replicating online yankee auctions to analyze auctioneers’ and...
TRANSCRIPT
Replicating Online Yankee Auctions to Analyze Auctioneers’ and Bidders’ Strategies
Ravi Bapna1
Paulo Goes
Dept. of Operations & Information Management,
U-41 IM, School of Business Administration,
University of Connecticut,
Storrs, CT 06269
Alok Gupta ([email protected])
Dept. of Information and Decision Sciences
3-365 Carlson School of Management
University of Minnesota
Minneapolis, MN 55455
April 2001
(Under Revision)
*This research was supported in part by TECI - the Treibick Electronic Commerce Initiative, OPIM/SBA, University of Connecticut. Third Author’s research is supported in part by NSF CAREER grant # IIS-0092780. Author names in alphabetical order. Please do not quote without authors' explicit permission.
2
Replicating Online Yankee Auctions to Analyze Auctioneers’ and Bidders’ Strategies
Abstract
Dynamic price setting mechanisms such as online auctions typify the new generation of
mercantile processes being used on the WWW. However, arbitrary pricing mechanisms can
result in loss of revenue and social capital in markets characterized by tight margins. This work
presents a relatively risk-free and cost-effective approach to managing innovation in the area of
web-based dynamic price setting processes. We focus on Yankee auctions, which sells multiple
identical units of a good to multiple buyers using an ascending and open auction mechanism.
This mechanism has its roots in the traditional English auction; however, significant new rules
make it an interesting mechanism to study. This study presents a multi-agent simulation
approach to manage the optimization of sellers’ revenue. It is based on the theoretical revenue
generating properties of the Yankee auctions and utilizes data from real auctions to instantiate
the simulation’s parameters. Based on the observed consumer bidding strategies in real online
auctions, three classes of bidding agents were developed and deployed. The validity of the
simulation model is established and subsequently the simulation model is configured to change
the values of key control factors, such as the bid increment. Our analysis indicates that the
auctioneers are, most of the time, far away from the optimal choice of bid increment, resulting in
substantial losses in a market with already tight margins. We discuss the challenges and key
constructs of the model development. We also demonstrate the extended capabilities of the
simulation tool by examining hybrid-bidding strategies derived as a combination of the original
strategies used by online bidders.
Keywords: dynamic pricing, online auctions, simulation
3
1.0 Introduction and Background
Dynamic pricing mechanisms or processes, in which the consumers become involved in the
price-setting process, are now an integral part of the web economy. A myriad collection of price-
setting processes such as traditional first-price auctions (e.g., Ebay and Onsale), reverse auctions,
name-your-price mechanisms (e.g., Priceline.com), quantity discounters (e.g., Mercata.com), and
methods using derivative based pricing for consumer goods (e.g., Iderive.com) emerged in the
new economy. Some continue to flourish (e.g. Ebay and Onsale) while others have floundered
(Mercata and Iderive). Despite the innovativeness of these pricing approaches and the excitement
surrounding them, little attention has been paid to the their effectiveness. Essentially,
directionless entrepreneurship, at times fueled by overzealous venture capitalists, replaced
scientific enquiry and rigor when it came to examining the efficacy and viability of candidate
mechanisms. This paper describes the details of a powerful yet cost-effective simulation
approach that can be used to evaluate and manage the successful deployment of dynamic pricing
mechanisms on the web.
We focus our attention on a specific dynamic pricing mechanism, namely online auctions.
However, the approach we develop is applicable and useful in examining the microstructures of
other e-market mechanisms as well. Online auctions represent a model for the way the Internet is
shaping the new economy. In the absence of spatial, temporal and geographic constraints these
mechanisms provide many benefits to both buyers and sellers. However, significant research is
still needed in designing new and better mechanisms, as well as examining the efficacy of
existing ones in the contexts of the markets they serve. This study, and the tool developed in it,
uses the observed insights obtained from tracking real-world online auctions to instantiate the
parameters needed to simulate the real-world process. Additionally, in a risk-free and cost-
effective manner it leverages the computational power of today’s desktops to provide direction to
the online auctioneers.
Online auctions are a testimony to the increasing participation of consumers in the price-
setting process. No longer is the time-tested posted-price mechanism the only choice available
4
for the exchange of assets. In consumer oriented markets online auctions offer a dynamic pricing
alternative to the age-old posted pricing mechanism. Consumers can now experience the thrill of
‘winning’ a product, potentially at a bargain, as opposed to the typically more tedious notion of
‘buying’ it. The growing dynamic-pricing phenomenon on the web has led to researchers asking
whether fixed prices are a thing of the past (Kauffman and Riggins, 1999). For sellers these
mechanisms bring access to newer markets, help clear aging or perishable inventory, and provide
experiential and at times viral marketing capabilities.
Traditional auction design and bidding strategies have been extensively studied in the
economics literature [6, 7, 8]. However, the significant changes brought about by the Internet on
this area are yet to be studied. Van Heck and Vervset, 1998 recently called for examining the
pervasive impact of advanced electronic communications on the well-established theory of
auctions.
In this paper, we concentrate on using theoretically motivated simulations, which use
real-world empirical data, to study the drivers in one of the auction mechanisms prevalent in the
online setting: the Yankee auction. Yankee auction is a special case of multi-item English
auction. Here, multiple units of the same product are sold to multiple bidders. The auction is
progressive in nature; however, each new bid does not have to be strictly greater than the
previous bid since there are multiple units available. The set of winning bids consists of the top
N bids, where N is the number of units up for auction. A new bid either has to be equal to the
minimum bid that is among the winning bids (if the set of winning bids has a cardinality of less
than N) or it has to be at least equal to minimum winning bid plus a pre-specified minimum bid
increment. The auction terminates on or after2 a pre-announced closing time and each of the
winning bidders pay the amount they last bid to win the auction. Note that in multi-item settings
this often leads to discriminatory pricing with consumers paying different amounts for the same
item. Such auctions are used on a variety of auction sites on the WWW such as Egghead.com’s
2 Most auctions have a going, going, gone period such that the auction terminates after the closing time has passed and no further bids are received in the last five minutes.
5
Surplus Auctions, and Ubid.com. Before we present the approach and goals of this paper in
detail, let us present the classical and current set of knowledge and research relevant to Yankee
auctions.
1.1 General and Online Auction Research
While a complete literature review on auction theory is beyond the scope of this paper,
we present some key and relevant findings.
The classical approach to analyze auctions, perhaps unconsciously constrained by the
physical limitations of traditional auctions, has been to use game theoretic models (see, for
example, McAfee and McMillan, 1987; Milgrom and Weber, 1982; and Milgrom 1989 for
detailed literature review and analysis). These game theoretic models are typically used for
analyzing auction of a single item and the objective of the analysis is to characterize Nash
equilibrium (Nash 1950) for the auction. Unfortunately, the results for single-item auctions do
not apply to multi-item auctions (Rothkopf and Harstad, 1994). Further, the game theoretic
approach is notoriously difficult and, often, analytically intractable with a large number of
bidders bidding for multiple units. The number of cases that need to be examined grow
exponentially both with the number of items and the number of bidders.
Perhaps due to inability of applying game theoretic approaches to such auctions, several
researchers such as Lee and Mehta, 1999; Vakrat and Seidmann, 1999a; and Pavlou and Ba,
2000, have taken an empirical approach in analyzing the effectiveness of online auctions. Vakrat
and Seidmann, 1999b, developed a stochastic model of bidder arrival process to make lot size
decisions. Using purely empirical data creates, at least, the limitation of not being able to test the
data against a benchmark of what "should have happened," i.e., no normative insights are created
into the auction process itself.
In this paper we describe an approach to analyze and optimize the auctioneer's revenue by
manipulating controllable factors. We present a simulation tool that is motivated by the
theoretical results of Bapna, Goes, and Gupta, 2000a, who develop an incomplete information
model to analyze online bidding activity in Multiple-Item Progressive Auctions (MIPEA), same
6
as popularly known online Yankee auctions. The simulation uses the theoretical insights
generated by Bapna et al.'s model and uses the data collected by monitoring real-world online
auctions to demonstrate the validity of the tool. The simulation tool was developed to satisfy the
following criteria:
• Given the data from an observed on-line auction, the simulation should replicate the auction
with observed revenue being statistically equivalent to the simulated auctions.
• The parameters of the auctions can be changed to test the effects of changing the
environment of a given auction.
The rest of this paper is organized as follows. In Section 2, we briefly review the theoretical
results of Bapna, Goes, and Gupta, 2000a and explain the motivation behind this paper. In
Section 3, we describe the data collection and discuss the various consumer-bidding strategies
observed in such online auctions. In Section 4 we describe the characteristics of the multi-agent
based simulation model. In Section 5, we present the results of the simulation study that show
that our simulation model successfully replicates the online auction environment and that the
simulation model can be used to improve the revenue of an auctioneer. In section 6 we discuss
some future research directions and discuss how the simulation model can be used to evaluate
other interesting issues such as consumer bidding strategies. Finally, we conclude in Section 7.
2.0 Theoretical Basis
Bapna, Goes, and Gupta, 2000a, developed a stylized model of equilibrium bid
characteristics with a minimal set of assumptions. This characterization is based on incomplete
information and with no assumptions regarding the distribution of the bidder valuations. While
this characterization did not produce a closed form expression for auctioneer's revenue, it
provided an upper and lower bound on the revenue based on the marginal bidder's valuation. A
marginal bidder is characterized as the losing bidder with the largest bid. The interesting aspect
of this characterization is that while the marginal bidder’s valuation is not known (without
making distributional assumptions), the marginal bid can be observed in practice. Further, the
7
difference between upper and lower bound is purely based on the number of items on bid and the
minimum bid increment. Let, δ denote the fraction of a bid increment k measuring the distance
between the marginal consumer's valuation V and the nearest lower feasible bid. N represents the
total number of items for sale. Finally, assume that bidders follow the pedestrian bidding
strategy, i.e., they always bid the lowest required bid. Rothkopf and Harstad (1994) note that
this strategy is optimal for English auctions with relatively small bid increments. Proposition 1
below provides an expression for the lower and upper bound on revenue of an auctioneer in
terms of marginal bidders value, V.
Proposition 1: (see Bapna, Goes, and Gupta, 2000a) Let V be the marginal consumers valuation,
and δ be a segment of the bid increment k that measures the distance between the marginal
consumer's valuation V and the nearest lower feasible bid. Then the lower bound and the upper
bound on the revenue of a seller selling multiple units under MIPEA are respectively N(V-δ) and
N(V-δ+k).
An interesting corollary of this proposition is that the range of revenue (upper bound -
lower bound) is N*k, which does not depend on V or δ. A legitimate question to ask is: given a
pre-specified Yankee auction with a given N, can the manipulation of bid increment, k, yield a
higher revenue for an auctioneer?
The following example illustrates, for the same valuations, how the temporal ordering of
two separate bidding sequences can result in either the lower or the upper bound on the revenue.
We also illustrate how these bounds are affected if the bid increments are changed.
Numerical Example 1 - Consider the following hypothetical scenario. Let N=3 and k= 5. Let
there be four bidders, say A, B, C, D with valuations of 56, 61, 62, 62 respectively. Let A be the
marginal customer and let the opening bid be $1.
8
• The lower bound occurs if we observe the following sequence of progressive bids:
D(46)--C(46)--B(46)--A(51)--B(51)--C(51)--D(56)--C(56)--B(56)--STOP because A will
have to, and will not, bid 61 to get in now. Revenue = $168. The upper bound occurs if
we observe the following sequence of progressive bids: B(51)--C(51)--D(51)--A(56)--
D(56)--C(56)--B(61)--C(61)--D(61)--STOP because A will have to, and will not, as per
observation 1 ,bid 61 to get in now. Revenue = $183.
• Now consider the case where we change the bid increment to 3. Note that if both
sequences above reach the level $51 (by starting the bidding from a different point) then
both lower and upper bound will move downwards to $162 ($54 paid by each winner)
and $177 ($57 paid by each winner), respectively.
Anecdotal evidence suggests that auctioneers realize the importance of k since we
routinely observed in online auctions that similar items are auctioned, at different times, using
different k. In this paper we investigate whether the seemingly randomly chosen values of k
were in the particular auctions were optimal or close to optimal. One approach to study this
would be to auction the same or similar items with different k. However, this is a costly
endeavor and, in the era of rapid obsolescence of computer products, it will still not answer the
question whether the auctioneers used appropriate bid increments. We use the word appropriate,
instead of optimal, because k affects the number of rounds of bidding required to reach the same
levels of revenue and an auctioneer may choose a different level of k for a faster or slower
convergence to the desired point. For example, a bid increment of 5 will take twice as many
rounds of bidding as compared to the bid increment of 10 starting from the same opening bid.
However, the implemented level of k should be such that it should not significantly affect the
revenue since the margins in items sold through these auctions are quite small.
We chose an alternative strategy to investigate the optimality of k by developing a
simulation model that can replicate a given observed auction once it has been provided the
observed bidding activity and then will allow us to manipulate the value of k to study the effect
9
of changing k. This involved three steps; i) Observe a significant number of online auctions and
collect complete set of bidding data for each auction, ii) Analyze the data to recognize consumer
bidding strategies, and iii) Develop the simulation model that uses the bid data and replicates
consumer behavior in real auctions such that the resulting revenue is statistically equivalent to
the observed auctions' revenue. In the next section, we describe the data collection process and
the simulation model.
3.0 Data Collection, Consumer Strategies
3.1 Data Collection
Initially, a survey was carried out of various Internet auction sites to identify a subset that
could be polled on a regular basis. Sites examined ranged from large public corporations such as
onsale.com that attract thousands of bidders to rarely visited auction sites such as artrock auction.
We selected on-line retailer Egghead Corporation's surplusauction.com based on the following
criteria:
• The company profile of an innovative on-line retailer with an expanding repertoire of
mercantile processes ranging from posted-price based catalogs to Yankee auctions,
• The significant volume of merchandise being auctioned every day, and
• The significant consumer interest, as measured by web-traffic, in these auctions.
Data collection was carried out by an automatic agent that was programmed to download, at
frequent intervals of 5 to 15 minutes, the html document containing a particular auction's product
description, minimum required bid, lot size and current high bidders. Subsequently, the series of
html files were parsed to condense all the information pertinent to a single auction, including all
the submitted bids, into a single data file.
We further screened the collected data to make sure that there was no data loss even during
the transient periods of the auctions so that we have the complete bidding activity. Specifically,
we were interested in the highest bid posted by each bidder even if they did not "win" the
10
auction. This information is necessary to replicate an auction via a discrete event simulation.
These auctions were carefully selected with respect to the important parameters of lot-size and
bid increment so as to ensure a statistically sound sample.
While we collected equilibrium (final bids) data for over 150 auctions, after careful
analysis we found that in a substantial number of cases we may have missed some bidding
activity in the transient stages of the auction. The bidding activity was missed due to either a
failure of the software agent to connect to the site at a given time or due to faster than expected
bidding activity. We found that we have 86 auctions where we are reasonably certain that we
have complete bidding activity. While, at first, it may seem that bid information from
individuals that did not win the auction is not important, it is important to note that even a single
bidder can alter the revenue path resulting in a different equilibria than without a bid from that
bidder. In the next section we provide results from our simulations that validate the capability of
the simulator in replicating real-world auctions and provides insights into the choice of optimal
bid increment for a given auction.
We were also able to identify, in the data we collected, broad consumer bidding strategies
that we describe next.
3.2 Observed Consumer Bidding Strategies
As identified by Bapna et al., 2000a, the bidders in online Yankee auctions can be categorized in
the following 3 broad categories according to their bidding strategies:
1. Evaluators - early one time high bidders who have a clear idea of their valuation and execute
a single bid, often during the early phases of the auctions and this bid is significantly greater
than minimum required bid at that time. In essence, the strategy here is to achieve the highest
time-priority (that is part of most Yankee auctions) for their personal maximum bid level.
These consumers are willing to pay a potential premium for the higher priority at their bid
level. From another perspective, they are reasonably certain that the marginal bid will be
close to their bid level and, thus, they want to be the first to enter at that level. Surely, such
11
bidders would be rare in traditional auction settings, where the cost of physically getting to
an auction site to make just a single bid would be a significant deterrent.
2. Participators - consumers who derive some utility form the process of participating in the
auction itself. They typically, make a low initial bid equal to the minimum required bid and
progressively monitor the progress of the auction and make ascending bids. These bidders
follow the pedestrian approach described in Section 2.
3. Opportunists - consumers who by nature are looking out for bargains and who buy when
they see one. They typically place minimum required bids just before the auction closes.
Note that, the maximum price penalty a bidder of this type pays is equal to the bid increment
k of a particular auction (Bapna, Goes, and Gupta, 2000b).
In our data we identified the number of each type of bidder for each auction. This
information is then used in the simulation model as a set of parameters, with each bidding
strategy being represented by a bidding agent that was coded to exhibit corresponding behavioral
characteristics. The next section details the simulation model.
4.0 Simulation Model
The objective of developing the simulation model is to test the effect of changing
controllable factors such as the bid increment and to examine whether the revenue generated
through an auction can be improved. The results of simulation can only be trusted if the
simulation replicates an online auction's result with its original parameters. Further, the
simulation should be easily configurable to run under any given auction. Therefore, instead of
hard-coding the bidding data, it is read from an input file.
The first task that needed to be accomplished was the creation of bidding agents that
would behave as the three types of bidders identified in Section 3.1.
12
4. 1 Creation of bidding agents to represent consumer-bidding strategies
To replicate an auction we have to replicate the three broad strategies chosen by the bidders. We
achieve this by creating three classes of bidding agents, each embodying the behavioral
characteristics of the bidding strategy they represent. These bidding agents fall within the
conventional definition of software agents which according to Nwana et al. (1998) comprises of
“software entities that have been given sufficient autonomy and intelligence to enable them to
carry out specified tasks with little or no human supervision.” In order to execute tasks on behalf
of a business process, computer application, or an individual, agents are designed to be goal
driven, which in our settings translates to maximizing their net worth by winning the object at
lowest possible price. We next define the specific attributes that we associate with each of the
three classes of bidding agents:
a) Participatory Agents: These agents arrive at an auction throughout the duration of an
auction. The key characteristics of these agents is that they never place a bid higher than
the minimum required bid to enter the winners’ list of an auction. In addition, they
continue to participate in the auction until the minimum required bid exceeds their
valuations.
b) Evaluatory Agents: These agents place just one bid that is equal to the highest feasible
bid level below their valuation. Such agents do not participate in the auction on an
ongoing basis and if there are enough bids above their bid, they are removed from the
winners list.
c) Opportunistic Agents: These agents enter into the bidding process only towards the end
of the auction. We operationalize this feature, in the simulation, by letting these agents
bid only when the value of the lowest winning bid has reached 90% of the marginal bid
level.
The next subsection provides simulation details.
13
4. 1 Simulation of Yankee Auctions
The entire simulation process is illustrated in the flowcharts of figure 1. Broadly, these steps
can be summarized as follows:
i) The observed final bid placed by each individual during a given auction is read from
a file. The bids for each type of bidders are placed in separate files, so the input is
read from three different files. The information is stored in a 2-dimensional array
containing bid value and the type of a bidder.
ii) The values of the original bid increment (k), simulated bid increment (k’), starting bid
level (r), and the number of units for sale (N) are also provided as an input to the
simulator.
iii) Based on the final bid of each bidder, a valuation is generated, for that bidder, by
adding a random number drawn from U(0, k). In online Yankee auctions, a bidder is
not allowed to bid between 2 successive feasible bid levels. For example, suppose the
starting bid for an auction was $3 and the minimum bid increment was $5, then the
bidders are only allowed to bid at levels of $3, $8, $13, $18... If a bidder bids $11,
their bid is automatically rounded to $8. Therefore, a person that has bid $103, may
actually have a true valuation of $108 (at $108, person may or may not bid because
she is indifferent) given that the minimum bid increment is $5. Therefore, if the jth
bidder’s highest bid was Bj, then his true valuation may be anywhere between Bj and
(Bj + k).
iv) The valuations array is then scrambled to remove the input bias since data is read
from files that store bids in sorted order and the data for each type of bidder is
bunched together in the valuations array. We use an innovative linear algorithm to
perform this scrambling; the details of this procedure are presented in the next
subsection. The original scrambled array forms the starting point for all the
replications for a given auction.
14
v) A copy of the scrambled value array is generated for each auction replication. Then,
the auction process starts by choosing an eligible bidder from the valuation array. An
eligible bidder is a bidder having a valuation higher than the current bid level; in
addition, an eligible bidder cannot be among the winners list at that moment. We use
a variation of our scrambling algorithm to ensure that at each bidding level we only
have to draw exactly N random numbers (same as the number of items for sale) to
find N eligible bidders at that level. The details of this approach are provided in
subsection 4.3 and, as shown there, it reduces the computational burden of the
simulator significantly. In practice, a simulation run, that would otherwise take 25-30
minutes on a Pentium 750 machine, takes less than a minute with our approach.
vi) While the evaluator and participator agents can arrive at any time during the auction,
the opportunists can only arrive towards the end of an auction. Since there is no one-
to-one correspondence between simulated time and actual time, we used a surrogate
measure to detect the last few rounds of an auction. We allowed opportunistic agents
to participate only after the 90% of the marginal bid level (lowest level to win
auction) has been reached in the simulated environment.
vii) The simulation stops when there are no more eligible bidders in the valuation list.
Note that, by doing so we are modeling the automatic extension of the auction
duration as implemented by many of the auction sites using Yankee auctions.
viii) When simulating an environment, it is recommended that independent replications of
the simulations be done to provide statistically robust results (see, for example, Banks
and Carson, 1984). We can specify the number of replications in our simulation
model, each starting with an independent random number stream. However, as
mentioned earlier, we start with the same scrambled value array as created in the first
replication. In addition, the other simulation parameters, i.e., the lot size, bid
increments, and the starting bid values are kept the same. In other words, during
15
different replications only the order in which bids from different bidders arrive is
different.
Figure 1 – The Simulation Process
The simulation model was developed using Visual Basic 6.0. The front end of the
simulator is shown in Appendix 1. All the parameters of interest such as the starting bid, the lot
size and the original and simulated bid increments are the input to the simulation model. We
created original and simulated bid increment as two separate inputs so that we can run the
Initialize Parameters -a) Input N, k, opening bid, total bidders,marginal bid, actual revenue
Generate random valuationsa)For Participators and Opportunistsdraw randomly from U[V, V+2k]b)Read observed final bid values forLosers and Evaluators from file andadjust by adding a random value fromU[0, k]
M inimizeConsiderationSet
Start Simulation
Opportunists_Allowed = FALSEi = 0
bid increment = kcurrent bid increment = 0
Seed random number generator
Get equilibrium( bid increment )
i = i + 1
i < maxiterations
yes
current bid increment =current bid increment + 1
No
current bidincrement < max
bid increment
Stop Simulation
Yes
Scramble valuations array
Get equilibrium( bidincrement )
Randomly pick bidder from valuationsarray
Opportunists_Allowed = TRUE
Place bid (bidder type)
ConsiderationSet = NULL
No
Return winning bids
bid increment = current bid incrementOpportunists_Allowed = FALSE
i=0
minimum reqd. bid >Opportunists threshold
Yes
Yes
No
A
A
Simulation Outline
Procedure to obtain equilibrium illustrating use ofprocedures Scramble and MinimizeConsiderationSet
Section4.2
Section4.3
Section4.4
16
simulations with alternative values of bid increments. As noted above, original value of k is
needed because the bidder valuations generated for an auction still need to be based on original
data as described earlier in this subsection.
In the next 2 subsections, we describe some of the details of the simulation model
including some innovative techniques that we have developed and applied in the simulation
program. In subsection 4.2.1 we provide the details of an efficient linear O(I) procedure to
scramble the valuations array to reduce input bias. In section 4.2.2 we describe the process of
selecting an eligible bidder by using an advanced version of the scrambling approach developed
in subsection 4.2.1. We will show both in terms of theoretical properties and through an
analytical comparison with purely random approach that our approach increases efficiency of the
simulator by requiring it to draw a minimal number of random numbers (a costly operation)
during a given replication.
4.2.1 Randomizing the sequence of the bidders valuations
Our objective is to create a mix of bidding agents representing the real-world consumers’
strategies and valuations. It is important from the perspective of simulation that these bidders
arrive in a random fashion and the likelihood of any type of bidder with any valuation at a given
point in simulation is truly random. As mentioned earlier, since the simulation program reads
the bids from 3 distinct files in a sequential order and stores it in an array, the valuations of each
type of bidder agents are bunched together in the array. Further, within each category of bidder
agents the bids are in sorted order. This can be a potential source of bias in the simulation. For
example, if all evaluators have relatively high index values (as compared to participators and
opportunists) in the array, then for an evaluator to be picked as a next eligible bidder a high
random number has to be generated.
We developed a linear O(I) algorithm, referred to as Scramble here, that randomizes the
valuations array and hence reduces the potential source of bias.
17
Proposition 2: Procedure Scramble randomizes an array of size I in O(I) time.
Proof: We present the pseudocode for the procedure that randomizes the array in linear time.
Procedure Scramble Begin
For l ranging from 1 to I j = generate_random_number [1, I-l] if not last element { store the j'th value-type pair as temporary value copy the last available(N-l+1)th slot’s value-type pair to the j’th slot replace the last available slot’s values with the temp values } loop l }
End
Let us explain the procedure my means of a numerical example.
Numerical Example 2: Assume that there are 50 values in the array to be randomized. The first
time number drawn j will be randomly drawn between 1 and 49. In that case, we will take the j'th
value-type pair and put it in the 50th slot and move values in 50th slot to j'th position. In the next
iteration j will be drawn between 1-48, and we will take the j'th value and copy it to 49th slot and
values from 49th slot to jth position and so forth. As i increases the random number generated
will be drawn from a smaller range. In summary, we are taking the last available slot (initially
the actual 50th slot, then 49th, and so forth) and copying its value to the j'th randomly selected
slot.
In the next subsection we describe an efficient mechanism created based on the procedure
above to reduce the number of random numbers that need to be drawn during a simulation run.
4.2.2 Enhancing the Simulation Efficiency by Minimizing Random Number Generation
During a simulation run, the next eligible bidder is chosen by randomly drawing an integer
between 1 and I, where I is the total number of eligible bidders. Let the integer drawn be j. The
jth eligible bidder then places a bid according to the strategy they follow. Note that bidders who
18
are already in the winners list or have a valuation less than the current bid level are not eligible
bidders. In terms of implementation, if the array of individual valuations is used directly then we
may need to draw a random number several times before we find an eligible bidder since many
in the array will be ineligible to bid. This creates inefficiency in the simulation process due to
the need of drawing several random numbers to find a single eligible bidder; the problem
accentuates towards the later parts of an auction where majority of the bidders may have become
ineligible. Random number generation is a costly operation and such wastage can result in
unnecessarily long simulation runs.
In order to maximize the computational efficiency of the simulation we created an
algorithm that is designed to utilize the minimum possible number of random numbers before
convergence to equilibrium. Specifically, we need to draw exactly N numbers at each bid level
and no more. Let the set I represent the set of all the bidders and let the set C ⊆ I, be the
candidate set of bidders, those who valuations exceed the minimum required bid, at any instance
of the auction. We describe the process of finding the minimum eligible set and arranging their
valuations in the array so that there is no need to draw multiple random numbers to find the next
eligible bidder below. We describe the process from the perspective of an ongoing auction
where the initial conditions (finding the first N eligible bidders) are already met since that part is
algorithmically trivial.
Figure 2 represents our approach pictorially. Essentially, we divide our valuations array
in 3 parts:
i) The set of eligible bidders. The cardinality of this set after the initial N
customers are in the winners list is, |C| ≤ (|I| – N), where I is the cardinality of
the whole set or the total number of bidders. In the array this set is arranged
and maintained as the first |C| elements. The auction stops when |C| is equal to
0.
19
Figure 2– Computational Efficiencies through Iterative Minimization of Consideration Set
ii) The set of current winners. The cardinality of this set is always N, as long as |I|
> N. In the array this part is arranged as the elements (|C|+1) to element
(|C|+N). This part is arranged in such a way that an element can be inserted at
any position from (|C|+1) to (|C|+N) but the element being pushed out is
always the element (|C|+N).
iii) The set of bidders whose valuations are less than the required current minimum
bid. These customers cannot participate in the auction anymore. The
cardinality of this set, L ≤ (|I| – N), increases as the auction progresses. In the
array, these bidders are positioned in as elements (|C|+N+1) to |I|.
Given this division, at any point in time the cardinality of the eligible set is given by the
following expression:
|C| = |I| – N – L (1)
To choose a new bidder, we draw a random number between 1 and |C|. Let the chosen
number be j. The jth eligible bidder is then simply chosen and places a bid based on their
strategy. The bid is placed in the appropriate place in the winner stack. This pushes out the
Eligible Set Winners Low Values
j
Bid of Randomly Chosen Bidder Push the new bid in the appropriate place in the Stack. Push out the last bidder in the stack.
Displaced Bidder Takes the jth place in the Consideration set
20
bidder who was in the (|C|+N)th position. If the valuation of this bidder is greater than the current
required bid, the bidder being pushed out is put in the jth position in the eligible bidder set,
otherwise |C| is reduced by one by moving the elements in (j+1)th to |C|th position to jth to (|C|-
1)th position. Whenever the minimum bid requirement increases by k, we scan the eligible bid
set and remove any bidders who may no longer be able to bid at the new level. Therefore, the
eligible bidder set keeps on shrinking as the auction progresses and the auction stops when the
eligible bidder set becomes a null set. By keeping the index of the eligible bidder set we only
have to draw a single random number for finding an eligible bidder. If there are N items for sale
then we need to draw, at the most, N random numbers at each bidding level.
In order to provide insight into the efficiency gained in this process versus a purely random
approach of finding an eligible bidder, let us provide an analytical comparison of our approach to
the purely random approach. First, we introduce some notation. Given that the set I represents
the set of all the bidders and the set C ⊆ I is the candidate set of bidders, define s as the
likelihood of finding a bidder in the candidate set from amongst all the bidders. Observe that, s =
Prob(finding i ∈ C) = IC
. Recall, that N represents the lot size. Let the bid increment k = αN,
where α > 0.
Proposition 3: If the marginal bidder's valuation, V is from a uniform distribution such that V ≈
U[0, xN], where x ≥ 1. The expected total number of random variables drawn with purely
random draws from a set of I bidders is ∑−
−
= −−
11
0 )1(
α
α
x
i ixxN . Our approach requires a maximum of
αNx )1( − random numbers to be drawn.
Proof: First, note that given our construction k = αN, the number of bidding cycles to
convergence = N
NxNα
− = α
1−x , where the numerator accounts for top quintile with N bidders.
21
Since our approach requires drawing a maximum of N random numbers in each bidding cycle,
the maximum required number of random draws is α
Nx )1( − .
With purely random draws, the expected number of times we need to draw a bidder i
belonging to the consideration set is binomially calculated as:
E(Number of times to draw i ∈ C) = ssjj
j∑∞
=
−−1
1)1( = s ∑∞
=
−−1
)1(j
jsdsd
Upon simplification we obtain,
E(Number of times to draw i ∈ C) = s
sdsds )1( −− =
s1 .
Extending to N winners,
the E(Number of times to draw N i’s ∈ C) = sN .
At the jth bidding cycle jαN bidders will have valuations less than the minimum required bid. In
addition there will be N bidders who are already in the winner’s list and thus no longer belong to
C.
Thus, we can re-express the probability of finding a bidder belonging to the candidate set
s = xN
NNjxN −− α = xjx 1−− α .
This implies that αjx
xs −−
=)1(
1 . Hence, summing over all the bidding rounds ranging from 0
through α
1−x we obtain:
The expected total number of random variables drawn is ∑−−
= −−
11
0 )1(
α
α
x
i ixxN .
Numerical Example 3: Assume x=20, α=0.5, Vmax = $200, k=5, and N=10.
Based on Proposition 3, the expected total number of random numbers drawn to reach
convergence = 20N ∑= −
37
0 5.0191
j j = 20N(
5.01...
5.181
191 ++ ) = 169.12 N.
Because in our approach, we eliminate all the ineligible bids from the candidate set at
every iteration, we are guaranteed to progress the auction at every iteration, hence the expected
22
number of random numbers drawn is simply equal to the number of bidding cycles to
convergence, i.e. α
1−x = 38N. Thus our approach results in an expected improvement in
efficiency of approximately 445%.
5.0 Simulation Results
Our primary objective in building the simulator was to be able to replicate the 86 real-
world auctions so that we can test the effect of theoretical, empirical, and heuristic rules on the
auction process. This is a challenging task since the complex real-world strategies used by
individual bidders in a given auction, which affect the revenue, cannot be coded in a general
simulation tool. Even though we use the 3 broad bidding strategies we identified in Section 3,
the effect of using hybrid strategies cannot be completely predicted. The problem of establishing
mechanism equivalence is further confounded by the fact that the observed data from actual
auctions is a single point (a realization of a bidding sequence), which could be any of the
multiple equilibrium points defined in Proposition 1. In addition, the revenues generated are
discrete because of the discrete nature of bid levels. Therefore, instead of using mean revenue to
establish equivalence, we establish equivalence by using either a chi-square goodness-of-fit test
for the distribution of winning bids or by using a binomial test depending upon whether there are
more than 2 levels of bids among the bids. In the next subsection we present the simulation
results and the results of the equivalence tests.
5.1 Test of bid distribution fit
We test the robustness of the simulator by using a goodness-of-fit procedure. Recall that
a typical multi-item auction, with bidders employing the strategies described in Section 3.2, ends
with winners at multiple bid levels. Given identical starting parameters, our test procedure
examines whether the observed frequencies of the various bid levels in the real world auctions
matches, in a statistical sense, the expected frequencies of the same bid levels that are generated
23
by the multiple replications of the simulation process. Intuitively, we test whether there is a
favorable likelihood that the real world auction itself can be generated by the distribution of the
winning bid levels created by the simulation. Consider the following distribution of winning bid
patterns for an auction we tracked and simulated.
Bid Level Distribution Fit(Auction xxx20A-yy4294)
0
1
2
3
4
5
Bid Level 49 54 59
ExpectedFrequencyActual Frequency
Figure 4- Do the simulated and actual distributions of the bid levels fit?
Figure 4 shows an auction that terminated with winning bidders at 4 different bid levels.
We then test the goodness of fit between the expected (simulated) and the observed distributions.
Thus our hypothesis of interest is:
H0: The observed real-world auction and its simulated replications belong to the same
underlying distribution with regards to its revenue generating properties.
Against the alternative hypothesis:
Ha: The observed real-world auction and its simulated replications belong to the different
underlying distributions with regards to their revenue generating properties.
Statistically, the chi-square test is ideally suited for this purpose. Particularly important to us is
the fact that it is non-parametric, does not assume any prior distribution, and applies to nominal
data, such as frequencies. Following are some of the reasons that dissuaded us from using a
parametric statistical procedure:
24
i) We only have a point estimate of the revenue of real auctions with no distributional
information. Further, the observed revenue could be from any place in the distribution
and there is no reason to believe that it indicates the central tendency of its underlying
distribution. Testing an empirical distribution (generated by the simulation) with
observed revenue, as the indicator of central tendency (mean or median) is inappropriate.
ii) The empirical distributions did not seem to be bell shaped and were quite flat, making the
parametric t-test inappropriate.
Another reason for not using parametric test is that given the restriction of bidders being
able to only bid in multiples of the bid increment, the revenue of these auctions is a discrete
variable. For example, suppose there are 5 items for sale and each winning bid is $100 (for total
revenue of $500). If the minimum bid increment is $10 then the next bid can only be $110 and
the revenue level $510.
The chi-square test has certain minimum requirements with respect to frequency of every
cell (≥ 5) and in those cases where this requirement is not satisfied there are two recourses. First,
if the number of cells, possible bid levels in our case, is exactly two (and frequency < 5) then the
binomial signs test should be resorted to. Otherwise, if the number of possible outcomes is
greater than two (and frequency < 5) then either, the multinomial signs test is recommended, or
cells should be merged to get cell frequencies greater than 5. For sake of completeness, in those
cases when we had two bid levels and cell frequencies < 5 we adopted the binomial signs test.
For all other cases we used the chi-square test, resorting to merging of cells in the case when
there were more than two bid-levels and the cell frequencies were less than 5.
The 86 auctions ranged in observed revenue from $40 to $24,790, in the number of items
for sale from 3 to 100, and in the number of bidders from 4 to 437. The complete sets of results
with detailed descriptions of the parameters of the auctions are listed in Tables 1a and Table 1b.
The columns provide the masked auction number, the revenue that was observed in the real
auction, range of the simulated auctions, the test statistic, and the p-value, respectively. Each
auction was replicated 31 times using independent random number seeds for the bidding
25
sequence. However, during all these runs the bidder valuations were kept constant, so the
revenue variability is only the result of randomizing sequence of bid arrivals.
Auct
ion
Num
ber
Obs
erve
d Au
ctio
n R
even
ueSi
mul
ated
M
inSi
mul
ated
M
axC
hi
Squa
rep-
valu
eAu
ctio
n N
umbe
r
Obs
erve
d A
uctio
n R
even
ueSi
mul
ated
M
inSi
mul
ated
M
axC
hi
Squa
rep -
valu
e1
$16,
190
$16,
210
$16,
270
10.2
10.
176
44$2
,007
$1
,987
$2
,077
1.
120.
572
$24,
790
$24,
670
$24,
990
0.78
0.37
647
$703
$6
63
$713
1.
190.
552
3$1
0,91
5 $1
0,77
5 $1
0,97
5 9.
530.
121
49$1
,012
$1
,012
.00
$1,0
52.0
0 1.
230.
542
4$1
0,77
0 $1
0,85
0 $1
0,99
0 2.
210.
3353
$1,5
53
$1,5
43
$1,5
93
0.5
0.77
95
$19,
836
$19,
436
$19,
616
4.34
0.11
454
$647
$6
27
$667
1.
840.
398
6$1
6,06
8 $1
6,16
8 $1
6,62
8 0.
780.
376
55$5
57
$527
$5
47
0.11
0.94
97
$11,
633
$11,
653
$11,
813
9.53
0.12
156
$1,1
47
$1,1
37
$1,1
57
1.67
0.43
58
$14,
093
$14,
033
$14,
113
0.76
0.69
59$5
90
$590
$6
30
0.03
0.85
79
$11,
610
$11,
340
$11,
600
4.34
0.11
461
$1,3
13
$1,1
93
$1,2
73
0.52
0.77
110
$17,
792
$17,
452
$17,
952
37.7
70.
313
64$8
13
$813
$8
53
0.22
0.89
411
$10,
694
$10,
754
$11,
194
0.81
0.93
865
$1,1
04
$1,1
04
$1,1
34
1.4
0.49
612
$15,
190
$15,
050
$15,
150
0.52
0.77
167
$2,8
00
$2,7
85
$2,8
50
5.77
0.57
513
$16,
922
$16,
662
$16,
762
0.17
0.91
968
$3,9
70
$3,9
45
$3,9
95
1.62
0.65
414
$13,
660
$13,
430
$13,
690
2.86
0.23
969
$1,3
47
$1,3
67
$1,4
02
1.29
0.52
615
$14,
968
$14,
908
$14,
988
0.62
0.98
970
$1,9
50
$1,9
05
$1,9
85
3.24
0.19
816
$18,
308
$18,
168
$18,
408
50.
172
71$2
,592
$2
,532
$2
,582
1.
930.
412
17$1
2,51
5 $1
1,05
5 $1
1,19
5 84
072
$2,6
45
$2,7
40
$2,8
20
3.75
0.15
318
$23,
473
$23,
273
$23,
533
0.67
0.71
673
$982
$9
62
$1,0
12
13.4
90.
002
19$3
,486
$3
,426
$3
,506
0.
260.
879
74$9
47
$977
$1
,002
3.
430.
1820
$11,
043
$10,
823
$11,
043
2.99
0.22
575
$1,5
46
$1,5
36
$1,5
56
2.8
0.24
723
$5,2
33
$5,1
13
$5,3
13
1.89
0.38
976
$1,8
11
$1,8
06
$1,8
26
0.94
0.62
330
$3,4
96
$3,4
96
$3,5
36
0.75
0.68
777
$586
$5
91
$626
0.
250.
884
35$1
3,61
0 $1
3,09
0 $1
3,29
0 43
.52
078
$1,5
36
$1,5
06
$1,5
41
2.71
0.25
836
$1,8
33
$1,8
53
$1,9
03
2.17
0.33
879
$609
$5
89
$634
3.
680.
159
38$8
,340
$8
,290
$8
,410
0.
240.
887
80$1
42
$142
$1
47
0.47
0.92
540
$3,4
49
$3,3
79
$3,4
39
0.01
0.99
785
$738
$7
38
$773
1.
770.
412
41$3
,779
$3
,729
$3
,829
3.
110.
211
86$1
,780
$1
,785
$1
,785
4.
10.
129
42$2
,869
$2
,819
$2
,949
0.
50.
78p-
valu
e of
less
than
0.0
5 im
plie
s re
ject
ion
of n
ull h
ypot
hesi
s of
equ
ality
Auct
ion
Num
ber
Obs
erve
d Au
ctio
n R
even
ueSi
mul
ated
M
inSi
mul
ated
M
axC
hi
Squa
rep-
valu
eAu
ctio
n N
umbe
r
Obs
erve
d A
uctio
n R
even
ueSi
mul
ated
M
inSi
mul
ated
M
axC
hi
Squa
rep -
valu
e1
$16,
190
$16,
210
$16,
270
10.2
10.
176
44$2
,007
$1
,987
$2
,077
1.
120.
57
Auct
ion
Num
ber
Obs
erve
d Au
ctio
n R
even
ueSi
mul
ated
M
inSi
mul
ated
M
axC
hi
Squa
rep-
valu
eAu
ctio
n N
umbe
r
Obs
erve
d A
uctio
n R
even
ueSi
mul
ated
M
inSi
mul
ated
M
axC
hi
Squa
rep -
valu
e1
$16,
190
$16,
210
$16,
270
10.2
10.
176
44$2
,007
$1
,987
$2
,077
1.
120.
572
$24,
790
$24,
670
$24,
990
0.78
0.37
647
$703
$6
63
$713
1.
190.
552
2$2
4,79
0 $2
4,67
0 $2
4,99
0 0.
780.
376
47$7
03
$663
$7
13
1.19
0.55
23
$10,
915
$10,
775
$10,
975
9.53
0.12
149
$1,0
12
$1,0
12.0
0 $1
,052
.00
1.23
0.54
24
$10,
770
$10,
850
$10,
990
2.21
0.33
53$1
,553
$1
,543
$1
,593
0.
50.
779
3$1
0,91
5 $1
0,77
5 $1
0,97
5 9.
530.
121
49$1
,012
$1
,012
.00
$1,0
52.0
0 1.
230.
542
4$1
0,77
0 $1
0,85
0 $1
0,99
0 2.
210.
3353
$1,5
53
$1,5
43
$1,5
93
0.5
0.77
95
$19,
836
$19,
436
$19,
616
4.34
0.11
454
$647
$6
27
$667
1.
840.
398
5$1
9,83
6 $1
9,43
6 $1
9,61
6 4.
340.
114
54$6
47
$627
$6
67
1.84
0.39
86
$16,
068
$16,
168
$16,
628
0.78
0.37
655
$557
$5
27
$547
0.
110.
949
7$1
1,63
3 $1
1,65
3 $1
1,81
3 9.
530.
121
56$1
,147
$1
,137
$1
,157
1.
670.
435
6$1
6,06
8 $1
6,16
8 $1
6,62
8 0.
780.
376
55$5
57
$527
$5
47
0.11
0.94
97
$11,
633
$11,
653
$11,
813
9.53
0.12
156
$1,1
47
$1,1
37
$1,1
57
1.67
0.43
58
$14,
093
$14,
033
$14,
113
0.76
0.69
59$5
90
$590
$6
30
0.03
0.85
79
$11,
610
8$1
4,09
3 $1
4,03
3 $1
4,11
3 0.
760.
6959
$590
$5
90
$630
0.
030.
857
9$1
1,61
0 $1
1,34
0 $1
1,60
0 4.
340.
114
61$1
,313
$1
,193
$1
,273
0.
520.
771
10$1
7,79
2 $1
7,45
2 $1
7,95
2 37
.77
0.31
364
$813
$8
13
$853
0.
220.
894
$11,
340
$11,
600
4.34
0.11
461
$1,3
13
$1,1
93
$1,2
73
0.52
0.77
110
$17,
792
$17,
452
$17,
952
37.7
70.
313
64$8
13
$813
$8
53
0.22
0.89
411
$10,
694
$10,
754
$11,
194
0.81
0.93
865
$1,1
04
$1,1
04
$1,1
34
1.4
0.49
612
$15,
190
$15,
050
$15,
150
0.52
0.77
167
11$1
0,69
4 $1
0,75
4 $1
1,19
4 0.
810.
938
65$1
,104
$1
,104
$1
,134
1.
40.
496
12$1
5,19
0 $1
5,05
0 $1
5,15
0 0.
520.
771
67$2
,800
$2
,785
$2
,850
5.
770.
575
13$1
6,92
2 $1
6,66
2 $1
6,76
2 0.
170.
919
68$3
,970
$3
,945
$3
,995
1.
620.
654
$2,8
00
$2,7
85
$2,8
50
5.77
0.57
513
$16,
922
$16,
662
$16,
762
0.17
0.91
968
$3,9
70
$3,9
45
$3,9
95
1.62
0.65
414
$13,
660
$13,
430
$13,
690
2.86
0.23
969
$1,3
47
$1,3
67
$1,4
02
1.29
0.52
615
$14,
968
$14,
908
$14,
988
0.62
0.98
970
$1,9
50
$1,9
05
$1,9
85
3.24
0.19
814
$13,
660
$13,
430
$13,
690
2.86
0.23
969
$1,3
47
$1,3
67
$1,4
02
1.29
0.52
615
$14,
968
$14,
908
$14,
988
0.62
0.98
970
$1,9
50
$1,9
05
$1,9
85
3.24
0.19
816
$18,
308
$18,
168
$18,
408
50.
172
71$2
,592
$2
,532
$2
,582
1.
930.
412
16$1
8,30
8 $1
8,16
8 $1
8,40
8 5
0.17
271
$2,5
92
$2,5
32
$2,5
82
1.93
0.41
217
$12,
515
$11,
055
$11,
195
840
72$2
,645
$2
,740
$2
,820
3.
750.
153
18$2
3,47
3 $2
3,27
3 $2
3,53
3 0.
670.
716
73$9
82
$962
$1
,012
13
.49
0.00
217
$12,
515
$11,
055
$11,
195
840
72$2
,645
$2
,740
$2
,820
3.
750.
153
18$2
3,47
3 $2
3,27
3 $2
3,53
3 0.
670.
716
73$9
82
$962
$1
,012
13
.49
0.00
219
$3,4
86
$3,4
26
$3,5
06
0.26
0.87
974
$947
$9
77
$1,0
02
3.43
0.18
19$3
,486
$3
,426
$3
,506
0.
260.
879
74$9
47
$977
$1
,002
3.
430.
1820
$11,
043
$10,
823
$11,
043
2.99
0.22
575
$1,5
46
$1,5
36
$1,5
56
2.8
0.24
723
$5,2
33
$5,1
13
$5,3
13
1.89
0.38
976
$1,8
11
$1,8
06
$1,8
26
0.94
0.62
320
$11,
043
$10,
823
$11,
043
2.99
0.22
575
$1,5
46
$1,5
36
$1,5
56
2.8
0.24
723
$5,2
33
$5,1
13
$5,3
13
1.89
0.38
976
$1,8
11
$1,8
06
$1,8
26
0.94
0.62
330
$3,4
96
$3,4
96
$3,5
36
0.75
0.68
777
$586
$5
91
$626
0.
250.
884
30$3
,496
$3
,496
$3
,536
0.
750.
687
77$5
86
$591
$6
26
0.25
0.88
435
$13,
610
$13,
090
$13,
290
43.5
20
78$1
,536
$1
,506
$1
,541
2.
710.
258
36$1
,833
$1
,853
$1
,903
2.
170.
338
79$6
09
$589
$6
34
3.68
0.15
935
$13,
610
$13,
090
$13,
290
43.5
20
78$1
,536
$1
,506
$1
,541
2.
710.
258
36$1
,833
$1
,853
$1
,903
2.
170.
338
79$6
09
$589
$6
34
3.68
0.15
938
$8,3
40
$8,2
90
$8,4
10
0.24
0.88
780
$142
$1
42
$147
0.
470.
925
38$8
,340
$8
,290
$8
,410
0.
240.
887
80$1
42
$142
$1
47
0.47
0.92
540
$3,4
49
$3,3
79
$3,4
39
0.01
0.99
785
$738
$7
38
$773
1.
770.
412
41$3
,779
$3
,729
$3
,829
3.
110.
211
86$1
,780
$1
,785
$1
,785
4.
10.
129
40$3
,449
$3
,379
$3
,439
0.
010.
997
85$7
38
$738
$7
73
1.77
0.41
241
$3,7
79
$3,7
29
$3,8
29
3.11
0.21
186
$1,7
80
$1,7
85
$1,7
85
4.1
0.12
942
$2,8
69
$2,8
19
$2,9
49
0.5
0.78
p-va
lue
of le
ss th
an 0
.05
impl
ies
reje
ctio
nof
nul
l hyp
othe
sis
of e
qual
ity
42$2
,869
$2
,819
$2
,949
0.
50.
78p-
valu
e of
less
than
0.0
5 im
plie
s re
ject
ion
of n
ull h
ypot
hesi
s of
equ
ality
Tabl
e 1a
– C
hi-S
quar
e Te
st fo
r Sim
ilarit
y of
Bid
Dis
tribu
tion
26
The results presented in Table 1a show the application of the chi-square test. We fail to
reject our null hypothesis if we get a p-value >0.05. Of the 55 auctions shown in this table we
fail to reject our null hypothesis in all but 3 (auctions 17, 35 and 73) of the cases. Table 1b
shows similar data for those 29 auctions that failed to meet the assumptions required to use the
chi-square test. For these, we have resorted to using the binomial signs test.
Auction Number
Observed Revenue
Simulated Mean
Simulated Min
Simulated Max p-value
10 $ 17,792 $ 17,501 $ 17,452 $ 17,952 0.999601 21 $ 11,175 $ 10,894 $ 10,835 $ 10,955 0.999998 22 $ 1,977 $ 1,835 $ 1,797 $ 1,857 0.254037 24 $ 2,456 $ 2,429 $ 2,396 $ 2,456 0.969859 25 $ 1,525 $ 1,499 $ 1,485 $ 1,525 0.954083 27 $ 1,397 $ 1,400 $ 1,377 $ 1,417 0.663923 28 $ 2,156 $ 2,123 $ 2,076 $ 2,156 0.122549 29 $ 2,256 $ 2,271 $ 2,236 $ 2,316 0.695468 31 $ 3,135 $ 3,107 $ 3,095 $ 3,135 0.985681 32 $ 2,377 $ 2,350 $ 2,337 $ 2,377 0.8738 33 $ 4,973 $ 4,940 $ 4,913 $ 4,973 0.826703 34 $ 3,584 $ 3,547 $ 3,524 $ 3,564 0.912039 37 $ 10,665 $ 10,692 $ 10,635 $ 10,805 0.133456 39 $ 788 $ 778 $ 748 $ 818 0.608374 45 $ 1,885 $ 1,907 $ 1,905 $ 1,915 0.84408 46 $ 2,117 $ 2,036 $ 2,003 $ 2,083 0.968891 48 $ 788 $ 800 $ 758 $ 828 0.122059 50 $ 675 $ 666 $ 665 $ 675 0.418515 51 $ 897 $ 884 $ 877 $ 887 0.902318 52 $ 387 $ 378 $ 367 $ 397 0.181963 57 $ 1,247 $ 1,233 $ 1,227 $ 1,247 0.8738 58 $ 2,645 $ 2,623 $ 2,615 $ 2,635 0.990591 60 $ 1,506 $ 1,514 $ 1,506 $ 1,566 0.076501 63 $ 2,774 $ 2,718 $ 2,674 $ 2,794 0.297668 81 $ 1,397 $ 1,380 $ 1,377 $ 1,387 0.444137 82 $ 40 $ 20 $ 20 $ 20 0.885114 83 $ 1,289 $ 1,284 $ 1,264 $ 1,289 0.0625 84 $ 499 $ 507 $ 494 $ 519 0.927988 87 $ 1,431 $ 1,447 $ 1,431 $ 1,461 0.890625
Table 1b– Binomial Signs test for data with cell frequencies < 5 and 2 bid levels
27
In all the 29 cases above we expect no significant difference between the observed and
the expected distribution of the bid levels. Together with Table 1a’s success in 52 out of 55
auctions, this gives us great confidence in the ability of the simulator to replicate the bidding
strategies, and by extension the revenue generation dynamics of the real world auctions that we
tracked.
Further robustness of the simulation can be deduced by examining the range of the
simulated revenue. Observe that this is quite small with the maximum range being 9% of the
observed revenue and the median of the revenue ranges being only 2% of the observed revenue.
This is an important parameter to consider in simulations since excess variability can reduce the
implications generated from simulating a process.
Before we discuss the optimization of revenue for these auctions, let us briefly provide
insights for the reasons of failure of our simulation tool in replicating 3 of the 86 auctions, i.e.,
auctions numbered 17, 35 and 73.
5.2 Auctions that could not be replicated
We believe that in the auction numbers 17 and 35 all the winning bids are at the level that
coincides with the upper bound case in Yankee auctions since all the winning bids are at the
same level and the observed revenue is greater than the highest revenue generated by the
simulator. The probability of actually realizing the upper bound is very small since it requires
that the marginal bidder (the person with the highest losing bid) is the first one to bid at the
previous level . Such an occurrence requires that bid sequences from the start follow a specific
pattern. Therefore, we believe that indeed the simulation model does generate reasonable range
of revenues, however, in this case the observed revenue is the upper bound of possible revenues.
For auction 73, we have an unusual case when one of the bidders chose to bid for 61 out
of the 85 items, eventually winning 42. Observe that, multiple quantity bids are allowed in such
auctions, and are modeled in the simulation as multiple single-item bids of similar type, say
participatory. The overall success of the simulator in replicating the original auctions gives us
28
confidence that this is indeed a reasonable approximation of the real-world strategic behavior by
the bidders. However, this modeling assumption begins to get tested as the quantity becomes
unusually large, as in auction 73. Given that this was an unusually large quantity bid with respect
to the auctions we tracked, and it was difficult for the simulator to capture the strategy space of
all the 60 bidders who were modeled to behave as the large quantity bidder.
In the next subsection we present results where we try to maximize the revenue by
changing the bid increment and observing its impact on average revenue.
5.3 Investigating the effect of bid increment on auction revenue
Since the auctions number 17, 35 and 73 were not adequately replicated we did not use
them in further analysis. For rest of the auctions we ran the simulation program with different
minimum bid increments ranging from $1 - $20. During these simulation runs the valuations of
the bidders remained the same as in the case with the original bid increment. Figure 5 presents
some representative patterns of average revenue generated from these simulation runs. Note that
the revenue does not seem to be a monotonic function of the bid increment. In auction number 10
and 18 there are significant peaks and valleys with several local optima. One common and
interesting observation that can be made from these graphs is that the local optima seem to be at
the multiples of a given number. For example in auction number 10, the local optima seems to be
occurring at bid increments of 3, 6, 9, 12 and 15; in auction number 24 it seems to be at 4, 8 and
12; and in auction number 21 at 6, 12, and 18. Intuitively, this happens because the different
multiples of the same number generate an overlapping set of feasible bid-levels. Together with a
fixed marginal consumer’s valuation V, this leads to identical values of Bmax, the marginal
consumers bid, leading to the regular patterns of local optima that we observe.
Table 2 presents the results of our study. The second column presents the observed bid
increment and the third column presents the optimal bid increment, i.e., the bid increment that
yielded the highest average revenue. We also provide, in the fourth column, the largest bid
increment that had a statistically equivalent average revenue at 5% significance. The motivation
29
for providing this information is that a smaller bid increment implies more rounds of bidding
activity to reach the similar equilibrium as achieved by a larger bid increment. Therefore, it may
be preferable to use the largest bid increment that provides the highest level of revenue. The
fifth column presents the actual auction revenue and the sixth presents the minimum revenue
produced by the optimal bid increment. It should be noted that in 44 out of the 86 cases the
range of revenue from the simulation is such that the lowest revenue attained using the optimal
bid-increment is higher than the revenue obtained with the actually used real-world bid
increment. In these cases, using the optimal bid-increment is a dominant strategy and it virtually
assures that the observed revenue will be higher than that with the bid increment used.
Additionally, looking across the 86 auctions, we observe an average 1.42% percent increase in
revenue by adopting the optimal bid-increment. In light of the fact that the items sold through
these auctions are often sold at fractional margins this represents a substantial gain in revenue.
Only 14 out of the 86 cases have support for bid increments greater than or equal to the original
bid increment k, with the original bid increment k being among the highest revenue generator for
only four auctions.
30
Auction 10
$17,440.00$17,460.00$17,480.00$17,500.00$17,520.00$17,540.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Auction 19
$3,350.00
$3,400.00
$3,450.00
$3,500.00
$3,550.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Auction 21
$10,800.00$10,820.00$10,840.00$10,860.00$10,880.00$10,900.00$10,920.00$10,940.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Auction 24
$2,340.00$2,360.00$2,380.00$2,400.00$2,420.00$2,440.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Figure 5 – Some Patterns of Average Revenue with Different Bid Increments
31
Simulation Number
Bid Increment (k)
Optimal k*
Recommended k
Auction Revenue
Min Revenue using k*
SimulationNumber
Bid Increment (k)
Optimal k*
Recommended k
Auction Revenue
Min Revenue using k*
1 20 4 15 16190 16218* 44 10 3 10 2007 2016* 2 20 13 13 24790 24990* 45 10 5 5 1885 1905* 3 20 3 12 10915 10979* 46 10 12 12 2117 2017 4 20 1 1 10770 11151* 47 10 9 15 703 693 5 20 6 11 19836 19576 48 10 4 14 788 800* 6 20 1 9 16068 16460* 49 10 5 5 1012 1017* 7 20 1 14 11633 11869* 50 10 13 13 726 712 8 20 7 7 14093 14117* 51 10 1 1 675 689* 9 20 4 11 11610 11564 52 10 3 3 897 882 10 20 20 20 17792 17428 53 10 2 15 387 375 11 20 1 4 10694 11129* 54 10 1 2 1553 1572* 12 20 1 1 15190 15201* 55 10 5 11 647 632 13 20 3 9 16922 16741 56 10 10 10 557 537 14 20 4 6 13660 13702* 57 10 3 12 1147 1143 15 20 2 2 14968 15178* 58 10 13 13 1247 1236 16 20 1 15 18308 18320* 59 10 1 5 2645 2624 17 20 11 15 12515 11191 60 10 4 13 590 614* 18 20 8 14 23473 23369 61 10 4 15 1506 1508* 19 20 3 3 3486 3510* 63 10 9 10 1313 1206 20 20 1 10 11043 11032 64 10 1 7 2774 2700 21 20 5 11 11175 10895 65 10 3 5 813 822* 22 20 8 15 1977 1833 66 10 1 4 1104 1123* 23 20 1 11 5233 5213 67 5 1 1 10960 11113* 24 20 20 20 2456 2396 68 5 1 2 2800 2872* 25 20 2 13 1525 1531* 69 5 1 1 3970 4019* 26 20 1 3 2065 2080* 70 5 1 1 1347 1422* 27 20 20 20 1397 1377 71 5 1 3 1950 1969* 28 20 15 20 2156 2116 72 5 1 1 2592 2594* 29 20 2 12 2256 2266* 73 5 1 1 2645 2901* 30 20 5 11 3496 3521* 74 5 1 1 982 991* 31 20 9 14 3135 3110 75 5 1 1 947 1005* 32 20 15 20 2377 2352 76 5 1 2 1546 1553* 33 20 9 9 4973 4939 77 5 5 5 1811 1806 34 20 5 15 3584 3544 78 5 1 1 586 630* 35 10 1 3 13610 13334 79 5 1 4 1536 1529 36 10 1 1 1833 1884* 80 5 1 4 609 605 37 10 3 9 10665 10751* 81 5 5 15 142 142 38 10 5 5 8340 8305 82 5 5 7 1397 1377 39 10 4 7 788 792* 83 5 1 1 40 24 40 10 5 15 3449 3394 84 5 1 1 1289 1287 41 10 5 6 3779 3769 85 5 1 1 499 509* 42 10 1 3 2869 2913* 86 5 1 1 738 764* 43 10 1 11 2509 2498 * indicates that the revenue generated with optimal k is statistically greater than that of implemented k. Recommended k presents the largest k that generates statistically equivalent revenue to that with optimal k.
Table 2 – Effect of Bid Increment on Auction Revenue
32
6.0 The Simulator as a Risk-Free Cost-Effective Decision Tool
One of the main advantages of having a reliable and robust simulator of a real-world
process, such as online auction market, is the potential of cost-effectively testing a variety of
strategies that would be otherwise too risky to test in a real-world setting. A wide range of
sensitivity analysis can be performed and comparative-statics can be generated for parameters of
interest. While the full demonstration of these capabilities is beyond the scope of this paper, we
present some of our initial thoughts in these directions that should whet the appetite of the
interested reader.
6.1 Hybrid Bidding Strategies
Consider the possibility of a bidder employing a combination of the bidding strategies
that we earlier described in Section 3.2. One such hybrid strategy could be a participator who
does not bid at the required minimum bid level. Instead, she jumps the bid by a certain number
(say one) of bid increment(s), with the expectation that she will be able to exploit the time-
priority and be the early entries at the next bid level. For instance, if in a 6-item auction the bid
increment is $10 and the current minimum required bid is $100, the hybrid participator chooses
to bid $110. Further, she may do this heuristically, employing a decision rule of the form, if the
number of bidders currently winning at a level higher than the minimum required bid is greater
than half the lot-size then jump bid. Relating this to the example above, the hybrid participator
will jump bid if there are 3 or more winners currently winning at values greater than $100, such
that she feels that there is high likelihood of the bid-levels going to next higher bid level, i.e,
$110. The decision to do this when half the winning bids are at high-levels, as opposed to some
other fraction, is the heuristic part of this rule. In future research, we will be treating the optimal
determination of this fraction as an interesting analytical and computational challenge.
For demonstrative purposes we randomly chose one bidder, following a participatory
strategy in the original auction, in a subset of the auctions we tracked to follow the hybrid
participatory strategy. To isolate the impact of the new bidding strategy, we compared the
33
performance of the same bidder adopting the two different strategies keeping all other
parameters the same. The metric used for the comparison was the average price paid by the
bidder over the 30 repetitions of the simulation, with lower being better. The results are
presented in Table 3.
Bidder Strategy Auction Number Bid Increment Participator Hybrid Participator Hybrid T-Stat Mean Price paid Variance (1-tail)
72 5 $ 74.00 $ 72.33 0.00 14.71 1.8439** 40 10 $ 307.67 $ 309.00 11.95 0.00 -2.1122**18 20 $ 710.33 $ 718.33 102.99 47.82 -3.5681*** 10 20 $ 1,420.33 $1,421.00 13.33 0.00 -1.009 20 $ 571.00 $ 581.00 103.45 0.00 -5.3851***
Table 3 – Hybrid v. Participatory Bidding Approaches
We randomly chose 5 auctions with different bid increments and found that the evidence
was of a mixed nature. In auction 72 the hybrid strategy was significantly better than the
participatory strategy, whereas in auctions 9, 18, and 40 the participatory strategy was
significantly better. In the case of auction 10 there was no significant difference in the two
strategies. Another interesting feature we observed was that in auction 72 the adopting the hybrid
strategy gave the bidder a 10% more chance of winning the auction as compared to the
participatory strategy. Recall that in the simulation it is not guaranteed that a participator, who
won the real-world auction, will also emerge as an eventual winner. There are potentially many
bidders whose valuations are clustered around the marginal consumer’s valuations and any of
these could eventually win. The detailed mechanics of this process are explained in Bapna, Goes,
Gupta (2000).
The evidence of Table 3 leads us to believe that this is an interesting area of future
research. Among the issues of interest will be the enumeration of the hybrid strategy space that
can be employed by the bidders resulting from a combination of the three core strategies we have
identified. Subsequently, can we find dominant strategies that are either pure or contingent in
nature?
34
7.0 Summary and Conclusions
In this paper we have presented a cost-effective tool in the form of a simulation model
that replicates and subsequently optimizes the design of online Yankee auctions. Tools such as
these can be used ex-ante in a dynamic marketplace, potentially avoiding many of the pitfalls
that can emerge from costly entrepreneurial ventures that resemble uncontrolled field
experiments.
The simulation model uses the observed bids from real online auctions to instantiate the
parameters and implements broad bidding strategies for replicating a given auction. Our results
indicate that the simulation model works very well, with 83 out of the 86 auctions successfully
being replicated.
The simulation model can be used to change the controllable parameters such as bid
increment, starting bid amount, and other rules of auctions to investigate the impact on the
auctioneers' revenue. In addition, as shown, the effect of alternative strategies by the bidders and
its impact on their surplus can also be studied.
This paper investigated the impact of changing the minimum required bid increment on
the auctioneers' revenue. Our result indicated that for the majority of the auctions, the
auctioneers used a significantly higher value for the minimum bid increment than the optimal
value. While the magnitude of the impact on revenue was small for the majority of the cases, in
light of small margins on the items usually sold through these auctions, the impact on profit may
be greater than 100%.
In summary, the simulation model developed in this paper is a powerful and cost-
effective tool, which can be used by the auctioneers to investigate a variety of issues. These
range from setting different control parameters and rules for a given auction to investigating the
impact of bidding strategies on consumer surplus as well as the auctioneers' revenue. We believe
that by using a calibrated model as a benchmark to compare the results, the simulation model can
be used to generate more trustworthy results as compared to the models using hypothetical
distributions to generate user valuations.
35
References
1. Banks, J. and Carson, J. S., Discrete-Event System Simulation, Prentice-Hall, Inc.,
Englewwod Cliffs, New Jersey, 1984.
2. Bapna, R., Goes, P., and Gupta, A., "A Theoretical and Empirical Investigation of Multi-
item On-line Auctions," Information Technology and Management, Jan 2000, 1(1), 1-23.
3. Bapna, R., Goes, P., and Gupta, A., "Online Auctions: Insights and Analysis,"
Forthcoming in Communications of the ACM, 2001.
4. Kauffman, R. J., and Riggins, F. J., "Information Systems and Economics,"
Communications of the ACM, August 1998, Vol 41, 8, 32-34.
5. Lee, B., Mehta, K., Efficiency Comparison in Electronic Market Mechanisms: Posted
Price Versus Auction Market, WISE Conference 1999, Charlotte, NC.
6. McAfee, R. P., and McMillan, J., "Auctions and Bidding," Journal of Economic
Literature, 25 (1987), 699-738.
7. Milgrom, P., “Auctions and Bidding: A Primer," Journal of Economic Perspectives, 3
(1989), 3-22.
8. Milgrom, P., and Weber, R, “A Theory of Auctions and Competitive Bidding,”
Econometrica, 50(1982), 1089-1122.
9. Nash, J., "The Bargaining Problem," Econometrica, 18, 155-162.
10. Nwana, H., Rosenschein, J., Sandholm, t., Sierra, C., Maes, P., & Guttmann, R. (1998).
“Agent-Mediated Electronic Commerce: Issues, Challenges and Some Viewpoints,”
Proc. Of 2nd Intl. Conf. On Autonomous Agents, Minneapolis, MN, May 9-13, pp. 189-
196.
11. Pavlou, P. A., Ba, S., "Does Online Reputation Matter? – An Empirical Investigation of
Reputation and Trust in Online Auction Markets," Proceedings of the AMCIS 2000
Conference, Long Beach CA.
12. Rothkopf, M. H., and Harstad, R. M., "Modeling Competitive Bidding: A Critical Essay,"
Management Science, 40 (3), 1994, 364-384.
36
13. Vakrat, Y. and Seidmann, A. "Can Online Auctions Beat Online Catalogs?," In P. De and
J DeGross (eds.), Proceedings of the 20th International Conference on Information
Systems (ICIS '99), Charlotte, NC, December, 1999a.
14. Vakrat, Y., and Seidmann, A., "Analysis and Design Models for Online Auctions,"
Proceedings of the INFORMS Conference on Information Systems and Technology
(CIST), May, 1999b, 64-78.
15. Van Heck, E., and Vervest, P., “How Should CIO’s Deal With Web-Based Auctions?”
Communications of the ACM, July1998, Vol 41, 7, 99-100.
37
Appendix 1 - The Simulation Front End