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Predicting Bidders’ Willingness to Pay in Online Multi-Unit Ascending Auctions:

Analytical and Empirical Insights

Ravi Bapna1 (ravi_bapna@isb.edu)

Paulo Goes2

(Paulo@sba.uconn.edu)

Alok Gupta3 (alokgupta@acm.org)

Gilbert Karuga4 (karuga@sba.uconn.edu)

Acknowledgement

Alok Gupta’s research is supported by NSF CAREER grant #IIS-0301239, but does not necessarily reflect the views of the NSF. Partial support for this research was also provided by TECI - the Treibick Electronic Commerce Initiative, OPIM/School of Business, University of Connecticut. The authors also thank four anonymous reviewers and an associate editor for their valuable suggestions that have resulted in significant improvements in the manuscript.

1. Indian School of Business, Hyderabad, India. 2. Dept. of Operations and Information Management, UConn School of Business, Storrs,

CT. 3. Kansas University, Lawrence, KS.

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Abstract

A large fraction of online auction activity deals with selling multiple identical units of an item,

using progressive discriminatory or Uniform pricing approaches. Our research focuses on the

progressive bid information, observable in Internet auctions, to infer bidders’ willingness to pay

(WTP) in real time. We derive a priori estimates of bidders’ maximum willingness to pay based

on a bidding strategy classification and a valuation prediction model. The classification model

identifies bidders as either adopting a myopic best response (MBR) bidding strategy or a non-

MBR strategy. Unlike prior econometric studies that utilize aggregate data to understand

underlying demand curves of products, we use an automated agent to capture observable micro-

data from bidding activity in several hundred Internet auctions. We validate our classification

and prediction model with bids made on online auctions from Samsclub.com (Uniform price)

and Ubid.com (Yankee). Our joint classification and prediction approach outperforms two other

naïve prediction strategies that draw random valuations between a bidder’s current bid and the

known market upper bound. Our prediction results indicate that we are able to estimate, on

average, within 2% of bidders’ revealed willingness to pay for Yankee and Uniform price multi-

unit auctions. We also extensively discuss how our results can facilitate mechanisms design

changes such as dynamic bid increments and dynamic buy-it-now prices.

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1. Introduction and Background

Online auctions exemplify the Internet’s ability to become a temporally and spatially

unconstrained market maker. Yet, while expanded in scale and scope, Internet auctions can

arguably be considered to be lacking in the skills of an expert auctioneer. Smith (1990) describes

how an expert auctioneer can be credited with maintaining the temporal order of the auction and

the movement of bids.

“ … $100,000, I have $100,000! $120,000! $130,000! I have $140,000 out back and

$150,000! Will you give me 175? 175! 200? 200!250? Will you give me 250?…”

This research is aimed at utilizing the enhanced information acquisition and

computational potential of today’s Internet auction environment to replicate the skills of an

expert auctioneer in a high volume, mechanized environment. We achieve this by developing

analytical models to infer bidders’ willingness to pay (WTP) in progressive multi-unit online

auctions. This is akin to an expert auctioneer capturing by gut and feel the underlying essence of

the auction room’s WTP. We highlight the enhanced computational capabilities that facilitate

real-time value discovery to suggest mechanism design changes such as dynamic bid-increments

and dynamic buy-it-now prices. Current online auction practice assumes bid-increments and buy-

it-now prices to be static choices, which are made prior to the commencement of the auction. Our

approach demonstrates how an auctioneer, equipped with an estimation model such as ours, can

utilize dynamic bid increments to achieve higher revenue and allocative efficiency, as well as set

dynamic buy-it-now prices.

A large fraction of online auction activity deals with selling multiple identical units of

items, such as aging computer hardware. Typically, these auctions use either a discriminatory

(Ubid.com) or a Uniform (Samsclub.com) pricing approach. Our research objective is to gain

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deeper insights into the price formation process of auctions that progressively reveal information

about bidders’ willingness to pay, and use those insights into deriving a priori estimates of the

expected willingness to pay of bidders.

We should point out the subtle distinction between the traditional notion of the valuation

of a product and our use of the phrase willingness to pay. Both reflect private information that

the bidder possesses, but given the online context, willingness to pay bounds valuations from

below. The motivation to use willingness to pay (WTP) comes from the observation that the

auctions under consideration post a “suggested” retail price, which effectively cap the WTP,

irrespective of the bidders’ valuations.1 Therefore, in the rest of this paper we use the phrase

willingness to pay instead of valuation.

While demand estimation has been the focus of many econometric papers, it has typically

been done using aggregate data, with results hinging on some tenuous assumptions about

consumers’ preferences. In contrast, we rely on the enhanced computational capabilities of

Internet enabled auctions to undertake real-time demand estimation using micro-data, making

empirically established, and less demanding assumptions about bidders’ behavior.

From a theoretical perspective, progressive ascending multi-unit auctions have received

only limited attention in the literature, usually under a set of assumptions that do not hold up in

the online context. For instance, bidders are assumed to be homogeneous, typically typed as

being symmetric, risk-neutral, and adopting Bayesian-Nash equilibrium strategies. While

tenable in the context of face-to-face single item auctions, this set of assumptions readily breaks

down in the vast majority of multi-unit online auctions. For such auctions, it is well known that

the computation of equilibrium bidding strategies is intractable (Nautz & Wolfsetter, 1997). A

1 In no case, in our extensive dataset of consisting of 78,014 bids from over 900 online auctions, did the final bid exceed 90% of the suggested retail price. Section 3 describes the data in more detail.

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key distinguishing feature of our work is the minimalist assumption of a myopic best response

(MBR) bidding that ties bidders’ revealed bids to their underlying WTP. MBR bidding agents’

bids conform to an equilibrium strategy with the assumption that the agents view the current

round as the last round of the auction and take prices as given (Parkes 2001). Recent work

(Bapna, Goes, Gupta and Jin 2004) on bidding strategies in Yankee auctions indicates that

approximately 66% of the serious bidders conform to this strategy.2 The remaining bidders are

“evaluators” in nature, placing non-MBR bids that are significantly higher than minimum

required bids and not revising them. We integrate the findings from Bapna et al. 2004 into our

classification and prediction scheme.

Using the knowledge that non-MBR type bidders do not revise their bids, we assume

their WTP to be the same as their initial bids; while for the MBR bidders we solve our analytical

inverse bid function estimation procedure. Thus WTP predictions are made for all bidders (MBR

and non-MBR) and the two sources of error are; a) the overall classification and b) the estimation

for the MBR types. Figure 1 summarizes our value discovery approach.

Figure 1- Classification followed by Prediction

2 A serious bidder was defined as a bidder whose highest bid was at least equal to 70% of the final auction price.

Classification

Type – Evaluator (Non-MBR) Type - MBR

WTP = Observed Bid

•Assume current bid maximizes bidders surplus

•Estimate inverse of bid function as WTP

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To the best of our knowledge, no other study has used the enhanced information

acquisition and processing capabilities of the online environment, where observance of the price

formation process can be used to infer bidder willingness to pay in real time.

The rest of this paper is organized as follows. In section 2, we provide an overview of the

related literature. In section 3, we provide insights into the market mechanism that we are

investigating and introduce the reader to our extensive multi-unit auction dataset. Section 4

focuses on the bidder strategy classification scheme and the development of an analytical model

for predicting MBR bidders’ willingness to pay. The prediction accuracy is tested empirically in

Sections 5. Section 6 demonstrates the utility of the WTP estimation in inferring the final price

prior to the close of the auction, and on setting dynamic bid increments as well as dynamic buy-

it-now prices. Section 7 concludes the paper by presenting directions for future work.

2. Relevant Literature

Given the vast body of auction literature [see, for example, McAfee and McMillan

(1987), Milgrom and Weber (1982), Milgrom (1989), Rothkopf and Harstad (1994) and

Menezes (1996) for a detailed literature review and analysis] it is instructive to begin by briefly

examining what, if anything, is new about online auctions. Arguably, online auctions have

expanded scope and scale, compared to their traditional counterparts. There is early evidence that

participation in online auctions is endogenously influenced, while the traditional assumption in

the literature takes the number of bidders at an auction as exogenously given [Paarsch (1992),

Laffont, Ossard and Vuong (1995)]. Bajari and Hortascu (2001) have shown, with data from

eBay, that modifying the mechanism affects the entry decisions. The expanded scale and scope

of the auction has made the participation of online auctions non-captive of its audience. The

bidders in online auctions come and go at will, while in traditional environments bidders are

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captivated through the close of the auction. Particularly relevant to our work are the enhanced

computational and networking resources that have made multi-unit auctions more feasible.

Multiple units can be sold simultaneously, not as a single lumped-product, but to multiple buyers

who exhaust the lot.

Another significant difference between the two auction environments is that the online

environment does not benefit from the skills and experience of human auctioneers. The online

auctions are propelled by static rules that govern the constitution and submission of valid bids,

while traditional auctions benefit from the experience of the auctioneer who pits bidders against

each other by skillfully assessing their utility and pacing the auction bidding accordingly. Our

research is aimed at marrying the best of both worlds. Our goal is to replicate the human expert

who can run a single auction expertly, and substitute it with a computationally intensive real-

time decision making tool that could, armed with the prediction information from this paper,

potentially support the simultaneous conduction of hundreds of online auctions in a more

efficient manner.

Note that a recurring theme in this study is the use of the information available on hand to

the online auctioneer. Thus, critical to our work is the open auction format in which value signals

are iteratively broadcasted to the participating agents. If understood correctly, these signals can

explain the underlying bidder valuations or reserve prices, and can form a vital input in

enhancing a mechanism’s capability to equitably allocate resources. Carare (2003) demonstrates

the utility of working with micro-data, observable in the online auction environment, by deriving

marginal valuations of the bidders for CPU specific variables. The goal of Carare (2003) is to

recover distributions of valuations for a specific product, namely computer processors, by using

all the data from completed auctions from discriminatory (pay-your-bid) online auctions. Our

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work, in contrast, attempts at modeling bidding behavior for real-time predictive purposes, for a

broad spectrum of products sold through both Uniform and discriminatory multi-unit online

auctions by using transient within auction data during a given auction.3

Cramton (1998) identifies the following benefits of progressive open multi-unit auctions

over their sealed bid counterparts: (i) efficiency of the price discovery process; (ii) revenue

maximization; (iii) reduction of the winners curse; and (iv) privacy and implementation. On the

other hand, Engelbrecht-Wiggans and Kahn (1997) and Engelbrecht-Wiggans et al. (1999) show

that multi-unit auctions, especially those that use a Uniform pricing scheme, give bidders an

incentive to reduce their demand, resulting in inefficient allocations. Ausubel (1997) proposed an

ascending-bid auction for multiple units that ameliorates the demand reduction incentive in

multi-unit auction by progressively and iteratively increasing the “asking” price of the auction as

the auction progresses. However, Ausubel (1997) does not show how auctioneers should

determine the increments of the “ask” price. The price increment aspect has implications on

auction efficiency and revenue. We posit that accurate prediction of bidders’ willingness to pay

can form the basis of dynamically determining optimal asks.

The utility of valuation prediction has also been recognized in the Artificial Intelligence

field, where automated agents employ value discovery models as components of bidding agents.

Parkes and Ungar (2000) use the notion of myopic best-response bidding strategies among

agents to illustrate how proxy bidders that embrace this strategy can be shielded from

manipulation. In their paper, myopic best response is described as a bidding strategy where

bidders submit bids that maximize their utility, given the prevailing prices. An initial research

challenge in adopting the Parkes and Ungar (2000) approach is determining whether such

bidders exist in the online environment. This argument connotates that bidders are perhaps non- 3 Both are widely used in the B2C online market. Ubid.com and Samsclub.com are representative popular sites.

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homogenous in their bidding approaches in the online environment, and is supported by the work

of Bapna, Goes and Gupta (2001) and Bapna et al. (2004) who identify at least three or more

different bidding strategies adopted in Yankee auctions. For a real-time prediction and

calibration approach to be applicable it is first necessary to understand the space of bidding

strategies used by the bidders. We can then use the information available during the course of an

auction to accurately identify a given bidder’s strategy.

Another study (Plott and Salmon, 2004) use a surplus maximization strategy to describe

bidding behavior in simultaneous ascending auctions. Although the auction mechanism is

different from the one studied by Parkes and Ungar (2000), the notion of myopic best response is

used as a way of tying bidders’ iterative type revelation, to their willingness to pay.

Attempts to predict buyers’ willingness to pay are made as an effort to increase the

efficiency of resource allocation mechanism. Even for fairly well developed markets such as the

exchange markets for financial instruments, predictions of agents’ valuations has been attempted

through the establishment of market pre-opening games that solicit bidder demands without

actual commitments. Using the Paris Bourse as a test bed, Biais et al. (1999) examined the

accuracy of valuation information derived from pre-opening market trade games. Their study

shows that although the information derived from such games is noisy in the early stages of the

game, there is some convergence to true market values as the market opening time approaches.

The approach of the price formation study by Biais et al. (1999) depicts environmental similarity

to our approach of predicting bidder willingness to pay in open ascending price auctions. The

initial phase of such auctions is equivalent to the pre-opening game at the Paris Bourse, and the

later stages of the auction can expect to witness more concerted and accurate revelations.

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Consequently, we expect our prediction results to improve as the auction progresses, a result we

demonstrate.

Other studies have approached value prediction as a learning activity, where the predictor

seeks to know the actual bidder valuation that is masked behind observed bids. Economic game

theory literature provides two dominantly used models of agent learning: the fictitious play and

reinforced learning model. Dekel, Fudenberg, and Levin (2001) provide insights of these two

learning models in the context of playing Bayesian games. They derive conditions necessary for

Nash equilibrium play in repeated games. A limitation of their study is the restrictive

assumptions necessary to justify the concept of Nash equilibrium. Additionally, the study

underscores the complexity of the problem with growing number of bidders and bidder

strategies. The study takes the context of a repeated game. Although an iterative auction provides

multiple opportunities for bidders to revise their bids, each bid revision occurs in a different

context from the previous one. The prices, the bidders, and essentially the auction environment

are different.

Another related paper is Bapna et al. (2002) which focuses on the auctioneer’s decisions.

The problem addressed in that paper is to compute a fixed optimal bid increment for an auction

to maximize the expected revenue given the tail distribution of highest N+1 bidders, where N is

number of units on sale. In contrast, in the current paper, we estimate the distribution of bidders’

willingness to pay (WTP) by individually estimating every bidder’s maximum WTP. In this

paper we develop a valuation prediction model, based on the behavior of an expected surplus

maximizing bidder. The technique developed in this paper can be used to estimate the highest

N+1 bidders’ WTP. These can be seen as a proxy for valuations, and can be used to maximize

the seller’s revenue using the model described in Bapna et al. (2002). Therefore this paper can

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be viewed as being complementary to Bapna et al. (2002). However, the applications of the

WTP prediction model developed in this paper are broad and not limited to the computation of

optimal bid increments. As we touch upon in Section 6, the auctioneer could use the prediction

model to develop dynamic bid increments suggested in Bapna et al. (2002), to infer a lower

bound on the final auction price during the early stages of an auction, and also to establish

dynamic buy-it now prices.

In the next section we describe multi-unit progressive online auctions and introduce the

reader to our empirical dataset.

3. Progressive Online Multi-unit Auctions

Our research deals with two popular online auction mechanisms in the wider B2C category of

auctions. These mechanisms offer consumers multiple units of the same item. Bidders compete

for the items, with each bidder submitting a bid indicating the quantity they desire and the per

unit price they are willing to pay. These auctions are conducted in an open format and bidders

can see the bids of competing bidders. Bidder participation in these auctions increases over time.

Bidders can join the auction at anytime during the auction duration, which could be several days

in length. Thus, although the auctions share some similarities to the traditional ascending

auctions, bidders are not captives of the auction process, as is the case in traditional ascending

auctions.

The auctioneer spells out auction rules that govern the bidding activity. The main sets of

rules guide the constitution and submission of bids are as follows:

The minimum required bid: All bidders are expected to submit bids that are at least as

high as the minimum required bid. This rule is important as long as the units supplied are fewer

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than demand. When demand exceeds the lot size, subsequent bidding is guided only by the bid

increment. The bid increment is the minimum increment by which a bidder must exceed the

minimum-winning bid in order to win an item in the auction. If a bidder exists that is willing to

bid at the new level, the minimum-winning bid is displaced from the winning list and replaced

by the new bidder. The new bid is placed in the winners’ list based on pre-specified ordering

rules. Bidders are not bound to bid strictly following the bid increment, and as noted by Easley

and Tenorio (1999), jump bidding is often observed in Yankee auctions. The jump-bidding

phenomenon is also witnessed in the Uniform price auctions.

The auction sites give the auction closing time. Some auctions sites extend the auction

duration if bidding activity is observed in the last few minutes of the auction. Samsclub.com

auctions refer to this design as Popcorn auctions.

Another common feature of online auctions is the suggested retail price, or a buy–it-now

price. With the suggested price, the auctioneer gives bidders an indicative price at which they

can acquire the same product. The buy-it-now price has a similar effect, but also affords the

bidders the chance of buying the product at the suggested price instead of participating in the

bidding process. Essentially, these variables cap the performance of the auction to some

suggested values, as rational bidders will not bid beyond the suggested retail price. When a buy-

it-now price exists, rational bidders will seize the opportunity to buy at this price once it becomes

eminent that the bidding will exceed the buy-it-now price. A summary analysis of final auction

prices relative to the suggested retail prices indicates that in no case, in our extensive dataset of

787 Uniform price online auctions, did the final bid exceed 90% of the suggested retail price.

Additionally, this holds regardless of the intensity of bidding as given by the ratio of bidders to

auction lot size. We can surmise from these results that the bidders’ willingness to pay is capped

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by the suggested retail prices. In the next sub-section we provide further details of our extensive

dataset.

3.1 Online Auction Data Collection

Our analysis uses data from two multi-unit online auctions.4 We deployed automatic

auction-tracking agents to observe and collect data on entire auction proceedings. One auction

site uses a Uniform price auction mechanism, while the other uses a Yankee auction mechanism.

Our automated agent was able to collect bidding data from 787 Uniform price auctions and 205

Yankee auctions, for a total of 78,014 bids or bid revisions. The auction-tracking agent was

programmed to visit the identified online auction’s web pages in intervals of 5-15 minutes, take

snapshots of the auction, and record the bidding history of the auction site. The auction-tracking

agent then compared the newly downloaded auction history with previously recorded history. If

differences in the history files were observed, the new activity of the auction was added to the

history file. With this technique, we were able to maintain a complete history of the auction,

noting each of the bids submitted and revisions made by each bidder in the auctions we tracked.

Appendix 2 provides a list of auction variables that our tracking agent collected data on.

After completing the data collection exercise, we investigated the data for completeness.

This required streaming the bidder arrival process through an auction program that replicated the

online auction, and making sure that the auctions concluded with the same winners as in the

actual auctions. Some of the auctions had significant chunks of missing data. In such cases, we

opted to drop them from our data set.

4 Samsclub.com and UBid.com

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3.2 Description of the data

Data on the Uniform Price auctions represent a total of 90 different products, ranging

from electronics, to apparel and sporting goods. The Yankee auctions data set contains products

that can mainly be classified as electronics and computing goods. Tables 1 provides summary

statistics about the data. The average number of bidders per auction in the Uniform price

auctions is 36, while the Yankee auctions on average attracted 47 bidders per auction. These

bidders compete for 13 items on average in both auction types. The average number of bids

submitted per auction is higher than the average number of bidders. This indicates that some

bidders submit several bids as the auction progresses.

Statistic

(number of bidders) Uniform Price Auction

Yankee Auction

Mean 36.16 47.04 Standard Error 2.26 4.35 Median 32 23 Mode 37 9 Standard Deviation 24.41 50.33 Range 102 226 Minimum 4 3 Maximum 106 229 Sum 4195 6304 Count 116 134

Table 1: Summary Statistics for the Number of Bidders per Auction

On average, bidders submit 1.3 (0.01) and 1.83 (0.05) bids for the Uniform and Yankee

auctions respectively. The range of number of bid revisions is 9 and 38 for the Uniform and

Yankee auctions respectively.

3.3 Bidding Strategy Classification

In a recent study, Bapna et al. (2004), used online auction data from 1999 and 2000, to

find a stable taxonomy of bidder behavior containing five types of bidding strategies pursued in

Yankee auctions. Bidders pursue different bidding strategies that, in aggregate, realize different

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winning likelihoods and consumer surplus. It is likely that similar categories exist in Uniform

price multi-unit auctions. However, no research has explicitly analyzed this. Our data set reveals

multiple bidding strategies. From the perspective of WTP prediction, we have to determine

whether the signals that bidders are providing through their bids conform to any surplus

maximizing strategic behavior, and whether this behavior is myopic. The latter would be evident

if there were bid revisions made by the bidders, implying that at the time of bid placement the

bidders’ were relying on information available to them on hand. Recall, that MBR bidders’

conform to an equilibrium strategy with the assumption that the bidders’ view the current round

as the last round of the auction and take prices as given (Parkes 2001). Bapna et al. (2004)

suggest that while the participatory and opportunistic strategies conform to this behavior, the

evaluatory strategy does not. Our task is to classify bids as they arrive into MBR and non-MBR

categories. Table 2 relates the MBR and Non-MBR strategies to the Yankee auction bidder

classification of Bapna et al. (2004).

Bidder Type Characteristics Bidder Class Early & Middle Evaluators Early one time high bidders;

clear idea on their willingness to pay; bids higher than minimum required

Non-MBR

Participators, Opportunists, Sip

and Dippers

Makes low initial bid, progressively monitor auction and make revisions

MBR

Table 2: Categories of Bidder Strategies

Given that our data consists of both Uniform (with un-established bidding strategies) and

Yankee auctions we conducted an initial empirical exploration of the bidding patterns,

summarized in Table 3.

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Uniform Price Auctions Yankee Auctions Bids greater than the minimum required bid

73 % 67 %

Bids smaller than the maximum current winning bid

84 % 88 %

Table 3: Summary of Bidding Patterns

Observe that two thirds of bids submitted are greater than the auctioneers’ minimum

required bid. This bidding pattern is observed in both the Yankee and Uniform price auctions.

Also, at any given time in the auction, the submitted bids are typically less than the current

highest bid recorded at that time. In the Uniform price auction, only in 16% of the bidding

instances incoming bids are greater than all current winning bids. Therefore, the majority of

bidders submit bids that lie between the minimum required bid and the maximum bid that has

already been submitted.

It is also worth questioning whether the classification of bidding strategies derived in

Yankee auctions (Bapna et al., 2004) holds in a different sample of Yankee auctions or in

Uniform price auctions. The latter issue is addressed empirically by Bapna, Chang, Goes and

Gupta (2005). They find a remarkable consistency in the mix of bidding classes across the two

auction types. The former issue has been addressed in Slavova (2005), who was able to replicate

our original classifications in an entirely different sample of Yankee auctions. This shows that

the classification in Bapna et al. (2004) is robust.

Let the current winning bids in a multi-unit ascending price be denoted by Nxxx ,...,, 21 ,

ordered by magnitude. From the bidding patterns presented in Table 3 above, we can deduce a

simple basic bidder strategy classification rule.

Definition 1-Basic MBR Classification Rule: Classify a bid that is lower than the largest winning

bid NX to be MBR, while a bid that exceeds the largest winning bid as non-MBR.

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Using this classification method, 16% of the bids in our Uniform auctions data set and

12% of the bids in our Yankee auctions data set would be classified as non-MBR bidders while

all the other bids are classified as MBR.

It is also conceivable that certain refinements to this rule, for example relaxing this rule

as the auction progresses or determining a threshold that is slightly different than the current

maximum winning bid would intuitively improve the strategy classification.

Let µ and σ be the mean and standard deviation of winning bids respectively. Suppose

b is next bid that is submitted into the auction. We computeσµ−

=bz , and use this value to

identify the bidder’s strategy. Intuitively, a bid that is significantly higher than current winning

bids, relative to the variance of the winning bids, does not conform to a MBR strategy. Note that

the distribution of the bids expected in this auction is truncated on the left end, because bids

submitted must exceed the minimum-winning bid. This calls for a truncation of the distribution

of winning bids. Since the new bid b will be greater than 1x , we shift the total probability mass

to values higher than 1x . Figure 2 below shows the belief revision process. Thus, rather than

using z as our strategy classification value, we compute a valueµ

σ−

=1

' x

zz .

Definition 2 -Normalized MBR Classification Rule: Classify a bidder with 'z greater than a (to

be determined) cutoff as not conforming to the MBR strategy. All other bidders are MBR.

The question remains, what level of cutoff to use to designate a higher bid as non-MBR.

We address this empirically, using a randomly sampled 20% of the dataset in the form of a

training dataset. Our strategy is to develop a classifier based on 20% and then judge its

performance against validation set. By definition, non-MBR bids are bids that are not

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subsequently revised, and all other bids are categorized as MBR. Against this test set we try out a

range of cutoff values for 'z and choose the value that gives us a pareto-optimal classification.

Figure 2: Truncated Distribution of Winning Bids

We choose a pareto-optimal cutoff level because in both the Yankee and Uniform price

auctions, there is an inverse relationship between the critical values of 'z that result in high

prediction accuracy among the bidders using the MBR and those that do not use the strategy.

Testing different cut–off points yields pareto-optimal 'z values of 0.9 and 0.5 for the Uniform

and Yankee auctions respectively. The value of 0.9 give a prediction accuracy of 46 % in the

Uniform price auctions, while the critical value of 0.5 yields 60% accuracy in the Yankee

auctions. The difference in the critical values between the two auction mechanisms underscores

the different incentives that each mechanism provides to bidders and the resultant influence on

the bidding strategies. Because the Uniform price auction provides incentives for bidders to

reveal their true WTP, the optimal critical value used to discriminate between MBR and non-

MBR bidders is higher in the Uniform price auction than in Yankee auctions, where bidders pay

a price equal to their bid.

By applying these critical values to the remaining 80% of our data, we realized accuracy

levels of 62% and 47% in predicting MBRS bidders in Uniform and Yankee auctions

respectively. With the ability to identify bidders who use an MBR strategy, we next seek to infer

the willingness to pay of for both types of bidders.

X1 µ X1 µ

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4. Prediction Model - Myopic Best Response Strategy

For non-MBR bidders, since by definition these bidders do not revise their bids, we

assume that the actual bids equal their WTP. The remainder of this section develops a prediction

model for MBR bidders WTP.

We begin with the model for the myopic best response bidding strategy. This strategy can

be interpreted as a surplus-maximizing bid calculated by a bidder in a given round of the

auctions assuming that all his competitors bids remain unchanged from the previous round. As

new arrivals come in, and bidders get displaced from the winning list, the myopic assumption

allows for belief revision by the same bidder, to account for the additional information that is

available. This results in a revision of the bidder’s willingness to pay each time a bidder revises

her bid.

Consider an auction for N units of an item. Let the current winning bids be denoted

by Nxxx ,...,, 21 , ordered by magnitude and within magnitude, by time of submission. Bidders

submitting these bids are assumed to have private willingness to pay (WTP) values equal to

NWWW ,...,, 21 respectively. These WTP values are bounded below by the bids already submitted,

that is NN xWxWxW ≥≥≥ ,...,, 2211 . When a new bid z is received, it must be greater than 1x ,

which is displaced from the winning list. We assume that the new bid z was determined to

myopically optimize the expected gain that the bidder will derive from the auction. The MBR

strategy forms the basis of our WTP prediction approach. Much like the myopic best-response

bidding strategy of Parkes and Ungar (2000), our strategy maximizes a bidder’s expected

surplus, given the already submitted bids and a belief on the actual WTP values of bidders who

submitted the earlier bids. The belief regarding other bidders’ WTP values is a probability

distribution with support in the range of the lowest winning bid and an upper bound, which can

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be set to a publicly known price for the item being auctioned, such as the suggested retail price

for the auction. Recall, that our empirical analysis of the suggested retail price (Section 3)

indicated that it was indeed an effective cap on the support of the distribution. The myopic

approach allows for belief revision as the auction progresses. That is, bidders who resubmit bids

revise their initial beliefs about others’ WTP values. Implicit in the above discussion is the

condition, that a significant percentage of bidders do indeed use this strategy.

Let the bidder who submitted the new bid z have a WTP value denoted byW . Suppose

that the new bid z is greater than k of the current winning bids. Therefore, the new sequence of

winning bids is Nkk xxzxxx ,...,,,,....,, 132 + . For the new bidder to win, given this state of the

auction, at least one of the k bidders whose bids are smaller than z must have a WTP value that

is less than z , assuming no new bidders join the auction. This search is conducted across known

potential bidders, those that have revealed some preference. As the approach allows for bid

revisions, the information signals of the new arrivals are, by design, captured ex post.

Let the new bidder’s belief about the WTP values of any of the current winners, be an

independent random variable with a density function if , and a distribution function iF , with

support in the range [ ]mx ,1 , where 1x is the smallest winning bid, and m is an indicative fixed

price for the item. The indicative fixed price could be assessed using price comparison agents

that are available on the Internet. Also, a number of online auction sites provide indicative retail

prices for items being auctioned5. Figure 3 below illustrates a generic belief function for a

specific bidder’s WTP value, conditional on the submitted bid.

5 Ebay has what is called a ‘buy-it-now’ price, and Ubid suggest a ‘maximum bid price.’

21

Figure 3: Belief Function of Bidders’ Willingness to Pay

The probability that a current winner’s WTP is greater than z is given as;

[ ] ( )zFdWWfzWP i

m

z iiii −=−=> ∫ 1)(1 (1)

Assuming that the WTP value for the bidders are independently distributed, the

probability that at least one of the currently winning bidders has a WTP value that is less than z

is,

( )( )∏=

−−k

ii zF

1

11 (2)

Observe that this is an asymmetric model in the support of the distribution of the individual

bidders, hence the use of the ∏ notation and the i subscript to the WTP distribution. Let the

price paid by the new bidder equal to dP and uP in Yankee and Uniform price auction. It is

obvious that dP = z , as each bidder pays a price equal to their bids in Yankee auction. On the

other hand x1 < uP ≤ z, and will be a function of the willingness to pay of the k bidders who are

outbid by the new bid and by the value of the new bid z itself. Thus uP ≡ ( )zWWW k ,,...,, 21ϕ .

xi z m Wi

fi(Wi)

22

If the auction uses a discriminatory pricing scheme, where bidders pay a price equal to

their bids, the new bidder will enjoy an expected gain equal to:

( ) ( )( )⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= ∏

=

k

ii zFzWGE

1

11*)( (3)

And the expected gain in a Uniform price auction will be given by:

( ) ( ) ( )( )⎟⎟⎠

⎞⎜⎜⎝

⎛−−−= ∏

=

k

iiu zFPWGE

1

11* (4)

We assume that the observed bid z optimizes the expected gain expressions given above

at equations 3 or 4, depending on the pricing scheme. Therefore, the observed bid should satisfy

the first and second order conditions for a maximum expected gain. Equations (5) and (6) show

the first order conditions for maximum expected gain under a Yankee and Uniform pricing

scheme respectively.

( )( ) ( )( ) ( ) ( ) ( )( ){ }

01111 11

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−=

∂∂ ∑ ∏∏

= −==

k

j

k

jiij

k

ii zFzfzWzF

zGE (5)

( )( ) ( )( ) ( ) ( ) ( )( ){ }

01111 11

' =⎟⎟⎠

⎞⎜⎜⎝

⎛−−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−−=

∂∂ ∑ ∏∏

= −==

k

j

k

jiiju

k

iiu zFzfPWzFP

zGE (6)

Where ( )z

zWWWz

PP ku

u ∂∂

=∂∂

=,,...,, 21' ϕ

After observing the bid z, and assuming that it was determined by the bidder to maximize

his expected gain, we can make inferences about the corresponding WTP value of the new

bidder. By solving equations 5 and 6 for W, we get the predicted WTP value of the bidder under

the respective pricing scheme. The expressions for WTP value prediction are given in equations

7 and 8.

23

( )( )

( ) ( )( ){ }

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

∑ ∏

∏

= −=

=

k

j

k

jiij

k

ii

Yankee

zFzf

zFzW

1 1

1^

1

11 (7)

( )( )

( ) ( )( ){ }

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+=

∑ ∏

∏

= −=

=

k

j

k

jiij

k

iiu

uUniform

zFzf

zFPPW

1 1

1

'

^

1

11 (8)

In the next subsection we present the prediction model assuming a Triangular distribution

of bidders’ WTP values. This distribution closely resembles the belief function of Figure 1, and

is analytically tractable.

4. 1 Distribution Specific WTP Prediction Model

For expositional clarity, we first present our prediction model for an auction with N=2

items on sale. Subsequently, we generalize the model for any N. Let the current winning bids be

1x and 2x with 21 xx ≤ and their corresponding WTP values of 1W and 2W respectively. Recall,

that the actual distribution of the bidder’s WTP is bounded above by m , a known fixed price for

the item being auctioned. Bidders’ WTP values are assumed to be independent and following a

triangular distribution, with support in the range ],[ 1 mx . It follows with certainty that the actual

WTP is greater or equal to the bid submitted. It is also realistic to expect that the chance of the

actual WTP value being greater than any point between the distribution support range, decreases

as the point of reference increases.

For the 2 unit case, Appendix 1 exhaustively enumerates the feasible auction outcomes as

a consequence of the third bid submission, as well as the likelihood of each outcome. Consider

the case, where 12 xxz >> . Under this scenario Appendix 1 (under the case 2xz > ) lists the six

possible price outcome, along with their respective likelihood, derived from the Triangular

24

distribution. Aggregating the feasible outcomes and their likelihood, the new bidder can expect

to pay a price equal to 2x with probability ( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−

− 21

221

xmxm or pay ( )

22 zx + with

probability ( ) ( )( ) ( )( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−−−

21

22

442

xmxmzmxm . The expected gain for the bidder would thus be as shown in equation

9.

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )( )⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−−−

⎟⎠⎞

⎜⎝⎛ +

−+⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−−= 21

22

4422

21

22

2 21

xmxmzmxmzxW

xmxmxWGE (9)

A bid that maximizes the expected surplus should satisfy the expression given in equation

10.

( )( ) ( ) ( )( ) ( )( )

( ) ( )( ) ( )( ) 04

221

21

22

32

21

22

442 =

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

−⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

−−−−=

∂∂

xmxmzmzx

Wxmxm

zmxmzGE (10)

Assuming that the received bid z maximizes the computed surplus, we solve for the

inferred WTP value W, and present the predicted WTP expression in equation 11:

( ) ( )( )( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛

−−−−

++= 3

442

2

^

421

zmzmxmzxW (11)

In the same spirit, the expression for expected surplus and predicted WTP values are

presented in Table 4 for all possible outcomes of a 2-unit auction.

25

Pricing Scheme

Bid Range

Expected Gain Predicted WTP Values

21 xzx ≤<

(i) ( ) ( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−−− 2

1

2

1xmzmzW ( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

+−zm

xmmz

213

21

Uniform Pricing 2xz >

(ii) ( ) ( )

( )( ) ( ) ( )

( ) ( )( ) ⎟⎟⎠⎞

⎜⎜⎝

⎛

−−−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−+⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−

−− 21

22

4422

21

22

2 21

xmxmzmxmzxW

xmxmxW

( ) ( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−−−

++ 3

442

2 421

zmzmxmzx

21 xzx ≤<

(iii) ( ) ( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−−− 2

1

2

1xmzmzW ( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

+−zm

xmmz

213

21

Yankee Pricing 2xz >

(iv) ( ) ( )

( ) ( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−−

−−− 2

12

2

4

1xmxm

zmzW ( ) ( ) ( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

−−−−−

+ 3

421

22

4 zmzmxmxmz

Table 4: Prediction of WTP Values for a 2 Unit Auction (Triangular Distribution)

Next we generalize the valuation prediction for model for any N.

4.2 Generalized WTP Prediction Model

Given a lot size of N items and an ordered list of currently winning bids, Nxxx ,...,, 21 , a

new bidder will displace bid 1x . The probability of the new bid z winning when it is greater

than k of the N currently winning bids is equal to the probability that at least one of the k bidders

has a WTP value that is less than z . Note that the results provided above for the two-item

auctions are valid for the generalized prediction model with a lot size when k =1 or 2. Table 5

contains the summarized results for the expected surplus and bidder’s WTP value prediction,

assuming that bidders’ actual WTP follow independent triangular distributions, with support in

[ ]mx ,1 .

26

Pricing Scheme

Expected Gain Predicted WTP Value

Uniform Pricing

( ) ( )

( ) ⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−−−

∏=

2

1

2

)( 1k

ii

k

N

xm

zmxW ( ) ( )

( )⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−−

−−

++∏= 1

221

21

2

2 k

k

ii

zm

xm

kzmxz

Yankee Pricing

( ) ( )

( ) ⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

−

−−−

∏=

k

ii

k

xm

zmzW

1

1 ( ) ( )

( )⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−

−−−+ −

=∏

11

k

k

i

ki

zmk

zmxmz

Table 5: Generalized Prediction of Bidder’s WTP (Triangular Distribution)

As N increases, it becomes intractable to exhaustively enumerate the state space, as we

did for the 2 unit case in Appendix 1. Hence the inferred WTP prediction in Table 5, represent a

first order approximation6. For the Uniform price auction, the price that a bidder pays is set at the

average price between the displaced and the incoming bid, while for Yankee auctions, the price

paid by bidders is equal to the bid itself.

In the next section we empirically test our classification and prediction model.

5. Empirical Validation of Prediction Model

Using data collected from the online auctions as described in section 3.1, we validate the

accuracy of the predicted willingness to pay. We created a program that can replicate and

manage the stream of the bid arrival process as it occurred in the actual auction. As each bid was

recorded, we used the WTP prediction models given in Tables 4 and 5 to predict the bidders’

willingness to pay. We compared the predicted WTP to the actual WTP as given by the final bid

of a specific bidder. We use the bidders’ final bids as proxies for their actual WTP. Note, that

the final bid is a realistic estimate of the actual WTP for the losers in both Uniform and

discriminatory multi-unit auctions, and a conservative estimate of the winners of such auctions. 6 To estimate the closing price of an Uniform auction, we approximate the Nth order statistic X(N) of the joint distribution of valuations by taking the expected value between x1 and z, the new bid.

27

5.1 Accuracy of the WTP Prediction Model

Tables 6a and 6b shows the absolute mean percentage difference7 between WTP as predicted by

our model and the actual WTP, as conservatively estimated by bidders’ final bids. The results

show the accuracy levels when the WTP model for MBR bidders is applied to all bidders

(without any classification), and subsequently when the two strategy classification models, the

Basic and the Normalized, are used.

The results show that with no classification and considering all bidders (winners and

losers) our predictions’ mean absolute error is 20.43% and 9.9% in Uniform price and Yankee

auctions respectively. As we incorporate the two classification schemes in increasing order of

sophistication, we observe that the prediction error reduces for both Uniform and Yankee

auctions, with the Normalized classification rule yielding little over 8 and 6% error for Uniform

and Yankee auctions respectively. This validates the utility of being able to classify bidders into

MBR and non-MBR categories, a point we further explore later in this section.

Table 6a of the Uniform auction indicates that the exclusion of winners reduces the

prediction error to 11.2%, however the same reduction is not evident in the case of Yankee

auctions (Table 6b).

Losers

Final bid as a fraction of Auction Price Classification Strategy

All bidders

All Losers > =.5

None 20.43% (40.69%)

11.20% (33.65%)

3.63% (14.83%)

Basic 15.41% (37.62%)

8.04% (31.52%)

2.36% (14.18%)

Average (Std. Dev.)

Normalized 8.48% (21.25%)

6.34% (22.65%)

2.45% (21.03%)

Table 6a: Predictions Accuracy of Bidders’ WTP – Uniform Price Auctions

28

Losers

Final bid as a fraction of Auction Price Classification Strategy

All bidders

All Losers > =.5

None 11.96% (21.29%)

13.43% (24.03%)

10.74% (12.81%)

Basic 11.41% (21.08%)

12.80% (23.83%)

10.31% (12.82%)

Average (Std. Dev.)

Normalized 7.79% (19.23%)

7.89% (21.53%)

7.19% (17.89%)

Table 6b: Predictions Accuracy of Bidders’ WTP – Yankee Auctions

We also sought to isolate and report the performance of our model on the set of bidders

whose final bid was at least 50% of the final auction price. The motivation to isolate these

bidders comes from the observation that there are many bidders who participate in the initial part

of the auction when the prices are very small and then drop out of the auction. Making

predictions for these types of bidders results in an over prediction that may not be representative

of the overall picture. When we isolated these early drop-outs we realized 3.63% prediction error

in Uniform price auctions and 10.74% for the Yankee auctions.

In order to get further insights into the workings of our prediction model we classify our

prediction mean absolute error according to the auction duration and compare against the random

model.8 The results are shown in Figures 4a, 4b, and 4c. In both Yankee and Uniform cases the

random model has prediction errors that are significantly higher than our proposed model in

every time interval. Additionally, the significance of the difference between the model and

random predictions of WTP increases over time (Figure 4c). This trend in noted for both Yankee

and Uniform Price auctions.

8 The random model was suggested by an anonymous referee.

29

( a ) C o m p a r i s o n o f A c c u r a c y T r e n d s B e t w e e n P r o p o s e d M o d e l a n d R a n d o m P r e d i c t i o n s o f W T P ( U n i f o r m A u c t i o n )

0 %

2 0 %

4 0 %

6 0 %

8 0 %

1 0 0 %

1 2 0 %

1 4 0 %

0.05

0.10.1

50.2

0.25

0.30.3

50.4

0.45

0.50.5

50.6

0.65

0.70.7

50.8

0.85

0.90.9

5 1

T i m e ( N o r m a l i z e d )

Perc

ent m

ean

abso

lute

err

or

P r e d ic t e dR a n d o m

( b ) C o m p a r i s o n o f A c c u r a c y T r e n d s B e t w e e n P r o p o s e d M o d e l a n d R a n d o m P r e d i c t i o n s o f W T P ( Y a n k e e A u c t i o n s )

0 %

1 0 0 %

2 0 0 %

3 0 0 %

4 0 0 %

5 0 0 %

6 0 0 %

7 0 0 %

8 0 0 %

0.05

0.10.1

50.2

0 .25

0.30.3

50.4

0 .45

0 .50.5

50 .6

0.65

0.70.7

50 .8

0 .85

0.90.9

5 1

T i m e ( N o r m a l i z e d )

Perc

ent m

ean

abso

lute

err

or

P r e d i c t e dR a n d o m

Figure 4: Comparison of Prediction Accuracy between the Proposed WTP Prediction Model and a Random Based Model

(c) Trends in the Significance of Difference of Prediction Errors Between Proposed Model and Random Predictions of WTP

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Time

P-va

lue

Uniform Auction

Yankee Auction

30

In summary, our analysis indicates that even with relatively low classification accuracy

(62% and 47% for Uniform and Yankee respectively) we get a fairly accurate estimate of

bidders’ WTP, often early in the auction. In the next sub-section we consider the case of an

auctioneer who has the added advantage of possessing historical data. For such cases, we ask

whether a multi-attribute classification scheme can improve the overall classification and

prediction performance.

5.2 Using Historical Data to Improve WTP Prediction Accuracy

In section 3.2, we provided two methods for classifying bidder strategies; a basic

approach and a normalized approach. Both these classification schemes are uni-dimensional, in

that they rely only on the bid information to classify bidders as MBR or non-MBR. Encouraged

by our empirical results from the previous section, we assessed whether adding further auction

attributes, in the form of independent variables, would enrich the classification scheme. In the

same token we also wanted to investigate whether the auctioneer could take advantage of

possessing historical auction progression data towards developing a multi-attribute classification

scheme.

We split our data set into two: the initial 20% of the auctions in our data set were used to

train a model that classifies bidders’ bidding strategies based on the outcome of our prediction.

In this historical data set, a bidder for whom prediction was higher than the final bid was tagged

as non-MBR, otherwise the bid was tagged as MBR. Then, using general auction variables, we

developed a model that was capable of predicting this classification. For the model to be viable

for prediction, it should rely on information that is available to the auctioneer, in real time, when

a bidding instance occurs. We use the estimated parameters from 20% of the data to classify the

31

bidders’ strategies in the subsequent 80% auctions, and measure the accuracy of our WTP

predictions.

5.2.1 Dependent Variable - At each bidding instance our prediction model estimates the

consumer’s willingness to pay for a product. By definition, the final bid made by MBR bidder

represents a conservative estimate of their actual willingness to pay. So if our predicted WTP at a

particular bidding instance is no more than the final bid of a specific bidder, we assume that the

bid that led to the prediction was constituted using a MBR strategy. Otherwise we consider the

bidder to be non-MBR.

5.2.2 Explanatory Variables - We consider variables for which an auctioneer can acquire data

at the time of making a prediction on the bidders’ WTP. The explanatory variables used in the

logit model were based on the analytical model, as well as some observable strategic behavior or

aggressiveness measures. From the analytical model we hypothized that the upper bound on the

expected bid price for an item m and the lot size of the auctions N are likely to influence the

bidding strategy adopted. These are captured in X1i and X3i respectively in equation 12, the best

fit logit regression model:

iiiiiii

i XXXXXP

PLog εβββββα ++++++=⎟⎟⎠

⎞⎜⎜⎝

⎛− 55443322111

(12)

Where:

iP - Probability that the bidder is an MBR

NKX i =1 - Ratio of number of current winning bids that are smaller than the current bid, to

lotsize

iX 2 - Number of times bidder has revised his bid

32

iX 3 - Standard deviation of current winning bids

iX 4 - Average of current winning bids

iX 5 - Normalize elapsed auction time.

X1i and X2i capture the strategic behavior of the bidders. High values of X2i would indicate that

the bidder was a participator. A ratio close to 1 for X1i would suggest an evaluatory, non MBR,

type of bidder.

The estimates of the model coefficients and their statistical significance are shown in Table 7a

and 7b for the Uniform and Yankee auction respectively. All the model variables, except X2i in

the Uniform case, are significant in explaining the predicted classification. Appendix 3 presents

correlation matrices for the variables used in the model. The values support independence among

the predictor variables.

Coefficient Estimate S.E. Wald Df Sig. 1β -7.487 0.737 103.119 1 0.000

2β -0.436 0.402 1.177 1 0.278

3β -0.057 0.026 5.034 1 0.025

4β -0.017 0.005 9.312 1 0.002

5β 2.421 0.557 18.879 1 0.000 Constant 3.767 0.597 39.867 1 0.000

Table 7a: Strategy Classification Model’s Coefficient estimates – Uniform Auctions

Coefficient Estimate S.E. Wald Df Sig. 1β -9.981 0.500 398.303 1 0.000

2β 1.883 0.368 26.158 1 0.000

3β -0.004 0.001 9.414 1 0.002

4β -0.003 0 46.283 1 0.000

5β 1.340 0.449 8.897 1 0.003 Constant 1.654 0.402 16.893 1 0.000

Table 7b: Strategy Classification Model’s Coefficient estimates – Yankee Auctions

33

The accuracy of the Logit classification model is higher than the Normalized

classification rule. With the sample data set (20 percent of the first auctions in our dataset) that

was used for the parameter estimation, the fitted model yielded a 81.82% and 92.7% prediction

accuracy on the Uniform price and the Yankee respectively. For the test on 80% of the data, a

bid that yields a probability Pi > 0.5 on the model given by equation 12 was labeled as a MBR.

Note that all the variables are available to the auctioneer at the time a bid is submitted. In

subsequent auctions (80% of the auctions that followed the sample data) the strategy prediction

model gave prediction accuracies of 81% and 90 % and for the Uniform price and the Yankee

auctions respectively.

Using the Logit model to classify bidders’ strategies, we repeated the prediction of

bidders’ WTP and reported the prediction accuracy. Table 8 indicates that this approach

significantly lowers the variance between predicted WTP and actual WTP. The average percent

deviation of the predicted WTP from the actual WTP is only 2.63 % and 1.89% for the Uniform

and Yankee auctions respectively.

Losers Final bid as a fraction of Auction Price Deviation of Predicted

WTP from Actual WTP All bidders

All Losers > =.5

Average 2.63% 2.31% 1.89% Uniform Standard

Deviation 16.59% 17.90% 15.21% Average 1.89% 2.03% 2.16%

Yankee Standard Deviation 6.97% 8.34% 7.27%

Table 8: WTP Prediction Improves Using the Logit Model for Strategy Classification

5.3 Performance of the Prediction Model

To show the performance of our WTP prediction model, we compared the results of our

predictions with two other naïve prediction models that are similar level in their information

34

requirements. In other words, we would like to compare the performance of our model with other

prediction methods that an auctioneer can use with every bid received.

One potential method is to use the bids themselves as proxies for the WTP. This method

requires that we assign a predicted WTP that is equal to the bid submitted. Using this method the

auctioneer errs only by under predicting the bidders’ actual WTP. This method does not provide

the auctioneer with any additional information that can be used calibrate the auction process. It

provides an absolute lower bound on the WTP. The second method that we consider assumes

that bidders’ WTP will be a random variable, between the bid submitted and a known fixed price

for the product being auctioned. For consistency, we assume the random values from a

Triangular distribution. This method has two-sided risk of both over and under predicting the

actual WTP. It also lacks rationality, but certainly pulls a value from a feasible range.

Our hypothesis about the accuracy of the prediction model is that the predicted WTP is

equal to the actual WTP. Using the final bid that a bidder submits in each auction as a proxy for

his actual willingness to pay, we conduct a comparison of the predictions from these three

models with the proxy for actual wiliness to pay. This is an ex-post analysis, as the final bids are

only observed after the auctions close.

We find that both for Uniform and Yankee auctions, the first time a bidder bids, any

prediction (using our method as well as the other two) is not statistically equivalent to true WTP

as measured by the bidders' observed WTP. It should be noted, however, that due to extremely

large number of observations (4752 for Uniform price and 3872 for Yankee auctions) the power

of the test is extremely high and even minor deviations from the observed WTP results in

rejection of null hypothesis of equality.

35

Table 9: P-Values for the Variation Between Predictions and Actual WTP for Alternative Prediction Methods. Values less than 0.05 indicate significant differences between

predicted and actual WTP.

Our prediction is statistically as well as qualitatively closest to the observed WTP as

compared to other approaches even with the first bids. As we move to second and third bids

made by the bidders our prediction starts making accurate WTP estimates as indicated by the p-

values of paired T-tests in Table 9. However, the other two naive prediction methods including

the random prediction results in rejection of null hypothesis with high significance. This provides

strong support for our method in comparison to other ‘ad-hoc’ prediction approaches.

To further examine the efficacy our prediction model, we ask whether the actual

predictions made from the different approaches are indeed significantly different. This approach

offers a more direct comparison of the three approaches, as they are not measured against the ex-

post actual bid. Another way to phrase this is to question whether the bidders are indeed

strategizing? If they are, the model designed to pick up this phenomenon should be

distinguishable from other naïve approaches. Table 10 shows the results of paired means tests

that compare predictions of our model with the other two naïve approaches. The results provide

overwhelming support to reject a hypothesis that equates predictions of our model to predictions

from the naïve methods.

P-value: Hypothesis: Predicted WTP = Actual WTP

Prediction Method

Comparision bid

Uniform Price Auctions Yankee Auctions

1st Bid 3.33279E-26 2.0827E-1522nd Bid 0.001082154 0.078658Our

Prediction 3rd Bid 0.798344 0.145662241st Bid 4.99491E-20 1.00126E-292nd Bid 0.00090675 4.82E-18Current Bid

Proxy 3rd Bid 0.087145 7.8489E-081st Bid 6.2769E-168 1.74903E-872nd Bid 4.18539E-23 0.0007637Random

Prediction 3rd Bid 7.848E-05 2.69E-12

36

P-value: Hypothesis: Our Predicted WTP = Alternative WTP estimates

Alternative Prediction Methods Uniform Price

Auctions Yankee Auctions

Random Prediction 1.24E-179 1.28248E-65 Current Bid Proxy 2.93E-50 4.277E-244

Table 10: P-Values for the Comparison Between Predictions from our proposed model and Alternative Prediction Methods. Values less than 0.05 indicate significant differences

between our predicted and alternative predictions of WTP

6. Applications of the Prediction Model

6.1 Inference on Final Auction Revenue

It is logical to assume that if the predictions are accurate, an auctioneer should be able to

infer a lower bound on the final auction revenue from the predicted willingness to pay. Such

inference can be done prior to the close of the auctions, as soon as the number of bids exceeds

the lot-size offered. Conservatively, we can assume that the current estimates correspond to the

final willingness-to-pay. In other words we can relax the myopic assumption on the bidding

strategy and define, at any stage of the auction, the predicted revenue for the Uniform and the

discriminatory auctions, say Ru and RD respectively, as:

Ru = N *W(N) (13)

RD =∑=

N

iiW

1)( (14)

Where W(i) represents the ith highest estimated WTP, at that time. The earlier such an estimate

can be made, with accuracy, the more utility it has for the auctioneer to use it to dynamically

calibrate the mechanism.

Figure 5 displays a comparison of the progression of the actual auction revenue to the

predicted revenue (as per equations 13 and 14), by the proportion of bids received. Observe that

we are able to estimate, on average, the final auction revenue with just 85 percent of the bids that

37

were submitted in Uniform price auctions and with just 70 percent of the bids that were

submitted in the discriminatory Yankee auctions.

0%

20%

40%

60%

80%

100%

120%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Fraction of Bids

(a) Uniform Price Auctions

% o

f Fin

al R

even

ue

Actual RevenuePredicted Revenue

0%

20%

40%

60%

80%

100%

120%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Fraction of Bids

(b) Yankee Auctions

% o

f Fin

al R

even

ue

Actual RevenuePredicted Revenue

Figure 5: Predicted and Auction Revenue from the Final auction Revenue

38

When coupled with the enhanced computational capabilities inherent to Internet auctions,

it is interesting to consider mechanism design opportunities that rely on the WTP estimation.

Next, we give some practical applications of the capability to predict bidders’ WTP.

6.2 Dynamic Bid Increments

In one of the auctions in our data set, a Uniform price auction was conducted for a

DeWalt Cordless Trim Saw/Driver/Drill Combo. This auction used a fixed bid increment of one

dollar. Using the same stream of bids as were received in the actual auction, we conducted a

simulated auction with dynamic bid increments. The bid increments were determined as the

difference between the minimum winning bid value and the (Nth + 1) highest predicted WTP. In

this particular auction, there were six units of the product. Thus the bid was set as the difference

between the minimum winning bid value and the seventh largest predicted WTP. In Figure 5, we

show the variability in the minimum bid increment. We also compare the revenue formation in

the two auctions under the respective bid increment setting methods.

39

0

2

4

6

8

10

12

14

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Bidding Instance

Bid

Incr

emen

ts ($

)

Dynamic Bid IncrementFixed Bid Increment ($1)

0

200

400

600

800

1000

1200

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45

Bidding Instance

Auc

tion

Rev

enue

($)

Fixed Bid IncrementDynamic Bid Increments

Figure 6: Comparison of Dynamic and Fixed Bid Increment Setting (a) Bid Increments (b) Revenue

40

The first panel shows that based on the predicted WTP values, the estimated minimum

bid will differ from fixed bid increments. In the second panel of Figure 6, we record a marginal

gain in revenue, which is realized through a reallocation that is forced by the dynamic bid

increments. Interestingly, from the perspective of the social planner, this re-allocation accounts

for an increase in the allocative efficiency9 of the auction from 99.8% to 100 %. This anecdotal

example is meant to give the reader a flavor of our ongoing work in calibrating multi-unit

auctions in real time. Note that such changes in bidders’ strategy may have an endogenous

impact on auction dynamics, however, investigation of such endogenous impact is beyond the

scope of this paper and will be focus of future research.

6.3 Dynamic Buy-it-Now Prices

A cursory look at current online auctions reveals that auctioneers have modified their

auction models to cater for buyers who are interested in a quick deal, in lieu of waiting for the

auction to close. The eBay auction calls it a buy-it-now price. At any time during the auction, the

auctioneer offers the bidders a fixed price offer, and bidders may opt to buy the product at that

price instead on continuing to participate in the auction. In the case of single item auctions this

also causes the auction to terminate. While similar features (namely suggested retail prices) are

offered on multi-unit auctions, anecdotal evidence, presented in Section 1, suggests that the

current implementation is not effective, in that a vast majority of auction close below the

suggested prices.

In setting a buy-it-now price the auctioneer has to achieve a balance between setting a

buy-it-now price that is too high to be effective and setting a price that is too low such that it

9 Allocative efficiency measures to what extent the goods get allocated to the bidder’s that value them the most.

41

loses expected revenue. We propose that auctioneers use the predicted WTP to dynamically

adjust the buy-it-now prices in accordance with the bidders’ demand functions for specific

auctions. The key to buy-it-now prices should be to avoid cannibalization, i.e., not setting a price

that is too low such that a person who will not win otherwise ends up buying. Our predictions

allow auctioneers to tailor buy-it-now prices in accordance with their risk profile. These prices

could range from a risk-seeking Nth highest valuation estimate to a more conservative highest

valuation estimate, as well as all the interim possibilities.

We also expect that dynamic buy-it-now prices will decrease the duration of multi-unit

auctions, a benefit that is already being reaped by single unit eBay auctions with offer the buy-it

now option. Setting optimal dynamic buy-it-now prices in multi-unit settings remains a

promising area of future work.

7 Conclusion and Future Research

Our work is motivated by the opportunity of bringing back the skills of an expert

auctioneer in the physical world, capturing by gut and feel the underlying essence of the auction

room’s WTP, into a high volume, mechanized but computationally powerful online environment.

We present a bidding strategy classification and valuation prediction model that estimates

bidders’ WTP in Uniform and Yankee multi-unit auctions. Our results can be classified under

two information categories, with and without the use of historical data, as shown in Figure 7.

Figure 7 - Information Richness Enhances Prediction Accuracy

No Classification

Moderate PredictionAccuracy

Normalized Classification

High Prediction Accuracy

Logit Classification

Very High PredictionAccuracy

No Historical Data

20% Historical Data Usedto Calibrate Multi-Attribute Logit Model

Uni-Dimensional

42

As is evident, prediction accuracy improves in going from no-classification to performing a one-

dimensional classification with no historical data. The use of a relatively small amount of

historical data to perform a multi-attribute classification further reduces both the average

percentage error and its variance. In addition, we find that our joint classification and prediction

technique outperforms alternative approaches that draw random valuations between a bidder’s

current bid and the known market upper bound.

Our findings enable inference on final auction prices prior to the close of the auction,

which in turn can be used to make real time mechanism design changes to increase auctioneer’s

revenue, maximize allocative efficiency and potentially, through smart agents, bidders’ surplus.

We expect future research to find more interesting uses of the prediction model developed here.

Bapna, Goes and Gupta (2003a) and Engelbrecht-Wiggans (1999) demonstrate that

progressive multi-unit auctions have multiple equilibria, some of which are more desirable than

others, from a revenue perspective. Further, Bapna, Goes and Gupta (2003b) have also shown

that online auctioneers are often far away from optimal mechanism design choices that could

increase their likelihoods of obtaining the desirable equilibria. In this context, real-time value

discovery tools, such as the one demonstrated in this paper, will provide the foundation for

dynamically calibrating the online auction mechanism, so as to maximize their likelihood of

obtaining the desirable equilibria. They can also serve as building blocks for designing the next

generation of smart bidding agents whose incentives are aligned with bidders.

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46

(Appendices are to be treated as online supplements) Appendix 1.: Exhaustive Summary of Feasible Auction Outcomes

When 21 xzx << Price Probability

Uniform Yankee

zWzW >< 21 , z z

( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−− 2

1

2

1*1xmzm

When 2xz > Price

Case Uniform Yankee Probability

zWxxW <<< 2221 , 2x z ( )

( )( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−− 2

1

22

22

2

11xmxm

xmzm

zWxzWx <<<< 2211 ,

( )2

2 zx + z ( )

( )( ) ( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−−−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−− 2

1

222

22

2

1xm

zmxmxmzm

zWxzW <<> 221 , ( )2

2 zx + z ( )

( )( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−− 2

1

2

22

2

1xmzm

xmzm

zWxW >< 221 , 2x z ( )

( )( )( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−2

1

22

22

2

1xmxm

xmzm

zWzWx ><< 212 , ( )2

2 zx + z ( )

( )( ) ( )

( ) ⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−−−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−2

1

222

22

2

xmzmxm

xmzm

zWzW >> 21 , W W ( )

( ) ( )212

2

4

xmxmzm−−

−

47

Appendix 2. Online Auction Data Variables Monitored by Tracking Agent:

Variables Description Auction variables Lot Number Product Description Current Bid Bid Increment Number of Bids Quantity Opening Bid Retail Price Open Date Close Date Auction Type

A unique ID that identifies each auction Product description details The current minimum winning bid level Auction’s pre-set bid increment Number of bidders who have already submitted bids. Number of items being sold Pre-set minimum starting bid Displayed retail price The time the auction begun Pricing method (Yankee or Uniform)

Bidder variables Member ID Bid Amount Quantity Won Bid Date Status

A unique ID that identifies each bidder The amount a bidder tendered The quantity bid for The quantity allocated to the bidder The time the bid was submitted Winning or losing status.

48

Appendix 3: Corrleation data A: Yankee Auctions Dataset

X1 X2 X3 X4 X5 X1 1 X2 0.05 1 X3 0.235 -0.081 1 X4 0.189 -0.151 -0.084 1 X5 -0.467 -0.655 0.000 -0.145 1

B: Uniform Price Auctions Dataset

X1 X2 X3 X4 X5 X1 1 X2 0.101 1 X3 0.133 0.109 1 X4 0.237 -0.077 -0.501 1 X5 -0.355 -0.344 0.233 -0.45 1

Variables:

X1- Ratio of number of current winning bids that are smaller than the current bid, to lot size X2 - Number of bid revisions. X3- Standard deviation of current winning bids X4- Average of current winning bids X5- Normalize elapsed auction time.