy. At each interval, the top bids and

bidder’s initials and hometown are displayed.

Winners are posted after each closing, and the goods are shipped soon after. Most auctions are

English, ascending auctions, in which the customer can view the current bids for the item, and if he

wishes, may submit a competitive bid. If subsequent bids knock a customer’s bid out of contention for

the item, the customer may revise his bid up until the closing of the auction. At close, the items are sold

to the highest bidders, usually for the actual price they bid. Often when an auction of multiple items

closes, each winner pays a slightly different price.4 Figure 2 shows an example of the data on an Onsale

screen from October 9, 1996. There is one additional important field, that of comments from each

bidder, which has been omitted from this screen due to space constraints.

4 This differs markedly from multiple identical unit English auction theory, which predicts all winners will pay the sameprice. This will be addressed more in sections which follow.

Figure 2: Portion of Onsale Auction Screen

Jensen CD5100 AM/FM CD Receiverwith Detachable Security Panel 40 Watts Total

List Price: $429.95 Minimum Bid: $1.00Quantity Available: 7 Bid Increment: $2.00Auction closes at or after Fri Oct 11, 1996 11:06 am PDT .Sales Format: Yankee Auction(TM)Last Bid occurred at Wed Oct 9, 1996 7:10 am PDT.The current high bidders are:

1. TK of Louisville, KY, Tue Oct 8, 8:24 pm ($95.00, 1)

2. GB of Indpls, IN, Wed Oct 9, 6:26 am ($95.00, 1)

3. GH of Bend, OR, Wed Oct 9, 2:46 am ($89.00, 1)

4. RP of Arlington, TX, Tue Oct 8, 2:41 pm ($85.00, 1)

5. DB of KailuaKona, HI, Wed Oct 9, 2:25 am ($81.00,1)

6. JB of Thornton, CO, Tue Oct 8, 8:05 pm ($79.00, 1)

7. RD of Edmonds, WA, Tue Oct 8, 10:58 pm ($79.00, 1)

Item 2050

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Onsale functions as an electronic catalog engaging in semi-automatic negotiation. Its negotiation

strategy is relatively straightforward, and predetermined by the rules of the auction. It receives bids,

posts them if they are within the top echelon of bidders, and when the auction closes, notifies winners and

ships the materials. It is negotiating with human beings, who view the pages and make their bid

decisions. The ontology difficulty has been solved because the descriptions of the items are relatively

complete; this is additionally helped by the fact that many items are electronics equipment, which lends

itself well to specifications this way. The strategy difficulty, from the perspective of the seller, has also

been solved; by announcing its strategy ahead of time (“Sell to the k highest bidders, at the prices bid”),

the catalog has solved problems of privacy and strategy inference.

To summarize this section, then, Onsale provides a case study of an electronic catalog which

engages in basic, semi-automatic negotiation behavior with its customers over the Internet, using the

tools of electronic commerce.

4.2 Auctions: A Brief Overview

Auctions have been around for thousands of years. The word itself comes from the Latin root

augere, which means “to increase,” and the practice dates back to at least 193 AD, when Didius

purchased the Roman emperor’s crown for 6250 drachmas. Today, Sotheby’s and Christie’s auction off

paintings, wine, and other fine goods. And with perhaps less glamour but with a larger dollar volume, the

United States Treasury auctions off its 91-day and 182-day Treasury bills. For the source of these

anecdotes and much more, see Cassady [2].

Auction theory is a complex economic subject, and only a brief treatment can be given here; for

more rigorous discussion, see the seminal paper by Vickrey [30], introductions and overviews by

Milgrom [13], Smith [28], and McAfee and McMillan [12], a more theoretical treatment by Milgrom and

Weber [14], and a strategic analysis with a game-theoretic perspective by Wilson [31].

In keeping with standard terminology, we define the seller’s reservation price as the lowest

acceptable sale price for the item. The buyer’s reservation price (or valuation) is the maximum price he is

willing to pay for the item. The independent private values model states that the customer’s valuation is

an iid draw from a common distribution, and hence statistically independent of the valuations of other

customers. The common values model states that the customer’s valuation of the item depends

additionally on at least one common objective variable, such as resale value or amount of oil in the tract.

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The common variable introduces statistical dependence to bidders’ valuations of the object, which, in

turn, allows a bidder to infer information from other bidders’ bids.

The English auction is the open outcry, oral, ascending auction, and it is the most common type of

auction. All bidders gather at the same time in the same place to bid, and the auctioneer solicits

progressively higher oral bids from the audience until only one bidder is left. The winner claims the item,

at the price he last bid. During the English auction, bidders have the opportunity to dynamically update

their reservation prices as they learn more about the reservation prices of the other bidders. The English

auction is efficient, in that it grants the item to the bidder who values it the most; the sale price will be the

valuation of the second-highest bidder. Art, antiques, and used automobiles are a few examples of items

commonly sold this way.

The first-price sealed bid auction uses a different set of rules. Bidders submit a single, irrevocable

sealed bid. The bids are opened simultaneously, and the winner is the highest bidder, who claims the item

at the price he bid. Less information is available than in the English auction. Because there is only one

bidding stage, the first-price sealed-bid auction does not give bidders the opportunity to dynamically

update their reservation prices or their bids based upon the bids of others, and hence bidders must choose

their bids with both the value of the item and the likely behavior of other bidders in mind.5 This auction is

not necessarily efficient; depending on the bidders’ strategies, it is possible the bidder who values the item

most highly will not win it. Items commonly sold this way are oil and mineral rights.

The optimal strategies for the bidders are different for the English and the first-price sealed-bid

auction. Define the bidder surplus to be the difference between the bidder’s valuation of the item and the

price he actually pays for it if he wins, and zero otherwise. The bidder wants to maximize his expected

surplus in both auctions. In the English auction, his optimal strategy is to bid up to his (revised)

reservation price, and then to drop out of the bidding.6 In the first-price sealed-bid auction, the bidder’s

optimal strategy is to shade his bid by some amount.7 Let n be the number of bidders participating in an

auction. If the distribution of valuations is uniform and the lowest possible valuation is zero, and bidders

5 If all bidders are symmetric and perfectly rational, game theory is able to provide some insight and some optimal biddingstrategies; however, in the real world, all bidders are seldom equally well-informed, equally skilled, and perfectly rational.6 Some work has been done with respect to optimal bid increments; in general, the smaller the increment, the smaller theprobability that the winner will bid more than minimally necessary to win the item. See Rothkopf and Harstad [20].7 If he wins the auction, the bidder must pay the full amount he bid. If he bids exactly his valuation for the item, theexpected payoff is zero: if he wins, his expected surplus is zero; if he loses, his expected surplus is zero. By shading hisbid, he increases the expected surplus if he wins, but also decreases his probability of winning. The problem is to determinethe optimal amount by which to shade the bid.

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are symmetric, risk-neutral, and hold independent private values, McAfee and McMillan [12] show that

the bidder’s optimal bid is (n-1)v/n; in other words, the bidder bids a fraction n-1/n of his valuation. As

the number of bidders n →∞, the optimal strategy converges to bidding exactly the bidder’s valuation.

4.3 Onsale in Terms of Auction Theory

This section will place the workings of the auctions at Onsale, Inc., in terms of auction theory.

The auctions conducted in Onsale currently have the following characteristics:

• English/first-price hybrid: for the first portion of each auction, it’s an open, continuous,

ascending auction in which each bidder may repeatedly gather information about others’ bids

and, if so desired, update his own. For the last few minutes, when the bidder cannot be sure

his email will be received and processed in time, the auction is a first-price sealed-bid auction.8

• Multi-item auction: most of these auctions are for small (1-10 items) lots of identical items.9

• Differential pricing: if there are k objects for sale, they are sold to the k highest bidders, at

the k highest, possibly different prices bid.10 Whereas English auctions for multiple items will

sell the items to the k highest bidders at the k+1th highest bidder’s price, first-price sealed-bid

auctions of multiple items will exhibit differential pricing. We submit that the pricing structure

formed is a function of the new hybrid auction rules and cover price formation in more detail

in Section 8.

• Simultaneous and sequential elements: each individual auction is a simultaneous auction;

the different auction rounds of the same item are sequential.

• No royalties: the bids completely determine the price paid for the item. There are no post-

sale royalties to be paid by the buyer.

In addition, we have made the following assumptions about the auction:

8 Onsale describes these auctions as “Yankee” auctions; while this is not currently an auction theory term, it may wellbecome one.9 While the same item can, and often does, repeat auction after auction, this characterization refers to the sale of multipleitems in the same auction.10 Assuming all items are sold. Occasionally not all items offered will sell.

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• Independent private values: This assumption states that customers are purchasing the

merchandise for personal use, not for resale; moreover, the maximum willingness-to-pay of a

customer (and therefore his highest bid) is independent of the maximum willingness-to-pay of

other customers. If the items were valued according to the common values model, the resale

value of the item as well as other bidders’ opinions about the item’s value would have to be

taken into account.

• Risk-neutral bidders: it is assumed that the bidders are risk-neutral. For the dollar amounts

in this auction, most of them well under $200 for electronic equipment, this seems to be a

reasonable assumption.

• Symmetric bidders: it is assumed that the bidders are symmetric in terms of information and

the distributions from which their valuations are drawn.

• At most 1 item per bidder: it is assumed that bidders wish to purchase at most one item.

5. Onsale as an Optimization Problem

The new electronic commerce technology presents many different optimization problems.

This section will formulate one as an optimization problem and show how dynamic programming

can be used to solve it.

5.1 Background and Assumptions

Consider an auction of several identical physical items from the seller’s point of view. The

seller receives shipments of the items every T days, and each shipment contains N identical items.

Assume there is no ordering cost. The seller auctions off the items in batches. The seller must

decide on the size of each batch (between 0 and N) to sell in each of the t time periods between

t=0 and T-1. The seller may sell different sized batches in different time periods. The length of

each time period is s, 1 ≤ s ≤ T. All of the t time periods are equally long. For now, the seller

may not choose s or T.

Customers arrive according to a Poisson Process with rate λ. The customers are

independent; there are many customers, and the probability each will bid during a particular

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auction period is relatively small and stationary, and so hence the Poisson distribution is a

reasonable one.

The auction is an English/first-price hybrid as described above.

All k items offered will be bid upon and sold. There is an implicit dependency on λ here,

which assumes that a sufficiently large number of customers visit the auction and bid on the item

so that all k items will be sold. This is a potential weakness in the model, and will be discussed

further in later sections. For now, we assume that the customer arrival rate is sufficiently high that

there will be almost surely enough customers to demand each of the k ≤ N items for sale in any

given period.

The seller has a reservation price r = 0. The items here are end-of-life technology

equipment which has been sent to Onsale for salvage value. If it is not sold here, it is not sold.

When an auction closes at the end of period t, the winners are announced; the items are

shipped immediately, and another auction commences for period t+1.

Customers have a willingness to pay which is independently and identically distributed ~

U[a,b], and it is reflected in their final bids. Since the customers are unknown to each other,

collusion is not possible. Since they are individual people, it seems reasonable that their

preferences and willingnesses to pay are independent. And since the goods will be used by the

customer (rather than resold), the valuation of the item to each customer follows the independent

private values model. The uniform distribution makes calculation and solution tractable.

Moreover, customers’ bids are continuous random variables, so that no two will be

exactly equal. This is an assumption which differs from the practice of the Onsale auction, in

which bids are usually required to be integer numbers of dollars, and in which at times several of

the k winning bids are at exactly the same price.

5.2 Variables

Let t = index of time periods, t = 0 ... T.

The decision variable is:

k(t) = the number of items the seller will optimally auction off in time period t.

Known, fixed inputs are:

14

s = length of each time period; time periods are of equal length

h = holding cost per item per time period; constant

Variables whose probability distributions are known and whose expected values will be calculated

are:

nt = number of customers who arrive and bid during time period t. The

parameter to estimate here is λ, the customer arrival rate.

ρit = revenue to seller from selling the ith - highest priced item in time t.

(for example, ρ1t is the revenue from selling the highest priced item at time

t; ρ2t is the revenue from selling the second-highest priced item at time t.

Necessarily, ρ1t > ρ2t , and i=1 ... kt. Note for now we are assuming that λ

is sufficiently large that all kt items will be almost surely sold. ) The

parameters to estimate here are a and b, the boundaries on customers’ final

maximum bids.

5.3 Formulation

The problem is formulated as a dynamic program. Let F(t, I) = the highest profit the seller

could have earned at T, given it is currently period t and there are I items remaining in inventory.

F(t, I) = max F(t +1, I - k(t)) I h E( ) j=1

k(t)

k t I jt( ) ...=− +

∑0

F(T, *) = 0; F(t, 0) = 0.

The solution, for the parameters N, T, s, a, b, and λ, is given by F(0, N) and the vector

k(t). The dollar amount of the solution is given by F(0, N), and the vector k(t) specifies how many

units to offer for sale in each period t.

All of the terms in the recursion are relatively straightforward except for the calculation of

E[ρjt], the expected revenue to be gained from the sale of the jth item. Its calculation follows;

more detailed derivations can be found in the Appendix in Section 10.1.

E[ ] E[E[ |n ]] = E n] Pr(n = n) = (n 1 j)(b a)

n 1a

e ( s)n!

jt jt t j tn=0

s n

n=0

= ⋅+ − −

++

∞ −∞

∑ ∑[

15

6. Solution and Data Analysis

This section applies the actual data from Onsale to the model outlined in previous

sections.

6.1 Data Collection Technique

The data was presented here was collected on a daily basis from September 23 until

October 11, 1996. Each day, the highest bids for car audio and computer memory/CPU

components were copied off of the Onsale Web site and saved in a data file. On days auctions

closed, the winners were copied down and saved as well. These text data files, spaced

approximately 24 hours apart, shed some light into the auction dynamics of these auctions.

The data collection methodology is far from perfect; ideally, we would have access to the

Onsale company data files and to the logs of all accesses of the auction pages. The biggest

weakness of the current methodology is that it gives only snapshots of the auction dynamics taken

24 hours apart. It is possible a bidder would bid, be displayed on the screen, and be displaced all

in the space of the 24 hours, hence never registering as a potential bidder in the data presented

here at all. It is also virtually certain that many potential bidders view the auction when it is more

advanced, and hence the price has advanced beyond their willingness to pay. These bidders will

never put forth a bid, remaining silent and never registering as a potential bidder at all. Moreover,

manually collecting the data and loaded it into spreadsheets is slow and tedious, and puts limits on

the amount of data which can be analyzed in a reasonable amount of time. Automating the data

collection process would allow much more data to be analyzed, giving more reliable results.

16

Despite these shortcomings,

however, the data presented here

reveal some interesting patterns.

Figure 3 shows the bid arrival process

as represented in the data collected.

Complete data from five

separate auctions has been collected

and analyzed; more data is available

and must be processed. Item 2050, a

40-Watt Jensen CD5100 AM/FM CD

Receiver with Detachable Security

Panel, was chosen as a representative

item because it was sold in each of the

five consecutive auctions, in lots of 7 identical units. This indicated that Onsale perhaps had more

than 7 units available in inventory when it ran the first auction, and was thus consciously choosing

batch-size offerings. Additionally, the CD Receiver received a fair amount of bidding attention

from bidders, and sold at around $135.00 in most of the auctions, well less than the list price of

$429.95. This price is large enough to command bidder attention, and yet small enough that

buyer risk aversion is not likely to play a huge role.

Figure 3: Bid Arrival Process Over TimeAuction ending 10/4/96; Item 2050

$80.00

$90.00

$100.00

$110.00

$120.00

$130.00

$140.00

$150.00

10/1/9612:00PM

10/2/9612:00AM

10/2/9612:00PM

10/3/9612:00AM

10/3/9612:00PM

10/4/9612:00AM

10/4/9612:00PM

Date and Time

Bid

in d

olla

rs

17

6.2 Parameter Estimation

To fit the Onsale case study into the model presented requires fitting parameters to the

data. The following

parameters pertain to data

collected for the five

consecutive auctions of Item

2050, the CD Receiver,

which closed between

September 27 and October

11, 1996. Figure 4 shows a

summary of these

parameters, and detailed

calculations of these parameters follow in 6.2.1 through 6.2.3.

6.2.1 Estimation of s, h, N, and TThe estimation of s, h, N and T is relatively straightforward.

s = length of each time period. This is given at 3.5 days because there are 2

auctions per week. Auctions close Tuesdays and Fridays.

h = holding cost per item per time period; constant. For now, this is not

known, but is assumed to be the cost of capital, perhaps 10% per dollar per year. For an item

which sells at an average price of $135.00, this would be about $0.13 per 3.5 day period. This

variable is only an estimate; it is completely unknown and does influence the optimal solution.

N = total number of items in inventory at t. For these five auctions, which each

sold 7 units, N = 35. It is possible the number of items was larger, but for the current calculations

we assume only 35 items.11

T = the total number of time periods available, in this case, 5 auctions.

11 Indeed, it is likely the number of items was larger. Item 2050 was sold in the auction immediately after the lastone analyzed here as well, again in the quantity of 7 units.

Figure 4: Summary of Parameters Estimated for Item 2050

Parameter Definition Estimated values length of auction 3.5 daysh holding cost per item per day $0.13 per item per

dayN initial inventory 35Τ total number of auctions run 5λ bidder arrival rate per auction 13.6 bidders per

auctiona lower limit of bidder maximum

bids$75.00

b upper limit of bidder maximumbids

$150.00

18

6.2.2 Estimation of λλThe estimation of the bidder arrival rate is somewhat more complex. The number of

customers who arrive and bid during time period t is modeled as a Poisson random variable with

parameter λ. The parameter is difficult to estimate because under current data collection

methods, all the bidders do not register on the site; those who view the current auction and decide

the price level is above their willingness to pay do not show up in our current data collections

methods, and indeed, may never submit a bid. When the going price of the item is very low, in the

very beginning of an auction, almost all customers who would have submitted a bid will register;

when the going price is very high, in the very end of the auction, only record bidders will register.

For the current estimate of λ, the number of unique bidders who registered at any point

during the auction was used. A bidder who updated his bid several times over the course of a

single auction was counted as a single arrival. Counting each auction as a separate sale, there

were a total of 89 distinct bidders, as identified by initials and hometown. Looking at the overall

picture, 21 of the bids were repeats from unsuccessful bidders in earlier auctions, yielding 68

completely distinct bidders.

This is a relatively large number of repeat bidders, and merits some discussion. The

Poisson arrival process assumes independence between the different auctions, and this is not the

case for this data. Despite the contradiction, the independence assumption may be somewhat

justified by noting that most repeat bidders were unsuccessful in previous auctions. Once a bidder

has won an item, he is unlikely to bid again, and hence will likely drop out of the auction

dynamics. Moreover, unsuccessful bids, while they affect the arrival rate and general dynamics of

the auction while it is underway, do not affect the final outcome or the final revenue received by

the seller. So, while the Poisson assumption of independence between consecutive auctions is not

borne out in this data, the impact is softened by the hypothesis that once a bidder has won, he is

indeed likely to drop out, making consecutive auctions independent.

The estimate of λ is taken as 68 / 5 = 13.6 distinct bidders per auction. This is a slower

arrival rate, and thus a more conservative estimation in three important respects. First, the higher

19

the arrival rate, the larger the expected revenue from the sale of the top k(t) items, and so a lower

arrival rate would tend to give more conservative estimates of sales revenue.12

Second, recall the earlier assumption that there will almost surely be enough bidders to

purchase all items. The higher the arrival rate, the more solid this assumption will be. At λ =

13.6, the probability that the number of bidders arriving is too low to purchase all the items

exceeds the 1% level at 6 items; with λ = 25.4 (calculated based on 89 distinct auction-level

bidders, rather than 68 distinct 5-auction bidders) these probabilities hold until 14 items are

offered.

Third, recall the imperfection in the data collection methods mentioned earlier in this

section. It is very likely that many bidders did not register in the data here, either because they

auction going price was already too high by the time they viewed the auction, or because of the

24-hour window between data collections. There is not currently a way to estimate by how much

these factors would increase the value of λ, but it is probably not a negligible amount For the

analysis conducted in the next sections, the more conservative estimate of λ = 13.6 bidders per

auction was used.

6.2.3 Estimation of a and bRecall the values of the customers’ final bids were assumed to be ~U[a, b]. It is important

to stress that these are the customers’ final, and hence, maximum bids; given a maximum bid is a

certain dollar amount, for the purposes of parameter estimation here the path taken to arrive at

that bid is irrelevant. Different customers will play the English auction different ways; some may

enter a low bid initially and nudge it upwards as required by the competition; others may wait

silently and enter a single, larger bid near the end of the auction. It is the distribution of these

final bids, not the path taken, which is being estimated here.

The customers’ final bids for the items were tallied. When a customer was among the

winners, the final bid was taken directly as the customer’s winning bid. When a customer had

participated early in the auction and then dropped out, the customer’s last standing bid which

12 E[ρj] is nondecreasing in λ. To see this, note that the calculation of E[ρj] weights E[ρj|n] by Pr(nt = n), theprobability there will be n arrivals in the period. Pr(nt = n) is nondecreasing in λ. Keeping j constant, E[ρj|n] = a+ (b-a)(n+1-j)/(n+1), which is nondecreasing in n. Finally, E[n] = λ, which is increasing in λ. Hence, E[ρj] isnondecreasing in λ.

20

registered in the data collected was assumed to be the maximum possible bid of the customer.

This assumption may somewhat underestimate the customer’s true maximum bid. A customer

whose last registered bid was $85 may return the next day to see that the bidding is now at $115.

If the customer’s true maximum bid was $100, she will not participate further in the bidding, and

the data captured will have her maximum bid at $85, rather than $100.

As shown in Figure 5,

the customer’s maximum bids

on Item 2050 did follow a

pattern somewhat close to a

uniform distribution.

Graphically, the parameters

were estimated at a=$75 and

b=$150. These parameters

mean that all customers are

assumed to be willing to pay up

to $75 for the CD player, and

none are assumed to be willing to pay more than $150 for it. As more data is analyzed, perhaps

these limits will change, but they seem to fairly well encapsulate the maximum bids found so far

with current data collection methods.

6.3 Solution Based on Model and Parameters

This section shows solutions obtained by using the parameters estimated above to

calculate an answer to the dynamic program.

As previously stated, the problem is specified by the parameters s, h, N, T, λ, a, and b.

The solution is specified by F(0, N) and the vector k(t).

The solution to this problem is:

F(0, 35) = $4454.35

k(t) = (7, 7, 7, 7, 7).

While the actual profit figures of Onsale for this particular series of auctions are not

known, the Onsale offering policy is exactly what the model predicts it should be: to offer seven

Figure 5: Distribution of Maximum Bids for Item 2050

Maximum Bidder Bids, Item 2050, 5 Auctions

0

20

40

60

80

100

120

140

160

Bidders sorted by maximum bid

Max

imum

bid

in d

olla

rs

21

items per auction over each of the five auctions. Intuitively, this makes sense. As long as the

holding cost is not too high, the seller would want to spread the items out as much as possible,

thereby selling fewer items each auction, and hence selling to the highest bidders possible. The

actual revenue that Onsale gathered from the sale of these 35 items over the period of time was

$4739.00, and if the estimate of holding cost here holds, the total holding cost was $25.48,

leaving a gross profit of $4713.52.13

The actual results of the five auctions are shown in Figure 6 below. The expected values

of the top 7 bids have been included for comparison in the last row.

One difference between the actual results and the model can be seen by comparing the

average winning bids to their respective order statistics. While the order statistics are certainly “in

the ballpark” as far as estimating the winning bids, the order statistics tend to overestimate the

highest bid and underestimate the lower winning bids. Additionally, the order statistics are spread

out more widely than the actual bids; this may reflect the actual bids’ tendency to “clump

together” as each bidder tries to bid the minimum amount possible and still win the item. Another

explanation is that the estimate of the bidder arrival rate, λ, is too low. A higher estimate for λ

would raise the expected number of bidders arriving in each period; this would change the

13 Of course, this gross profit figure does not cover the cost of goods. The cost of goods sold is not a decisionvariable here, and is presumed to be the exact same independent of when the items were sold. It does not affect themodel, and so here is left out.

Figure 6: Winning Bids from Five Auctions, Item 2050

Auctionending

High bid Secondhighestbid

Thirdhighestbid

Fourthhighestbid

Fifthhighestbid

Sixthhighestbid

Svnthhighestbid

TotalRev-enue

9/27/96 $145 $145 $145 $143 $141 $141 $141 $100110/1/96 $137 $135 $135 $133 $133 $133 $133 $93910/4/96 $139 $139 $139 $139 $139 $137 $135 $96710/8/96 $133 $129 $127 $127 $127 $125 $125 $89310/11/96 $135 $135 $135 $135 $133 $133 $133 $939

Average $137.80 $136.60 $136.20 $135.00 $134.60 $133.80 $133.40

E[ρj] $144.49 $138.97 $133.45 $127.90 $122.26 $116.43 $110.19

22

expected values of the order statistics. With more bidders expected per auction, the expected

values of the order statistics will be closer together and also closer to the top of the distribution.

6.4 Properties of the Optimal Solution

This section analyzes some properties of the optimal solution which provide insight into

the workings of this problem.

The optimal k(t) is nonincreasing in t if the other parameters remain constant. This can be

shown by contradiction. Assume that at optimality, k(t) is increasing in t, so that k(t) < k(t+1).

This means that k(t+1) = k(t) + x, x > 0 and integer.14 Since λ, a, and b remain the same period

to period, the expected revenue from selling the jth highest priced item, j=1...k(t), each period

remains the same. Hence, the expected revenue from selling the first k(t) units in each period is

the same. Selling the additional x units at either t or t+1 brings exactly the same expected

revenue, since in each period k(t) units have already been sold. However, selling the x units at

period t+1 carries one additional period’s holding cost = xh. Since x > 0 and h > 0, this cost is a

positive number, and hence lowers expected profit. It would be better to sell the x item in period

t, saving the holding cost. Therefore, this cannot be the optimal solution; rather k(t) ≥ k(t+1), and

hence k(t) is nonincreasing in t at optimality, all other parameters held constant.

Define a switch of an item from period u to period v, u > v, to be a change in policy from

k = (k(1), ... ,k(v), ... ,k(u), ... ,k(t)) to k = ((k(1), ... ,k(v)+1, ... ,k(u)-1, ... ,k(t)). Hence,

switching one item from period u to period v means selling one more unit in period v, one less

unit in period u, and keeping all other policies the same. A switch is for exactly one item.

A switch is profitable if the marginal revenue from the switch exceeds the marginal cost of

the switch. Consider two policies, k = (k(1), ... ,k(v), ... ,k(u), ... ,k(t)), and ks = ((k(1), ...

,k(v)+1, ... ,k(u)-1, ... ,k(t)), u > v, where they are identical except ks has switched one unit from

period u to period v. Such a switch is profitable if the expected gain in marginal revenue from

making the switch exceeds the expected marginal cost of making the switch.

Let i = k(v) and let j = k(u).

14 Recall that k(t) and k(t+1) are integers here, also.

23

The expected marginal revenue gained is the expected revenue from selling the item as the

i+1th highest priced item in period v, which is E[ρi+1, v]. Assuming stationarity in t, this can be

abbreviated to E[ρi+1].

The expected marginal cost of making the switch involves the holding cost saved by

selling the item (u-v) periods earlier and the loss of expected revenue from selling the item as the

jth highest priced item in period u. Since the holding cost is a savings, it has a negative impact on

the cost, and hence a negative sign. The expression for the marginal cost is -h(u-v) + E[ρj,].

The switch will be profitable if E[ρi+1] > -h(u-v) + E[ρj,]. The optimal k policy would

have no possible profitable switches. Moreover, holding other variables the same, the farther

apart u and v are, the more profitable the switch will be.

For the data parameters estimated here, the optimal k(t) vector changes from k =(7, 7, 7,

7, 7) to (8, 7, 7, 7, 6) when holding cost reaches about $1.80 per unit per period. This is the net

effect of a switch of one unit from period 4 to period 0. The values for the decision are shown in

Figure 7.

Figure 7: Effect of Holding Cost on Optimal Allocation of Items

Parameter values: a = 75, b = 150, λ = 13.6 arrivals per period, N=35 items, T=5 auctionsHoldingcost h perunit perperiod

k beforeproposedswitch

k afterproposedswitch

MR = E[ρi+1] = E[ρ8]

MC = -h(u-v) + E[ρj,]. = -h(4-0) + E[ρ7]

Is switchprofitable?

0.13 (7, 7, 7, 7, 7) (8, 7, 7, 7, 6) 103.30 (-0.52)+ 110.18 = 109.66 No1.70 (7, 7, 7, 7, 7) (8, 7, 7, 7, 6) 103.30 (-6.80) + 110.18 = 103.38 No1.72 (7, 7, 7, 7, 7) (8, 7, 7, 7, 6) 103.30 (-6.88) + 110.18 = 103.30 Indiff1.80 (7, 7, 7, 7, 7) (8, 7, 7, 7, 6) 103.30 (-7.20) + 110.18 = 102.98 Yes

24

7. Information System Requirements

The practical consideration of the information system requirements merits

discussion here. In order to use this methodology to actually make and evaluate business

decisions, the business’ information system must include an inventory system which can

track the number of items in inventory on a periodic basis, every s units of time. Since

units are shipped to the winners very soon after each auction closes, the order entry

system would need to be integrated into the inventory warehouse tracking system.

The probability distributions of nt and ρit are assumed to be stationary, and

therefore can be calculated ahead of time. However, it is possible to update the estimates

of the probability density functions, and hence the expected values, with real-time data

should it be desired.

More capable information systems would allow more capabilities. For example, if

the information system could track the current bidding dynamics and compare them to the

history of previous auctions of the same item, an especially competitive bidding session

could be detected. In the data presented above, the session which ended September 27

was more competitive than the other sessions, and resulted in more total revenue. If the

company could detect such a session while it was still “live,” it could choose to offer one

additional item during those highly competitive auctions. The reasoning would be that

receiving the 8th highest price for an item in a highly competitive auction might give the

seller more revenue than the 7th highest price in a not-so-competitive auction.

8. Price Formation and Bidder Strategy

Possibly the most interesting aspect of the current operations is the price formation

in the English/first-price hybrid structure. Bidders may participate in the auction

continually over the life of the auction, or they may dial in immediately before the auction

closes and submit a single bid. What are the implications of this new structure for price

formation?

The final price formation differs from the English and the first-price sealed-bid

price structures. In a true English auction selling k identical items, the k highest bidders

will not bid any more than absolutely necessary to secure the items. Since the items are

25

identical, it does not matter which winner receives which item, and hence each winner will

pay exactly the same price, the price of the k+1th highest bidder. The drawback for

revenue maximization is that the seller caps his revenue at the k+1th highest reservation

price, when there were k bidders willing to pay more than that amount.

The English auction makes much more information available to the bidders than

does the first-price sealed-bid one. Milgrom and Weber [14] define the concept of

affiliation, which, loosely rephrased, holds that bidders’ valuations of an item are not

negatively correlated. The knowledge another bidder values the item highly may raise

one’s valuation, or leave it unchanged, but it will not reduce it. Milgrom and Weber also

show that more information tends to raise the average sale price of the item.15 The

English auction takes advantage of affiliation and the information effects by making

information about other bidders’ reservation prices publicly available, and thereby raising

the expected sales revenue from the item.

Additionally, raising the number of bidders raises the expected revenue from the

English auction. The rationale is relatively straightforward: the more bidders there are,

the higher the expected value of the top k bidders. Moreover, the more bidders there are,

the more information is revealed; the information effect tends to raise the sale price as

well.

In a pure first-price sealed-bid auction, the bidders must put together a strategic

bid with no additional information; for the conditions described above, the optimal bid is

(n-1)v/n, where n is the number of bidders and v is the bidder’s individual valuation of the

item. The number of participants is not known to either the seller or the bidders, but must

be assumed to be very large; in that case, as n grows very large, the optimal first-price

strategy is to submit one’s own valuation of the item.

For the bidder, the optimal strategy may depend on how much he enjoys “playing

the game.” A bidder who values the interaction and entertainment of the auction screens

may want to play the English portion of the auction, inching his bid up in small increments.

A bidder interested in procuring the item at a minimum of time and cost may be better off

26

participating in the first-price sealed-bid component, by dialing in immediately before the

auction closes and bidding his full reservation price. As long as there are sufficiently large

number of bidders participating in the English section to lubricate the market and keep

information flowing, the first-price player could use the collected accumulated information

without having to provide any of his own.

The English/first-price hybrid very well may offer the best of both worlds for the

seller interested in raising the expected sale price. In its first phase, the English auction

characteristics, including the bidders’ prices and comments, are publicly available; per

Milgrom and Weber [14], this information raises the expected revenue. In the last phase,

the optimal strategy for a first-price sealed-bid auction with a large number of bidders is to

bid one’s own valuation for the item. Since the valuations are likely to differ for different

bidders, this breaks the uniform ceiling of the k+1th highest bidder’s valuation, which

would have been seen for the pure English auction.

9. Summary and Conclusions

This paper has brought together three separate fields of study: electronic

commerce, auction theory, and optimization techniques. It has shown how an online

auction can be the first step towards fully automated electronic negotiations, and

furthermore how the new electronic commerce technology provides additional degrees of

freedom when running auctions. These additional degrees of freedom introduce new

optimization problems, and the question of the optimal item offering schedule was

addressed here. It was shown that a dynamic program can be used with some parameters

to give a solution to the problem; furthermore, the optimization can be re-run each time

period if desired.

This area holds much promise of future research. One prominent area is more

rigorous mathematical analysis of the cost and profit functions, extending the work done

on the properties of the optimal k(t) and the switching analysis in Section 6.4. It is likely

this analysis will yield insight into the problem structure, suggesting efficient solution

15 The information may come from the other bidders, from the seller, or from a third party. The concept isthat more information reduces the uncertainty in the bidder’s estimation of the valuation of item, and

27

algorithms for a variety of problems. In addition to more mathematical analysis of the

existing problem, there is a much broader range of problems to solve.

The item offering schedule is not the only optimization problem which is presented

by this new technology. The dynamic nature of the auctions offers another opportunity

for the seller. If a particular item has a known sales history, the seller can monitor the

current auction and determine if it is unusually slow or unusually “hot.” If it is an

unusually high-priced auction, the seller could introduce additional items into it midway

through the bidding process, hoping to gather the additional revenue for these extra items

as well.

The seller also has the opportunity to place “phantom” bids for just a few dollars

more than the highest current bid. Since bidders are only identified by initials and

hometown, it is relatively easy to introduce a fictitious bidder. The seller runs the risk that

no bidders will top the fictitious bid, in which case the seller has lost potential revenue

from a single item; however, the potential benefit is that the fictitious bid will “jump-start”

the bidding process by encouraging the competitive spirit among the bidders.

Bidders also have more options available with this setup. A bidder may decide he

wishes to purchase a certain product and is willing to pay at most $100 for it; and program

a software agent to continuously bid up to $100 in each and every auction for the next six

months, notifying the bidder if he wins. This strategy is consistent with the bidder’s

optimal strategy for the auction as outlined above, and moreover is simple enough that a

current software agent could execute it. It is also proof against seller or other buyer

inference, in that in the course of the auction information about the bidder’s reservation

price would likely be revealed anyway. (This strategy could produce a secondary industry

composed of virtual catalogs which troll virtual auctions for low-priced items offered in

“slow” auctions, and then purchase and re-sell the items at a profit later.)

And, finally, there is the entire field of information effects. Milgrom and Weber

[14] identify the additional information as a crucial component of the English auction, and

a key reason its revenue tends to be higher than the information-poor sealed-bid auction.

How can the seller leverage the information effect? Should only the current winning bids

hence he does not need to “hedge his bets” as much with more information.

28

be posted online? How much auction history should be revealed? These questions require

quantification of the information effect.

In this paper we have shown how new technology poses new questions, and have

suggested a method by which the tools of optimization can help manage the new degrees

of freedom granted here.

10. Mathematical AppendixThis appendix contains detailed solutions to the problem presented in the body of thepaper.

10.1 Order Statistics DerivationDerivation of the following equation:

E[ ] E[E[ |n ]] = E n] Pr(n = n) = (n 1 j)(b a)

n 1a

e ( s)n!

jt jt t j tn=0

s n

n=0

= ⋅+ − −

++

∞ −∞

∑ ∑[

Since the bid arrival process is stationary in t, we can suppress the t subscript. Given nbids arrived during the period:

Let r1…rn be the actual bids received during the period, unordered. Recall that r1…rn

are iid random variables, ~U[a,b].Let ρ1…ρn be the ordered bids received during the period, so that ρ1 is the highest bid,

ρ2 is the next highest bid, etc. Then ρ1…ρn are distributed as the order statisticscorresponding to r1…rn .

We can model the order statistic distribution as a binomial random variable as follows:The event {ρi ≤ x} = the event {0 or 1 or ... or i-1 bids are ≥ x}.Model this as a binomial random variable, and consider it a success if ri ≥ x.Then the probability of success is (1-F(x)), and the probability of failure is F(x).And the event {ρi ≤ x} = the event {up to i-1 successes in n trials}.

( )( )So F | n is Pr( F

And f | n is f(x) (Fi

ρ

ρ

ρ

ρ

i ij

i j n j

i

n i i

x nn

jx F x

x nn

n i i F x x

≤ =

= = − −

=

− −

− −

∑| ) ( ) ( ))

Pr( | )!

( )!( )! ( ( )) ( ))

0

1

1

1

Since ri ~U[a,b],F(x) = (x-a)/(b-a) ; f(x) = 1/(b-a) ; 1-F(x) = (b-x)/(x-a);

Substituting these into the expression above for fρ|n gives:

f | n is n!

(n j)!(j 1)!x ab a

1b - a

b - xb - aj

n j j 1

ρ ρPr( | )j x n= = − −−−

− −

29

E[ n] = Pr( x n) x dx

=n!

(n - j)!(j -1)!x ab a

1b a

b xb a x dx

=n!

(n - j)!(j -1)!1

b a (x a) (b x) x dx

j jx=o

n j j 1

x=a

b

n

n j j 1

x=a

b

ρ ρ| |=

−−

−

−−

−

− −

∞

− −

− −

∫

∫

∫

Calculating this integral16 gives

E[ n] = n!

(n - j)!(j -1)!

(n +1- j)(b - a)

+ a.

jρ |( )!( )!( )

!( )( )

−

− − −

+ − −+

+

=+

1 1 11

1

b aj n j b a

nn j b a

na

n

n n

And substituting back into the original expression for E[ρjt] gives:

E[ ] E[E[ |n ]] =(n 1 j)(b a)

n 1 ae ( t)

n! jt jt t

t n

n=0ρ ρ

λλ

=+ − −

+ +

−∞∑ .

10.2 Lemma 1Before solving the larger integral, a lemma is needed.

Lemma 1: I = (x - a) j -1

n - (j -1) I j

n- j

x=a

b

j-1( )b x dxj− =−∫ 1 .

Proof:

16 Detailed calculations of this integral can be found in Sections 10.2 and 10.3.

30

Let b x dx

dx

v

n j

I

b xx an j

x an

j

j ab

jn j

a

b n j

I = (x - a)

Integrate by parts:

Let u = (b - x)

du = (j -1)(b - x)

(x - a) dx

dv = (x - a)

= uv - v du

-

jn- j

x=a

b

j-1

j-1-1

n- j

n-(j-1)

a

b

( )

( )

( )

( )( )

( )( )( ) ( )

−

−

=

− −

= −−− −

−−

−

−− − − =

∫

∫

1

11 1

1

1 ( )( )( ) ( )

( ) ( )

( )

( ) ( )

jj b x dx

x a b x dx

a

bj

n j

a

bj

−

− − −

= − −

=

∫

∫

− −

− − − −

11 1 1

1 1 1 0 + j -1

n - (j -1)

j -1

n - (j -1) I j-1

( )

Expanding I out gives

I = (j -1)!(n - j)!

n!(j -1)!(n - j)!

n!

j

jnI b a1 = −

31

The recursion between j -1 and j has been established.To close the induction, it must be shown for j is 1.

At j is 1, I = (x - a)

(x - a)

(b - a)

n

At j is 2, I (x - a) integrate by parts with u = b - x and dv = (x - a)

0 - (x - a)

1

n -1 (x - a)

1

n -1 (b - a)

j -1

n -

jn-1

a

b

n

n

jn-2

a

bn-2

n-1

a

b

n

n

( )

.

( ) ;

( )

b x dx

n

b x dx

ndx

n

n

a

b

a

b

−

=

=

= −

=−

−

=

=

=

−∫

∫

∫

1 1

1

1

(j -1) Therefore, the induction hypothesis is proved.I j −1.

10.3 Solution to the Larger IntegralThe problem is to calculate

E[ n] = n!

(n - j)!(j-1)!1

b a(x a) (b x) x dx

Let G = n!

(n - j)!(j-1)!1

b a

j

nn j j 1

x=a

b

n

|

−

− −

−

− −∫

E[ n] = G x dx

E[ n] = x dx

Introduce a change of variable:Let y = x - a; then dy = dxLet c = b - a

jx=a

j

x=a

| ( ) ( )

|( ) ( )

x a b x

Gx a b x

n j jb

n j jb

− −

− −

− −

− −

∫

∫

1

1

32

E[ n] = (y + a) dy

= dy + a dy

j

y=0

y=0 y=0

|( ) ( )

( ) ( ) ( ) ( )

Gy c y

y c y y c y

n j jc

n j jc

n j jc

− −

+ − − − −

−

− −

∫

∫ ∫

1

1 1 1

Applying Lemma 1 to both of these integrals yields:

( )

= (j -1)!(n +1- j)!c

(n +1)! +

a(j -1)!(n - j)!cn!

(j -1)!(n - j)!(b - a)

n!

E[ n] = G (j -1)!(n - j)!(b - a)

n!

n+1 n

n

j

n

=+ − −

++

+ − −+

+

( )( )

|( )( )

n j b an

a

Son j b a

na

11

11

33

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35

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