Title: Modeling the Dynamics of Online Auctions Using a Functional Data Analytic Approach
1Modeling the Dynamics of Online Auctions Using a
Functional Data Analytic Approach
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- Galit Shmueli ( Wolfgang Jank)
- Dept of Decision Information Technologies
- Robert H. Smith School of Business
- University of Maryland, College Park
December 2004
2Overview
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- Online auctions
- Importance
- How they work
- Classical empirical research and new
opportunities - Where are the statisticians?
- Using FDA for
- Representing auctions
- Studying auction dynamics
- Comparing auctions
- Exploring relations with other variables
- Current Future directions
3Online Auctions
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- Central in the eMarket place (eBay, Yahoo!,
Amazon.com) - High accessibility, low transaction costs
- eBay has more than 27M active users (from over
61M registered). Every moment there are 10M
items across more than 43,000 product categories
amounting to nearly 15 billion in gross
merchandise sales (BusinessWeek, 2003)
4Online Auctions
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- The focus of much empirical research
Players IS and economists
5eBay.com
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- Is by far the largest C2C auction site
- Buy/sell anything imaginable
- (Almost) anyone can buy/sell. You need a credit
card to register (free). - In lots of countries
6How eBay auctions workSelling an item
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Set some auction features (duration, opening
price,) Describe item Bells whistles
more info on shipping, text description,
payment options, etc.
7How eBay auctions work Bidding on an item
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- Choose auction
- Proxy bidding
- Place max bid
- eBay bids for you
- Price increases by one increment
- Highest bidder pays 2nd highest bid
- Highest bid is not disclosed!
8Bidding on an item cont.
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- Auction theory bid your max and leave
- In practice lots of sniping
- Sniping agents (wow more data!)
9Research Qs Asked by Economists and IS
researchers
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- Auction design mechanisms mostly regressions on
final price - Lucking-Reiley et al Opening Bid, Number of
Bidders, Number of Bids, Length of Auction,
Reputation of Seller - Bapna et al Bid increments
- Winners Curse structural model prior
- Winner likely to over-pay (Bajari Hortacsu)
- Bid Shilling t-tests
- Fraudulent price-pushing by the seller
(Kauffman Wood) - Reputation and trust regression, probit model
- Seller rating effect on price or P( rating)
(Wood et al Ba Pavlov) - Bid Sniping bid time CDF
- Last minute biding to increase chances of success
(Roth Ockenfels) - But early bidding also prevalent
- Bidding strategies k-means clustering
- 3 strategies Participators, evaluators,
opportunists (Bapna et al.)
10No statisticians playing the game!
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11Why? Data Accessibility?
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- eBay displays data for all auctions completed in
the last 30 days. - Millions of auctions (how do you sample?)
- Data are on in HTML format!!!!
- Researchers use spiders (web agents)
- People usually write their own code
- eBay changes the rules and formats
- eBay does NOT like spiders
- You really need some programming expertise
- Commercial software (Andale, Hammertap)
- data directly from eBay
- limited (mostly aggregates)
- Expensive, unreliable
12Lots of opportunities there!
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- No statistical framing (sample/pop, type of data,
etc) - No data visualization
- Mostly traditional statistical methods
- Ignoring data
- Sampling issues
- and more.
13Unstated assumptions in current (static) approach
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- An auction is an observation from a population of
eBay auctions (US market, certain time-frame,
etc.) - Sample collected by web-spider is random and
representative of population. - Data structure multivariate, with a fixed set of
measurements on each auction - Auctions are independent
14Visualizing Online Auction Data
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- Lots of empirical research, but no-one is LOOKING
at the data! - Ordinary displays not always useful
Shmueli Jank, Visualizing online auctions,
JCGS, forthcoming
15Enlightening Visualizations
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- Detecting Fraud (color seller rating)
16Advanced visualizations for interpreting modeling
results
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- Surplus from eBay auctions (Bapna, Jank,
Shmueli, 2004) - Data from sniping agent gives highest bid
- What are factors that affect surplus?
- Advanced, interactive visualizations help learn
the multidimensional structure of the data and to
interpret results of complicated models! - Beats heavy statistical software like SAS
17Understanding complicated results surplus model
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18Back to current research
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- Almost exclusively static
- Auction Snapshot at end
- response price, bids,
- But eBay does show complete bid histories!
19Our new dynamic approach
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- Auction complete bid history
- Response
- Price over time
- of bidders over time
- Average bidder rating over time
- Interested in auction dynamics!
- Car/horse race
20Data Structure Challenges
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- Each bid history time series measured at
unequally-spaced time points, closed interval. - Bidding is usually sparse at mid-auction and
dense at auction end - Different auctions
- Different number of bids, placed at different
times - Different durations
- Much variability across auctions
- We have LOTS of auctions!
- How to represent an auction?
21Alternative representation Curves!
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- Functional Data Analysis is a modern statistical
approach suitable for modeling objects (curves,
3D objects, etc), not just scalars/vectors. - Made famous by the two monographs of
Ramsay Silverman - http//ego.psych.mcgill.ca/misc/fda
22Example of FDA Handwriting
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- Possible goal detect fraudulent signature
- Twenty traces of writing fda by same person
- We can think of these traces as functions with
X,Y coordinates - Use FDA to explore and model similarities and
differences between the 20 traces.
23FDA for bidding data
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- Bids from single auction are represented by
single entity - Assume a very flexible underlying curve for all
auctions - Storage and computation represent each auction
by some basis function and a set of coefficients - Perform statistical analyses on
- the coefficients, or
- a grid taken on the curves
24The bidding path (the functional object)
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- An auction is represented by its bidding path, a
continuous function relating (or other!) over
time - In practice, bidding paths are observed at random
discrete time points. These are in the observed
bid histories - We aim to reconstruct the unobservable continuous
profile from the observed discrete bid history
25Recovering the bidding path
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- Use smoothing to recover the bidding path
- One useful smoother is the Penalized Smoothing
Spline - Piecewise polynomial with smooth breakpoints
- Penalize curvature by minimizing
26Smoothing Splines for recovering bidding paths
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- Strengths
- Good tradeoff between fit and local variability
- Computationally cheap ( numerically stable)
well approximated by a finite set of Bspline
basis functions
- For smooth derivatives penalize higher order
derivatives - Challenges
- Must determine l and knots
- Requires prior interpolationsmoothing
- Curves not necessarily monotone
27From bid histories to bidding paths potential
enhancements
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- Use live-bids rather than proxy-bids
- Use monotone splines (non-decreasing)
- Integrate auction theory into curve requirements
(knot positions, polynomial order, etc)
28Learning about Auction dynamics (the auction as a
car race)
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- 1st derivative velocity, 2nd acceleration,
3rd?
Auction 1
29A sample of auctions
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- 158 auctions for new Palm M515 PDAs
- 7-days, new ? 250
30And their derivatives
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31Curve fitting Sensitivity Analysis
- Smoothing splines pre-smoothing ? monotone
smoothing splines - Choice of knots hardly influential
- Smoothing parameter chosen ad-hoc
32Smoothing spline vs.Monotone smoothing spline
33Basis function expansions
- Splines linear combination of B-splines
- Monotone The ratio can be
approximated by a linear combination of basis
functions j - Fitted function
34Exploratory analysis of curves Auction Explorer
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35Handling the curves
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- Two approaches
- Functional datum (fd)
- Use curve coefficients directly in analysis
- When linear representation linear operations
- Grid
- Use a set of discrete values from a grid taken on
the curves. - When nonlinear operations and nonlinear
representation (e.g. monotone splines)
36Exploring Modeling The Auction Curves
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- Summaries of curves
- Average curve
- 95 CI for curve
- Bid paths and/or derivative curves
- Compare subsets of auctions
37Exploratory analysis Auction Clustering
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- Using the bidding curve coefficients we apply
cluster analysis (k-medoids)
Early bidding
Sniping
38Comparing cluster dynamics Phase-plane plots
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Sniping
Early bidding
39Characterizing the 2 Profiles
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Opening Bid Seller Rating Bidder
Rating Bids Early 46.01(7.94) 908.16
(106.08) 101.86 (10.42) 7.04 (0.52)
Late 22.31(6.94) 1171.54 (292.89) 94.29
(13.29) 11.13 (0.83)
- Two profiles diverse wrt Opening Bid
- Investigate this influence dynamically via
Functional Regression
40functional-PCA When do auctions behave
differently?
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Principal components as perturbations of the mean
- When during the auction do bid curves deviate
most/least? - PCA varimax
- 300 premium wristwatches
41Functional Regression Models
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- Involve a curve as a response/predictor
- In our case, response bidding path
- Predictors
- Static opening price, seller rating, etc.
- Dynamic current bidders, current avg bidder
rating - Grid fit a regression model at each grid point
and then interpolate the coefficients
42Functional Regression of Bidding Path vs. Opening
Bid
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Estimated Parameter Curve
43Functional Regression of Bidding Acceleration vs.
Opening Bid
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Estimated Parameter Curve
44Interpretation Opening Bid and Auction Energy
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Value of Item
Value of Item
Potential Market Energy left in the auction
Open Bid
Open Bid
45Current Future Directions
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- Real-time forecasting of bidding paths of ongoing
auctions - Representing an auction in 2D (price bids over
time) - Modeling other aspects of auction data
- Consumer surplus with Ravi Bapna
- Bid arrival process with Ralph Russo (Iowa)
- New predictors currency, category, and dynamic
ones - Effects of auction design changes
- eBay addiction
- Other eCommerce and IT applications
- Papers http//www.smith.umd.edu/ceme/statistics
46Extras
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47Smoothing Spline Parameters
- Order of the Spline
- cubic spline popular, provides smooth fit 2nd
derivative (curvature), no breakpoints - To obtain m smooth derivatives, use spline of
order m2. - Knot locations (breakpoints)
- The more knots, the more flexible (wiggliness)
- Tradeoff between data-fit and variability of
function - Smoothness penalty parameter l
- l? 0 fit approaches exact interpolation
- l? ? fit approaches linear regression
48Alternatively bspline basis functions
- B-splines on fixed grid of knots (s1lts2ltsq) give
good approximation to most smooth functions - Computational aspect numerical stability,
especially for irregularly distributed
time-points - They form a set of natural cubic splines with
limited support
Basis function i
coefficients