Optimal Release and Empirical Analysis of Online Auctions Fredrik Odegaard Department of Statistical

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Optimal Release and Empirical Analysis of Online
AuctionsFredrik OdegaardDepartment of
Statistical Actuarial SciencesThe University
of Western Ontario20091112
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Research Objective
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Outline
Motivation Background
Empirical Analysis
Auction Release Model
Empirical Analysis
Final Remarks
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William Vickrey (1914 1996)
Counterspeculation, Auctions, and Competitive
Sealed Tenders, The Journal of Finance 1961

• An even more rapid procedure could be developed,
  with relatively little increase in the apparatus
  required, if each bidder were provided with a set
  of dials or switches which could be set to any
  desired bid, with the electronic or relay
  apparatus arranged to search out the two top bids
  and indicate the person making the top bid and
  the amount of the second bid. (p.23)

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A survey of e-commerce The Economist May 15th
2004 (p.12)
The companys eBay strategy is to build a
global trading platform that operates at the
beginning and the end of the typical bell-curve
of a products life. For instance, when a
successful new product is launched, demand for it
may be so high that it becomes scarce, so it can
be auctioned for more than the recommended
price. . And once its first owner has finished
with it, a used market of uncertain values
emerges.
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Dell_Financial_Services at eBay
163 Dell laptop 1.4GHz Pentium M, 512MB, 40GB
274 Dell desktop 2.4GHz Pentium 4, 256MB, 40GB
(no monitor)
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Desktop (274)IP4 2.4GHz, 256 MB, 40GB
Boxplots of Selected Products
Laptop (163) IPM 1.4GHz, 512MB, 40GB
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Online Auction Release Problem

• A seller with N items to sell, wishes to release
  them in a sequence of auctions so as to maximize
  profit
• Trade-Off
• Holding cost per item per period, h
• Fewer ongoing auctions ? higher expected final
  price
• What is the optimal release policy?

N












day 0
day T
day t
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Markov Decision Process (MDP) Formulation
t
t1
t2
(X1, Y1 0, 0, 1)
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MDP Formulation cont.

• Transition Probabilities
• Price transitions of individual auction with z
  auctions underway
• Two Cases
• Guaranteed Successful Auctions, ?0,0z 0
• Possibly Unsuccessful Auctions, ?0,0z gt 0

Discrete Prices
Continuous Prices
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Price Transition Assumptions

• Assumption 1 monotone price transitions
• The probability that the price transition
  exceeds a given level is increasing in the
  current price.
• Assumption 2 cannibalization effect
• At a given price level, the probability the
  price transition exceeds a given level decreases
  when there are two ongoing auctions.
• Assumption 3 diminishing cannibalization
• The difference in probability of price
  transitions exceeding a given level, between
  having one versus two ongoing auctions, is
  decreasing in the current price.

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Guaranteed Successful Auction - Single Listing

• Value Function, Vt(p1,t10,0,1)
• -2h max?Vt(q,t110,0,1)?q1
  Vt(q,t11r,1,2)?q2?r2
• max-2h?Vt(q,t110,0,1)?q1
  R(p1,t10,0,2)
• Proposition Monotone Value Function
• The value function is increasing in the current
  price of each auction at each time period.
• Theorem Optimal Release Policy
• The optimal release policy is
• a control limit in the current
• price for each time period.

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Possibly Unsuccessful Auctions - Multiple
Re-Listing

• Proposition
• Problem is equivalent to a Negative Dynamic
  Program.
• V(0,t10,0,1) -2h max?V(q,t110,0,1)?q1
  ??V(q,t11r,1,2)?q2?r2
• max-2h?V(q,t110,0,1)?q1
  V(0,t110,0,1)?01
• R(0,t10,0,2) V(0,t20,0,1)?(0,0)
• Proposition
• Optimal solution exist and policy and value
  iteration converges.
• (cf. Bertsekas and Tsitsiklis 1991, Puterman
  1994)
• Proposition
• Once a positive bid arrives the problem reduces
  to the single listing case, and optimal policy is
  a control limit policy.
• Remark If X1 0 then the optimal policy need not
  be monotone.

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Within Period Price Transitions - Laptop
0h ? 12h
12h ? 24h
24h ? 36h
60h ? 72h
48h ? 60h
36h ? 48h
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Probability Distribution of Price Increments
Price in 12 hours
probability density
Current Price
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Price Increments - Zero-Inflated Gamma
Distribution
x - Current Price, y - Elapsed Auction Time, z -
of Ongoing Auctions
Log(? /1-?) ß0 ß1x ß2z ß3xz ß41y24
ß51y36 ß61y48
Log(µ) ?0 ?1x ?2z ?3xz ?41y24
?51y36 ?61y48
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Probability of Price Increment - Bernoulli Event
?
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Positive Price Increment Gamma Distributed
µ
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Expected Price Transition
x - Current Price, y - Elapsed Auction Time, z -
of Ongoing Auctions
Assumption 1 Monotone Price Transitions
Assumption 2 3 Cannibalization Diminishing
Cannibalization
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Goodness-of-Fit Laptop
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Final Remarks

• Discussed an MDP model for deciding when to
  release an online auction
• Discussed an Statistical model for analyzing
  price dynamics over discrete periods of online
  auctions
• Empirically validated assumptions that ensure
  optimal release policy is of a threshold type
• Extensions Real estate market, DVD release,.
• The N item Case
• Price-Dependent Cannibalization
• Link bidding behavior model with optimal release
  model

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Back-Up Slides
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Possibly Unsuccessful Auctions - Multiple
Re-Listing
(0,00,0,1)
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Multiple Re-Listing Numerical Example
let ? 2, Xi 0, 10, 20
20
10
0
Y1
0
1
2
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Zero-Inflated Gamma Distribution
x - Current Price, y - Elapsed Auction Time, z -
of Ongoing Auctions
2.421 - .295z Log(? /1-?) 1.671 - .007x -
.140z (1.299)1y48 na
ns Log(µ) 5.155 - .003x - .123z .004xz
- (.186)1y36 5.959 - .005x