Analytical and Empirical Models of Online Auctions by Fredrik Ødegaard B.Sc. Arizona State University 1996 M.Sc. Stanford University 1998 M.Sc. Stanford University 1998 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Business Administration) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2008 c Fredrik Ødegaard, 2008 Abstract This thesis provides a discussion on some analytical and empirical models of online auctions. The objective is to provide an alternative framework for analyzing online auctions, and to characterize the distribution of intermediate prices. Chapter 1 provides a mathematical formulation of the eBay auction format and background to the data used in the empirical analysis. Chapter 2 analyzes policies for optimally disposing inventory using online auctions. It is assumed a seller has a fixed number of items to sell using a sequence of, possibly overlapping, single-item auctions. The decision the seller must make is when to start each auction. The decision involves a trade-off between a holding cost for each period an item remains unsold, and a cannibalization effect among competing auctions. Consequently the seller must trade-off the expected marginal gain for the ongoing auctions with the expected marginal cost of the unreleased items by further deferring their release. The problem is formulated as a discrete time Markov Decision Problem. Conditions are derived to ensure that the optimal release policy is a control limit policy in the current price of the ongoing auctions. Chapter 2 focuses on the two item case which has sufficient complexity to raise challenging questions. An underlying assumption in Chapter 2 is that the auction dynamics can be captured by a set of transition probabilities. Chapter 3 shows with two fixed bidding strategies how the transition probabilities can be derived for a given auction format and bidder arrival process. The two specific bidding strategies analyzed are when bidders bid: 1) a minimal increment, and 2) their true valuation. Chapters 4 and 5 provides empirical analyzes of 4,000 eBay auctions conducted by Dell. Chapter 4 provides a statistical model where over discrete time periods, prices of online auctions follow a ii zero-inflated gamma distribution. Chapter 5 provides an analysis of the 44,000 bids placed in the auctions, based on bids following a gamma distribution. Both models presented in Chapters 4 and 5 are based on conditional probabilities given the price and elapsed time of an auction, and certain parameters of the competing auctions. Chapter 6 concludes the thesis with a discussion of the main results and possible extensions. iii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Research Objective and Overview of Thesis . . . . . . . . . . . . . . . . . . . 1 2. Brief Background on Auctions and Online Auctions . . . . . . . . . . . . . . . 4 3. The eBay Single-Item Auction Format . . . . . . . . . . . . . . . . . . . . . . 7 4. The Data and Dell Financial Services . . . . . . . . . . . . . . . . . . . . . . . 19 Chapter 2. Optimal Release of Inventory Using Online Auctions: The Two Item Case 38 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2. Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3. Guaranteed Successful Auctions - Single Listing . . . . . . . . . . . . . . . . . 63 4. Possibly Unsuccessful Auction - Multiple Re-Listing . . . . . . . . . . . . . . 88 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 iv Chapter 3. Fixed Bidding Strategies: The Two Auctions Case . . . . . . . . . . . . 100 1. Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2. Auction Rules and Bidder Attributes . . . . . . . . . . . . . . . . . . . . . . . 104 3. Minimum Bid Increment Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 110 4. Truthful Bidding Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Chapter 4. Empirical Analysis of Within Period Price-Increments . . . . . . . . . . 138 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2. Zero-Inflated Gamma Distributed Price-Increments . . . . . . . . . . . . . . . 139 3. Structural Properties For Optimal Auction Release Policy . . . . . . . . . . . 145 4. Generalized Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5. Model Formulation and Selection for DFS Data . . . . . . . . . . . . . . . . . 167 6. Results for DFS Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Chapter 5. Empirical Analysis of Online Bidding Behavior . . . . . . . . . . . . . . 204 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 2. Descriptive Statistics of Bids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 3. Analysis of Bid-Increments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 5. Analysis of Timing of Bids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 6. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Chapter 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 1. Chapter 2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 2. Chapter 3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 3. Chapters 4 and 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 v Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Appendix A. Proofs for Results in Chapter 2 . . . . . . . . . . . . . . . . . . . . . 257 Appendix B. Proofs for Results to Minimal Bid-Increment Strategy in Chapter 3 . 282 Appendix C. Supplement to Truthful Bidding Strategy in Chapter 3 . . . . . . . . 287 Appendix D. GLM Output and Residual Plots for the Models in Chapter 4 . . . . 306 Appendix E. Linear Regression Model of Price-Increments in Final 12hours of an Appendix F. Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Supplementary Material for Empirical Analysis in Chapter 5 . . . . . 316 vi List of Tables 1.1 Minimum Bid Increments on eBay . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Descriptive Statistics - Entire Data Set . . . . . . . . . . . . . . . . . . . . . . . 25 1.3 Auction and Product Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.4 Distribution of Desktop Configuration . . . . . . . . . . . . . . . . . . . . . . . . 30 1.5 Distribution of Laptop Configuration . . . . . . . . . . . . . . . . . . . . . . . . 32 1.6 Product Configuration of Subsets Analyzed . . . . . . . . . . . . . . . . . . . . . 32 1.7 Descriptive Statistic of Final Price . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8 Cross-Tabulation of Desktop D1 Configuration . . . . . . . . . . . . . . . . . . . 35 1.9 Cross-Tabulation of Laptop L1 Configuration . . . . . . . . . . . . . . . . . . . . 37 2.1 Expected Profit of Open Loop Policies - Numerical Example . . . . . . . . . . . 65 2.2 Optimality Equations - Discrete Prices, τ = 3 . . . . . . . . . . . . . . . . . . . . 70 2.3 Optimal Decision vs Holding Cost - Numerical Example . . . . . . . . . . . . . . 80 2.4 Expected Profit vs Holding Cost - Numerical Example . . . . . . . . . . . . . . . 82 2.5 Release Threshold vs Holding Cost - Numerical Example . . . . . . . . . . . . . 83 2.6 Optimality Equations for Multiple Re-Listing . . . . . . . . . . . . . . . . . . . . 93 2.7 Optimal Decision vs Holding Cost (Multiple Re-Listing) - Numerical Example . 98 4.1 Product Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 4.2 Within Period Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.3 Number of Zero and Positive Price-Increments . . . . . . . . . . . . . . . . . . . 171 vii 4.4 Estimated Coefficients for Final Models . . . . . . . . . . . . . . . . . . . . . . . 179 4.5 Residuals and Gamma Shape Parameter (First Period) . . . . . . . . . . . . . . 190 4.6 Residuals and Gamma Shape Parameter (Middle Periods) . . . . . . . . . . . . 190 4.7 Residuals and Gamma Shape Parameter (Final Period) . . . . . . . . . . . . . . 190 4.8 Expected Loss due to Cannibalization (Middle Periods) . . . . . . . . . . . . . . 196 4.9 Expected Loss due to Cannibalization (Final Period) . . . . . . . . . . . . . . . 197 5.1 Descriptive Statistics of Bids and Bidders . . . . . . . . . . . . . . . . . . . . . . 209 5.2 Aggregate Descriptive Statistics of Auctions . . . . . . . . . . . . . . . . . . . . . 210 5.3 Results for Base Model of Bid-Increments in First 71 Hours . . . . . . . . . . . . 225 5.4 Results for Base Model of Bid-Increments Final Hour . . . . . . . . . . . . . . . 227 5.5 Coefficient Estimates for Bid Increments - Up To Final Hour . . . . . . . . . . . 231 5.6 Coefficient Estimates for Bid Increments . . . . . . . . . . . . . . . . . . . . . . . 234 5.7 Deviance Residual and Gamma Shape Parameter Estimate . . . . . . . . . . . . 238 A.1 Optimality Equations - Single Listing Case . . . . . . . . . . . . . . . . . . . . . 260 D.1 Results of Final Model - D1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 D.2 Results of Final Model - D3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 D.3 Results of Final Model - D4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 D.4 Results of Final Model - L1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 D.5 Results of Final Model - L4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 D.6 Results of Final Model - L5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 viii E.1 Results for Normal Linear Regression in Final Period . . . . . . . . . . . . . . . 315 F.1 Results for Laptop - Base Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 F.2 Results Desktop - Base Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 ix List of Figures 1.1 Screen-shot of an eBay Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Time Series of Final Price - Entire Data Set . . . . . . . . . . . . . . . . . . . . . 26 1.3 Histogram of Final Price - D1 and L1 . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4 Distribution of Final Price by Product Specification - D1 . . . . . . . . . . . . . 36 1.5 Distribution of Final Price by Product Specification - L1 . . . . . . . . . . . . . 37 2.1 Time-series of Final Price - D3 and L4 . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 Expected Profit vs Holding Cost - Numerical Example . . . . . . . . . . . . . . . 41 2.3 Time-line for Two Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.4 System State Transitions - Numerical Example . . . . . . . . . . . . . . . . . . . 50 2.5 Within Period Price Transitions - L4 . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.1 Distribution of Final Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.2 Time-line for Bidding Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.3 Minimal Bid Increment Strategy - Numerical Illustration . . . . . . . . . . . . . 112 3.4 Distribution of Price-Transitions - Minimal Bid Increment Strategy . . . . . . . 127 3.5 Distribution of Price-Transitions - Truthful Bidding Strategy . . . . . . . . . . . 135 4.1 Time-line for Auction with Three Periods . . . . . . . . . . . . . . . . . . . . . . 139 4.2 Expected Price-Transition and Feasible Region - Numerical Example . . . . . . . 149 4.3 Illustrations of the Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . 162 x 4.4 Box-plot of Current Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 4.5 Feasible Region and Expected Final Price-Transition . . . . . . . . . . . . . . . 182 4.6 Illustration of Cannibalization Effect - D1, D3, L1, L4, L5 . . . . . . . . . . . . . 188 4.7 Price-Increment vs Current Price - D1 and L1 . . . . . . . . . . . . . . . . . . . 198 4.8 Price-Increment vs Current Price - D3 and D4 . . . . . . . . . . . . . . . . . . . 199 4.9 Price-Increment vs Current Price - L4 and L5 . . . . . . . . . . . . . . . . . . . . 200 4.10 Price-Increment vs Average Number of Ongoing Auctions - D1 and L1 . . . . . 201 4.11 Price-Increment vs Average Number of Ongoing Auctions - D3 and D4 . . . . . 202 4.12 Price-Increment vs Average Number of Ongoing Auctions - L4 and L5 . . . . . 203 5.1 Histogram of Bids and Bid-Increments . . . . . . . . . . . . . . . . . . . . . . . 207 5.2 Bid-Increments vs Current Price . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 5.3 Boxplot of Bid-Increments in Final Hour . . . . . . . . . . . . . . . . . . . . . . 217 5.4 Bid-Increments vs Elapsed Auction Time (Final Hour only) . . . . . . . . . . . 220 5.5 Histogram Timing of Bids and Inter-Arrival Time . . . . . . . . . . . . . . . . . 242 D.1 Deviance and Response Residuals - D1 and L1 . . . . . . . . . . . . . . . . . . . 310 D.2 Deviance and Response Residuals - D3 and D4 . . . . . . . . . . . . . . . . . . . 311 D.3 Deviance and Response Residuals - L4 and L5 . . . . . . . . . . . . . . . . . . . 312 F.1 Current Price vs Elapsed Auction Time . . . . . . . . . . . . . . . . . . . . . . . 317 F.2 Deviance and Response Residual Plots - up to final hour . . . . . . . . . . . . . 318 F.3 Deviance and Response Residual Plots - final hour . . . . . . . . . . . . . . . . . 319 F.4 Number of Bids per Bidder vs Number of Auctions per Bidder . . . . . . . . . . 320 F.5 Inter-Arrival Time vs Elapsed Auction Time and Current Price . . . . . . . . . . 323 xi Notation Time, decision epoch, and period t Length of planning horizon T Duration of an auction τ Starting price of an auction and bidders’ lower bound on the valuation of item p Maximum price of an auction and bidders’ upper bound on the valuation of the item P Bidders’ valuation of the item V Index of auction or item i Current price of auction i Xi High-bid of auction i Hi Elapsed discrete time of auction i (used in Chapters 2 and 4) Yi Elapsed continuous time of auction i (used in Chapters 3 and 5) ti Current price of auction i after Y periods (Xi,t in Chapters 3 and 5) (Xi , Yi ) or Xi,Y Number of ongoing auctions Z Cost per period of holding one unit of inventory h Indicator function; equals 1 if statement in brackets is true, 0 otherwise xii 1{·} Acknowledgements My first and foremost gratitude goes to my advisor and mentor Marty Puterman. His faith and high expectation in taking me on as a PhD student, his analytical and moral (and financial) support, the many hours of stimulating discussions, and the unparalleled inspiration he has provided, is something that I will always remember and never be able to reciprocate. My deepest thanks also goes to my committee members Ivar Ekeland, Mahesh Nagarajan, and Maurice Queyranne for their time, support, and many insightful comments and critiques. A special thanks to Mahesh for always keeping me on track, and for raising challenging questions during our daily Socratic coffee-breaks. Thanks also to Andrew Gray for his outstanding work and programming code; without which this thesis would have been incomplete. I would also like to extend. . . . . . my deepest gratitude to my mentor and friend Pontus Roos for his constant and positive encouragement. . . . a warm hug to Dodie Katzenstein for her care and love of my family. . . . thanks to the Faculty, Staff, and Students at the OpLog Department at Sauder. Special thanks goes to Daniel Granot for providing encouraging, inspirational and always upbeat motivation. A special thanks also goes to Elaine Cho at the PhD program office for all her support. . . . thanks to my four colleagues in the Management Science area group at Ivey for their xiii unwavering support and patience. In particular to Greg Zaric for his invaluable advice which sent me to UBC in the first place. . . . thanks to all my friends, OPs, and family members (through blood and law): I cannot list you all so next round is on me. Last, I would like to thank my parents Dr. Gunni Ødegaard and Dr. Knut Ødegaard for their love, nurture and support over all the years. A special thanks to my brother Karle Ødegaard for keeping the tunes and my girls jamming. Finally, I would like to thank the three most important people in my life, for their love, encouragement, and relentless patience, - Elisa, Sofia, Isabella: Lo Hicimos! Partial funding for my PhD studies and thesis was generously provided by the E.D. MacPhee Memorial Fellowship, Sauder School of Business, Faculty of Graduate Studies, and Professor Martin L. Puterman’s NSERC Discovery grant. xiv Dedication xv CHAPTER 1 Introduction 1. Research Objective and Overview of Thesis Auctions provide an important and integral part of commerce. One of the main appeals with auctions is that they can help solve the challenging pricing decision. If a seller sets the price too high he runs the risk of not selling the product, while if he sets it too low he might forfeit revenue. Similarly, though of a different nature, deciding how much to be willing to pay for a product is a difficult decision. Reasonably, there does not exist a price that is too low for a buyer, while clearly paying too much is either not feasible or not worthwhile. In order for a transaction to occur, the two parties’ problems must be resolved to a mutual agreement. The trade mechanism of auctions provides a relatively easy to implement and often efficient solution. Generally speaking, an auction allocates a product to the buyer who values it the most, while generating the best possible revenue for the seller. This thesis aims to further our understanding of how auctions work, and provide a framework for sellers, buyers, and auctioneers to make better decisions. The main objective is to provide a framework for analyzing the dynamic and stochastic nature of online auctions. There are two main departures from the traditional auction theory framework. First, online auctions are modeled as stochastic processes characterized by various parameters. In other words, the analysis does not follow the standard auction/game theory framework and derive properties of a bid strategy (Bayesian Nash) equilibrium. Second, rather than focusing on the distribution of the final price, the thesis centers on the distribution of intermediate prices of ongoing auctions. Specifically, the analysis mainly 1 considers, the conditional intermediate price-transition probabilities, given certain parameters. It should be clear that the analysis of intermediate prices enables the analysis of the final price, but not necessarily vice versa. An important managerial decision and the main research question addressed is: how should a seller release items for auction if he wishes to maximize his profit? That is, given an inventory of N items, and restricted to sell each item using a single-item auction, what is the optimal release policy? Should he release the N items for auctions 1) simultaneously, 2) as a series of N non-overlapping sequentially started auctions, or 3) according to a policy that depends on the ongoing auctions? The objective of Chapter 2 is to address this issue. The problem is modeled as a discrete time Markov Decision Process (MDP), where each period auctions evolve according to a stochastic process. As a non-trivial constraint, a trade-off between a holding (or depreciation) cost and a ‘cannibalization’ effect among competing auctions is imposed. Though Chapter 2 only focus on the two item case (N = 2), the framework, analysis, and results give insight to the general N -item case. One of the main results is that given certain structural properties of the price-transition probabilities, the optimal release policy is of a threshold type. Specifically, for the two item case, in each period there exists a price such that if the first auction is above this price, then it is optimal to release the second item for auction. However, if the first auction is below the price threshold, then it is optimal to defer the release at least one more period. The insights and extensions for the general N item case are discussed in Chapter 6. An underlying assumption of Chapter 2 is that the auction dynamics can be summarized by a set of conditional price-transition probabilities. Chapter 3 illustrates, with two 2 examples, how these can be derived for a given auction format, bidder arrival process, and fixed bidding strategy. Although the two bidding strategies discussed, under certain conditions, result in a Bayesian Nash equilibrium, it is not argued that this is the case for the setting discussed in Chapter 2. In fact, in the implicit setting of Chapter 2 or eBay the two bidding strategies would not result in an equilibrium outcome. The objective of Chapter 3 is only to illustrate how the conditional price-transition probabilities can be derived from a given bidding behavior. Since the framework for analyzing auctions presented in Chapter 2 is new, two empirical analyses for model validation are included in Chapter 4 and 5. The data for both empirical analyses come from the eBay auctions of Dell Financial Services, and consist of more than 4,000 auctions and 44,000 bids. More details regarding the data are presented in Section 4 below. The objective of the first empirical analysis is threefold. First, to present a statistical model that can characterize the stochastic process by which auctions evolve over discrete time periods. In other words, to provide a data driven or statistical methodology to characterize the stochastic process, and estimate the conditional intermediate price-transition probabilities. Second, to provide structural properties on the statistical model such that the main results from Chapter 2 hold. Third, to illustrate and validate the empirical model by fitting it to eBay auction data. The second empirical analysis focuses on the individual bids. The objective is to propose and test a model regarding bidders’ underlying bid strategies. Specifically, Chapter 5 provides a statistical analysis of bidders’ bid-increments, i.e. the amount above the current price of an auction. Both Chapter 4 and 5 focus on the conditional probabilities given various auction parameters, and are based on Generalized Linear Models (GLM). 3 Chapter 6 concludes the thesis with an overall discussion and extensions for future work. The remainder of Chapter 1 provides a brief background to auctions and online auctions, a mathematical description of eBay’s auction mechanism, and a description of the data used for the empirical analysis. Comments Regarding Notation. For ease of discussion, sellers and bidders/buyers will be referred to as he, with no gender bias intended. The word ‘seller’ is used rather than ‘bid-taker’ (which is more common in the auction literature). Furthermore, the terms ‘bidders’ and ‘buyers’ are used interchangeably. The term ‘auction’ will be used instead of ‘online auction’. Mathematical functions that are non-decreasing (non-increasing), are referred to as increasing (decreasing). Non-overlapping sequentially released auctions, are referred to as simply sequentially released auctions. Finally, throughout the thesis the pronoun ‘we’ is used. 2. Brief Background on Auctions and Online Auctions Auctions as a formal commercial mechanism date back to antiquity [29, p.5],[14, p.1], and are today used for a wide variety of commodities, products, and services. Two of the more familiar products auctioned, or at least two that receive much attention in the news, are expensive art objects and radio (wireless) spectrum. Some interesting anecdotal stories, regarding extreme failures and successes of radio spectrum auctions, are provided in Tim Harford’s The Undercover Economist (2005). Another type of auction that has received much attention over the last decade is the Internet auction. Despite their relatively short history, Internet based auctions, or online auctions, have quickly become an integral part of modern eCommerce. From having been mainly regarded as Internet based fleamarkets for the Consumer-to-Consumer (C2C) markets, their importance and presence in 4 the Business-to-Consumer (B2C) markets has and continues to grow rapidly. Today many well-established firms operate online auctions, not only as alternative sales channels, but also as strategic tools in pricing and product introduction decisions. Examples of large corporations using eBay include Sears, IBM, Fujitsu, and Dell.1 Companies and organizations that host their own auctions include Dell, Major League Baseball, shopNBC, and Comet.2 One common use of online auctions is as alternative salvage channels. For example, both Dell and Fujitsu use their online auction channel to sell refurbished products from returned and remaining inventory. Another important application of online auctions include the Business-to-Business (B2B) markets, such as online exchanges based on principles of combinatorial auctions (cf. [14, Ch.16]). Though many web-sites that hosted online auctions no longer exist, including auctions.yahoo.com and auctions.amazon.com, there are still many online auction sites remaining, e.g. ubid.com, bidz.com, and ebid.net. However, the most dominant online auction ‘house’ was and still is ebay.com. Consequently, one of the most common yardsticks for measuring the importance and growth of online auctions are the annual figures of active users and sales volume on eBay. eBay defines active users to be those that at least once either placed a bid, bought, or listed something during the year. The sales volume is the value of all successfully closed listings and reported by eBay as Gross Merchandise Volume (GMV). Note that not all listings that make up the GMV figures are auctions. Over the last four years, the number of active users were: 83M (2007), 82M (2006), 72M (2005), 56M (2004). The sales volume over the same years were: $56B (2007), $52B (2006), $44B (2005), $34B (2004).3 To put those figures in perspective, the US Census Bureau estimated 1http://stores.ebay.com/Sears; http://stores.ebay.com/IbmFactoryOutlet; http://stores.ebay.com/Fujitsu-Scanner-Outlet; http://cgi3.ebay.com/ws/eBayISAPI.dll?ViewUserPage&userid=dell_financial_services 2 http://www.dellauction.com; http://auction.mlb.com/; http://auctions.shopnbc.com/; http://www.clearance-comet.co.uk/ 3eBay Annual Report 2007, 2006, 2005, 2004. Available at www.ebay.com. 5 the 2007 eCommerce segment of US retail sales to account for close to $127B (about 3.1% of total US retail sales).4 Another comparison that may give additional perspective on eBay’s sales volume, is with the annual revenue of the largest US retail stores. In 2006, the 6th largest retail company Sears Holdings reported annual revenue of $53B, followed by Walgreen with annual revenue of $47B.5 Despite the fact that auctions have been used for centuries, it was not until the late 1950s that formal analysis of auctions started. Most people attribute the first auction theory paper with the 1961 seminal and Nobel Prize winning work of William Vickrey [14, p.ix].6 However, Michael Rothkopf refers to Lawrence Friedman’s paper ‘A Competitive Bidding Strategy’ from 1956 (Operations Research, vol.4), as the earliest formal analysis on auction and bidding theory [26, p.369]. In fact, according to Rothkopf, the first PhD in Operations Research was Lawrence Friedman’s dissertation on competitive bidding from Case Institute of Technology in 1957 [25, p.1]. Since then, auction theory has flourished and resulted in an enormous body of literature. Two papers that deserve special attention, are the independent work from 1981 of Myerson (Optimal Auction Design, Mathematics of Operations Research, vol.6), and Riley and Samuelson (Optimal Auctions, American Economic Review, vol.71). Both of these papers generalize some of the ideas presented in Vickrey’s original work from 1961. In particular, they prove the so-called revenue equivalence principle [14, p.36]. In addition, Myerson’s paper includes the celebrated revelation principle [14, p.81].7 For a formal and comprehensive account of auction theory, including the revenue equivalence and revelation principles, see V. Krishna’s Auction Theory (2002). For a summary 4 US Census Bureau News CB08-72 5http://www.stores.org/pdf/07TOP100Chart.pdf 6William S. Vickrey, born June 21, 1914, in Victoria, BC (Canada), was together with James A. Mirrlees awarded the Nobel prize on October 8, 1996. He passed away three days later on October 11, 1996. 7Roger B. Myerson, together with Leonid Hurwicz and Eric S. Maskin, received the 2007 Nobel prize. 6 and critique of some of the main auction theory results, see Rothkopf and Harstad (1994). In the last ten years there has been an almost equally large proliferation of literature regarding online auctions. Despite some fundamental differences between traditional auctions and online auctions, most notably the context and time dimension, most researchers choose to analyze online auctions using the standard auction or game theoretical framework. As pointed out by Rothkopf this may or may not be the most useful or appropriate approach [25, p.8-9]. An advantage with online auctions is that they provide a great source of data for empirical analysis. This probably explains the huge proliferation of studies and PhD dissertations on online auctions, as predicted by Steven E. Landsburg in 1999.8 Online auctions have also resulted in many experimental studies. Researchers can use the Internet as a laboratory, and run experiments and field tests to investigate various issues. Though a bit premature, given the infancy of online auctions at the time, the two early survey papers Pinker, Seidmann and Vakrat (2003), and Bajari and Hortacsu (2004), provide a good overview of some important issues regarding online auctions. 3. The eBay Single-Item Auction Format In the western world, the term ‘eBay’ has become a household name. Most people are familiar with eBay, and know, for instance, that it is an online auction web-site. However, not everyone is aware of the exact price mechanism behind eBay auctions. In particular, there tends to be some confusion regarding the auction rules dictating the final price. At first it may seem that eBay auctions are first-price auctions, meaning that the bidder with highest bid wins and pays the amount he bid. This is not the case. In fact, eBay auctions 8www.slate.com/id/22998 7 are in effect more like second-price auctions, and almost seem to have been inspired by the following quote from the seminal 1961 auction paper by Vickrey, ”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.” [32, p.23] Note that Vickrey’s paper precedes eBay by 35 years. Part of the confusion is that eBay does not provide a clear explanation for the auctions rules and what happens in certain specific situations. The objective of this section is to explicitly characterize the price mechanism of eBay’s single-item auctions. The rules for multi-item auctions, auctions with multiple identical items, are a bit different. For a discussion on the differences and similarities between eBay and Vickrey auctions, see Chapter 2 of Stieglitz’s Snipers, Shills, & Sharks (2007). A screen-shot of a typical eBay auction is displayed in Figure 1.1. An eBay auction is characterized by five pieces of information: (1) Item - Each auction includes a description of the item being auctioned and often pictures. Shown in Figure 1.1 is an auction for a ‘DUAL SIM CNET IPhone Touch Screen PDA Mobile Phone.’ (2) Seller - The items are not sold by eBay but by a private seller. Information about each seller includes a user-id (proxy for the seller’s name), their geographical location, a feedback rating score, and comments he has accumulated from previous transactions. The seller of the auction in Figure 1.1 is ‘menzies1978,’who is located in the United Kingdom, and has a ‘99.9% Positive’ feedback rating. (3) Time - Each auction lasts for a pre-specified length of time. When a potential bidder visits the site he can see when the auction started, when it will end, and 8 Figure 1.1. Screen-shot of an eBay auction. how much time is remaining. The auction in Figure 1.1 ended on July 17, 2007, at 03:31:14 Pacific Daylight Time. At the time the screen-shot was taken there were 4 days and 11 hours remaining for the auction. (4) Price - Each auction consists of a starting bid, current bid, and a minimal bid increment. In addition, some auctions also have a hidden reserve price and/or a Buy-it-Now price. The auction in Figure 1.1 has a current bid of AU$102.00 (approximately US$87.86). (5) Bid List - Each auction also displays how many bids have been submitted, and a list of the corresponding time-stamps and the amount of each bid. The only exception is the highest bid placed, which is only shown as the minimum increment above the second highest bid. Prior to 2007, the list also included the user-id of 9 each bidder. However, in 2007 this changed and now only the seller is able to see the user-ids of the bidders. The bids are listed in ascending order. In case of a tie, the earlier bid take precedence. In Figure 1.1, the current high-bidder is ‘richnju.’By clicking on the link ‘7 bids,’the history of bids is shown. The information about the item and seller is fairly straight forward and requires no further discussion. A brief comment, however, is that many people not familiar with eBay are surprised that people would feel comfortable buying something that they cannot physically inspect, or is from someone that they have little information about. Today eBay supports auctions of almost everything and anything. This includes cars, baseball cards, jewelry, consumer electronics and real estate, to mention a few categories. Though fraud does exist on eBay, the overall sales volume and statistics speak for themselves. People have adapted to web-based shopping, and are not hesitant to buying something solely based on the description and picture a ‘stranger’ provides on a web-site. One method that eBay employs for building trust between sellers and buyers, is through their feedback rating system. After an auction has ended and the item and payment transactions have been made, the seller and buyer can report a feedback score and comment about each other. An early paper regarding how the feedback rating affects the final price is Lucking-Reiley et al. (2007). For a discussion on bidders’ trust regarding a seller, see Chwelos et al. (2005) and Chwelos and Dhar (2005) . Next we discuss the information regarding the listed bidders as it applied prior to January, 2007. Currently eBay handles the information regarding bidders differently as explained below. During an auction, the time-stamp and amount for all non-winning bids are disclosed. For the high-bidder only the time-stamp is displayed. In other words, the actual amount of the high-bidder’s bid is not revealed (until of course he is out-bid). Information 10 from expired auctions are available on eBay for a few weeks. This includes the amount of all non-winning bids, and prior to 2007, the user-id of the bidders. However, for privacy and anti-fraud purposes, in 2007, eBay changed the format and once an auction reached $200 the user-id was represented by a generic ‘Bidder #’. Currently eBay uses a different disguise which takes in effect from the start of an auction. The data collected for this thesis is therefore unique, in that it tracks all individual bidders for almost all auctions offered by Dell Financial Services from December, 2005, until February, 2007. Section 4.2 provides more details. The main difference with traditional auctions and online auctions, is that the latter lasts for a pre-determined length of time. When a seller starts an auction he must choose the auction length. The current options on eBay are 1, 3, 5, 7 or 10 days. We define the length of an auction as τ . There are three time-stamps provided by eBay. One for when the auction starts, one for when the auction ends, and one for the remaining time of the auction (which is continuously updated). Due to the speed of internet technology and a bidding strategy called sniping, the time-stamps are defined down to the second (Figure 1.1). Bids can only be submitted while the auction is ongoing. In particular, eBay auctions close firmly at the announced ending time regardless of any bidding activity. Other auctions sites, for instance, the former auctions.amazon.com, and dellauction.com, offer a going, going, and gone ending rule. There the auction end-time is extended by 10 minutes for every bid in the final 10 minutes. For an analysis on the impact of the two different ending rules, see Roth and Ockenfels (2002). In this thesis, rather than focusing on the remaining time of an auction, we focus on the elapsed time of an auction. We define the elapsed auction time by t, and t+ as the instantaneous moment after a bid has been placed. In other words, a potential bidder arrives at 11 an auction after t time units have elapsed. If he decides to submit a bid then the moment immediately following his submission is denoted by t+ . There are five different price variables in eBay auctions: starting bid, current bid, bid increment, reserve price, and high bid. In addition there are of course, the bids as well. We define the bid submitted at time t by Bt and refer to it as the bid at time t. It can be noted that eBay auctions are standard [14, p.15], [26, p.366], meaning the highest bid submitted wins the auction, and given the reserve price was met, is guaranteed the item. Starting bid, defined by pmin , is the minimum allowable first bid as decided by the seller. An auction is initially priced at zero, and the first bidder must bid at or above pmin . With slight abuse of notation, B1st ≥ pmin . This is not the same as reserve price, but rather the initial price the auction will jump to once a bid greater that it has been submitted. To illustrate, suppose pmin = $20, and the first bidder bids B1st = $30, then the current bid of the auction will jump to $20. The second price variable is the current bid, which we define by Xt . This is the amount that the high bidder would have to pay, if the auction were to end immediately. That is, if no more bids are submitted, then the high bidder only has to pay Xt . Though Xt is indexed by t, to indicate the price at time t, it is not a function of time but strictly a function of submitted bids. The dynamics of Xt will be discussed shortly. The third price variable is the bid increment which we define by kX . This is the minimum amount that a potential bidder must be willing to bid above Xt . In other words, if a potential 12 bidder arrives at time t, and decides to place a bid Bt then, (1.1) Bt ≥ Xt + kX ∀t ∈ [0, τ ) Though kX is a function of the current price, which on eBay varies according to Table 1.1, we suppress the subscript X and simply write k. Current Price - Xt ($) Bid Increment - kX ($) 0.01 - 0.99 0.05 1.00 - 4.99 0.25 5.00 - 24.99 0.50 25.00 - 99.99 1.00 100.00 - 249.99 2.50 250.00 - 499.99 5.00 500.00 - 999.99 10.00 1000.00 - 2499.99 25.00 2500.00 - 4999.99 50.00 5000.00 ≤ 100.00 Table 1.1. Minimum bid increments on eBay (June 2008) The reserve price, defined by vr , is the minimum price for which the seller will award the item. Note that vr ≥ pmin . In other words if the auction ends below vr , then the seller is not obligated to award the item. The seller chooses the reserve price. In auctions with no reserve price, the bidder who bids the most is guaranteed to be awarded the item. On eBay, the actual amount of vr is not disclosed, instead there is a message stating whether the reserve price has been met or not. Finally there is the high bid, defined by Ht , which is the amount of the highest bid placed after t time units has elapsed. Unlike pmin and Xt , the highest bid is never displayed as long as it remains the highest bid. That is, up to the time when a bid Bt > Ht , is submitted. Then Ht is revealed and the new high bid Ht+ = Bt , remains hidden. Naturally the bidder who submitted the high bid knows the actual amount. However, potential bidders do have some information regarding Ht , since clearly Ht ≥ Xt . In fact, due to the dynamics of the pricing mechanism, potential bidders have even more information about Ht , namely, if 13 Ht ≥ Xt + k. Next we discuss the price dynamics of eBay auctions. The most straightforward dynamic regards the high bid. When a bid, Bt , is submitted the new high bid, Ht+ , is simply the maximum of Bt and Ht , for t ∈ [0, τ ), (1.2) Ht+ = max {Bt , Ht } In case of a tie, the current high-bidder will remain as high-bidder. The above relationship also holds true when the bidder is the current high-bidder, i.e. when the high-bidder revises his current high-bid. However, if a high-bidder revises his bid then nothing happens to the current bid, i.e. Xt+ = Xt . The following discussion applies to cases when a bidder who is currently not the highbidder arrives and places a bid. The dynamics for the current bid are also straightforward when Xt = 0 and the first bid arrives. For Xt = 0, Bt ≥ pmin , t ∈ [0, τ ), (1.3) Xt+ = pmin 1{pmin ≤Bt <vr } + vr 1{Bt ≥vr } where 1{·} is the indicator function which equals 1 if the argument in the bracket is true, and 0 otherwise. Equation (1.3) states that if the first bid is less than the reserve price, then the current bid will jump to the minimum price (the bid has to, of course, be above the starting bid). If the bid is above the reserve price, then the current bid will jump to the reserve price. Note that equation (1.3) is well-defined even when vr = pmin . When Xt ≥ pmin , the dynamics are a bit more complicated. Therefore, we first consider the case 14 with no reserve price (vr = pmin ). For Xt ≥ pmin , Bt ≥ Xt + k, t ∈ [0, τ ), Xt + k Ht < Xt + k Xt+ = min {Bt + k, Ht } Bt ≤ Ht Ht ≥ X t + k min {Ht + k, Bt } Bt > Ht which we write as, for Xt ≥ pmin , t ∈ [0, τ ), (1.4) Xt+ = (Xt + k)1{Ht <Xt +k} + (max {min {Bt + k, Ht }, min {Ht + k, Bt }}) 1{Ht ≥Xt +k} We now consider the situation when a seller has included a reserve price vr > pmin . As mentioned above, the actual amount of vr is not disclosed, and only a message stating whether vr has been met is displayed. That is, if vr has not been met, then Ht < vr , i.e. Ht < vr if and only if Xt < vr . Note that the dynamics for Ht are not affected by vr > pmin and still follow (1.2). For Xt ≥ pmin , Bt ≥ Xt + k, t ∈ [0, τ ), Xt + k Ht < Xt + k min{Bt + k, Ht } Ht ≥ Xt + k, Bt ≤ Ht Ht < vr , Bt < vr or Ht ≥ vr Xt+ = min{Ht + k, Bt } Ht ≥ Xt + k, Bt > Ht max{Xt + k, vr } Ht < Xt + k Ht < vr , Bt ≥ vr max{H + k, v } H ≥ X + k t r t t which we write as, for Xt ≥ pmin , bt ≥ Xt + k, t ∈ [0, τ ), (1.5) Xt+ =(1 − 1{Ht <vr ≤Bt } )[1{Ht <Xt +k} (Xt + k) + 1{Ht ≥Xt +k} max{min{Bt + k, Ht }, min{Ht , Bt }}] + 1{Ht <vr ≤Bt } [1{Ht <Xt +k} max{Xt + k, vr } + 1{Ht ≥Xt +k} max{Ht + k, vr }] Note that, if vr = pmin , then (1.5) simplifies to (1.4), since Ht ≥ pmin , and 1{Ht <vr ≤Bt } = 0. 15 We summarize the main mechanisms of eBay as follows, (1) An auction starts with a fixed time horizon τ , starting price pmin , and possibly a hidden reserve price vr . (2) The first bidder must bid at or above pmin . (3) When a bidder arrives to the auction site he observes,9 (i) How much time t has elapsed, t ∈ [0, τ ]. (ii) If he is the high-bidder. (iii) The current price Xt . (iv) The list of previous bids, i.e. all non-winning bids. (v) If the reserve price vr has been met or not, i.e. is Xt ≥ vr . (4) If he decides to bid, then Bt ≥ Xt + k (if the bidder is the high-bidder, then Bt ≥ Ht ). (5) After a bid is submitted, Ht and Xt are updated accordingly. (6) When the auction expires at t = τ , the person with the high-bid, Hτ , pays Xτ and receives the item, provided Hτ ≥ vr . However, if Hτ < vr , then the seller is not obligated to award the item. Though the format of an eBay auction seems to resemble a mix of a first-price sealed-bid and open English (ascending) auction, since all non-winning bids are displayed and Xt is continuously updated. In effect, it is a second-price auction. Or more appropriately one could define eBay auctions as second-price+k censored-English auctions. We say ‘secondprice+k’ since the highest bid wins, but only has to pay the second highest bid plus the minimum increment k. We say ‘censored English’ since although all non-winning bids are disclosed, the high-bid is never displayed. Another reason we say eBay auctions are secondprice auctions is seen in the case of k = 0 for all Xt . If we ignore reserve price (vr = pmin ), 9There are of course many other variables a prospective bidder might consider in his decision whether to bid and/or the amount, but here we only point out the main time and price variables. 16 we notice that with k = 0, the following dynamics apply, for Xt ≥ pmin , Bt ≥ Xt , t ∈ [0, τ ), Xt+ = min {Bt , Ht } The above equation is exactly the price dynamic of a strictly second-price auction. Similarly if vr ≥ pmin and k = 0, for Xt ≥ pmin , Bt ≥ Xt , t ∈ [0, τ ), Xt+ = (1 − 1{Ht <vr ≤Bt } ) min{Bt , Ht } + 1{Ht <vr ≤Bt } vr 3.1. Buy-it-now auctions. There is one additional feature we have not included in the discussion above, namely the Buy It Now price, which we define by pbuy . As the name suggests, pbuy is a pre-set price at which the seller is willing to end the auction and award the item immediately. In other words, if someone bids pbuy , then the auction terminates. Naturally pbuy is shown and not hidden as vr . However, pbuy is only available as long as Xt < vr . That is, if 1) pmin = vr , Xt = 0, and Bt < pbuy , or 2) pmin < Xt < vr , and vr ≤ Bt < pbuy , then the Buy It Now option is removed at t+ . Note that pbuy > vr and therefore pbuy can be regarded as a maximum price or ‘list price’ of the item. However, this list price is only available until the first bid arrives, or until Bt ≥ vr arrives. The requirement Bt < pbuy in the second case above is a bit redundant, since if Bt = pbuy , then t+ indicates the end of the auction and pbuy is also ‘removed’. As a consequence of the Buy It Now feature the dynamics of the auction change slightly. Let us again consider the dynamics described earlier. Case 1) vr = pmin For t ∈ [0, τ ), the auction length is defined as follows, τbin = min{τ, inf{t | Xt = 0, Bt = pbuy }} 17 and the dynamics of the high bid and current bid are given by, Ht+ = min{Bt , Ht } Xt+ pmin = pbuy 1 Bt < pbuy Xt = 0 Bt = pbuy {Ht <Xt +k} (Xt + k) + 1{Ht ≥Xt +k} max{min{Bt + k, Ht }, min{Ht + k, Bt }} Xt ≥ pmin Case 2) vr > pmin (recall that Ht < vr if and only if Xt < vr .) For t ∈ [0, τ ), the auction length is, τbin = min{τ, inf{t | Xt < vr , bt = pbuy }} and the dynamics of the high bid and current bid are, Ht+ = min{Bt , Ht } Xt+ pmin Bt < vr vr vr ≤ Bt < pbuy max{min{Bt + k, Ht }, min{Ht + k, Bt }} Bt < vr = max{Xt + k, vr } vr ≤ Bt < pbuy pbuy Bt = pbuy max{min{Bt + k, Ht }, min{Ht + k, Bt }} Therefore, for Xt = 0, Xt+ = 1{Bt <vr } pmin + 1{vr <Bt <pbuy } vr + 1{Bt =pbuy } pbuy 18 Xt = 0 Xt < vr vr ≤ Xt and for Xt > pmin , Xt+ =1{Xt <vr ,Bt =pbuy } pbuy + (1 − 1{Xt <vr ,Bt =pbuy } )[ (1 − 1{Ht <vr ≤Bt } )[1{Ht <Xt +k} (Xt + k) + 1{Ht ≥Xt +k} max{min{Bt + k, Ht }, min{Ht + k, Bt }}] + 1{Ht <vr ≤Bt } (1{Ht <Xt +k} max{Xt + k, vr } + 1{Ht ≥Xt +k} max{Ht + k, vr }] Next we discuss the background for the data used in the empirical analysis of Chapter 4 and 5. 4. The Data and Dell Financial Services The data we analyze in this thesis comes from the eBay listings of Dell Financial Services L.P. (DFS). DFS is a joint venture between Dell Inc. and CIT Group Inc. (CIT), that provides financing of Dell products to various customer groups, including home, education, small, medium, and large businesses. Dell, which was founded in 1984, is one of the largest computer system manufacturers and sellers in the world. CIT, which was founded in 1908, provides financial solutions for both commercial and consumer clients. For more information regarding Dell and CIT, visit dell.com and cit.com. Since founded in 1997, DFS has ”originated more than $18 billion in finance transactions.”10 After products return from leasing programs, DFS selects the highest quality products, refurbishes them and sells them via their private online channels, dfsdirectsales.com and dellauction.com, as well as on ebay.com. In other words, the three online channels provide DFS with alternative salvage channels. How long DFS has been using these channels or how DFS allocates the products to the different channels is not specified. However, from their eBay profile we know that DFS has been an eBay member since April 2001 (eBay user-id: 10www.dfsdirectsales.com 19 dell financial services). There are a few comments to make. First, DFS only sells to the US market. A customer must have a US based credit card and a US delivery address. Second, DFS does not operate an eBay store, but lists items on eBay as an individual member. Third, for unknown reasons, since February 26, 2007, DFS has ceased to list products on eBay. They still, however, use their own online channels. Finally, Dell, and not DFS, also employs an online channel called Dell Outlet for selling returned and refurbished products at fixed but discounted prices.11 From the perspective of the customers, the three channels provide opportunities to purchase used and refurbished Dell products at a discount. Issues like taxes, shipping and handling, warranties, and return policies vary for each product sold. In general, the products are covered under Dell’s general warranty. However, since products are returned after various lease durations, the extent of the remaining warranty differs from product to product. The specifics of the remaining warranty is available online for a potential buyer to verify before having to commit to a purchase or bid. Moreover, DFS also provides a return policy of 30 days, for a credit of the purchase price (excluding shipping and handling). It should be noted, that unlike Dell’s direct business model where customers tailor the product configuration according to their needs or budget, the configurations of items listed by DFS are fixed and DFS does not provide any upgrade or modification services. The three channels each have different pricing mechanisms. At dfsdirectsales.com products are sold at a fixed list price. How DFS determines the list-price of each product or configuration is not specified. As a rough estimate, the list-price seems to be about half of the price of a new product, as sold on dell.com. At dellauction.com and eBay.com, products are predominantly sold using the online auction version of a second-price auction. 11http://www.dell.com/content/segmenter.aspx?c=us\&l=en\&s=dfo 20 The main difference between the two auction channels is that, unlike the hard auction end on ebay.com, at dellauction.com, DFS employs a going, going, ...., gone end. Similar to the former ending rule at auctions.amazon.com, dellauction.com auctions are automatically extended by 10 minutes for each arriving bid in the last 10 minutes.12 This is presumably to provide less of an incentive for the strategy of sniping [24]. A common attribute among the three channels is that the picture and description of each product listing are standardized. For all listings, DFS uses a generic picture of the product that is being sold or auctioned. In other words, DFS does not provide digital images of the individual item that is sold or auctioned. In addition, DFS uses a standardized template to describe the product. The information provided includes a description of the main features, but does not give detailed description regarding, for instance, cosmetic appearance. It would seem reasonable to assume that by using generic pictures and standardized templates, DFS reduces the administrative cost of using online channels. The effect this has on the final price is not immediate. Some related research topics include how to allocate products to the different channels, and how the allocation affects the overall revenue and profit. A third research area is investigating how the fixed list-prices and auction prices affect each other, and the impact on overall revenue and profit. One paper that address the latter issue is Caldentey and Vulcano (2008). 4.1. Reasons for Selecting DFS. Besides eBay’s huge commercial success and user popularity, eBay has also become one of the most popular sources of data for auction research. With millions of listings and data from completed auctions made public for a few weeks, eBay provides an unparalleled rich source for empirical investigations. However, though the data is available, it is not a trivial task to extract large data sets. Automating a 12cf. ‘Auto Bid Extend’, Glossary at www.dellauction.com 21 data extraction process requires Internet and eBay experience, a computer code or software technology, and an understanding of how to use the code or software. In the next section we give a brief explanation of how our data were extracted. Most researchers tend to focus on a specific item, and then simply download as much auction data as possible for that particular item. We decided to take a different approach and only focus on a particular seller, namely DFS, and then download all auctions they listed on eBay over a given time-period. The main reason for this, was that we wanted to control for the effect of the seller’s reputation, trustworthiness, and feedback rating. Studies have indicated that the trustworthiness and feedback rating may effect the final price of an auction [16]. Therefore, to control for this issue, we decided to only consider a specific seller, and to take a well-established company whose reputation and trustworthiness would in general not be considered suspect. Another reason we chose DFS was that we wanted to maximize the chance that there would be an abundance of auctions over an extended period of time. In addition, we wanted to maximize the chance of having many auctions of identical or near identical products. An unforeseen benefit of the DFS auction data, was that the description and picture provided was standardized, thereby providing control for the effect wording, description and pictures may have on the auction dynamics. It should, however, be noted that over time DFS did make some changes to the template used. 4.2. Data Extraction and Handling. Extracting and analyzing the data involved both manual and automated steps. We will not provide a detailed account of each step, but instead give an overview regarding the main components. All data extraction, handling and 22 analysis was executed on a standard PC with Internet connectivity. The first three steps of data extraction only needed to be done every three weeks. (1) The first step was to log on to eBay and perform a manual search of all completed listing for Dell Financial Services. This step required an eBay user-id and password, and knowledge of how to use their search function. (2) The next step was to save the entire list as a html-file using the ‘Save As’ function provided by the web-browser. (3) The third step was to run a Perl script that automatically scanned the saved list and downloaded into flat files (text files) all the relevant information for each individual auction. This step produced eight files: two auction data files and one bid data file per PC category (laptop and desktop), one file with non-PC and recalled auction listings (monitors, docking stations, and test listing), and one file for any error messages. In the auction data files, each row corresponded to an individual auction and each column corresponded to a different variable. See Table 1.3 for the list of variables. In the bid data file each row corresponded to an individual bid for a specific auction. The script worked as follows. First, it searched through the saved list and picked out the individual listing ID for all new auctions since previous data extraction. It then queried ebay.com regarding each auction and collected the pre-specified information. Section 4.3 lists the specific information collected. Perl is a general purpose programming language, and to run the script requires a compiler. We used ‘ActivePerl’ available at no cost at www.activestate.com. The script was written by UBC student Andrew Gray and consists of 540 lines. (4) After data had been extracted into flat files, it was populated into an Excel spreadsheet by a Visual Basic (VBA) macro. The same VBA macro was also used to 23 perform data cleaning, data selection as well as numerous computations. The script was also written by Andrew Gray and each user function was executed inside Excel. (5) After a subset was selected and the variables of interest had been computed, the resulting Excel sheet was exported to a tab-delimited text-file. (6) The resulting text-files was then imported into the statistical software ‘R’ [22], for various statistical and graphical analysis. ‘R’ is available at no charge at www.r-project.org, and is an extension to the statistical software ‘S’. 4.3. The Data. Data were collected from mid December, 2005, until the end of February, 2007. More specifically we obtained complete information regarding 6,683 auctions, with start and end date between December 12, 2005, and February 26, 2007. As mentioned above, DFS has since ceased their activity on eBay. Only auction listings for PC desktops and PC laptops at Dell Financial Services eBay-site were collected. In Table 1.2 some aggregated statistics are summarized. There were 3,802 desktop auctions and 2,881 laptop auctions, which combined for a total sales of $1,979,240. The average final price, average number of bids, and average number of bidders for the desktop auctions were $205, 13.54, and 7.89, respectively. For laptop auctions the corresponding values were $416, 17.09, and 9.37. In other words, on average a laptop PC sold for about twice that of desktop PC, and attracted about three more bids and one more bidder. We also note the great variation in final price, bids and number of bidders. In particular, the standard deviation in final price for desktop and laptop auctions is quite large, at $66.81 and $86.24 respectively. This illustrates the huge uncertainty in final price a seller is faced with in selling via online auctions. It would not seem reasonable to assume that the variation is strictly a result of the variation in bidders’ valuation. In Section 4.4 and Chapter 5 we analyze this issue further. The variation in bids and number of bidders is also interesting to note. Often in the auction theory literature it is assumed the number of bidders is fixed. 24 From Table 1.2 it is clear this is not the case here. Furthermore, this would intuitively seem to indicate at least one source of variation regarding the final price. Chapter 3 discusses a model for the distribution of final price that incorporates the variability in the number of bidders. Total Dell Product Auctions Mean (s.d.) Bids Sales Final Price $780,114 Desktop 3,802 51,495 Laptop 2,881 49,242 All 6,683 100,747 $1,979,240 Bids Bidders 205.20 (66.81) 13.54 (5.53) 7.97 (2.32) $1,119,126 416.20 (86.24) 17.09 (6.79) 9.37 (2.68) Table 1.2. Descriptive statistics for the all auctions. Figure 1.2 provides a time series of the final price for all auctions coded by PC category Desktop and Laptop. The horizontal axis is time, and the vertical axis represents the final price of the auction. Each circle represents an individual auction, where black circles are laptop auctions, and grey circles are desktop auctions. There are at least three rather apparent features. First, the desktop auctions’ final prices are about half those of laptop auctions, as noted above. Second, laptop auctions appear to have been sold at two levels. There is a clear divide of laptop final prices before and after July, 2006. During July, 2006, there is a clear decline in final prices, after which they seem to stabilize again. Desktop final prices seem to remain steady throughout the observation period. Third, as already noted, there is a considerable fluctuation in final prices for both laptops and desktops. Some of these fluctuations might be due to the difference in product configurations. In Section 4.4 below we briefly analyze this issue. However, the fluctuations also reflect the stochastic nature of the online auction revenue stream. The main motivation for this 25 800 600 400 0 200 Final Price Dec ’05 Apr ’06 Jul ’06 Oct ’06 Jan ’07 Figure 1.2. Final price of each auction for the entire data set (6,683). The horizontal axis represents time. The vertical axis represents the final price of an individual auction. Black circles represents laptop auctions (2,881), and grey circles desktop auctions (3,802). thesis, is to provide insight for making better decisions given such a stochastic environment. Table 1.3 below displays the auction and product information recorded for each product. The first ten variables are standard to all eBay auctions and provide information regarding the auction state and bidding history. Variables 8, 9, and 10, are recorded for each successful bid (registered bid) in the bid history. The last 11 variables are information that DFS 26 decided to include in their listing. They provide the specifications of the product being auctioned. As mentioned above, DFS or bidders do not choose the specific product configuration. In other words, though DFS does have the option of not listing a particular product with a specific configuration, the last 11 variables are not decision variables. Therefore, the only decision variables are ‘Description’, ‘Start time’, ‘End time’, and ‘Starting bid’. The ‘Start time’ is the time-stamp when DFS lists the auction, and therefore reflects the decision if and when to release an item for auction. The ‘End time’ is the time-stamp when an auction will end, and reflects the decision how long an auction should last. In addition to the information listed in Table 1.3, DFS also includes a generic picture of a new product, some additional information regarding what is included (AC adapter, pointing device/mouse, keyboard, etc.), instructions how to verify any remaining warranty, and various shipping options. Shipping cost range from $20 to $80 depending on service requested by the auction winner. A few comments follow. (1) The standardized template that DFS uses to list the last 11 variables was modified over time. (2) It seemed that DFS was not consistent in always including the information regarding all 11 variables. More specifically the only variables consistently reported were: Category, Brand, Processor Type, Processor Model, Processor Speed, Memory (RAM), and Hard Drive Capacity. Therefore in the ensuing statistical analysis we only focus on these variables and ignore the others. (3) Unlike the PC laptops, which naturally came with a screen, none of the PC desktops included a monitor. (4) Though DFS also auctions items such as monitors and docking stations, we exclusively restricted our data extraction and analysis for PC desktops and laptops. 27 (5) All auctions that ended with the Buy-It-Now option were excluded. The reason for this is that once a bid arrives or the reserve price is met, the Buy-It-Now option disappears. Since the information, if an auction was initially listed with a BuyIt-Now price, is not recorded, we decided to ignore all Buy-It-Now transactions. It can, however, be noted that the number of Buy-It-Now transactions was less than one percent of all auction listings. With regard to the previous comment, it can also be noted that all auctions were listed without a reserve price, and almost exclusively had a starting bid of $.99. Though this might appear to be rather risky, both anecdotal and research evidence supports this use. The listing fees are less for auctions without a reserve price and a low starting bid. In addition, and more importantly, low reserve price and starting bid tends to lead to higher expected revenue [4]. (6) DFS only sells and ships to US based customers. (7) Though data were extracted every 2-3 weeks, some auctions may have been missed. In other words, some completed auctions may have been removed by eBay before we had a chance to download the information. Therefore, the data set does not exclusively cover all DFS eBay desktop and laptop auctions. However, there is no reason to believe that the excluded auctions had special features or exhibited any unusual auction dynamics. We now describe the data more in depth. 4.3.1. Desktop Data. In total there were 3,802 successful desktop auctions with a total value of $780,114. The mean final price over the entire study period was $205.20. As mentioned above, since not all variables listed in Table 1.3 were consistently reported, we chose to focus on the main characteristics that were reported: Category, Brand, Processor Type, Processor Model, Processor Speed, Memory (RAM), and Hard Drive Capacity. All 28 1. 2. Name Item ID Description 3. 4. 5. 6. 7. Winning bid Start time End time Starting bid Number of bids Description Example eBay listing ID number 6831091024 Brief description of item DELL WINXP Latitude D600 1.6 GHz 1024MB CDRW DVD NR Amount of final price 576.87 Time stamp of auction start 12-Dec-05 10:07:39 Time stamp of auction end 15-Dec-05 10:07:39 Amount first bid has to exceed 0.99 Total number of bids that arrived in auction 19 8. 9. 10. User ID Bid Amount Date of Bid eBay User ID of bidder Amount of bid Time stamp of bid 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. Category Brand Processor Type Processor Model Processor Speed Bundled Items Memory (RAM) Hard Drive Capacity Operating System Primary Drive Condition Type of PC product (desktop or laptop) Brand of PC product (all Dell) Brand of processor (all Intel) Model specification of processor Speed specification of processor Included software (operating system) Specification of internal memory Specification of hard drive capacity Description if/what operating system is included Description of CD or DVD drive One word describing condition Table 1.3. The auction and product variables collected. frittikanada 127.32 11-Feb-06 16:46:23 Laptop Dell Intel Pentium M 1.6GHz WINXP 1024MB 60GB Yes CD-RW Refurbished auctions consisted of Dell PC with an Intel processor. In Table 1.4 the distribution of the other variables are listed. Out of the five processor models, the Intel Pentium 4 (IP4) was the most common, accounting for 87% of all listings. There were 18 different processor speeds, ranging from .866 to 3.2GHz. The most common was 2.0GHz which accounted for 32% of all listings, and the five most common, 1.8, 2.0, 2.2, 2.3, and 2.4GHz, accounted for 83% of all listings. There were eight different memory (RAM) sizes, where 256 and 512MB were the most common covering respectively 46% and 27% of all listings. There were 13 different hard drive capacities, ranging from 6 to 200GB. The two most common were 40 and 20GB, which accounted for 57% and 34% of sales each. The last variable in Table 1.4 is the auction duration measured in days. Though DFS seemed to experiment with a few 1 day auctions, specifically 459 or 12% of all listings, they predominantly used an auction duration of 3 days, accounting for 88% of all listings. The 29 eight auctions that do not have one of the standard eBay auction durations, i.e. the ones with the decimal extensions, are auctions that DFS for unknown reasons simply chose to end early and award to the high bidder at that time. It should be noted that these are not Buy-It-Now auctions, as these have been removed. They are also not auctions that were at a particular high price, such that it might seem unnecessary to let them proceed the full auction duration with little probability of seeing more bids. In fact almost the opposite, all eight auctions were at rather low prices. No further explanation as to why they were ended is provided. In the ensuing analysis these auctions will therefore be removed. Processor Model Processor Speed Memory Hard Drive IP4 3,309 0.866GHz 2 128MB 144 224MB 11 IP4 Xeon 2 1.3GHz 1.4GHz 1.5GHz 1.6GHz 1.7GHz 1.8GHz 256MB 320MB IPM 1 6 27 15 144 402 1,760 541 6GB 1 10GB 15 18GB 4 34 2.0GHz 1,231 2.2GHz 238 2.3GHz 308 2.4GHz 973 2.5GHz 142 512MB 768MB 1000MB 40GB 60GB 80GB 1,021 25 22 2,179 78 162 IPIii 2 2.6GHz 54 2.7GHz 102 2.8GHz 136 2.9GHz 18 Celeron 455 3.1GHz 2 3.2GHz 1 1024MB 278 20GB 1,279 120GB 30GB 64 160GB 33GB 2 200GB 36GB 6 Duration 0.51699d 1 1.68313d 1 1d 459 (days) 0.82015d 1 1.72839d 1 3d 3,334 1.00861d 1 1.91704d 1 5d 1 1.12295d 1 2.03098d 1 Table 1.4. Counts of product configurations and auction duration for Dell PC desktop auctions. 9 2 1 the 3,802 4.3.2. Laptop Data. In total there were 2,881 successful laptop auctions for a total value of $1,119,126. The mean final price over the entire study period was $416.20. Similar to the desktop auctions we will only focus on the main characteristics that were consistently reported. All auctions were for a Dell PC with an Intel processor. In Table 1.5 the distribution of the laptop configurations are listed. The two most common processor models were Intel Pentium 4 (IP4) and Intel Pentium M (IPM), which accounted for 49% and 46% of all 30 listings respectively. There were 22 different processor speeds, ranging from .866 to 3.2GHz. The five most common, which accounted for 87% of all listings, were: 1.4 (22%), 1.6 (15%), 1.7 (7%), 1.8 (18%), and 2.0GHz (25%). Out of the eight different memory (RAM) sizes the most common were 256, 512 and 1024MB, covering 18%, 57% and 20% respectively. There were 13 different hard drive capacities, ranging from 5 to 80 GB. The three most common were 20, 30, and 40GB, which accounted for 18%, 42% and 33% respectively. Similar to the desktop auctions the most common auction duration was 3 days, which accounted for 91% of all listings. Furthermore, as with the desktop auctions, there are a few listings that have non-standard auction duration and are not Buy-It-Now auctions. The two special listings with an auction length of 2.95833 days, are auctions that were ended exactly one hour prior to the auction ‘End time’, and were released on the same date (but not at the same time). The reason and mechanism to do this is not known, but perhaps there was a glitch with the data extraction or error with the eBay listing. In total there were ten auctions released that day, and only two that did not elapse the full 3 days. In the ensuing analysis the four auctions with non-standard duration have been removed. 4.4. Data Selection and Some Descriptive Analysis. In order to control for the effect of product configuration on the analysis in Chapter 4 and 5, only six subsets are used. These are chosen by selecting the main categories of processor model, processor speed, memory, hard drive capacity, and auction length. More specifically, only the six product configurations listed in Table 1.6 were analyzed in Chapter 4 and 5. The products were chosen to limit the analysis to the cases with the most data. The aggregated ‘products’ D1 and L1 were selected by choosing the attributes listed in Table 1.4 and 1.5 with the most cases. The remaining products were chosen by looking at cross-tabulation counts from subsets D1 and L1. That is, D3 and D4 are subsets of D1, and L4 and L5 are subsets of L1. 31 Processor Model Processor Speed IP4 1,422 0.866GHz 3 0.9GHz 2 Memory 128MB 224MB 63 10 IPM 1,327 1.0GHz 34 1.1GHz 2 1.2GHz 78 1.3GHz 58 1.4GHz 624 256MB 320MB 506 14 IPIii 104 Celeron 28 1.5GHz 1.6GHz 1.7GHz 1.8GHz 1.9GHz 76 427 203 507 3 2.0GHz 708 2.5GHz 2.2GHz 42 2.6GHz 2.3GHz 3 2.8GHz 2.4GHz 75 2.9GHz 3.0GHz 3.2GHz 1024MB 563 512MB 768MB 1000MB 60GB 80GB Hard Drive 1,652 18 55 150 28 5 18 8 3 1 1 5GB 1 20GB 517 10GB 9 30GB 1,212 12GB 2 40GB 962 Duration .73461d 1 1d 251 (days) 2.32205d 1 3d 2,626 2.95833d 2 Table 1.5. Counts of product configurations and auction duration by category for the 2,881 Dell PC laptop auctions. Category Desktop Laptop Subset Processor Processor Memory Hard Drive Duration Name Model Speed (GHz) (MB) (GB) (Days) D1 IP 4 1.7-2.8 256, 512, 1024 20, 40 3 D3 IP 4 2.4 256 40 3 D4 IP 4 2.0 512 40 3 L1 IP 4, IP M 1.4-2.0 256, 512, 1024 20, 30, 40 3 L4 IP 4 1.8 512 30 3 L5 IP M 1.4 512 40 3 Table 1.6. Product configuration for the subsets analyzed. Count 2,072 274 167 2,046 172 163 See Table 1.8 and 1.9 below. The last column in Table 1.6 includes only those listings with a ‘Starting bid’ of $.99. For Laptop auctions there was one listing that started at $501 and received one bid. For Desktop auctions there were four auctions that had a ‘Starting bid’ of $227, $304, $374, and $382 respectively, and where each received one bid. Note that these are not Buy-It-Now auctions. A notational comment is that throughout the thesis we refer to D1, D3, D4, L1, L4, and L5 as ‘products’, even though D1 and L1 span several product configurations. Table 1.7 lists some descriptive statistics regarding the final price of each product. We note that the mean and median final price for product D3, is slightly below the mean and median for the aggregated subset D1, while for product D4 the reverse is true. This is most 32 D1 D3 D4 Mean 208.20 193.65 230.59 (55.74) (36.90) (36.78) (s.d.) Median 200.00 190.71 230 L1 412.20 (74.33) 405.00 Min 93.92 127.50 129.20 10.50 501.00 338.50 348.26 700.00 Max Count 2,072 274 167 2,046 Table 1.7. Mean (st.dev), Median, Minimum, and selected subsets. L4 427.98 (68.45) 424.44 L5 409.59 (73.63) 405.00 266.00 280.99 613.00 596.99 172 163 Maximum Final Price for likely because D4 has twice the memory (RAM) than D3 (though D3 has a faster processor). Similarly, we note that the mean and median for the product L5, is slightly less than for the aggregated subset L1, while L4 has values above L1. The only difference between L4 and L5, is with regard to the processor. L4 has a 1.8GHz Intel Pentium 4 processor, while L5 has an 1.4GHz Intel Pentium M. Another interesting observation is that the variance is less for the specific products D3, D4, L4, and L5, than the aggregated products D1 and L1. However, there is still considerable variation in final price for the four specific products. Figure 1.3 provides histogram of the final price for D1 and L1. Though both are skewed to the left, the distribution for L1 is more symmetrical than for D1. In the following sections the variation for the various product configurations are analyzed further. A final comment, is that consecutive bids in a short time period by the same bidder were removed. Where ‘short time’ was defined as 10 minutes, meaning that if a bidder places another bid within 10 minutes of his previous bid, then the first bid is removed. In other words, only the last bid a bidder placed in one ‘session’ is considered. If a bidder waits more than 10 minutes to place another bid then this is defined as a second ‘session’ and two bids are recorded. The result was that 14,202 out of 100,747 bids were removed. We conclude this chapter with a brief statistical analysis regarding the price variation for D1 and L1. Chapter 4 and 5 provides further analysis regarding the variation of final 33 250 200 150 0 50 100 Frequency 300 200 Frequency 100 0 0 200 400 600 800 0 Final Price 200 400 600 800 Final Price Subset D1 - Desktop (N=2,072) Subset L1 - Laptop (N=2,046) Figure 1.3. Distribution of the Final Price for subset D1 (Desktop) and subset L1 (Laptop). price. In each box-plot the lower and upper edge of the box represents the 25th and 75th percentiles. The line inside the box represents the median final price. The dashed lines, or ‘whiskers’, from each box, are drawn to the observation furthest away, but within a factor of 1.5 × IQR, from the edge of the box. The IQR is the inner quartile range. Circles outside the whiskers are observations that would be classified as extreme and potentially outliers. The notches inside the box indicates a range around the median. An informal statistical test if the median from two box-plots are different, is if the notches overlap. Desktop - D1. For D1 the mean and standard deviation of the final price is $208.20 and $55.74, and the median, minimum, and maximum were $200.00, $93.92, and $501.00 respectively. Table 1.8 displays the cross-tabulation counts for the selected categories. There is a clear cluster of products with a processor speed ranging from 1.8 to 2.4GHz, memory 34 size of 256 or 512MB, and hard drive capacity of 20 or 40GB. Figure 1.4 shows the distributions of the final price by hard drive capacity, memory, and processor speed. There are two noteworthy observations. First, within each product specification there is considerable variation. For example, the median final price of D1, with hard drive capacity 40GB, is about $200, with the 25th and 75th percentiles at about $180 and $250 respectively. Second, the median price is increasing in the product specification. This is most visible for hard drive capacity and memory, where the median price is clearly increasing in the respective specification. An informal test for a significant change in median price, is if the notches of two box-plots overlap. For memory, there clearly is no overlap in the notches, thus indicating a significant increase in median final price. For hard drive capacity the notches also do not overlap, though not by a great margin. For processor speed, overall the median final price seems to increase. However, the increase is not monotonic in the processor speed. Intel Pentium 4 256 512 1024 20 40 20 40 20 40 1.7 40 1 3 1 0 0 1.8 110 52 33 50 3 9 2.0 165 225 107 167 4 51 2.2 0 37 0 11 0 1 2.3 143 4 42 48 3 8 2.4 91 274 44 102 6 55 2.5 9 26 4 22 2 1 2.6 0 20 2 5 0 3 2.7 3 6 0 9 0 6 2.8 2 20 3 21 2 16 Table 1.8. Cross-Tabulation of selected Desktop product configurations. Laptop - L1. For L1 the mean and standard deviation of the final price is $412.20 and $74.33, and the median, minimum, and maximum were $405.00, $10.50, and $700.00 respectively. The minimum value of $10.50 is the clear outlier as seen in both Figure 1.3 above and Figure 1.5 below. The outlier a ‘DELL WINXP Latitude C640 1.8 GHz 1024MB CDRW DVD’ listed between 25-Oct-06 21:29:47 and 28-Oct-06 21:29:47, and received three 35 500 Final Price 200 300 Final Price 100 200 100 40 256 512 Memory (MB) Hard Drive Capacity (GB) Hard Drive (GB) 300 400 500 400 500 400 300 Final Price 200 100 20 Memory (MB) 1024 1.7 1.8 2 2.2 2.3 2.4 2.5 2.6 Processor Speed (GHz) Figure 1.4. Distribution of the Final Price for D1 auctions by the three main categories. bids from two bidders (eBay listing number: 160044957851). Removing the outlier only marginally changes the mean, median, and standard deviation, but drastically increases the minimum to $233.50. Table 1.9 displays the cross-tabulation counts for the selected categories. There are two clear clusters. One for Intel Pentium 4, 1.8-2.0GHz, 512MB, and 20, 30, or 40GB. And one for Intel Pentium M, 1.4-1.6GHz, 512MB, 20, 30, or 40GB. Figure 1.5 shows the distribution of the final price by processor model, processor speed, hard drive capacity, and memory. Similar to D1 we see that within each product specification there is considerable variation. For instance, the median final price of L1, with hard drive capacity of 40GB, is about $400, with the 25th and 75th percentiles at about $380 and $450 respectively. However, unlike D1, the median final price does not increase as drastic in the product specification. In fact, for hard drive capacity and processor model, the median final price seems to be the same across the possible values. For memory, we see that auctions for L1 with 256MB, the median price is ‘significantly’ less than for L1 with 512 or 1024MB. However, between the two higher memory sizes, there does not seem to be any difference. For processor speed, the median final price, though fluctuating, is not monotonic. An interesting topic for further empirical 36 2.7 Processor Speed (GHz) 2.8 Intel Pentium 4 Intel Pentium M 512 1024 256 512 1024 20 30 40 20 30 40 20 30 40 20 30 40 20 30 0 2 2 0 0 0 23 8 34 49 166 163 2 5 1 0 2 0 0 0 1 0 12 2 7 7 0 0 1 2 4 0 0 0 14 10 3 36 105 125 1 27 2 2 2 0 0 0 3 2 2 0 32 18 0 0 38 172 44 2 83 8 0 0 0 0 1 3 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 40 169 106 9 93 33 0 0 0 0 0 0 0 0 Table 1.9. Cross-Tabulation of selected Laptop product configurations. 500 600 200 300 400 Final Price 500 400 Final Price 300 100 200 0 100 0 0 1.4 1 1.5 1.6 1.7 1.8 1.9 2 Processor Speed (GHz) Processor Model 600 500 400 Final Price 400 0 100 200 300 0 100 200 300 500 600 700 Processor Speed (GHz) 700 Processor Model (IP 4 - left, IP M - right) Final Price 40 31 32 18 3 0 0 0 600 700 1.4 1.5 1.6 1.7 1.8 1.9 2.0 700 256 20 30 40 2 0 0 0 2 0 2 2 0 5 3 1 44 25 7 0 1 0 62 52 27 256 512 1024 20 Memory (MB) 30 40 Hard Drive Capacity (GB) Memory (MB) Hard Drive (GB) Figure 1.5. Distribution of the Final Price for L1 auctions by the four main specification categories. research would be a hedonistic pricing analysis of the various product specifications. This will not be pursued in this thesis. 37 CHAPTER 2 Optimal Release of Inventory Using Online Auctions: The Two Item Case 1. Introduction The objective of this chapter is to provide a model for selling a fixed inventory using a sequence of single item auctions, and to derive structural properties regarding the optimal release policy. More specifically, how should a seller, given N identical items, optimally release each individual item for auction in order to maximize total profit. We assume all auction parameters, such as auction duration, starting price, bid increment, etc., have been fixed and that the only decision to make is regarding the timing to release each item for auction. The problem is modeled as a discrete time Markov Decision Problem (MDP) with focus on sufficient conditions to ensure optimal monotone policies. The imposed trade-off to make the problem interesting is that, on the one hand, the seller incurs a holding cost for each period an item remains unsold, while on the other hand, the more ongoing auctions the seller has, the lower the expected final price in each of those auctions. In other words, we assume competing auctions ‘cannibalize’ on each other. The holding cost represents, in addition to the usual components, such as cost of capital, insurance, and space, the value depreciation of an item, and can therefore also be regarded as a depreciation factor. An illustration of the depreciation is provided in Figure 2.1, which depicts the final price for the D3 desktop (bottom circles) and L4 laptop (top solid circles) auctions. The vertical axis is the final price, and the solid and dashed line represents the least square linear regression for L4 and D3 respectively. We see that for the L4 laptops the average selling price decreased 38 600 500 400 300 0 100 200 Final Price Dec ’05 Apr ’06 Jul ’06 Oct ’06 Jan ’07 Figure 2.1. Final price as a function of the ending date, for the D3 (bottom circles) and L4 (top solid circles) auctions at the eBay site of Dell Financial Services. All auctions lasted for three days and took place between 15th of December, 2005, and 30th of June, 2006. The solid and dashed line represents the least square linear regression for L4 and D3 respectively. more than $200 over 2006. The decrease for D3 was not as drastic. More detail empirical analysis, including a validation of the cannibalization assumption, appears in Chapter 4. The optimal release policy will mainly be driven by the effect from the holding cost and the cannibalization effect. If the holding cost is ‘very low’ then it will never be optimal to have more than one auction underway at any time. The reason for this is because the fewer 39 the number of ongoing auctions the higher the expected final price for each of the auctions, due to the assumed cannibalization. That is, the optimal release policy is to wait until the current auction is completed before releasing the next item, i.e. to hold N non-overlapping sequential auctions. On the other hand, if the holding cost is ‘very high’ then it will never be optimal to delay the release of an item and instead all items should be released immediately. The reason for this is that the additional holding cost from deferring will exceed the gain in expected final price by having fewer ongoing auctions. The optimal release policy is to hold N simultaneous auctions (all overlapping and note that this is different from one N -item auction). Our main interest are situations where the holding cost has some strategic implication and the optimal policy is not one of the extreme policies. Furthermore, we will show that the optimal release policy is a state dependent or closed loop policy, in contrast to a state independent or open loop policy. More details regarding the definitions of open and closed loop policy will be given in Section 3.1. Note that the two extreme open loop policies are the sequential and simultaneous release policies, and that open loop policies are special cases of closed loop policies. A numerical example illustrating the above discussion is provided in Figure 2.2. The figure depicts, for two items (N = 2) and auction length of three periods, the expected total profit (value) as a function of the per period per item holding cost. See Section 3.4 for details regarding formulation and computation of the example. The four dashed lines labeled A through D, represent the value for the four open loop policies; non-overlapping sequentially released (A), released with two day overlap (B), released with one day overlap (C), and simultaneously released (D). The solid line that lays above them represents the total expected profit for the optimal policy, a closed loop policy, which we describe how to compute below. We see that for ‘low’ holding cost the optimal policy is to release the auctions sequentially, while if the holding cost is ‘high’ the optimal policy is to release the 40 75 B 70 C Expected Total Profit 65 D ¬ Optimal Policy 60 55 50 45 A 40 2 2.5 3 3.5 4 4.5 Holding Cost 5 5.5 6 6.5 Figure 2.2. Two item, three period numerical example of expected total profit as a function of the per item per period holding cost. The four dashed lines represent the following open loop policies: (A) non-overlapping sequentially release, (B) release with two day overlap, (C) release with one day overlap, and (D) simultaneously release. The solid line that lays above them represents the total expected profit for the optimal policy, which is closed loop. See Section 3.4 for details regarding formulation and computation. auctions simultaneously. For cases in between, the optimal closed loop policy is adaptive and depends on the current auction price. Though it may appear to be an oversimplification, this chapter will only consider the case when N = 2. The reader will see that the two item case provides sufficient complexity to be both interesting and give rise to some surprising results. Furthermore, this will enable the discussion to focus on the governing trade-off, between releasing and deferring the release, and not become convoluted by the combinatorial complexity and curse of dimensionality of the N item case. It should also be noted that this problem has not yet been addressed in neither the existing auction theory or inventory literature. Previous research 41 has mainly focused on the analysis of an isolated single auction, either single-item or multiitem auctions, and in the multiple auctions case only considered non-overlapping sequential auctions. In addition, most research has dealt with the (optimal) specification of various auction parameters, e.g. reserve price, bid increment, auction length, lot size, etc. The novelty of this chapter is that it provides a framework for analyzing the issue of strategic timing of auctions when auctions compete or cannibalize on each other. This problem falls into the third category of open research areas as outlined by Pinker, Seidmann and Vakrat (2003). Namely how could (or should) a firm integrate online auctions into their business model. Bajari and Hortacsu (2004) stated that more research needs to be done regarding ”the analysis of markets with multiple simultaneous auctions.” The ambition is that the ensuing discussion provides a structural framework, insights and results regarding this issue. 1.1. Literature Review. In recent years auction theory has come to play an important role in the management science and revenue management field, resulting in a wide spectrum of applications. However, given the voluminous literature on inventory management and dynamic pricing, relatively little has been written with regards to inventory management using online auctions. Two papers that consider the impact auctions have on the inventory re-ordering policy are Vulcano and van Ryzin (2004), and Huh and Janakiraman (2008). Vulcano and van Ryzin focus on how a seller should optimally choose the auction format and how this decision will affect the optimal inventory re-ordering policy. They formulate the problem as an infinite horizon dynamic program and show the optimal joint auction-format and replenishment policy. Huh and Janakiraman show that using auctions as a sales channel, conditions to ensure that (s, S) policies are optimal are satisfied. Vulcano, van Ryzin, and Maglaras (2002) have previously analyzed a problem that is similar to the one we address. There they consider a seller, who given a fixed inventory and fixed time-horizon, has to optimally ‘auction’ off the goods. The underlying ‘auction’ mechanism 42 they consider is in the spirit of www.priceline.com where people place ‘bids’ and sellers can choose to accept or reject the offers. They model each multi-item ‘auction’ as a separate period and perform the symmetric equilibrium analysis for each period (auction). One of the main results is that the seller should not employ a standard auction format. An auction is standard if the highest bidder is guaranteed to be awarded the item [14, p.29]. Another related paper is by Pinker, Seidmann, and Vakrat (2001), who analyze the problem of disposing a given inventory using a sequence of non-overlapping multi-item online auctions. Based on the symmetric equilibrium analysis and uniform valuations, their objective is to categorize the optimal number of multi-item auctions and the optimal unit to release in each auction. In contrast to these papers, we permit the auctions to overlap and analyze the auction dynamics as a Markov chain. The above papers all use a game theoretic approach. A paper which uses a different analysis methodology is Bertsimas, Hawkins and Perakis (2003). The problem they address is how a seller should optimally set the auction control parameters starting price, reserve price and auction length, in order to maximize revenue. They model the problem as a MDP and based on over 17,000 eBay auctions determine the optimal parameters. Bapna, Goes, and Gupta (2003) also address the issue of optimal auction control parameters in a revenue management context. The main focus of their analysis is to highlight the importance and structural implication of the bid increment in a first-price multi-item auction. Using data from 90 online auctions they empirically validate their findings. The two common elements of the above literature is that they focus on the optimal setting of auction parameters and analyze each auction in isolation. In contrast, we model the optimal release or timing of auctions given fixed auction parameters and a dynamic interaction between competing auctions. 43 A paper that analyzes the dynamics between competing auctions is Peters and Severinov (2006). They consider the case when all auctions are simultaneously released. They present a model with M bidders and N single-item auctions, where both M and N are fixed, and derive the Bayesian-Nash equilibrium for the final price of the N auctions. In particular they show that the final price will be the same for all auctions, namely one increment above the M −N highest valuation. Meaning that, if there are 10 bidders and 5 items, the price in all 5 auctions will be one increment above the 6th highest valuation. Though they are implicitly assuming an online setting, there is nothing explicit in their model that incorporates the special dynamics of online auctions, such as the arrival rate of bidders or fixed auction dead-line. In contrast to the above papers, our framework is more in line with the model presented by Segev, Beam and Shantikumar (2001), where online auctions are modeled as Markov chains. The main focus of their paper is to characterize the distribution of the final price given a specific arrival rate and bidding strategy. Overview of Chapter 2. The remainder of this chapter is organized as follows. In section 2 we formulate the problem and general model. In section 3 we discuss the case when the auctions are guaranteed to be successful, and hence the seller only has to list an item once. While in section 4 we discuss the case when there is a positive probability an auction receives zero bids and the seller has to re-list items from unsuccessful auctions. In section 5 we summarize our conclusions and provide ideas for future research. 44 2. Problem Formulation We are considering a seller who, over a planning horizon T , intends to sell two identical items using a pair of single-item auctions. Each auction is assumed to have the same fixed and finite time-length τ . We divide τ into a sequence of discrete periods such that each auction period coincide with the length of the discrete time period that constitute T . The seller decides at the start of each period whether or not to release an item for auction. It is important to emphasize that an ongoing auction does not have to be completed before the next auction is started; auctions may overlap each other. We model the seller’s problem as a discrete time Markov Decision Problem (MDP) with the objective of maximizing expected total profit. Two cases regarding the time-horizon will be considered. In the first case we assume the auctions are guaranteed to be successful and hence the seller only has to list an item once. Since the seller only has two items, the seller is faced with a finite planning horizon 2τ . In the second case, we assume there is a positive probability that an auction is unsuccessful, meaning that no bids arrived, and that the seller has to re-list unsold items. Consequently the seller is faced with an infinite planning horizon. The reason for separating the two cases is that they require different models and analysis. An important aspect to keep in mind is that we do not model the individual bidders or their bidding strategy. Each auction is modeled as a Markov chain, where the state of an auction evolves according to certain dynamics. Chapter 3 illustrates, with two fixed bidding strategies, how the Markov chain transition probabilities can be derived from the individual bidding behavior. In Chapter 4 we discuss an empirical model for how a seller can capture the Markov chain transition probabilities from real auction data. We will throughout the chapter assume two fundamental aspects regarding the seller. The first is that the seller would only be interested in selling via auctions if the accumulated holding cost over the duration of an auction is compensated by the expected final price. 45 We summarize this in the following lemma and refer to it as the positive expected profit assumption. Lemma If the expected revenue from an auction does not exceed the holding cost accumulated over the auction duration, then it is optimal to immediately dispose of the items. The second assumption is that the seller is vigilant in keeping track of how many items he has released for auction and how many that are remaining, and that there will not be any reason to speculatively hold inventory. This is summarized as follows and referred to as the vigilant seller assumption. Lemma If the price dynamics of an auction are independent of time and the holding cost is positive, then it will always be optimal to have at least one auction underway while there still is remaining inventory. In other words, if there are no auctions underway but the seller still has items remaining he should always start at least one auction. Thus at the start of the planning horizon, he should always start at least one auction. This lemma is equivalent to Lemma 1 in Pinker, Seidmann and Vakrat (2001), where a proof is provided. 2.1. Markov Decision Problem Formulation. To formulate the seller’s problem as an MDP, we require the following elements. Decision Epochs, t = 0, 1, . . . , T We assume discrete time periods of equal length and that decisions are made at the beginning of each period. We are implicitly thinking of T as a fixed number of days and that 46 decisions are made on a daily basis. However, for a general framework where, for instance, decisions are made more frequently, T could be increased to reflect the appropriate planning horizon. We will consider two cases: T < ∞ and T = ∞. The finite planning horizon case arises if the seller only lists an item once, while an infinite planning horizon formulation is required when there is a positive probability an item does not sell and the seller has to re-list it. State Space At each decision epoch t, the system state, S = ([X1 , Y1 ; X2 , Y2 ], Z), consists of the state of each auction, [Xi , Yi ]i=1,2 , and the number of ongoing auctions Z. Each auction i, i = 1, 2, is defined by the pair of random variables current price (bid), Xi , and elapsed auction time, Yi . We will consider both discrete and continuous prices. For the discrete case Xi ∈ {0, p, p + k, p + 2k, . . . , P }, where p, k and P are positive, finite integers. While for the continuous case Xi ∈ {0} ∪ [p, P ], where [p, P ] ⊂ +. In both cases, p is the starting price of the auction, P the upper limit of what any bidder would be willing to bid, and for discrete prices, k is the price-increment. We assume Yi is discrete and finite, Yi ∈ {0, 1, . . . , τ } ∪ {δ}, where τ < ∞. The symbol δ is used to indicate that the auction is completed and the item awarded. We will interchangeably use the notation Xi,Yi and (Xi , Yi ) to denote the state of auction i, i = 1, 2. For instance, Xi,τ is the final price of auction i. The notation XY is used to represent an auction that has elapsed for Y periods. At the start of an auction Yi = 0 and Xi = 0. For each additional period an auction is underway Yi increases by one. When an auction has successfully been completed, that is Xi,τ ≥ p, the item is awarded and payment received. In this case, the state of the auction 47 ([0, 0; 0, 0], 1) ([x1 , y1 ; x2 , y2 ], z) ([x1 , y1 ; x2 , y2 ], z ) ✲ t=0 t=1 t=2 t=3 Figure 2.3. Time-line for two auctions. At t = 0 the first auction is started automatically, and the non-trivial decision is to decide whether to start auction 2. If auction 2 is started at t = 0, then at t = 1, x1 , x2 ≥ 0, y1 = y2 = 1, z = 2 (assuming τ > 1), and there is no decision to make. If auction 2 is not started at t = 0, then at t = 1, x1 ≥ 0, y1 = 1, x2 = 0, y2 = 0, z = 1, and the non-trivial decision is whether or not to start auction 2. evolves as follows, for p ≤ xi ≤ xi , . . . =⇒ (xi , τ − 1) =⇒ (xi , τ ) =⇒ ∆i =⇒ ∆i =⇒ . . . where ∆i = (Xi , δ). We let ∆ denote the absorbing state when both items have been sold, ∆ = ([∆1 ; ∆2 ], 0). If an auction is unsuccessful, that is Xi,τ = 0, the auction returns to the initial state (0, 0), that is the transitions follow, (q, τ ) w. prob. Pr{Xi,τ = q | Xi,τ −1 = 0} . . . =⇒ (0, τ − 1) =⇒ (0, τ ) ≡ (0, 0) w. prob. Pr{Xi,τ = 0 | Xi,τ −1 = 0} Though it may appear redundant we include a counter Z of the number of ongoing auctions. The number of ongoing auctions at time t will be defined by Zt . In order to avoid issues with Zt in decision epochs where an auction will be started by the vigilant seller assumption, we define Zt to be the number of ongoing auctions in the instantaneous moment before decision epoch t, before any price jumps have occurred and before the seller has made a non-trivial or relevant decision. For instance, at the start of the planning horizon Z0 = 1. See Figure 2.3 for an illustration of the time-line. The reader familiar with auctions or auction theory, may notice that we have not included a reserve price. Section 3.6 discusses the implication of including a reserve price and shows that it imposes no change to the results. As a minor notational convention, we will avoid double parenthesis for functions where the state space is the only argument, that is 48 we write f ([X1 , Y1 ; X2 , Y2 ], Z) instead of the strictly correct f (([X1 , Y1 ; X2 , Y2 ], Z)). Actions The only non-trivial decision facing the seller is to decide when to release an item provided that the current auction has not yet been successfully completed. In other words, nontrivial decision only pertain to states where Yi < τ and Yj = 0, i = j. Under all other conditions, the seller either does not have any decision to make or will release an item due to the vigilant seller assumption. At each decision epoch, the actions a = 1 corresponds to releasing the remaining item, and a = 0 not to release it. Furthermore, because the items are identical, one can without loss of generality, define the remaining item to be item 2. For the finite time-horizon this should be fairly obvious. However, for the infinite time-horizon, due to that when an auction is not successful and has to be re-listed, this may not be as obvious. We will revisit this issue in Section 4. Consequently the action space is, for s = ([x1 , y1 ; x2 , y2 ], z), As = {0, 1} y1 < τ and y2 = 0 {0} y1 = τ, δ or y2 > 0 In Figure 2.4 a simple example, with Xi = 0, 1 and τ = 2, illustrates how the system state may evolve.States enclosed in a box indicate situations with non-trivial decisions. Transitions due to the non-trivial decision of releasing the second item are represented by the dashed lines. Transitions due to not releasing or releasing due to the vigilant seller assumption are represented by the solid lines. Note that there are four possible loops: ([0, 0; 0, 0], 1) ([0, 1; 0, 1], 2), ([0, 0; 0, 0], 1) ([0, 1; 0, 0], 1), and ([1, δ; 0, 0], 1) ([0, 1; 0, 0], 1), ([0, 1; 0, 0], 1) ([1, δ; 0, 1], 1). And that there is one absorbing state ∆ = ([1, δ1 ; 1, δ2 ], 0), which is represented by the bold dashed line. 49 ❄ ([0,0;0,0],1) ❄ ([1,1;1,1],2) ❄ ❄ ([0,1;1,1],2) ❄ ([1,1;0,1],2) ([0,1;0,1],2) ❄ ❄ ([1,2;1,2],0) ❄ ❄ ❄ ([1,2;0,0],1) ([1,δ;1,1],1) ❄ ([1,1;0,0],1) ([1,2;0,1],1) ❄ ([1,δ;0,1],1) ✛ ❖ ✖ ❄ ✠ ([0,1;0,0],1) ❄ ([1,2;1,1],1) ❄ ✲ ([1,δ;0,0],1) ✕ ❄ ✲([1,δ;1,2],0) ✛ ❄ ✲ ([1,δ;1,δ],0) ✛ Figure 2.4. Example of system state transitions when Xi = 0, 1 and τ = 2. States enclosed in a box indicate situations with non-trivial decisions. Solid lines represents transitions due to not releasing or release by vigilant seller assumption; dashed lines represents transitions due to non-trivial release decisions; bold dashed line represent absorbing cycle. Rewards For each period in which an item has not been sold, the seller incurs a positive holding cost h. When an auction is successfully completed the seller receives the payment and awards the item. After an item has been sold and the state (Xi , Yi ) = ∆i , i = 1, 2, the seller will in perpetuity neither incur any cost nor receive any payment for that item. Let rt (s) denote the reward in period t given a state s ∈ S. It is given by, rt ([x1 , y1 ; x2 , y2 ], z) = x1 1{y1 =τ } − h1{y1 <τ } + x2 1{y2 =τ } − h1{y2 <τ } Transition Probabilities Each period in which an auction is underway the price transitions follow the dynamics of an exogenously given stochastic process. In other words, we assume that there is some underlying bidder arrival process and bidding behavior, which can be completely summarized by a probability distribution regarding the one period price transitions. For discrete prices, 50 these are represented by the following transition probability matrices, π0,0|1 0 Π1 = . .. 0 π0,p|1 ··· πp,p|1 ··· .. . .. 0 ··· . π0,P |1 πp,P |1 .. . πP,P |1 π0,0|2 0 Π2 = . .. 0 π0,p|2 ··· πp,p|2 ··· .. . .. 0 ··· . π0,P |2 πp,P |2 .. . πP,P |2 Πz , z = 1, 2, is the one-period transition probability matrix for an individual auction when there are z auctions underway, for y < τ , x ≤ q, and z = 1, 2, (2.1) Pr{Xi,y+1 = q | Xi,y = x, Z = z} = πx,q|z i = 1, 2 In the case of continuous prices, we assume the price transition dynamics can be represented by a conditional cumulative distribution function, for y < τ , x ≤ x , and z = 1, 2, (2.2) Pr{Xi,y+1 ≤ x | Xi,y = x, Z = z} = FXz y+1 |Xy (x |x) = x x z fX (q|x)dq y+1 |Xy z where fX (·|x) is the one-period conditional transition probability density function for y+1 |Xy an auction which after y periods is at a price x, and there are z auctions underway. Using the Chapman-Kolmogorov equations (cf. [23, Ch.4.2]), the n-period transition probabilities for a single auction can be derived. To illustrate, assume prices are discrete and we are interested in the two- and three-period transition probabilities, and that there are z1 , z2 , and z3 auctions underway in the ensuing three periods respectively, for y ≤ τ − 3, q Pr{Xy+2 = q | Xy = x, Zt = z1 , Zt+1 = z2 } = πx,j|z1 πj,q|z2 j=x q q Pr{Xy+3 = q | Xy = x, Zt = z1 , Zt+1 = z2 , Zt+2 = z3 } = πx,j|z1 πj,k|z2 πk,q|z3 j=x k=j 51 Consequently, to derive the probability distribution of the final price we simply multiply the transition probability matrices accordingly. For instance, suppose τ = 3 then the top row in Πz1 Πz2 Πz3 ≡ Πz1 ·z2 ·z3 provides the unconditional probability distribution of the final price for an item with z1 , z2 , and z3 auctions in the first, second, and third period respectively. For continuous prices and y ≤ τ − 3, z1 ·z2 fX (x |x) = y+2 |Xy z1 ·z2 ·z3 fX (x |x) = y+3 |Xy x x v x x x z2 z1 fX (x |u)fX (u|x)du y+2 |Xy+1 y+1 |Xy z3 z2 z1 fX (x |v)fX (v|u)fX (u|x)dudv y+3 |Xy+2 y+2 |Xy+1 y+1 |Xy In order to simplify the notation we occasionally omit the subscript ‘Xy+1 |Xy ’ and write F z (·|·) and f z (·|·), i = 1, 2, with the implicit assumption that y < τ . 2.2. Assumptions Regarding the Transition Probabilities. We will next provide some additional assumptions regarding the transition probabilities. These assumptions, which can be seen as a reflection of the bidding behavior, will ensure that certain structural results will follow. The assumptions should not be regarded as categorical statements about all bidders, but rather as a statistical reflection of what the bidding behavior is like in the majority of auctions. In Chapter 4 a statistical model to derive the transition probabilities and validate the assumptions is provided. The validation is based on eBay auctions from Dell Financial Services (DFS) that ran between December 2005 to February 2007. More information regarding DFS and the data was discussed in Chapter 1. When two auctions are underway we assume that the auction prices evolve independently. That is, the price in one auction does not affect the transition dynamics of the 52 other auction. In other words, for discrete prices and yi < τ , i = 1, 2, Pr{Xi,yi +1 = q | X1,y1 = x1 , X2,y2 = x2 , Z = 2} = Pr{Xi,yi +1 = q | Xi,yi = xi , Z = 2} = πxi ,q|2 While for continuous prices and yi < τ , i = 1, 2, Pr{Xi,yi +1 ≤ x | X1,y1 = x1 , X2,y2 = x2 , Z = 2} = Pr{Xi,yi +1 ≤ x | Xi,yi = xi , Z = 2} = F 2 (x |xi ) Implicitly this assumes that bidders choose a bid-amount only based on the current price and elapsed auction time of the auction they are placing a bid in. Chapter 6 provides a discussion regarding extensions to correlated price-transitions, i.e. where the price-transitions also depend on the price of the competing auction. Consequently, with two auctions underway, the transition probability for the system state is the product of the individual transition probabilities. For discrete prices and y1 , y2 < τ , Pr{St+1 = ([q, y1 + 1; r, y2 + 1], z ) | St = ([x1 , y1 ; x2 , y2 ], 2)} = (πx1 ,q|2 )(πx2 ,r|2 ) Since the main interest pertains to the states s ∈ S such that As = {0, 1}, we can explicitly state the system state transition probabilities. In the discrete case we define π(s |s, a) to be the one period system state transition probability, for s = ([x1 , y1 ; 0, 0], 1) and y1 < τ , π π(s |s, a) = x1 ,q|1 a=0 (πx1 ,q|2 )(π0,r|2 ) a = 1 where for a = 0, s = ([q, y1 + 1; 0, 0], 1), and for a = 1, s = ([q, y1 + 1; r, 1], z) with z = 1, 2. The extension to the continuous case is straight forward, though we need to define the notation a bit more carefully. We define F (s |s, a) to be the one period system state 53 transition distribution function. That is, for s = ([x1 , y1 ; x2 , y2 ], z), s = ([x1 , y1 + 1; x2 , y2 + 1], z ), F (s |s, a) ≡ Pr{X1,y1 +1 ≤ x1 , X2,y2 +1 ≤ x2 , Zt+1 = z | St = s} Similar to the discrete case we are mainly interested in states s ∈ S such that As = {0, 1}, for s = ([x1 , y1 ; 0, 0], 1) and y1 < τ , F (s |s, a) = F 1 (q|x1 ) a=0 F 2 (x1 |x1 )F 2 (x2 |0) a = 1 where for a = 0, s = ([x1 , y1 + 1; 0, 0], 1), and for a = 1, s = ([x1 , y1 + 1; x2 , 1], z) with z = 1, 2. We assume bids are non-retractable, which for the case of discrete prices implies that Πz , z = 1, 2, are upper-triangular (πq,xi |z = 0 for q < xi ). While for continuous prices, we require that F z (x |x) = 0, for x < x, z = 1, 2. Consequently, the current price of an auction is increasing. Though strictly speaking on, for instance, eBay, bidders may retract a bid, it is very rare. Transition probabilities are assumed to be stationary with respect to both: 1) calender time t, and 2) elapsed auction time Yi , i = 1, 2. The former assumption, which was included in the vigilant seller assumption, is mainly for ease of notation and to ensure the model is tractable. In reality, the dynamics of Xi may depend on calender time. For instance, at night, weekends or certain weekdays there might be less bidding activity. We will for simplicity ignore this and strictly consider stationary transition probabilities with regard to calender time. Likewise we will ignore non-stationary transitions with regard to the elapsed auction time. A well-established phenomena of online auctions, is that the price dynamics or bidding behavior is dramatically different toward the end of auctions. One reason for 54 this, is because some bidders try to place their bids as close as possible to the end of the auction, thereby leaving no time for others to counter-bid. This is referred to as sniping. Roth and Ockenfels (2002), and Shmueli, Russo, and Wolfgang (2004) analyze different aspects regarding non-stationary bidding activity. In Section 3.7 we will discuss how the change in auction dynamics over time can be incorporated. The next set of assumptions play a more crucial role in the ensuing analysis. Each is stated for discrete and continuous prices, and make use of Leibnitz Rule, ∂ ∂y β(y) β(y) f (x, y)dx = α(y) α(y) ∂f (x, y) ∂ ∂ dx + f (β(y), y) β(y) − f (α(y), y) α(y) ∂y ∂y ∂y Assumption 2.1. The probability of making a jump to the higher prices is increasing in the current price. Discrete prices: for x < P , z = 1, 2, P P πx+1,q|z πx,q|z ≤ (2.3) ∀r ≤ P q=r q=r Continuous prices: for y < τ , x ≤ x ≤ P , and z = 1, 2, (2.4) x F z (x |x) = f z (u|x)du is decreasing in x x Equivalently, ∂ z F (x |x) = ∂x x x ∂ z f (u|x)du − f z (x|x) ≤ 0 ∂x Assumption 2.1 reflects that bids are increasing in the current price. In other words, the likelihood of placing a ‘high’ bid is increasing in the current price. This holds for example if bid increments were independent of the current price. In reality, however, bid increments tend to be decreasing in the current price, and it is therefore not immediate that Assumption 2.1 holds. Empirical evidence supporting Assumption 2.1 and showing that bid increments 55 are decreasing in the current price, can be seen in Figure 2.5 below. They depict the pricejumps at 12 hour intervals for the L4 auctions. Each dot represents an individual auction. Auctions along the 45 degree line are auctions in which the price remained unchanged 12 hours later (no price-transition took place). Note that in the final period all auctions had strictly positive price-increments. The feature supporting our assumption is that in all figures the price-jumps form an upward sloping ‘band’. A counter indication to our claim would be if there was a large number of auctions that at low prices (≈ $0 − 150) made jumps to the high prices (≈ $500 − 600). More details are provided in Chapter 4 Assumption 2.2. The probability of making a jump to higher prices decreases when there are two ongoing auctions. Discrete prices: for x ≤ P , P (2.5) P πx,q|2 ≤ q=r πx,q|1 ∀r ≤ P q=r Continuous prices: for y < τ and x ≤ P , (2.6) F 1 (x |x) ≤ F 2 (x |x) ∀x ≤ P This assumption formalizes how we model the cannibalization effect. In other words, with two ongoing auctions, each auction will experience more ‘modest’ price-transitions. For empirical support see Chapter 4. An alternative to Assumption 2.2 is, for discrete prices, if P q=r πx,q|1 ≤ P q=r πx,q|2 , for all x, r ≤ P . That is, with two auctions you are more likely to see higher price jumps in each individual auction than when only one auction is underway. However, then the problem of releasing the second item becomes trivial. Since if it is better to have two auctions underway and the holding cost is positive, then it will always be optimal to release the second item 56 300 400 500 600 200 300 Price after 48hours 400 500 500 400 300 Price after 36hours 200 100 100 0 0 200 0 100 200 300 400 500 600 0 100 200 300 400 Price after 12hours Price after 24hours Price after 36hours 12 → 24h 24 → 36h 36 → 48h 600 400 0 0 100 100 200 300 Final Price (72hours) 500 500 400 300 200 Price after 60hours 600 600 600 500 400 300 Price after 24hours 200 100 0 100 600 0 0 100 200 300 400 500 600 0 100 200 300 400 Price after 48hours Price after 60hours 48 → 60h 60 → 72h 500 600 Figure 2.5. Price transitions at 12 hour intervals for the L4 laptop auctions. The horizontal axis represents the price at various 12 hour intervals, while the vertical axis represents the price 12 hours later. Each circle represents the price-transition for an auction. Observations on the 45 degree line represents auctions that received no bids for that period. Note that in the final period all auctions received bids. immediately. Assumption 2.3. The difference, in probability of making jumps to the higher prices, between having one versus two ongoing auctions, is decreasing in the current price. Discrete prices: for x < P , P (2.7) P (πx,q|1 − πx,q|2 ) ≥ q=r (πx+1,q|1 − πx+1,q|2 ) q=r 57 ∀r ≤ P 500 600 Continuous prices: for y < τ and x ≤ P , (2.8) F 2 (x |x) − F 1 (x |x) is decreasing in x Equivalently, ∂ 2 ∂ 1 F (x |x) ≤ F (x |x) ∂x ∂x This states that the cannibalization effect is diminishing in the current price. In other words the closer the current price is to the upper bound P the less of a difference there will be between having one or two auctions underway. Qualitatively, we see in the graphs of Figure 2.5, that the closer the price is to P ≈ 600 the less ‘room’ there is for the price-transitions, and hence the less cannibalization there can be. Again, a more rigorous empirical analysis is provided in Chapter 4. 2.3. Examples. At this point it may be natural to inquire about the existence of transition probability matrices and conditional cumulative distribution functions, that satisfy the above assumptions. We next provide conditions under which of some common probability distributions satisfy them. Namely, Uniform - discrete and continuous, Bernoulli, and Exponential. In addition, we later illustrate the assumptions and implications with numerical examples. Discrete Uniform Distribution Without loss of generality let p = k = 1. Suppose that in periods when there is only one auction underway there is an equal probability of jumping to any of the remaining prices, for pi ≤ P , πpi ,q|1 = πpi = 1/(P + 1 − pi ) for all q ∈ [pi , P ]. Furthermore, suppose when two auctions are underway the probability of remaining at the same price increase with κ and that the probability of jumping to P decrease with κ, as shown in the transition probability matrices below. 58 1 P +1 0 . ΠU 1 = .. 0 0 1 P +1 ··· 1 P +1 1 P ··· 1 P .. . .. . .. . 0 ··· 1 2 0 ··· 0 1 +κ P +1 1 0 P .. .. ΠU 2 = . . 1 0 2 1 0 1 P +1 1 P +1 1 P − κ −κ .. . −κ 1 ··· 1 P +1 1 P +1 +κ ··· 1 P 1 P .. . .. .. . 0 ··· 0 ··· . 1 2 +κ 0 1 2 The next results summarizes that the above transition probability matrices support Assumptions 2.1, 2.2, and 2.3. Proposition 2.4. If 0 ≤ κ ≤ 1 P +1 U then ΠU 1 and Π2 satisfies Assumptions 2.1, 2.2 and 2.3. Proof Proposition 2.4 - See Appendix A. A modification to ΠU 2 is to have κ be dependent on the price. In which case for Assumption 2.3 to hold we require 1 P +1 ≥ κ0 ≥ κ1 ≥ . . . ≥ κP −1 . Continuous Uniform Distribution An example with continuous prices and uniform distributed price-transitions, can be constructed as follows. Assume the starting price p = 0, and the maximum price P = 1, i.e. Xi ∈ [0, 1]. Assume that when there is only one ongoing auction that the price-transition is uniformly distributed between the current price and the upper limit 1. Furthermore, assume the ‘cannibalization’ effect is such that with two ongoing auctions, the price-transition is 59 triangularly distributed between the current price and the upper limit. Specifically, let, 1 (2.9) f (q|x) = (2.10) 0 2 f (q|x) = 1 1−x x≤q≤1 o/w 2−2q (1−x)2 0 x≤q≤1 o/w Proposition 2.5. If prices are continuous and price transitions are distributed according to (2.9) and (2.10) then Assumptions 2.1, 2.2, and 2.3 holds. Proof Proposition 2.5 - See Appendix A. Bernoulli Distribution Suppose that for each period and every price level there are only two possible transitions remain at same price or jump up by one increment. This bidding process is the core of the auction dynamics analyzed by Segev, Beam, and Shantikumar (2001). In this scenario the maximum price P ≡ τ , and consequently the size of the transition probability matrices are (τ + 1) × (τ + 1). Let Π1 and Π2 be defined as follows, ΠBe 1 1 − π 0 0 = .. . 0 0 π0 ··· 0 1 − π1 ··· 0 .. . .. .. . 0 ··· 1 − πτ −1 0 ··· 0 . 0 0 .. . πτ −1 1 60 ΠBe 2 1 − ρ 0 0 = .. . 0 0 ρ0 ··· 0 1 − ρ1 ··· 0 .. . .. .. . 0 ··· 1 − ρτ −1 0 ··· 0 . 0 0 .. . ρτ −1 1 In other words, Pr{Xy+1 = q + 1|Xy = q, Z = 1} = πq , and Pr{Xy+1 = q + 1|Xy = q, Z = 2} = ρq . Due to the special structure of Bernoulli price transitions an adjustment to Assumption 2.3 is required: we assume, for x < P − 1, (πx,x+1|1 − πx,x+1|2 ) ≥ (πx+1,x+2|1 − πx+1,x+2|2 ). Without this modification Assumption 2.3 would not hold, since P q=x+2 (πx,q|1 − πx,q|2 ) = 0 ≤ P q=x+1 (πx+1,q|1 P q=x+2 (πx+1,q|1 P q=x+1 (πx,q|1 − πx+1,q|2 ), while − πx,q|2 ) ≥ − πx+1,q|2 ) = 0. The adjustment to the condition will not alter any of the structural properties for the Bernoulli distributed price transition. In order for ΠBe 1 and ΠBe 2 to satisfy Assumptions 2.1, 2.2, and 2.3(mod.), we require, (2.11) π0 − ρ0 ≥ π1 − ρ1 ≥ . . . ≥ πτ −1 − ρτ −1 ≥ 0 Inequalities (2.11) reflects the diminishing ‘cannibalization’ effect, and ensures that Assumptions 2.2 and 2.3(mod.) holds. We summarize the result in the following proposition. Be Proposition 2.6. If price transitions are distributed according to ΠBe 1 and Π2 , and (2.11) holds, then Assumptions 2.1, 2.2, and 2.3(mod.) are satisfied. Proof Proposition 2.6 - See Appendix A. Below a numerical example for the Bernoulli distributed price transitions is provided. It can be verified that (2.11), and hence that Assumptions 2.1, 2.2 and 2.3(mod.) holds. ΠBe 1 .4 0 = 0 0 .6 0 .5 .5 0 .7 0 0 0 0 .3 1 ΠBe 2 .5 0 = 0 0 .5 0 .6 .4 0 .8 0 0 0 0 .2 1 A special case of ΠBe z , z = 1, 2, is when the transition probabilities are independent of the current price, that is, when πq = π and ρq = ρ for all q = 0, 1, . . . , τ − 1. This special case 61 has some interesting consequences which are discussed in Section 3.3.1. Exponential Distribution The following example will focus on the price-increment and not on the price-transitions. Assume prices are positive and unbounded, Xi ∈ +, and that the conditional within period price-increment C, given Xi = x, is exponentially distributed with rate λz (x), z = 1, 2. That is, for c = x − x, x, x ∈ +, Pr{Xi,y+1 ≤ x |Xi,y = x, Z = z} = Pr{C ≤ c|Xi,y = x, Z = z} = GzC (c|x) = (2.12) 1 − exp(−λz (x)c) c ≥ 0 0 o/w The rate λz (·) is a function both of the current price and the number of ongoing auctions. The expected price increment is 1/λz (x), which it would seem natural to assume, is decreasing in the current price. Therefore, we require λz (x) to be increasing in x. In other words, the higher the current price the smaller the expected price-increment. Though technical conditions on λz (x) could be imposed, such that the three assumptions hold, they would make the problem both less intuitive and less informative. The main problem is due to the shape of the exponential distribution which, for instance, prevents Assumption 2.1 to hold. Therefore, we impose conditions to ensure that the expected price-transition has certain properties. Specifically, we assume, for x ∈ +, (2.13) 1/λz (x) is decreasing in x, z = 1, 2 (2.14) x + 1/λz (x) is increasing in x, z = 1, 2 (2.15) 1/λ2 (x) ≤ 1/λ1 (x) (2.16) 1/λ1 (x) − 1/λ2 (x) is decreasing in x 62 With these conditions the ensuing structural results in Section 3.3 holds. An example of a rate function λz (x) for which the above conditions hold includes, 1/λz (x) = exp β0 − β1 x − β2 1{z=2} where β0 , β1 , β2 ≥ 0, and β1 is such that, β1 exp β0 − β1 x − β2 1{z=2} ≤ 1 ∀x, z In Chapter 4 a more general version with gamma distributed price-increments is discussed and fitted to the eBay auction data from DFS. 3. Guaranteed Successful Auctions - Single Listing The first case we consider is when the auctions are guaranteed to be successful, and hence the seller only has to list an item once. This could occur when the items are such that it is certain a positive bid will arrive (e.g. π0,0|z = 0, z = 1, 2), or when the seller decides in advance to immediately salvage items remaining from unsuccessful auctions. An illustration of the former includes the 6,000 laptop and desktop eBay auctions of DFS. Out of all auctions with a starting price of $.99, not a single auction was unsuccessful. Due to the additional assumption that auctions are guaranteed to be successful, we can simplify the MDP model. Decision Epochs As a consequence of the vigilant seller assumption there is no reason to consider a planning horizon beyond two sequential auctions, hence T = 2τ . Furthermore, provided the second item has not been released, non-trivial decisions can only be made in periods t = 0, 1, 2, . . . , τ −1. At t = τ the vigilant seller assumption requires that the second item is released immediately, if it has not already been released. 63 State Space Since we assume the items will at least sell for p, we omit the 0 state. Thus for the discrete case Xi ∈ {p, p + 1, . . . , P } while for the continuous case Xi ∈ [p, P ]. Rewards In order to facilitate the ‘accounting’, and since we are not assuming discounting, we assume the seller receives the payment at t = T . Therefore, the reward rt (s) for a given s ∈ S and period t is as follows, −h1 rt ([x1 , y1 ; x2 , y2 ], z) = {y1 <τ } − h1{y2 <τ } x1 + x2 t = 0, 1, . . . , T − 1 t=T Transition Probabilities Since we assume Xi ≥ p > 0 and the items are guaranteed to be awarded, we define the transition probability to start at p instead of 0, e.g. the entries in the top row of Πz is πp,q|z for q ∈ {p, p + 1, . . . , P }, z = 1, 2. It should, however, be noted that from a behavioral point of view the bidding process may be different if the starting price or even current price is 0 rather than p. For instance, suppose we have two auctions, a and b, which both have elapsed for yi periods and both with current price of $p, but where auction b started at $p and still has not received any bids while auction a has reached $p after some bid activity. Then there is anecdotal ‘evidence’ to suggest that the bidding dynamics for the two auctions will be different. Auction a is more likely to receive more bids. To read accounts from eBay sellers on this issue, search the terms ‘low starting price’ at the eBay discussion boards for sellers,1 and newcomers.2 We will ignore such behavioral considerations. 3.1. Auction Release Policies. A Markov deterministic policy is a sequence of decision rules which determine what action to take in each decision epoch, possibly contingent on the state of the system but not on the past. Let γt (s) be the decision rule in period t 1http://forums.ebay.com/db2/forum.jspa?forumID=143 2 http://forums.ebay.com/db1/forum.jspa?forumID=120 64 given a state s ∈ S. As a consequence of the vigilant seller assumption, we only need to consider decision rules for t = 0, 1, . . . , τ , and hence, a policy γ is defined as follows, γ = (γ0 (s), γ1 (s), . . . , γτ (s)) γt (s) ∈ {0, 1}, ∀s ∈ S, t = 0, 1, . . . , τ If all the decision rules, γt (s), are independent of the price components of state s we refer to the policy γ as an open loop policy, while if the decision rules depend on both the price and time components of state s the resulting policy is referred to as a closed loop policy. Note that there are only τ + 1 open loop policies of interest. We write VO(j) to denote the total expected profit of releasing the second item j periods after the first, j = 0, 1, 2, . . . , τ . In Table 2.1 the four open loop policies and their respective total expected profit for discrete prices and τ = 3 are provided. In the table we see that although we incur an additional unit of h for each additional period we hold the second item, the expected final price for both items increase since there is an additional period when both auctions evolve according to Π1 instead of Π2 . The decision whether to release the second item or hold it one more period will depend on whether the increase in expected final price for both items will compensate the additional holding cost. j γ 0 (1,0,0,0) Total expected profit - VO(j) −6h + 2 P l=p l q=p l r=q l(πp,q|2 )(πq,r|2 )(πr,l|2 ) 1 (0,1,0,0) −7h + P l=p l q=p l r=q l(πp,q|1 )(πq,r|2 )(πr,l|2 ) + P l=p l q=p l r=q l(πp,q|2 )(πq,r|2 )(πr,l|1 ) 2 (0,0,1,0) −8h + P l=p l q=p l r=q l(πp,q|1 )(πq,r|1 )(πr,l|2 ) + P l=p l q=p l r=q l(πp,q|2 )(πq,r|1 )(πr,l|1 ) 3 (0,0,0,1) −9h + 2 P l=p l q=p l r=q l(πp,q|1 )(πq,r|1 )(πr,l|1 ) Table 2.1. Total expected profit for the four open loop policies of releasing item 2 j periods after item 1, for discrete prices and τ = 3. 65 3.2. Optimality Equations. Before we discuss the optimality equations we introduce some notation. When both items have been released we define E[Xi,τ |St ] to be the conditional expected final price of auction i, i = 1, 2, in period t, Discrete prices: (2.17) P E[Xi,τ |St = ([x1 , y1 ; x2 , y2 ], z)] = q Pr{Xi,τ = q|Xi,yi = xi , Zt = z, Zt+1 = z , . . . , Zt+(τ −yi ) = z } q=xi Continuous prices: P (2.18) E[Xi,τ |St = ([x1 , y1 ; x2 , y2 ], z)] = xi z·z ···z qfX (q|xi )dq τ |Xy i z·z ···z (q|x ) Where Pr{Xi,τ = q|Xi,yi = xi , Zt = z, Zt+1 = z , . . . , Zt+(τ −yi ) = z } and fX i τ |Xy i are derived using the Chapman-Kolmogorov equations discussed in Section 2.1. The main issue regarding the expected final price is that it only depends on the current price of an auction, and how many auctions will be underway for the duration of the auction. Once both auctions have been released we know how many auctions there will be for the remainder of each individual auction. As a consequence of the assumptions that auctions progress independently and Assumption 2.1, E[Xi,τ |St ] is increasing in xi and independent of xj , for i = j. We summarize this in the following result. Corollary 2.7. If auctions progress independently of price in other auctions and Assumption 2.1 holds, then the conditional expected final price, E[Xi,τ |St = ([x1 , y1 ; x2 , y2 ], z)], is increasing in xi and independent of xj , i = 1, 2, i = j. Proof Corollary 2.7 - The result regarding independence of the price in the other auction is immediate by the assumption that price transitions do not depend on the price in the other auction. Proof by induction on the number of remaining periods n = τ − yi . Without loss of generality, consider auction 1. For n = 0, E[X1,τ |St = ([x1 , τ ; x2 , y2 ], z)] = x1 , which is increasing in x1 . Assume the result holds for n = 0, 1, . . . , l − 1, i.e. for y1 = 66 τ, τ − 1, . . . , τ − (l − 1). Let n = l then y = τ − l, P E[X1,τ |St = ([x1 , τ − l; x2 , y2 ], z)] = E[X1,τ |St+1 = ([q, τ − (l − 1); x2 , y2 ], z )]πx1 ,q|z q=x1 P ≤ E[X1,τ |St+1 = ([q, τ − (l − 1); x2 , y2 ], z )]πx1 +1,q|z = E[X1,τ |St = ([x1 + 1, τ − l; x2 , y2 ], z)] q=x1 +1 where inequality holds due to Lemma 4.7.2 in Puterman (1994), the induction assumption and Assumption 2.1. The proof for continuous prices is basically the same but with the summation replaced by an integration. Although Lemma 4.7.2 in Puterman (1994) is with respect to discrete variables and infinite sequences, it can be adapted to continuous variables and/or finite sequences. Alternatively the results from Lemma 9.1.1 and Proposition 9.1.2 in Ross (1996) can be applied. Furthermore, when both items have been released, we define R(St ) to represent the total expected profit over the remainder of the planning horizon in period t, (2.19) R(St = ([x1 , y1 ; x2 , y2 ], z)) = −h(2τ − y1 − y2 ) + E[X1,τ |St = ([x1 , y1 ; x2 , y2 ], z)] + E[X2,τ |St = ([x1 , y1 ; x2 , y2 ], z)] There is a slight misuse of notation when yi = δ, i = 1, 2. In this case we implicitly define τ − δ = 0, since no holding cost will be incurred. Note that R(St ) is not necessarily increasing or decreasing in the elapsed time of the auctions. Though the incurred holding cost will decrease, the expected final price of the auctions will also decrease. It is this trade-off that is the crux of the problem regarding when to start the second auction. However, as summarized in the next result, R(St ) is increasing in x1 and x2 . 67 Corollary 2.8. If Assumption 2.1 holds then R(St ) is increasing in x1 and x2 , for all y1 , y2 , and z = 0, 1, 2. Proof of Corollary 2.8 - Each auction progress independently of the price in the other auction, the result is therefore immediate by Corollary 2.7. Lastly, we define g2 (St ) to be the gain in the expected final price of auction 2 by having delayed the release of item 2 for one period, (2.20) g2 (St = ([x1 , y1 ; x2 , y2 ], z)) = E[X2,τ |St+1 = ([x1 , y1 + 1; x2 , y2 ], z )] − E[X2,τ |St = ([x1 , y1 ; x2 , y2 ], z)] where z, z = 0, 1, 2 and by definition if y1 = τ, δ then y1 + 1 = δ. Due to Assumption 2.2 and that auctions progress independently, g2 (St ) ≥ 0 and independent of x1 . In other words the fewer periods remaining for the 1st auction, and regardless of the price in auction 1, the higher the expected final price for the 2nd auction. We summarize this in the following corollary. Corollary 2.9. If auctions progress independently of price in other auctions and Assumption 2.2 holds then g2 (St ) ≥ 0 and independent of x1 and x1 . Proof of Corollary 2.9 - See Appendix A. Next we present the optimality equations. Let Vt (s) denote the expected total future reward (expected total profit) given the system is in state s ∈ S in period t. For discrete 68 prices Vt (s), satisfies the following optimality equations, (2.21) Vt (s) = max rt (s) + a∈A(s) rT (s) Vt+1 (s )π(s |s, a) t = 0, 1, . . . , T − 1 s ∈S t=T For continuous prices Vt (s) satisfies, rt (s) + max (2.22) Vt (s) = rT (s) a∈A(s) S Vt+1 (s )f (s |s, a)ds t = 0, 1, . . . , T − 1 t=T Due to the vigilant seller assumption, the structure of the transition probabilities, and that auctions are guaranteed to be successful, the value function (2.21) and (2.22) can be summarized and explicitly evaluated according to the three cases listed in the following lemma. Lemma 2.10. If we assume a vigilant seller and that auctions are guaranteed to be successful, then the value functions of interest for discrete prices are as follows, Vt ([x1 , y1 ; x2 , y2 ], 0) = x1 + x2 t=T Vt ([x1 , y1 ; x2 , y2 ], z) = R([x1 , y1 ; x2 , y2 ], z) t=τ P Vt ([x1 , y1 ; x2 , y2 ], 1) = max{−2h + Vt+1 ([q, y1 + 1; p, 0], 1)πx1 ,q|1 , R([x1 , y1 ; p, 0], 2)} t<τ q=x1 If prices are continuous the only change is the final equation which becomes, P Vt ([x1 , y1 ; x2 , y2 ], 1) = max{−2h + x1 Vt+1 ([q, y1 + 1; p, 0], 1)f 1 (q|x1 )dq , R([x1 , y1 ; p, 0], 2)} t < τ Proof of Lemma 2.10 - See Appendix A. The above value functions can be computed using backward induction. In Table 2.2 the optimality equations for discrete prices and τ = 3 are listed. 69 V6 ([x1 , 3; x2 , 3], 0) = x1 + x2 V3 ([x1 , 3; p, 0], 1) = R([x1 , 3; p, 0], 1) V2 ([x1 , 2; p, 0], 1) = max{−2h + P q=x1 V3 ([q, 3; p, 0], 1)πx1 ,q|1 , R([x1 , 2; p, 0], 2) } V1 ([x1 , 1; p, 0], 1) = max{−2h + P q=x1 V2 ([q, 2; p, 0], 1)πx1 ,q|1 , R([x1 , 1; p, 0], 2) } V0 ([p, 0; p, 0], 1) = max{−2h + P q=p V1 ([q, 1; p, 0], 1)πp,q|1 , R([p, 0; p, 0], 2) } Table 2.2. Optimality equations for discrete prices and τ = 3. 3.3. Structural Results. Given the above MDP and the assumption that auctions are guaranteed to be successful, we derive three monotonicity properties: the optimal value function is increasing in the current price of the two auctions, the optimal policy is a threshold policy, and the threshold is decreasing in the holding cost. Note that though the proofs are for the case of discrete prices, the results hold for continuous prices as well. Proposition 2.11. If Assumption 2.1 holds and auctions are guaranteed to be successful, then the optimal value function, Vt ([x1 , y1 ; x2 , y2 ], z), is increasing in x1 and x2 , for t = 0, 1, . . . , T . Proof of Proposition 2.11 - By Lemma 2.10 there are only three cases to consider. Case 1) If t = T , then by Lemma 2.10, Vt ([x1 , y1 ; x2 , y2 ], z) = x1 + x2 , and the result is immediate. Case 2) If t < τ and z = 2, or τ ≤ t < T , then by Lemma 2.10, Vt ([x1 , y1 ; x2 , y2 ], z) = R([x1 , y1 ; x2 , y2 ], z), and the result follows from Corollary 2.8. Case 3) For t < τ and z = 1, by Lemma 2.10, P Vt ([x1 , y1 ; x2 , y2 ], z) = max{−2h + Vt+1 ([q, y1 + 1; p, 0], z)πx1 ,q|1 , R([x1 , y1 ; p, 0], 2)} q=x1 We establish the result using backward induction on t. Let t = τ − 1 and hence y1 + 1 = τ , then by Lemma 4.7.2 in Puterman (1994), Case 2) above and Assumption 2.1, P q=x1 Vt+1 ([q, τ ; p, 0], 1)πx1 ,q|1 is increasing in x1 , and by Corollary 2.8, R([x1 , y1 ; p, 0], 2) 70 is increasing in x1 . Since Vt ([x1 , y1 ; x2 , y2 ], z) is the maximum of two increasing functions it is also increasing in x1 and the result holds. Assume Proposition 2.11 holds for t = τ − (l − 1), . . . , τ − 2, τ − 1. Let t = τ − l and hence y1 + 1 = τ − (l − 1), and again by Lemma 4.7.2 in Puterman (1994), the induction assumption and Assumption 2.1, P q=x1 Vt+1 ([q, y1 + 1; p, 0], 2)πx1 ,q|1 is increasing in x1 , and by Corollary 2.8, R([x1 , y1 ; p, 0], 2) is increasing in x1 . Since Vt ([x1 , y1 ; x2 , y2 ], z) is the maximum of two increasing functions it is also increasing in x1 and the result holds. Similar to the proof of Corollary 2.7 the results from Lemma 9.1.1 and Proposition 9.1.2 in Ross (1996) can be applied. The proof for the continuous case is identical but with the summation replaced by an integration. In other words an increase in the current price of either item 1 or item 2 will increase the optimal expected total reward. Though this might seem natural and ‘obvious’ it is a result of the assumptions made, most notably that at a higher price-level the auction is more likely to advance to the higher prices than at a low price-level. And as discussed if bid-increments are decreasing in price then it is not immediate that this assumption holds. Chapter 4 contains examples from DFS’ eBay auctions where this result does not hold, as well as examples for which the result holds. Theorem 2.12. If Assumptions 2.1, 2.2, and 2.3 hold and auctions are guaranteed to be successful, then there exist optimal decision rules, γt ([x1 , y1 ; p, 0], 1), which are increasing in x1 , for t = 0, 1, . . . , τ − 1. Consequently, the optimal policy is a threshold policy in x1 . Proof of Theorem 2.12 - Let prices be discrete. Sufficient to show that Vt ([x1 , y1 ; p, 0], 1) − R([x1 , y1 ; p, 0], 2) is decreasing in x1 , for all t < τ . By Corollary 2.8 and Proposition 2.11, R([x1 , y1 ; p, 0], 2) and Vt ([x1 , y1 ; p, 0], 1) are increasing in x1 . We make use of the following 71 relationship, P (2.23) R([x1 , y1 ; x2 , y2 ], 2) = −h + R([q, y1 + 1; x2 , y2 ], z)πx1 ,q|2 − g2 ([x1 , y1 ; x2 , y2 ]) q=x1 Proof by backward induction on t. Let t = τ − 1, then y1 = τ − 1 and by Lemma 2.10 and (2.23), Vt ([x1 , y1 ; p, 0], 1) − R([x1 , y1 ; p, 0], 2) P = max{−h + R([q, τ ; p, 0], 1)(πx1 ,q|1 − πx1 ,q|2 ) + g2 ([x1 , τ − 1; p, 0]) , 0} q=x1 P = max{−h − hτ + q(πx1 ,q|1 − πx1 ,q|2 ) + E[X2,τ |([x1 , τ, p, 0], 1)] + g2 ([x1 , τ − 1; p, 0]) , 0} q=x1 by Corollary 2.7 and 2.9, E[X2,τ |([x1 , τ, p, 0], 1)] respectively g2 ([x1 , τ − 1; p, 0]) are independent of x1 , and since, P q(πx1 ,q|1 − πx1 ,q|2 ) q=x1 P = x1 P P (πx1 ,q|1 − πx1 ,q|2 ) + q=x1 (πx1 ,q|1 − πx1 ,q|2 ) + q=x1 +1 =0 P (πx1 ,q|1 − πx1 ,q|2 ) + . . . + q=x1 +2 ↓ in x1 by Ass.2.3 ↓ in x1 by Ass.2.3 (πx1 ,q|1 − πx1 ,q|2 ) q=P ↓ in x1 by Ass.2.3 the result holds for t = τ −1. Assume the result holds for t = τ −(l −1), τ −(l −2), . . . , τ −1, and define Λt (x) = Vt ([x, y; p, 0], 1) − R([x, y; p, 0], 2). Let t = τ − l. Then by Lemma 2.10 and (2.23), Vt ([x1 , y1 ; p, 0], 1) − R([x1 , y1 ; p, 0], 2) = P = max{−h + Vt+1 ([q, y1 + 1; p, 0], 1)πx1 ,q|1 − R([q, y1 + 1; p, 0], 2)πx1 ,q|2 + g2 ([x1 , y1 ; p, 0]) , 0} q=x1 P P = max{−h + Λt+1 (q)πx1 ,q|1 + q=x1 First show that P q=x1 R([q, y1 + 1; p, 0], 2)(πx1 ,q|1 − πx1 ,q|2 ) + g2 ([x1 , y1 ; p, 0]) , 0} q=x1 R([q, y1 +1; p, 0], 2)(πx1 ,q|1 −πx1 ,q|2 ) is decreasing in x1 . By Corollary 2.8, R([x, y; p, 0], 2) is increasing in x. Define α(x+1) = R([x+1, y; p, 0], 2)−R([x, y; p, 0], 2). 72 Therefore, P R([q, y1 + 1; p, 0], 2)(πx1 ,q|1 − πx1 ,q|2 ) q=x1 P P (πx1 ,q|1 − πx1 ,q|2 ) +α(x1 + 1) = R([x1 , y1 + 1; p, 0], 2) q=x1 (πx1 ,q|1 − πx1 ,q|2 ) q=x1 +1 =0 P P + α(x1 + 2) (πx1 ,q|1 − πx1 ,q|2 ) + . . . + α(P ) q=x1 +2 (πx1 ,q|1 − πx1 ,q|2 ) q=P P P ≥ +α(x1 + 1) (πx1 +1,q|1 − πx1 +1,q|2 ) +α(x1 + 2) (πx1 +1,q|1 − πx1 +1,q|2 ) q=x1 +2 q=x1 +1 =0 P + . . . + α(P ) (πx1 +1,q|1 − πx1 +1,q|2 ) q=P P = R([q, y1 + 1; p, 0], 2)(πx1 +1,q|1 − πx1 +1,q|2 ) q=x1 +1 where the inequality holds by Assumption 2.3. Therefore, πx1 ,q|2 ) is decreasing in x1 . Next show that P q=x1 P q=x1 R([q, y1 +1; p, 0], 2)(πx1 ,q|1 − Λt+1 (q)πx1 ,q|1 is decreasing in x1 . By the induction assumption Λt+1 (x) is decreasing in x, therefore define β(x + 1) = Λt+1 (x) − Λt+1 (x + 1), then, P Λt+1 (q)πx1 ,q|1 q=x1 P P = Λt+1 (x1 ) P πx1 ,q|1 −β(x + 1) q=x1 P πx1 ,q|1 − β(x + 2) q=x1 +1 πx1 ,q|1 − . . . − β(P ) q=x1 +2 πx1 ,q|1 q=P =1 P ≥ Λt+1 (x1 ) πx1 +1,q|1 −β(x + 1) q=x1 πx1 +1,q|1 −β(x + 2) πx1 +1,q|1 − . . . − β(P ) q=x1 +2 q=x1 +1 =1 P P P πx1 +1,q|1 q=P =1 P Λt+1 (q)πx1 +1,q|1 = q=x1 +1 where the inequality holds due to Assumption 2.1. Therefore P q=x1 Λt+1 (q)πx1 ,q|1 is de- creasing in x1 . Since g2 ([x1 , t; p, 0]) is independent of x1 , the result holds for all t < τ . 73 The proof for continuous prices follows the same logic but with slightly different arguments. Theorem 2.12 implies that, for each t there exists a pt such that if X1 ≥ pt then it is optimal to release the second item for auction, while if X1 < pt it is optimal to hold the second item at least one more period. The reason we are only considering t = 0, 1, . . . , τ − 1 is because we are only interested in those periods where non-trivial decision can be made. For t = τ the decision to release is immediate by the vigilant seller assumption. Also note that if Y2 > 0 the second item has already been released and no further decision needs to be made. The main assumption driving the result of Theorem 2.12 is the diminishing cannibalization effect of Assumption 2.3. Our next result summarizes the effect the holding cost has on pt . Corollary 2.13. For each t < τ , the control limit in Theorem 2.12, pt , is decreasing in the holding cost h. Proof of Corollary 2.13 - For a given decision epoch t we know that for X1 ≥ pt any additional holding cost by deferring the release is not compensated by the gain in expected final price for the two items. Therefore, if h increases and since the expected final prices remains the same, then any additional holding cost will still not be compensated (in fact it is even less compensated), and the result follows. Given these three properties it may be natural to ask if the threshold price, pt , is monotone in t or Y1 . With only the three assumptions stated above the answer is no. And it turns out that monotonicity over time will also depend on the holding cost h. In other words, depending on the holding cost pt ≤ pt+1 or pt ≥ pt+1 . See the numerical example in 74 Section 3.4 for an illustration. In Section 3.5 some bounds on the holding cost are discussed. 3.3.1. Examples. In Section 2.3 we provided four common probability distributions and conditions on their parameters that support Assumptions 2.1, 2.2, and 2.3. Consequently, we have the following results. Corollary 2.14. If price increments are distributed as discrete or continuous Uniform random variables, as specified in Section 2.3, then Proposition 2.11 and Theorem 2.12 holds. Corollary 2.15. If price increments are distributed as Bernoulli random variables as specified in Section 2.3 and (2.11) holds, then Proposition 2.11 and Theorem 2.12 holds. Proof of Corollary 2.14 and 2.15 - For each of the cases we have that Assumptions 2.1, 2.2, and 2.3 holds, it therefore follows that Proposition 2.11 and Theorem 2.12 holds. In the Bernoulli case with price-independent transition probabilities, πq = π and ρq = ρ for all q = 0, 1, . . . , τ − 1, the optimal policies simplify further. First note that due to the special structure of the transition probability matrices the n-period transition matrices are symmetric in the following sense, for τ = 3, Π1·2·2 = Π2·2·1 and Π1·1·2 = Π2·1·1 . As a result the expected value for the open loop policy of releasing the second item j periods after the first item has been released is given by, (2.24) VO(j) = −(2τ + j)h + 2(p + jπ + (τ − j)ρ) j = 0, 1, 2, . . . , τ Furthermore, the total marginal gain by deferring the release one period, 2(π − ρ), is independent of X1 and as a result closed loop policies are not required. Proposition 2.16. In the case of price-independent Bernoulli increments, the optimal policy is to release both items simultaneously if and only if h ≥ 2(π − ρ). If h < 2(π − ρ) then releasing the two auctions sequentially is the optimal policy. 75 Proof of Proposition 2.16 - See Appendix A. The two policies described in Proposition 2.16 are the only optimal policies - open and closed loop policies included. The interpretation of the condition h ≥ 2(π − ρ), is that if the holding cost exceeds the expected one-period gain for both auctions, by deferring the release it will never be optimal to defer the release of item 2. 3.4. Numerical Examples. Example 1 - τ = 2 The purpose of this example is illustrate that a closed loop policy may be optimal, that the optimal threshold for releasing an item need not be monotone in t or Y1 , and that in order to determine the optimal policy, the seller must solve the dynamic program by backward induction. Consider a two period auction, τ = 2, for an item valued at $10, $20 and $30, with the following transition probability matrices. 10 20 10 .6 Π1 = 20 0 30 0 .3 .6 0 30 .1 .4 1 10 10 .7 Π2 = 20 0 30 0 20 30 0 .65 .35 0 1 .3 Note that Assumptions 2.1, 2.2, and 2.3 hold. At decision epoch t = 2, the 1st auction has ended and the decision to release the 2nd item is immediate irrespective of X1 by the vigilant seller assumption. At time t = 1 the 1st auction has one period remaining and the seller can choose to either release the 2nd item or defer the release one period. If X1 = $10 76 then the seller’s total expected profit is the following: −4h + 15 + 19.2 a = 0 (defer release) V1 ([10, 1; 10, 0], 1) = max −3h + 13 + 17.7 a = 1 (release) We see that the marginal cost of deferring the release is h while the expected marginal gain is 3.5. If h > 3.5 it is optimal to release the 2nd auction while if h < 3.5 it is optimal to defer the release. If X1 = $20 then the seller has the following total expected profit: −4h + 24 + 19.2 V1 ([20, 1; 10, 0], 1) = max a = 0 (defer release) −3h + 23.5 + 17.7 a = 1 (release) In this case, the marginal cost is still h and the expected marginal gain is now only 2, i.e. if h > 2 then the optimal decision is to release else it is optimal to defer the release. If X1 = $30 then the seller has the following total expected profit: −4h + 30 + 19.2 a = 0 (defer release) V1 ([30, 1; 10, 0], 1) = max −3h + 30 + 17.7 a = 1 (release) While the marginal cost is still h, the expected marginal gain is now only 1.5, i.e. if h > 1.5 then the optimal decision is to release else it is optimal to defer the release. Therefore when we roll back one period and analyze the optimal decision at t = 0, we have four cases to consider: 1) h ≥ 3.5, 2) 3.5 > h ≥ 2, 3) 2 > h ≥ 1.5, and 4) 1.5 > h. Let t = 0 and X1 = $10, the seller’s expected profit is the following: V0 ([10, 0; 10, 0], 1) = max −2h + .6V1 ([10, 1; 10, 0], 1) + .3V1 ([20, 1; 10, 0], 1) + .1V1 ([30, 1; 10, 0], 1) a = 0 −4h + 2(16.15) a=1 77 Which for the four cases results in the following. Case 1) h ≥ 3.5: −5h + .6(30.7) + .3(41.2) + .1(47.7) a = 0 V0 ([10, 0; 10, 0], 1) = max = max −4h + 32.3 a=1 −5h + 35.55 a=0 −4h + 32.3 a=1 Case 2) 3.5 > h ≥ 2: −2h − .6(4h) − .4(3h) + .6(34.2) + .3(41.2) + .1(47.7) a = 0 V0 ([10, 0; 10, 0], 1) = max = max −4h + 32.3 a=1 −5.6h + 37.65 a=0 −4h + 32.3 a=1 Case 3) 2 > h ≥ 1.5: −2h − .9(4h) − .1(3h) + .6(34.2) + .3(43.2) + .1(47.7) a = 0 V0 ([10, 0; 10, 0], 1) = max = max −4h + 32.3 a=1 −5.9h + 38.25 a=0 −4h + 32.3 a=1 78 Case 4) 1.5 > h: −6h + .6(34.2) + .3(43.2) + .1(49.2) a = 0 V0 ([10, 0; 10, 0], 1) = max −4h + 32.3 a=1 −6h + 38.4 a = 0 = max −4h + 32.3 a = 1 In the three cases we can compare the expected marginal cost with the expected marginal gain for deferring the release. For case 1) the expected marginal cost is h while the expected marginal gain is 3.25. Since h ≥ 3.5 the optimal decision is to release the 2nd item immediately. For case 2) the expected marginal cost is 1.6h while the expected marginal gain is 5.35. Therefore we have two further sub-cases to consider: 2a)If 3.5 > h ≥ 3.34 then the expected marginal cost outweighs the expected marginal gain and it is optimal to release the 2nd item immediately, 2b) If 3.34 > h ≥ 2 then the expected marginal gain by deferring the release compensates for the expected marginal cost, and therefore it is optimal to defer the release. For case 3) the expected marginal cost is 1.9h while the expected marginal gain is 5.95, therefore it is optimal to defer the release. And similarly for case 4) the expected marginal cost is 2h while the expected marginal gain is 6.1, and since 1.5 > h it is optimal to defer the release. In Table 2.3 we summarize the above results. For instance, we see that if h = $3 and t = 0 or 1 and X1 = 10 it is optimal to defer the release. If, however, at t = 1 the price has jumped to either $20 or $30 then it is optimal to release the 2nd item. The main point of this example is to illustrate that a closed loop policy may be optimal, and that for a given h and X1 it may be optimal to release item 2 in one period though the optimal decision in a later period, at the same price, is to defer the release. Note, however, that for the cases when h ≥ 3.34 a seller that follows an optimal policy would not have to make a decision 79 t=0 t=1 X1 = 10 X1 = 10 X1 = 20 X1 = 30 h ≥ 3.5 Release Release Release Release 3.5 > h ≥ 3.34 Release Defer Release Release 3.34 > h ≥ 2 Defer Defer Release Release 2 > h ≥ 1.5 Defer Defer Defer Release Defer Defer Defer Defer 1.5 > h Table 2.3. Example 1 - Optimal decision as a function of the holding cost h (τ = 2) in period t = 1, but in order to derive the optimal policy, the optimal decision for period t = 1 has to be evaluated. Example 2 - τ = 3 Our next example illustrates the potential gain by using an adaptive closed loop policy versus an non-adaptive open loop policy. It also shows that depending on the holding cost the price threshold may be increasing or decreasing over time. Assume the auction length is three periods, τ = 3, the start price p = $10, the upper bound P = $60, the price increments are $10, and the transition probabilities are as follows. 10 20 30 40 50 10 .25 20 0 30 0 Π1 = 40 0 50 0 60 0 10 60 10 .3 20 0 30 0 Π2 = 40 0 50 0 60 0 .1 .25 .3 .25 .1 .1 0 .25 .4 .25 .1 0 0 .4 .45 .15 0 0 0 .6 .4 0 0 0 0 1 .25 .2 .1 20 30 40 50 .1 80 60 0 .35 .35 .2 .1 0 0 .4 .35 .2 .05 0 0 .5 .4 .1 0 0 0 .65 .35 0 0 0 0 1 .3 .25 .1 .05 It can be verified that the matrices satisfy Assumptions 2.1, 2.2, and 2.3. The expected profit for the four open loop policies can explicitly be computed, VO(0) = −6h + 2 × 40.76 VO(1) = −7h + 43.69 + 43.40 VO(2) = −8h + 46.09 + 45.82 VO(3) = −9h + 2 × 47.93 While for the closed loop policy we use backward induction to find the action that maximizes the value equations. In Table 2.4 below the value of each policy with respect to various holding costs are shown. The last two columns displays the difference between the closed loop policy and the best and worst open loop policy for a given h. As one can see if the holding cost is ‘low’ then it is better to release the two auctions sequentially, while if the holding cost is ‘high’ then it will never be worth holding the 2nd item an additional period so that the optimal policy is to release both immediately. The interesting cases are in between where we see that the closed loop policy performs better than any of the open loop policies. Though the gain at a given h for the optimal closed loop policy versus the best open loop policy is perhaps not that drastic, the gain versus the other open loop policy can be quite large. For instance, if h = $2.30 then the difference between using the optimal closed loop policy and open loop policy of simultaneous release is more than $7 (11% improvement). In Table 2.5 the critical price thresholds pt for various holding costs are summarized (in the table ‘n/a’ indicates that pt > P ). In the table we see that if, for instance, h = 4.00 and t = 1 then if X1 < 40 then it is optimal to defer the release of item 2, while if X1 ≥ 40 81 h OP0 OP1 OP2 .10 80.93 86.39 91.10 1.00 75.53 80.09 83.90 2.00 69.53 73.09 75.90 2.30 67.73 70.99 73.50 4.00 57.53 59.09 59.90 5.00 51.53 52.09 51.90 5.80 46.73 46.49 45.50 6.00 45.53 45.09 43.90 9.00 27.53 24.09 19.90 10.00 21.53 17.09 11.90 15.00 −8.47 −17.91 −28.10 Table 2.4. Expected profit Loop Policies OP3 Optimal Policy Max Gain(%) Min Gain(%) 94.97 94.97 17.9 0 86.87 15.0 0 86.87 77.87 77.87 12.0 0 75.17 75.21 11.0 .1 59.87 60.99 6.0 1.8 50.87 53.01 4.2 1.8 43.67 46.85 7.3 .3 41.87 45.53 8.7 0 14.87 27.53 85.1 0 5.87 21.53 266.8 0 −39.14 −8.47 vs holding cost comparison for the Open and Closed it is optimal to start the 2nd auction. Note that when h is ‘low’ then pt > P and hence it is always optimal to wait one more period before releasing the 2nd auction . Similarly, if h is ‘high’ then it is never optimal to defer the release. Another thing to note is that depending on the holding cost, the price threshold could be either increasing or decreasing. To illustrate, consider the case when h = 2.75, then p0 = $60, while p1 = p2 = $50. On the other hand, if, for instance, h = 5.50 then p0 = $20, while p1 = p2 = $30. The managerial consequence is that depending on the holding cost, the manager may become more or less ‘sensitive’ as to when release the second item as the first auction evolves. For instance, when h is relatively low, such that pt is decreasing in t, then the manager will lower his release threshold for each period and hence be less sensitive to the current price. On the other hand, when h is relatively high, such that pt is increasing in t, then the manager becomes more sensitive to the current price and requires a higher release threshold. We conclude by illustrating the optimal policy for two realizations. Let h = 4.00, at t = 0, since auction 1 starts at x1 = $10, which is less than the release threshold p∗0 = $40, it is optimal to defer the release. Suppose that after one period, auction 1 has jumped to x1 = $30, since this is less than the release threshold p∗1 = $40, it is optimal to yet again defer the start of auction 2. Suppose that after two periods, auction 1 has jumped to x1 = $40, 82 h 1.00 2.50 2.75 4.00 5.00 5.50 6.00 8.00 Table 2.5. Threshold t=0 t=1 t=2 n/a n/a n/a n/a 60 60 60 50 50 40 40 40 30 30 40 20 30 30 10 20 30 10 10 10 price level pt for various holding cost h since this is equal to the release threshold p∗2 = $40, it is now optimal to start the second auction. In other words, given the price-transitions of auction 1, it was optimal to wait until the start of the third period, or when auction 1 had elapsed 2 periods, before starting auction 2. Suppose instead that after one period, auction 1 had jumped to x1 = $40, then it would have been optimal to start auction 2 already at the start of the second period. 3.5. Holding Cost. As we have seen if the holding cost is ‘too low’ then it will never be optimal to release the 2nd item until the 1st item has been sold. And consequently, the optimal closed loop policy is same as the open loop policy of sequentially releasing the two auctions. A necessary and sufficient condition for this to hold, is if the holding cost is so low that the additional gain for the 2nd item by deferring the release alone compensates for the additional holding cost. Proposition 2.17. The optimal policy is to release the 2nd item for auction after the 1st auction is completed if and only if (2.25) h ≤ min {g2 ([x1 , y1 ; p, 0])} y1 <τ Proof of Proposition 2.17 - (⇐) If (2.25) holds then the additional gain in expected final price for item 2, by deferring the release one period, alone compensates for the additional holding cost. Since that gain holds for all periods that the 1st auction is still ongoing and independently of x1 , it is always optimal to release the 2nd auction after the 1st is finished. 83 (⇒) If (2.25) does not hold then there exist a period for which the gain in expected final price for item 2 does not compensate the additional holding cost. Consequently, in order for it to be optimal to defer the release in that period, there must be some gain in expected final price of item 1. However, this gain is dependent on x1 and hence the optimal policy is not independent of x1 . Therefore, the optimal policy might not be to release the items sequentially. Similarly, if h is ‘too high’ then there will be no incentive to hold the 2nd item any further since the expected total gain will not compensate for the additional holding cost. Consequently the optimal closed loop policy will be the same as the open loop policy of simultaneous release. However, the condition for h to be ‘too high’ is more complicated than the condition for ‘too low’. For instance, it is not sufficient that h is such that the best open loop policy is to release them simultaneously to ensure that this is the optimal closed loop policy (see Example 2 above). Also recall that though it may be optimal to release the two items simultaneously, it does not necessarily mean that in every period it is always optimal to release the 2nd item. It is possible that for t = 0 and X1 = p it is optimal to release the 2nd auction while for t = 1 and X1 = p it would be optimal to defer the release given that it has not been released (see Example 1 above). A ‘bound’ for h to be ‘too high ’ and consequently the optimal policy can never be to release the two auctions simultaneously is if, VO(1) − VO(0) ≥ 0 The implication on the holding cost h in the above inequality is better illustrated as follows, (VO(1) + (2τ + 1)h) − (VO(0) + 2τ h) ≥ h 84 In other words, if the expected gain in revenue by deferring the release one period is greater than h, then the optimal closed loop policy can never be to release the 2nd auction immediately. Therefore in order for h to be ‘too high’ it has to be large enough that the open loop policy of simultaneous release is better than the open loop policy of deferring the release by one period. Note though that this is only a necessary condition, it could be optimal to defer the release despite that the above inequality does not hold. To determine the threshold on the holding cost for which the optimal closed loop policy is to release the two items simultaneously one has to solve the dynamic program given that h is such that the above inequality holds. Example 2 (continued) - τ = 3 In the numerical example above we see that if h ≤ min {g2 ([x1 , y1 ; p, 0])} = 47.93 − 45.81 = 2.12 y1 <τ then the optimal closed loop policy is to release the 2nd auction after the 1st auction is over, i.e. if h ≤ 2.12 then pt > P for all t < τ . To find the upper limit of h for which the optimal closed loop policy is to release the two auctions simultaneously we solve for h by backward induction given that h ≥ (43.69 + 43.40) − (2 × 40.76) = 5.57, this results in that if h ≥ 5.88 then the optimal closed loop policy is to release the 2nd item immediately, i.e. if h ≥ 5.88 then p0 = p = 10. Note too that if, for instance, h = 6.00 then p1 = 20 and p2 = 30. 3.6. Reserve Price. In most online auctions the seller has the option of imposing a reserve price vr . If the final price Xi,τ < vr then the seller is not obligated to award the item. In eBay auctions the value of vr is not disclosed and the only information bidders have is if Xi,Yi ≥ vr and hence if the item is guaranteed to be awarded. A natural comment might be why not simply set the starting bid at the reserve price, i.e. set p = vr . Anecdotal 85 evidence suggest that the bidding behavior is different for auctions with low starting prices. The reasoning seems to be that most bidders prefer to participate in ‘active’ auctions, where the price has reached a level due to bidding activity, rather than auctions where no bidding has taken place though the price is exactly the same. In order for the seller to protect himself against the possibility of selling below his reservation value he can impose a reserve price. However, it is also anecdotally known that having a ‘too high’ reserve price deters bidding activity. Here the argument is that auctions that are guaranteed to be awarded, i.e. the reserve price has been met, attracts more bidding competition due to the guarantee that the highest bidder will win. Bertsimas, Hawkins and Perakis (2003) analyze some of these issues and determine the optimal start price and reserve price. In the following discussion we still assume the seller will at most list an item once, but now with an alternative salvage channel with a guarantee of vr (and not p as before). We will use the same MDP formulation as before but assume the value of an auction that ends at or below the reserve price is vr . In other words, the expected final price of an item is defined as follows, for i = 1, 2, Discrete prices: (2.26) E[Xi,τ |St = ([x1 , y1 , x2 , y2 ]z)] = vr vr Pr{Xi,τ = q|Xi,yi = xi , Zt = z, Zt+1 = z , . . . , Zt+(τ −yi ) = z } q=xi P Pr{Xi,τ = q|Xi,yi = xi , Zt = z, Zt+1 = z , . . . , Zt+(τ −yi ) = z } + q=vr +1 Continuous prices: vr (2.27) E[Xi,τ |St = ([x1 , y1 , x2 , y2 ]z)] = vr xi 86 z·z ···z fX (q|xi )dq + τ |Xy i P vr z·z ···z qfX (q|xi )dq τ |Xy i Note that the behavior issue regarding increased bidding activity or higher price jumps due to the reserve price having been met is consistent in the current MDP framework. The main monotonicity assumption for each transition probability matrix Πz , z = 1, 2, is Assumption 2.1. This assumption does not exclude experiencing higher price jumps once the current price has passed a given threshold vr ∈ [p, P ]. As the reader may notice, introducing a reserve price does not effect the main structural results. Proposition 2.18. If the auctions are guaranteed to be successful and the seller has imposed a reserve price vr , and the value of an auction is defined according to (2.26) or (2.27), then Proposition 2.11, Theorem 2.12, and Corollary 2.13 holds. Proof of Proposition 2.18 - see Appendix A. 3.7. Dynamic Price Transition Probabilities. We have assumed that the price dynamics, i.e. transition probabilities, are stationary with respect to calender time t and elapsed auction time Y . There is, however, anecdotal and statistical evidence to suggest the opposite. For example, anecdotal evidence suggest that eBay auction activity for certain products vary between the weekdays, as well as between weekdays and weekends. Similarly it is well established that the bidding activity is much higher toward the end of an auction. This phenomena includes the well known bidding strategy sniping. Papers that analyze the phenomena of dynamic bid arrivals with respect to Y , and its implication include Roth and Ockenfels (2002) and Shmueli, Russo and Wolfgang (2004). For ease of tractability and we ignore the issue of non-stationarity with respect to calender time t. Instead we focus on the implication of having non-stationary transition probabilities with respect to the elapsed auction time. That is when Πz or F z changes as the auction progresses. Let Πz,y be the transition probability matrix and let F z,y be the 87 transition distribution function, for an individual item when there are z auctions underway and the auction has elapsed for y periods. For instance, suppose prices are discrete and the price dynamics are the same for all periods except the last one, i.e. Πz,0 = Πz,1 = . . . = Πz,τ −2 = Πz,τ −1 . Even with dynamic transition probabilities, for the guaranteed successful auction case, the main structural results still hold which we summarize in the following proposition. Proposition 2.19. For the single listing case, if for y = 0, 1, . . . , τ − 1 and z = 1, 2, Πz,y and F z,y supports Assumption 2.1, and the relation between Π1,y and Π2,y , and between F 1,y and F 2,y supports Assumption 2.2 and 2.3, then Propositions 2.11, Theorem 2.12, and Corollary 2.13 still hold. Proof of Proposition 2.19 - None of the proofs involve arguments across different elapsed auction periods. Each argument is always with respect to a given elapsed auction period. Therefore including a subscript to indicate the elapsed auction period, on the transition probabilities or transition distribution functions does not impact the validity of any of the arguments in any of the proofs. Since the three assumptions are still valid for each elapsed auction period transition probability, there are no changes to the above results. 4. Possibly Unsuccessful Auction - Multiple Re-Listing We now consider the case when there is a positive probability an auction is unsuccessful, meaning that there is some chance an auction receives no bids, and that the seller does not have any alternative salvage channel. For discrete prices this means that π0,0|z > 0, z = 1, 2, while for continuous prices there is some point mass for having a zero price transition. An example of a distribution for continuous prices with this property is the ‘zero-inflated gamma’ distribution that will be discussed in Chapter 4. Reasons why an auction may not receive any bids are that the seller perhaps posted a too high starting 88 price or that the items simply do not generate enough interest. For instance, a quick search on the completed listing of the eBay stores Pokerstores (poker chips), The Sharper Image (consumer electronics), uptempoair (Nike sportswear), GlobalGolfUSA (used golf clubs) reveal that a large quantity of their listings do not attract a single bid. In contrast, to the 6,000 auctions from Dell Financial Services’s eBay store we have data on, where not a single auction with a starting price of $.99 was unsuccessful. Bertsimas, Hawkins and Perakis (2003) discuss and provide empirical evidence regarding optimal control of starting price and reserve price. For our purposes, even if the sample of companies listed above are doing something wrong in their administration of auction control parameters, we include this section for mathematical completeness. And it turns out that the managerial consequence is both important and interesting, since unlike the single listing case the optimal policy need not be a control limit policy. By the positive expected profit assumption, even though the seller at the end of an unsuccessful auction has incurred a total cost of τ h, it is still optimal to try to auction the item once again. Therefore the seller has to decide when to re-list items that remain from previous unsuccessful auctions. Since the items are identical we can classify the item that is waiting to be released as item 2, and the item which is currently up for auction as item 1. That is, the ongoing auction will be labeled as auction 1, and the auction that we are deciding whether to start or not as auction 2. This means that if both auctions are underway, auction 2 was started after auction 1, and auction 1 is unsuccessful, then auction 2 immediately becomes auction 1. Therefore Y1 ≥ Y2 and the system state transitions may form loops. Specifically it may loop back to the initial starting state. See Figure 2.4 in Section 2.1. Consequently the time when an auction is successfully completed is not known. To address this issue, we formulate the problem as an expected total-reward MDP [21, Ch.7]. However, the problem involves some subtleties such that neither the properties of positive or 89 negative dynamic programs directly apply. In particular, there need not be unique solutions to the optimality equations, and policy or value iteration need not converge without further assumptions [21, Ch.7], [3]. Our model has the following relevant structural properties, (1) The state space and actions are finite. (2) The expected one-period reward in each state is bounded above and below. (3) There is a single absorbing state ∆ under all policies. The third property holds due to the vigilant seller assumption, without which there would be an additional absorbing state, ([0, 0; 0, 0], 0), with total reward of negative infinity, resulting from the policy of never releasing an item for auction. Given these three properties, the problem can be converted to a negative dynamic problem with the following transformation. In each transient state subtract 2P from the one-period reward. As a consequence all rewards in the transformed problem are less than or equal to zero [21, Proof of Theorem 8.10.1.]. Therefore, optimal solutions exists, and value iteration and policy iteration converges (though a modification to policy iteration may be required [21, Ch.10.4.2]). Alternatively Assumption 1 and 2 of Bertsekas and Tsitsiklis (1991) holds (even without the vigilant seller assumption). For simplicity of notation in the remaining of this section we only consider the case of discrete prices. All results presented will hold for continuous prices as well. 4.1. Auction Release Policy. For the multiple re-listing (infinite horizon) case a Markov deterministic policy γ is defined by, γ = (γ0 (s), γ1 (s), γ2 (s), . . .) γt (s) ∈ {0, 1}, ∀s ∈ S, t ≥ 0 If γt (s) = γt (s), for all t = t , and for a given s ∈ S, the policy is referred to as stationary. In the multiple re-listing case, we only need to consider stationary policies and therefore 90 use the notation γ(s), and interchangeable refer to it as both the decision rule and policy [21, Theorem 7.3.6.], [3, Proposition 2]. In the multiple re-listing case, open loop policies do not apply. Instead we define two types of closed loop policies. We refer to a policy that only depends on Y1 as time-based closed loop, and policies that depends on (X1 , Y1 ) as price-based closed loop. The reason we need this distinction is because, unlike the single listing case, it is necessary to consider decisions even after the first decision to release item 2 has been made. The reason for this is that auctions may get out of ‘sync’ with each other. We illustrate with an example, suppose τ = 5 and the seller decides to start the second auction 2 periods after the first. And suppose further that the first auction is unsuccessful, which means that the seller must now decide whether, after the second auction has elapsed three periods, to re-list item 1, and if the decision is not to release what to do when the second auction has elapsed four periods. This reasoning generalizes for any τ , and any policy which specify not to release if Y1 < yr and release if Y1 = yr , where yr < τ /2, must also specify what to do when Y1 ≥ τ − yr . For policies defined such that the first release should occur when Y1 = yr > τ /2, the issue of what decision to make if the first auction is unsuccessful has already been addressed. In other words, though there are 2τ +1 time-based closed loop policies, many are infeasible. For example, if τ = 5 then the policy to only release item 2 after auction 1 has elapsed 1 or 3 periods, is basically the same as the policy to only release item 2 after auction 1 has elapsed 1 period, i.e. if auction 2 is to be released when auction 1 has elapsed for 1 period, then there will never be an opportunity to release item 2 after auction 1 has elapsed 3 periods. 4.2. Optimality Equation. Before analyzing the optimality equations for two items we begin by only considering one item. Let the pair of random variables (X, Y ) denote the state of an auction, where X ∈ {0, p, p + k, . . . , P } and Y ∈ {0, 1, . . . , τ } or δ, and E[Xτ |Xy = x] denotes the expected final price given X = x and Y = y. Let v(x, y) denote 91 the value (total expected future reward) given the system is in state (x, y). By the positive expected profit and vigilant seller assumptions, the item should be immediately re-listed following an unsuccessful auction. Therefore the value of an item in state (x, y) is, v(x, y) = −h(τ − y) + E[Xτ |Xy = x] + (πx,0|1 )(π0,0|1 )(τ −(y+1)) v(0, 0) y = δ 0 y=δ Note that πx,0|1 = 0 for all x > 0, and that we are not considering discounting. Consequently the expected value of an item continuously re-listed until the auction is successful is, (2.28) v(0, 0) = −hτ + E[Xτ |X0 = 0] 1 − (π0,0|1 )τ We now return to the two item case. In total there are 19 different cases for which the optimality equation needs to be evaluated. These appear in Table 2.6 below. Note that there are only non-trivial decisions in those periods for which y1 < τ and y2 = 0, namely cases 16, 17, and 19. Furthermore, note that under cases 13, 15 and 17 there is a positive probability of looping back to the initial state ([0, 0; 0, 0], 1). However, due to the positive expected profit assumption, we have that π0,0|z < 1, z = 1, 2, and therefore with probability one the system state will eventually reach the recurrent state ∆ = ([∆1 , ∆2 ], 0). Recall that ∆i = (Xi , δ) is the state of item i, i = 1, 2, when it has been awarded and hence will not incur any additional cost or generate any further revenue. From the discussion above, we note that any solution satisfying the optimality equations in Table 2.6 is an optimal solution [21, Proposition 7.3.4], [3, Proposition 2]. Though the optimal policy need in general not be a control limit policy (see the example below), under some of the cases the optimal policy is a control limit. 4.3. Structural Results. Similarly to the single listing case the optimality equations can be simplified and explicitly evaluated for some of the cases. 92 Case Item 1 y1 Item 2 y2 z V ([x1 , y1 ; x2 , y2 ], z) 1) δ δ 0 =0 2) δ τ 0 = x2 + V (∆) δ τ −1 x2 > 0 x2 = 0 1 3) 4) = −h + = −h + P r=x2 V ([x1 , δ; r, τ ], 0)πx2 ,r|1 P r=p V ([x1 , δ; r, τ ], 0)π0,r|1 + P r=x2 5) δ <τ −1 1 = −h + 6) τ τ 0 = x1 + x2 + V (∆) τ τ −1 x2 > 0 x2 = 0 1 7) 8) V ([x1 , δ; 0, 0], 1)π0,0|1 V ([x1 , δ; r, y2 + 1], 1)πx2 ,r|1 = x1 − h + = x1 − h + P r=x2 V ([x1 , δ; r, τ ], 0)πx2 ,r|1 P r=p V ([x1 , δ; r, τ ], 0)π0,r|1 + = x1 − h + P r=x2 9) τ <τ −1 1 τ −1 x2 > 0 x2 = 0 x2 > 0 x2 = 0 2 10) 11) 12) 13) τ −1 x1 > 0 x1 > 0 x1 = 0 x1 = 0 τ −1 x1 > 0 x1 = 0 <τ −1 2 14) 15) τ −1 x1 > 0 x1 = 0 0 16) 17) 18) <τ −1 <τ −1 2 = −2h + 19) <τ −1 0 1 = −2h + max{ V ([x1 , δ; 0, 0], 1)π0,0|1 V ([x1 , δ; r, y2 + 1], 1)πx2 ,r|1 P P P q=x1 P r=x2 ( P r=x2 V ([q, τ ; r, y2 + 1], 1)πx1 ,q|2 πx2 ,r|2 P q=p V ([q, τ ; r, y2 + 1], 1)π0,q|2 + π0,0|2 V = −2h + q=x1 r=x2 V ([q, τ ; r, τ ], 0)πx1 ,q|2 πx2 ,r|2 P P = −2h + q=x1 ( r=p V ([q, τ ; r, τ ], 0)π0,r|2 + π0,0|2 V ([q, τ ; 0, 0], 1))πx1 ,q|2 P P = −2h + r=x2 ( q=p V ([q, τ ; r, τ ], 0)π0,q|2 + π0,0|2 V ([r, τ ; 0, 0], 1))πx2 ,r|2 P P = −2h + q=p r=p V ([q, τ ; r, τ ], 0)π0,q|2 π0,r|2 + π0,0|2 π0,0|2 V ([0, 0; 0, 0], 1) P P + q=p V ([q, τ ; 0, 0], 1)π0,q|2 π0,0|2 + r=p V ([r, τ ; 0, 0], 1)π0,0|2 π0,r|2 = −2h + = −2h + ([r, y2 + 1; 0, 0], 1))πx2 ,r|2 1 = −2h + max{ = −2h + max{ P q=x1 P P P q=x1 r=0 V ([q, τ ; r, 1], 1)πx1 ,q|2 π0,r|2 } q=x1 V ([q, τ ; 0, 0], 1)πx1 ,q|1 , P q=p V ([q, τ ; 0, 0], 1)π0,q|1 + π0,0|1 V ([0, 0; 0, 0], 1) , P P q=p V ([q, τ ; r, 1], 1)πp,q|2 + π0,0|2 V ([r, 1; 0, 0], 1))π0,r|2 } r=0 ( P r=x2 P q=x1 V P q=x1 V ([q, y1 + 1; r, y2 + 1], 2)πx1 ,q|2 πx2 ,r|2 ([q, y1 + 1; 0, 0], 1)πx1 ,q|1 , P r=0 V ([q, y1 + 1; r, 1], 2)πx1 ,q|2 π0,r|2 } Table 2.6. Optimality equations for multiple re-listing case (infinte planning horizon) Lemma 2.20. If we assume a vigilant seller and that the first auction has received a bid, then the value functions for those states can be evaluated as follows, 93 (1) If x1 > 0, y1 = τ, δ and z = 0, 1, or x1 > 0, y1 , y2 < τ and z = 2, then V ([x1 , y1 ; x2 , y2 ], z) = R1 ([x1 , y1 ; x2 , y2 ], z) (2) If x1 > 0, y1 < τ, y2 = 0, z = 1 then P V ([x1 , y1 ; x2 , y2 ], z) = max{−2h + V ([q, y1 + 1; 0, 0], 1)πx1 ,q|1 , R1 ([x1 , y1 ; 0, 0], 2)} q=x1 where R1 (·) represents the value of having both items released and a positive price in the first auction, for x1 > 0, and z = 0, 1, 2, R1 (S = ([x1 , y1 ; x2 , y2 ], z)) = − h(2τ − y1 − y2 ) + E[X1,τ |S = ([x1 , y1 ; x2 , y2 ], z)] + E[X2,τ |S = ([x1 , y1 ; x2 , y2 ], z)] + (πx2 ,0|z )(π0,0|z )τ −y1 −1 (π0,0|1 )y1 −y2 v(0, 0) where E[Xi,τ |S = ([x1 , y1 ; x2 , y2 ], z)] is defined by (2.17) and v(0, 0) is defined by (2.28). Proof of Lemma 2.20 - see Appendix A. The implication of Lemma 2.20 is that once a bid arrives in the first auction, there are no loops back to the initial state. Therefore, the number of auctions in each of the periods an auction is underway and the expected final price of each auction can be determined. Hence the problem is reduced to the case of guaranteed successful auctions. Consequently the optimal decision when X1 > 0 and Y2 = 0 follows a control limit policy. This result also holds when both auctions are underway but the second auction has received a bid as summarized in the following result. Lemma 2.21. If we assume a vigilant seller and that both auctions are underway but only the second auction has received a bid, then the value functions can be evaluated as follows, for y1 , y2 < τ, x1 = 0, x2 > 0, z = 2, V ([x1 , y1 ; x2 , y2 ], z) = R2 ([x1 , y1 ; x2 , y2 ], 2) 94 where R2 (·) represents the value of having both auctions underway and a positive current price in the second auction, for y1 , y2 < τ , x1 = 0, x2 > 0, and z = 2, R2 (S = ([x1 , y1 ; x2 , y2 ], z)) = − 2h(τ − y1 ) + E[X1,τ |S = ([x1 , y1 ; x2 , y2 ], z)] + (1 − (πx1 ,0|z )τ −y1 )(−h(y1 − y2 ) + E[X2,τ |S = ([x1 , y1 ; x2 , y2 ], z)]) + (πx1 ,0|z )τ −y1 E[V (S )|S = ([x1 , y1 ; x2 , y2 ], z)] where S = ([X2 , y2 + τ − y1 ; 0, 0], 1), X2 > 0, V (S ) is calculated according to the second case of Lemma 2.20 and the conditional expectation is with respect to X2 , and E[Xi,τ |S = s] is defined by (2.17). Proof of Lemma 2.21 - see Appendix A. The implication of Lemma 2.21 is that the possible decision to re-list item 1, which happens with probability (π0,0|2 )τ −t1 , follows a control limit policy. For all other cases, in order to determine the optimal solution and policy, one has to solve the optimality equations either using value iteration or policy iteration. Below the resulting 8 cases of the optimality equation are summarized. The value function for those states where a positive bid has arrived have been separated from the states where neither auction has received a bid. The issue with the multiple re-listing case is exactly when no bid has arrived and the potential for looping back to the starting state exist. (1) y1 = τ, δ, z = 0, 1, or y1 , y2 < τ, x1 > 0, z = 2 V ([x1 , y1 ; x2 , y2 ], z) =R1 ([x1 , y1 ; x2 , y2 ], z) (2) y1 < τ, y2 = 0, x1 > 0, z = 1, P V ([x1 , y1 ; x2 , y2 ], z) = max{−2h + V ([q, y1 + 1; 0, 0], 1)πx1 ,q|1 , R1 ([x1 , y1 ; 0, 0], 2)} q=x1 95 (3) y1 , y2 < τ, x1 = 0, x2 > 0, z = 2 V ([x1 , y1 ; x2 , y2 ], z) = R2 ([0, y1 ; x2 , y2 ], 2) (4) y1 , y2 = τ − 1, x1 , x2 = 0, z = 2, P P V ([x1 , y1 ; x2 , y2 ], z) = − 2h + P R1 ([q, τ ; r, τ ], 0)π0,q|2 π0,r|2 + q=p r=p R1 ([q, τ ; 0, 0], 1)π0,q|2 π0,0|2 q=p P R1 ([r, τ ; 0, 0], 1)π0,0|2 π0,r|2 + (π0,0|2 )2 V ([0, 0; 0, 0], 1) + r=p (5) y1 = τ − 1, y2 < τ − 1, x1 = x2 = 0, z = 2, P P P V ([r, y2 + 1; 0, 0], 1)π0,0|2 π0,r|2 R1 ([q, τ ; r, y2 + 1], 1)π0,q|2 π0,r|2 + V ([x1 , y1 ; x2 , y2 ], z) = − 2h + r=p q=p r=0 + (π0,0|2 )2 V ([0, y2 + 1; 0, 0], 1) (6) y1 = τ − 1, y2 = 0, x1 = 0, z = 1, P V ([q, τ ; 0, 0], 1)π0,q|1 + π0,0|1 V ([0, 0; 0, 0], 1) , V ([x1 , y1 ; x2 , y2 ], z) = − 2h + max{ q=p P P P R1 ([q, τ ; r, 1], 1)πp,q|2 π0,r|2 + V ([r, 1; 0, 0], 1)π0,0|2 π0,r|2 r=p q=p r=0 + (π0,0|2 )2 V ([0, 1; 0, 0], 1) } (7) y1 < τ − 1, y2 < τ − 1, x1 = x2 = 0, z = 2, P P V ([x1 , y1 ; x2 , y2 ], z) = − 2h + R1 ([q, y1 + 1; r, y2 + 1], 2)π0,q|2 π0,r|2 q=p r=0 P + R2 ([0, y1 + 1; r, y2 + 1], 2)π0,0|2 π0,r|2 + V ([0, y1 + 1; 0, y2 + 1], 2)π0,0|2 π0,0|2 r=p 96 (8) y1 < τ − 1, y2 = 0, x1 = 0, z = 1, P V ([x1 , y1 ; x2 , y2 ], z) = − 2h + max{ V ([q, y1 + 1; 0, 0], 1)πx1 ,q|1 + V ([0, y1 + 1; 0, 0], 1)π0,0|1 , q=p P P P R1 ([q, y1 + 1; r, 1], 2)π0,q|2 π0,r|2 + q=p r=0 R2 ([0, y1 + 1; r, 1], 1)π0,0|2 π0,r|2 r=p + π0,0|2 π0,0|2 V ([0, y1 + 1; 0, 1], 2) } 4.4. Numerical Example. Next we illustrate this with a numerical example. This example shows that the optimal policy in the multiple re-listing case might not be a threshold policy. Let prices be discrete, τ = 2, p = k = 10, P = 30 and the transition probability matrices be defined as follows, 0 0 .5 10 0 Π1 = 20 0 30 0 10 20 .2 .2 .6 .3 0 .6 0 0 30 .1 .1 .4 1 0 0 .6 10 0 Π2 = 20 0 30 0 10 .2 .7 0 0 20 30 0 .3 0 .65 .35 0 1 .2 In Table 2.7 below the optimal policy, derived using policy iteration with V (∆) = 0, for various holding costs is shown. Note in particular that there are instances when it may be optimal to release the second item when X1 = 0 yet defer if X1 > 0. For example, if h = 2.75 and the first auction elapsed one period, Y1 = 1, then we see that it is optimal to release the item 2 if X1 = 0 (or X1 ≥ $20), but optimal to defer the release if X1 = $10. Note that this scenario can occur since at the start of the first auction it is optimal to defer the release of the item 2 and π0,0|1 > 0, but that this can not occur if, for instance, h = 3, since then the optimal decision at the start of the first auction is to release item 2, and hence there is no decision to be made when Y1 = 1. If, however, both auctions are 97 unsuccessful, which happens with probability (.6)2∗2 = .1296 then the problem is back to its original state at which it was optimal to release both items. 3.6 ≤ h 2.9 ≤ h ≤ 3.5 2.7 ≤ h ≤ 2.8 1.9 ≤ h ≤ 2.6 1.3 ≤ h ≤ 1.8 h ≤ 1.2 Y1 = 0 X1 = 0 X1 = 0 Release Release Release Release Defer Release Defer Defer Defer Defer Defer Defer Y1 = 1 X1 = 10 Release Defer Defer Defer Defer Defer X1 = 20 Release Release Release Release Defer Defer X1 = 30 Release Release Release Release Release Defer Table 2.7. Optimal decision as a function of various holding costs for numerical example. 5. Discussion This chapter has analyzed the problem of strategically releasing items for auction in order to maximize profit. The objective has been to provide a framework for modeling the dynamics of competing auctions and derive structural properties on the optimal auction release policy. The two main underlying assumptions that formed the basis for our analysis were: 1) each period an item remains unsold a holding cost is incurred, and 2) competing auctions ‘cannibalize’ each other and decrease the expected final price of each auction. Two scenarios were analyzed - guaranteed successful auctions and possibly unsuccessful auctions. For the first case the problem reduces to a finite horizon MDP, while the second case results in an infinite horizon negative dynamic program. Given certain structural assumptions on the transition probabilities, we were able to show that in the first case the optimal release policy is a control limit policy in the current price of the ongoing auction. Furthermore, we showed that the control limit is decreasing in the holding cost. However, for the case when there is a positive probability that an auction may be unsuccessful, the optimal policy does not have to be a control limit policy. The problem that arises is that the optimal decision when the ongoing auction has not received any bids may or may not be consistent with a 98 control limit policy. The main managerial insight and contribution of this chapter is that there is a significant value of understanding the special dynamics of online auctions, and that by using a price adaptive or closed loop policy a seller can improve his expected total profit. Because online auctions are rather inexpensive to conduct and administer, they are becoming more and more popular as alternative salvage channels. In industries where the value of ‘old’ items depreciates quickly, such as consumer electronics or fashion goods, being able to optimally sell excess inventory quickly can be of great importance. In addition, even though the gain on each individual item may be small, the overall impact can be quite substantial as the size of the inventory grows. In the concluding Chapter 6, an overview and some insights regarding the two most obvious and perhaps important extension are provided. Namely the general N -item case and the case when price-transitions also depend on the price of the competing auction. Other extensions, that will not be covered in this thesis, include continuous time decision making and incorporating the decision regarding other auction parameters. In particular, the decision regarding how long an auction should last. That is, when should an auction be started and for how long should it last. 99 CHAPTER 3 Fixed Bidding Strategies: The Two Auctions Case 1. Objective In the previous chapter the analysis focused on the within period price-transitions of auctions. We assumed auctions evolve according to a specified stochastic process, and that within a period, transitions of price follow a given probability distribution. Furthermore, structural properties on the distribution function were provided such that the optimal release policy is of a threshold type. In the succeeding Chapter 4, we describe a statistical model for analyzing the within period price-transitions based on the Dell auction data. The model there is based on that within a period, price-increments follow a zero-inflated gamma distribution. Chapter 4 also includes an empirical validation of the structural results of Chapter 2, based on data from 4,000 eBay auctions of Dell Financial Services (DFS). Neither Chapter 2 or 4 includes any specifics regarding what or how individual bidders behave. Since this approach is different from the traditional auction theory framework, the reader may be interested to know what underlying individual bidding behavior would result in stochastic processes such as the ones discussed. In particular, what game theoretic model might apply, and what would the within period price-transition probability distribution look like at a bid strategy equilibrium. In other words, does there exists a bid strategy (Bayesian Nash) equilibrium for a given set of auction rules and bidder attributes? And what is the resulting within period price-transition distribution function? To the disappointment of some, this chapter will not address these questions. Instead we analyze two fixed bidding strategies and derive the cumulative distribution function (CDF) 100 of the within period price-transitions. That is, the objective is not to provide an equilibrium analysis on the individual bidding strategies, but to discuss the resulting within period price-transitions from two specific bidding strategies: 1) bid the minimum increment above the lowest priced auction, 2) bid truthfully your valuation in the lowest priced auction. The former is the bid strategy proposed in Peters and Severinov (2006), while the latter is an extension of the bid strategy proposed in Vickrey (1961). The truthful bidding strategy is also what eBay promotes bidders to do.1 More details and discussion follow. In other words, the objective is to illustrate how the CDF for the within period price-transitions can be derived from the individual bidding behavior. Depending on the auction rules, assumptions of how bidders arrive and their possible strategies, the transition probabilities might be very difficult to derive. For simple bidding strategies and/or simple properties due to certain assumptions, closed form solutions to the CDF may exist. The first strategy we analyze provides an example of this. For more complicated situations, such as the second strategy analyzed, upper and lower bounds on the CDF can be given. For even more complicated situations, one may have to revert to simulation in order to estimate the CDF. One reason why the CDF of the within period price-transitions is of interest, is that it enables us to model how auctions progress, and consequently to make better informed decisions during an auction. For instance, a seller might be interested to know whether or not to start another auction, while a bidder might be interested to know if he should place a bid, and if so how much to bid. By understanding how auctions evolve both sellers and bidders are able to make better decisions. A second reason why we are interested in the CDF of the within period price-transitions, is that it provides a mechanism for analyzing the variation in final price of online auctions. In Figure 3.1 the distribution of the final price for six products auctioned by DFS on eBay from February 2005 to January 2007 are shown. In 1http://pages.ebay.com/help/buy/buying-ov.html 101 700 600 500 400 300 0 100 200 Final Price D1 D3 D4 L1 L4 L5 Product Figure 3.1. Distribution of final price for the six products analyzed. D1 and L1 represent a product line, while D3, D4, L4, and L5 represent a specific product. More information and descriptive statistics is provided in Chapter 1 and 5 each box-plot the lower and upper edge of the box represents the 25th and 75th percentiles. The line inside the box represents the median final price. The dashed lines (‘whiskers’) from each box, are drawn to the observation furthest away but within a factor of 1.5 × IQR, from the edge of the box. The IQR is the inner quartile range. Circles outside the whiskers are observations, which could be classified as extreme and potentially outliers. The notches inside the box indicates a range around the median. An informal statistical test if the median from two box-plots are different, is if the notches overlap. We note that for all six products the final price exhibits great variation. A natural question is: what drives this apparent variation? Furthermore, is it possible to characterize the conditional distribution of the final price given that an auction has elapsed for some time and is currently at some price. 102 The traditional auction theory answer is that the variation in final price is a direct result of the variation in bidders’ valuation. For instance, in a single sealed-bid second-price auction, where bidders follow the Nash equilibrium strategy of bidding truthfully, the distribution of the final price follows the order statistics of the bidder with the second highest valuation [14, p.15]. In the online auction setting, there would seem to be at least one more source of variation. Namely, the number of bidders. A key difference between traditional auctions and online auctions, is that in the former the number of bidders is fixed, while in the latter the number of bidders varies between auctions and over time. This chapter describes a method for incorporating the stochastic number of bidders into the CDF of the within period price-transition as well as final price. Though there are many papers that analyze the distribution regarding the final price of auctions, there are few that focus on the progression of prices or how auctions evolve. One paper that explicitly model the progression is Segev, Beam and Shantikumar (2001). They consider a single auction and model price as a Markov chain where the ‘active’ bidders form a queue. Their model is based on a first-price auction format, bidders arriving according to a Poisson process, and a bidding strategy identical to the minimal bid increment strategy. This chapter differs from their paper in that we consider a second-price auction format, and analyze the progression of price in two ongoing auctions. Overview of Chapter 3. The chapter is organized as follows. In Section 2 an overview of the auction rules, bidders’ valuation and arrival process is provided. Section 3 and 4 discuss the two bidding strategies. Section 5 summarizes the findings. 103 1.1. Notation. The notation in this chapter will be slightly different than the one used in the previous chapter. In Chapter 2, elapsed auction time was modeled as a discrete random variable and represented by Y . In contrast, we now define elapsed auction time as continuous and denote it by t. Suppose two auctions are underway simultaneously, then we index auctions by i, and define Xi,ti as the current price of auction i that has elapsed ti time units, i = 1, 2. The objective of the chapter is to derive the conditional cumulative distribution function of the price in auction i at the end of a time interval [t, t + ∆t], (3.1) FXz i,t +∆t |Xi,t (q|xi ) = Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , Z[t,t+∆t] = z} i i i, z = 1, 2 where ∆t is the length of the interval and Z[t,t+∆t] is the number of ongoing auctions during the time interval (including the one under consideration). Similar to Chapter 2, this chapter only considers the case of at most two ongoing auctions. Though the results for the first bidding strategy extends to the general N auction case, the notation becomes more convoluted with no substantial gain. For the second bidding strategy the extension to general N auctions becomes a lot more complicated. In order to simplify notation we write Fiz (q|xi ) instead of FXz i,t +∆t |Xi,t +∆t (q|xi ), and Z∆t instead of Z[t,t+∆t] . i i 2. Auction Rules and Bidder Attributes We consider a simplified version of the eBay auction format, namely a second-price auction format. That is, the highest bidder wins but only has to pay the amount of the second highest bid, or in the case of only one bidder the starting price p. We do not include a reserve price, and although a minimum bid increment could be included, omitting it simplifies the notation. Therefore, suppose an auction has elapsed for t time units, then the price of the auction, Xt , is either: 1) the second highest bid placed so far given that two or more bids have been placed, 2) the starting price p given that only one bid has been placed so far, or 3) 0 if no bids have arrived. Let B be the bid amount placed by a bidder, and 104 (j) Bt (2) be the j th highest bid amount so far, e.g. Bt is the second highest bid placed in an auction that has elapsed t time units. The price of an auction that has elapsed t time units is, 0 (3.2) Xt = p Bt(2) no bid has been placed one bid has been placed at least two bids have been placed If two bidders bid the same amount then the bidder that placed his bid first becomes the high bidder. We define Xi,ti as the price of auction i after it has elapsed for ti time units, and Hi,ti as the hidden or censored value of the highest bid, i = 1, 2. Both bidders and the seller can only observe Xi,ti , i = 1, 2. The state of the two auctions is defined by [X1,t1 , H1,t1 , t1 ; X2,t2 , H2,t2 , t2 ]. For example, suppose elapsed auction time is measured in hours, then [15, 31.01, 36; 5, 7, 12] denotes the system state where auction 1 has elapsed for 36 hours, is currently priced at $15 (the second highest bid placed so far in that auction), and has a current high-bid of $31.01, and auction 2 has elapsed for 12 hours, current price of $5, and a current high-bid of $7. We assume the seller does not keep track of who is currently the high-bidder or who the bidders have been. The information regarding who the high-bidder is, or rather if there has been a change of the high-bidder, does provide additional information that can be used to further derive information regarding the size of the high-bid in the two auctions. The derivations in Section 3 and 4 are strictly with respect to the current price of an auction. Bidders have unit demand and private valuation, V , which is independent and identically distributed across bidders and over time, with support on the interval [p, P ]. In other words, p is both the starting price of each auction and the lower bound of the valuation. We define 105 GV (v) as the distribution function of the valuation, for v ∈ , (3.3) GV (v) = Pr{V ≤ v} Given that a bidder’s valuation exceeds a threshold r, we define the conditional cumulative distribution function of a bidder’s valuation by GV (v|r), for v ∈ , (3.4) GV (v|r) = Pr{V ≤ v|V ≥ r} = GV (v) − GV (r) 1 − GV (r) The conditional CDF provides information regarding the amount of the censored highbid, and is required to characterize the probability distribution of the price-transitions. The density function associated with each distribution function is symbolized by a lower case letter, gV (v) = gV (v|r) = ∂ GV (v) ∂v ∂ ∂ GV (v) − GV (r) gV (v) GV (v|r) = = ∂v ∂v 1 − GV (r) 1 − GV (r) For simplicity of notation we omit the subscript V and write G(v) (g(v)) and G(v|r) (g(v|r)) instead of GV (v) (gV (v)) respectively GV (v|r) (gV (v|r)). We define V(j) to be the j th order statistic of the valuation of bidders that have arrived so far. For instance, the random variable V(3) denotes the third highest valuation of the bidders that have arrived. Next we describe the sequence of events depicted in Figure 3.2. The sequence of events are as follows. Suppose at time t two auctions are underway, the system state is ([x1 , h1 , t1 ; x2 , h2 , t2 ], 2), and a bidder with valuation V = v arrives. The bidder can only observe the price and elapsed time of the two auctions, i.e. he only observes x1 , t1 , x2 , t2 . If the lowest priced auction is above his valuation then he leaves and never returns, i.e. if v ≤ min{x1 , x2 }. If, on the other hand, his valuation is not below the lowest priced auction then he will bid in a given auction; the bid is prescribed by the 106 start t2 t2 t2 + ∆t t1 t1 t1 + ∆t start Auction 2 end Auction 1 end ✲ t t time t + ∆t Figure 3.2. Time-line for the sequence of bidding events. At time t a bidder with valuation V = v arrives and observes x1 , t1 , x2 , t2 . If v ≤ min{x1 , x2 }, then the bidder leaves, while if v > min{x1 , x2 }, then he will place a bid according to a fixed bidding strategy. The bidder will remain at the auction site and keep bidding until he either wins an auction or the price in both auctions exceed his valuation v. At time t another bidder, with valuation V = v , arrives and observes x1 , t1 , x2 , t2 . He too will follow the same fixed bid strategy, and remain at the auction site until either he wins an auction or both auctions exceed his valuation v . The objective of this chapter is to characterize the prices at time t + ∆t when the system state is ([X1 , U1 , t1 + ∆t; X2 , U2 , t2 + ∆t], 2), given an initial state ([x1 , U1 , t1 ; x2 , U2 , t2 ], 2) at time t. specific bidding strategy. A bid strategy is a function that specifies the auction and amount a bidder should bid upon arriving at the auction site. We define Bi as the amount a bidder bids in auction i, i = 1, 2, and B = (B1 , B2 ) as the pair of bid amounts for the two auctions. If a bidder chooses not to place a bid in auction i then Bi = 0, i = 1, 2. We assume no bidder will place a positive bid amount above P or below p, i.e. Bi ∈ {0} ∪ [p, P ], i = 1, 2. Given that bidders have unit demand, bidders will not place positive bids in both auctions at the same time. The two bidding strategies discussed in this chapter only depend on how many auctions are underway, the price of the auctions, and the valuation of the bidder, (3.5) B : {1, 2} ×({0} ∪ [p, P ], {0} ∪ [p, P ]) × [p, P ] −→ ({0} ∪ [p, P ], {0} ∪ [p, P ]) Z X1 X2 V B1 B2 Additional factors that make the bidding strategy more realistic include the elapsed time of the auctions, risk aversion, and bid history. In Chapter 5 an empirical analysis of 44,000 individual bids placed in 4,000 desktop and laptop auctions of DFS is provided. The objective there is to characterize the bid-increment and explore the relationship to various factors. For instance, how does the price of an auction and/or the price of a competing 107 auction affect the mean bid-increment. The bidder will remain at the auction site and keep bidding until he either wins an auction or the lowest priced auction exceeds his valuation v. That is, suppose the bidder placed a bid in auction 1, but that the amount he bid did not make him the current high-bidder, then he will again observe the prices of both auctions and follow the same logic once more, i.e. if the lowest priced auction exceeds v then he leaves, else he bids according to the given bid strategy. If a bidder does become the high-bidder then he remains at the auction site and observes the progression of the auctions. If at some point he is outbid, then he observes the prices in both auctions, and by the same logic as before, will leave if v is less than the lowest priced auction, and else bid according to the fixed bidding strategy. That a bidder ‘remains at the auction site’ does not necessarily mean that he is continuously observing the auction site. For instance, he could go about his affairs and when another bidder places a bid he receives notification. On eBay, for instance, you receive an email that you have been outbid. To illustrate, label the bidder arriving at time t with valuation v as Bidder 1. Suppose that at time t, Bidder 1 become the high-bidder in auction 1, and that the former high-bidder (the bidder that Bidder 1 replaced) decided to leave. Furthermore, suppose that at time t a second bidder (Bidder 2) with valuation V = v arrives. Bidder 2 observes x1 , t1 , x2 , t2 , and by the same logic, if v ≤ min{x1 , x2 }, then he will leave, and if v > min{x1 , x2 }, then he will place a bid according to the fixed bid strategy. Suppose Bidder 2 bids in auction 1, and replaces Bidder 1 as the high-bidder. Bidder 1 will now observe the prices in both auctions, and follow the same procedure as before, i.e. if v is less than the lowest priced auction then he leaves, else he bids according to the given bid strategy. 108 We assume that when placing a bid no time elapses. This implies that no time elapses between when a bidder arrives and he is the high-bidder in an auction or the price in both auctions exceeds his valuation. And consequently, since a bidder will remain at the auction site until he either wins an auction or the price in both auctions exceeds his valuation, at any given time there will be at most one bidder per auction present. Namely the high bidder of each auction. The assumption that no time elapses in placing a bid is clearly unrealistic and purely for modeling purposes. In addition, we assume that bidders do not speculate regarding how many auctions will be released in the future. More specifically, we assume bidders do not consider the possibility that additional auctions may start at a later time. That is, a bidder with valuation above the lowest priced auction, would never choose not to place a bid, and instead re-visit at a later time in hope for a lower priced auction. If the lowest priced auction is below a bidder’s valuation then he bids, otherwise he leaves and never returns. Furthermore, if only one auction is underway and a bidder is outbid he leaves and never returns to see if a second auction has started. Next we discuss the arrival process of bidders. We assume bidders arrive according to an exogenously given stochastic process that only depends on time. Let M[t,t+∆t] be the number of bidders that arrive in a time interval [t, t + ∆t], and, (3.6) ρM (m|[t, t + ∆t]) = Pr{M[t,t+∆t] = m} m = 0, 1, 2, . . . For simplicity of notation we write M∆t and ρM (m|∆t), instead of M[t,t+∆t] and ρM (m|[t, t + ∆t]). Other variables that might affect the the arrival rate to the auction site include the price and elapsed time of the auctions, and the number of ongoing auctions. As an example of an arrival process, bidders may arrive according to a non-homogeneous 109 Poisson process with rate λt . The distribution of the number of bidders arriving in the interval [t, t + ∆t] is then given by, ρM (m|∆t) = e−γ(t) (γ(t))m m! m = 0, 1, 2, . . . where t+∆t γ(t) = t λu du Next we show how the CDF of the within period price-transitions can be evaluated given the above auctions rules and two specific bidding strategies. 3. Minimum Bid Increment Strategy The first strategy we consider is when each bidder bids a minimal increment above the lowest priced auction. A bidder will stop bidding once he is the high-bidder in an auction or the price is above his valuation. If he is the high-bidder but subsequently is out-bid then he evaluates all auctions and follows the same bid strategy as before. In other words, the process is as follows, (1) A bidder with valuation V = v arrives to the auction site. (2) He observes the price in all ongoing auctions. (3) If the price in the lowest priced auction is below his valuation (v > min{X1 , X2 }), then he bids in the lowest priced auction an amount that is the lesser of a minimal increment k above the price or his valuation (B = min{X1 + k, X2 + k, v). If the auctions are priced equally, then with probability .5 he chooses either one. (a) If he is successful and becomes the high-bidder, then he continuously observe how the auction progress and if he is ever outbid returns to step (2). (b) If he is not successful and his bid does not make him the high-bidder, then he returns to step (2). 110 (4) If the lowest priced auction is above his valuation (v ≤ min{X1 , X2 }), then he leaves the auction site and never returns. Note that this implies that the amount of the high-bid in either auction at any given time is the minimal increment above the current price, i.e. Hi = Xi + k. However, the high-bidder will remain at the auction site until the auctions exceed his valuation or he wins an auction. We illustrate with a numerical example which is depicted in Figure 3.3. Suppose at time t, the system state is ([10, 11, 2; 8, 9, 1], 2), Bidder 1 with valuation V = 15 is the high-bidder in auction 1, Bidder 2 with valuation V = 17 is the high-bidder in auction 2, and that Bidder 3 with valuation V = 10 arrives to the auction site. Bidder 3 observes the prices in both auctions and places a bid in auction 2 for $9. This raises the price in auction 2 to $9, but does not make Bidder 3 the high-bidder, since Bidder 2 placed his bid first. Bidder 3 then observes the prices again, and places a bid for $10 in auction 2. Bidder 3 replaces Bidder 2 as the high-bidder, while the price in auction 2 remains $9. Bidder 2 now observes the two auctions, and places a bid for $10 in auction 2. The price in auction 2 jumps to $10 but due to that Bidder 3 placed his bid first, he remains as the high-bidder. Bidder 2 observes that the price in both auctions is $10, chooses one with probability .5, and places a bid of $11. Suppose that Bidder 2 chose auction 2 and replaced Bidder 3 as the high-bidder. Bidder 3 now observes that the price in both auctions is at $10, which is his valuation, and therefore decides to leave the auction site. We assume that no time elapsed from that Bidder 3 arrived at the auction site until he left the auction site. At time t the next bidder arrives or auction 1 has ended. This bidding strategy is discussed by Peters and Severinov (2006), who show that for M bidders and N simultaneously released auctions, this strategy leads to a Bayesian-Nash equilibrium. An implicit assumption in their model is that there is no ‘friction’ or ‘cost’ for 111 ([10, 11, 2; 8, 9, 1], 2) ([10, 11, 2; 9, 9, 1], 2) ([10, 11, 2; 9, 10, 1], 2) ([10, 11, 2; 10, 10, 1], 2) ([10, 11, 2; 10, 11, 1], 2) ✲ t t A1: Bid.1 (V = 15) A2: Bid.2 (V = 17) A1: Bid.1 (V = 15) A2: Bid.2 (V = 17) Bid.3 (V = 10) arrives Bid.3 (V = 10) and bids $9 in A2 bids $10 in A2 A1: Bid.1 (V = 15) A2: Bid.3 (V = 10) A1: Bid.1 (V = 15) A2: Bid.3 (V = 10) A1: Bid.1 (V = 15) A2: Bid.2 (V = 17) Bid.2 (V = 17) bids $10 in A2 Bid.2 (V = 17) bids $11 in A2 Bid.3 (V = 17) leaves Figure 3.3. Illustration of the sequence of bidding events for the minimal bid increment strategy. It is assumed that no time elapses from Bidder 3 arrives until he leaves. placing bids [18, p.223]. In other words, no time elapses when placing a bid. For instance, if placing a bid takes time and there is a positive probability that a bid will not get registered in the final moments (e.g. eBay auctions), then the above strategy would not result in an equilibrium. That is, a potential bidder who doubts that his valuation is or will be among the N highest, would be better off by not allowing the price of the auctions to raise to his valuation. Contrariwise, it could be more beneficial for such bidders to wait until the final moments of the auction and place one single bid. Regardless whether the strategy results in an equilibrium, we assume all bidders follow this strategy. Similar to Peters and Severinov (2006) we assume placing a bid takes no time, i.e. the system is ‘frictionless’. This results in that the current price follows the order statistics of the N + 1 highest valuation of the bidders [18, p.229]. In other words, if there is only one auction underway then the price at any given time is V(2) , i.e. the second highest valuation of the bidders that has visited so far. While if two auctions are underway, then the price of both auctions is V(3) , i.e. the third highest valuation of the bidders that has visited so far. An assumption that should be made explicit is that bidders are not time-sensitive. That is, a bidder’s decision is strictly based on price and not on how much time is remaining. A major difference between the setting analyzed in this thesis and, for instance, Peters and Severinov (2006) is that we include the possibility that one auction will end before the other. 112 t As a result it may seem more reasonable that the bidders would include the time aspect in their decision making. And this may further question whether this strategy results in an equilibrium. To derive the distribution function for the within period price-transitions we consider the case of one and two ongoing auctions separately. 3.1. One Ongoing Auction. We start by considering the case when there is only one ongoing auction, which without loss of generality we index as auction 1. Since there is only one ongoing auction and bidders are non-speculative regarding the possibility that a second auction will start, the CDF of the within period price-transition follows the distribution of the second highest valuation that arrives in the time-interval [t, t + ∆t]. Therefore, we can condition upon how many bidders arrive and determine the order statistics of the valuation of bidders. However, if there is a current high-bidder then he has a valuation above the current price and we first condition upon his valuation. Therefore, there are two possible starting states ([0, 0, t1 ; 0, 0, 0], 1) and ([x, x + k; 0, 0], 1). Recall that the system state is defined by the price, high-bid, and elapsed auction time of each auction, ([X1 , H1 , t1 ; X2 , H2 , t2 ], Z), and implicitly it is assumed the second auction has not started and no information regarding it is available to potential bidders. The results for the two possible initial states are summarized in the following lemmas. Lemma 3.1. If no bids have been placed and the system state is ([0, 0, t1 ; 0, 0, 0], 1), and auction 2 will not be released during time-interval [t, t + ∆t], then the CDF of the within period price-transition of auction 1 is, ∞ F11 (q|0) = Pr{X1,t+∆t ≤ q|X1,t = 0, Z∆t = 1, M∆t = m}ρM (m|∆t) m=0 113 where Pr{X1,t+∆t ≤ q|X1,t = 0, Z∆t = 1, M∆t = m} 0 = q<0 m=0 1 0≤q 0 = q<p (G(q))m + m(G(q))m−1 (1 − G(q)) m≥1 p≤q Proof Lemma 3.1 - See Appendix B. Lemma 3.2. If at least one bid has been placed and the system state is ([x, x+k, t1 ; 0, 0, 0], 1), and auction 2 will not be released during time-interval [t, t+∆t], then the CDF of the within period price-transition of auction 1 is, F11 (q|x) = P Pr{Xt+∆t ≤ q|Xt = x, V(1) = v, Z∆t = 1}g(v|x)dv x P ∞ = x Pr{Xt+∆t ≤ q|Xt = x, V(1) = v, Z∆t = 1, M∆t = m}ρM (m)g(v|x)dv m=0 114 where Pr{Xt+∆t ≤ q|Xt = x, V(1) = v, Z∆t = 1, M∆t = m} 0 q<x = 1 x≤q 0 = q<x G(q) 1 x≤q<v m=1 v≤q 0 = m=0 q<x m (G(q)) (G(q))m + m(G(q))m−1 (1 − G(q)) x≤q<v m≥2 v≤q Proof Lemma 3.2 - See Appendix B. Note that if x = p in Lemma 3.2, then g(v|x) = g(v). 3.2. Two Ongoing Auctions. Next we consider when two auctions are underway. Using the same approach as the one auction case, we condition upon the number of arriving bidders and the current high-bidders’ valuations. There are a two cases regarding when the two auctions were released. Case 1: Auctions Released Simultaneously If the auctions are released simultaneously then if at least two bidders arrive the price will be the same in both auctions [18, p.229]. This is a consequence that a bidder will bid up to his valuation and not leave the auction site until both auctions exceed his valuation. If on the 115 other hand only one bidder arrives then the price in one auction will be p while in the other it will be zero. And if no bidder has arrived, then the price is of course zero in both. That is there are only three possible states at time t: ([0, 0, t1 ; 0, 0, t2 ], 2), ([p, p + k, t1 ; 0, 0, t2 ], 2), ([x, x + k, t1 ; x, x + k, t2 ], 2). Below the CDF of the within period price-transition for each auction and each possible starting state is provided. Lemma 3.3. If no bids have been placed and the system state is ([0, 0, t1 ; 0, 0, t2 ], 2), and neither auction will expire during the time-interval [t, t + ∆t], then the CDF of the within period price-transition of auction i is, for i = 1, 2, ∞ Fi2 (q|0) = Pr{Xi,ti +∆t ≤ q|Xi,ti = 0, Z∆t = 2, M∆t = m}ρM (m|∆t) m=0 where Pr{Xi,ti +∆t ≤ q|Xi,ti = 0, Z∆t = 2, M∆t = m} 0 = 1 0 = .5 1 q<0 m=0 0≤q q<0 p≤q 0 = m=1 0≤q<p q<p (G(q))m + m(G(q))m−1 (1 − G(q)) + m≥2 m 2 (G(q))m−2 (1 − G(q))2 p≤q Proof Lemma 3.3 - See Appendix B. Lemma 3.4. If only one bid has been placed and the system state is ([p, p+k, t1 ; 0, 0, t2 ], 2) (or ([0, 0, t1 ; p, p + k, t2 ], 2)), and neither auction will expire during the time-interval [t, t + 116 ∆t], then the CDF of the within period price-transition of auction i is, for i = 1, 2, Fi2 (q|xi ) = P Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , Z∆t = 2}g(v1 )dv1 p P ∞ = p Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , Z∆t = 2, M∆t = m}ρM (m|∆t)g(v1 )dv1 m=0 where Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , Z∆t = 2, M∆t = m} 0 = = q < xi 1 xi ≤ q 0 q<p m=0 m=1 1 p≤q 0 = q<p (G(q))m + m(G(q))m−1 (1 − (G(q))) (G(q))m + m(G(q))m−1 (1 − (G(q))) + p ≤ q < v1 m 2 (G(q))m−2 (1 − (G(q)))2 m≥2 v1 ≤ q Proof Lemma 3.4 - See Appendix B. Lemma 3.5. If at least one bid in each auction has been placed and the system state is ([x, x+k, t1 ; x, x+k, t2 ], 2), and neither auction will expire during the time-interval [t, t+∆t], then the CDF of the within period price-transition of auction i is, for i = 1, 2, Fi2 (q|xi ) P v1 = x P x v1 Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , V(2) = v2 , Z∆t = 2}φ(v2 |x, v1 )g(v1 |x)dv2 dv1 ∞ = x x Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , V(2) = v2 , Z∆t = 2, M∆t = m}ρM (m|∆t)φ(v2 |x, v1 )g(v1 |x)dv2 dv1 m=0 117 where g(v1 |x) is the conditional density of the V(1) given that V(1) ≥ x, and φ(v2 |x, v1 ) is the conditional density of V(2) , given that x ≤ V(2) ≤ v1 , φ(v2 |x, v1 ) = ∂ G(v2 ) − G(x) ∂ G(v2 ) − G(x) = ∂v2 (1 − G(x)) − (1 − G(v1 )) ∂v2 G(v1 ) − G(x) and Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , V(2) = v2 , Z∆t = 2, M∆t = m} 0 q<x = 1 x≤q 0 G(q) = 1 1 q<x x ≤ q < v2 (G(q))m + m(G(q))m−1 (1 − (G(q))) (G(q))m + m(G(q))m−1 (1 − (G(q))) + m=1 v2 ≤ q < v1 v1 ≤ q 0 (G(q))m = m=0 q<x x ≤ q < v2 v2 ≤ q < v1 m 2 Proof Lemma 3.5 - See Appendix B. 118 (G(q))m−2 (1 − (G(q)))2 v1 ≤ q m≥2 Case 2: Auction 2 Started After Auction 1 If the auctions were not started simultaneously then there are two additional possible states. Namely the states where the price in auction 1 is higher than the price of auction 2. Since auction 1 started before auction 2, it could have reached a price x > p by the time auction 2 started. In this case there are two possible starting points for auction 2 to consider. Recall that bidders are not time sensitive and strictly focus on the price in an auction. Furthermore, bidders who have been outbid in auction 1 do not return to see if another auction has started. Below the distribution functions for the possible starting states are given. Lemma 3.6. If the system state is either ([0, 0, t1 ; 0, 0, t2 ], 2), ([p, p + k, t1 ; 0, 0, t2 ], 2) (([0, 0, t1 ; p, p + k, t2 ], 2)), or ([x, x + k, t1 ; x, x + k, t2 ], 2), where t2 < t1 , and neither auction will expire in the time-interval [t, t+∆t], then the CDF of the within period price-transitions are identical to the respective CDF when the auctions were released simultaneously. Proof Lemma 3.6 - See Appendix B. Lemma 3.7. If the system state is ([x, x + k, t1 ; 0, 0, t2 ], 2), where x > p and t2 < t1 , and if neither auction will expire in the time-interval [t, t + ∆t], then the CDF of the within period price-transition of auction i, i = 1, 2, is given by the following equations. For auction 1, F12 (q|x) P = Pr{X1,t1 +∆t ≤ q|X1,t1 = x, V(1) = v1 , Z∆t = 2}g(v1 |x)dv1 x P ∞ = x Pr{X1,t1 +∆t ≤ q|X1,t1 = x, V(1) = v1 , Z∆t = 2, M∆t = m}ρM (m|∆t)g(v1 |x)dv1 m=0 119 where Pr{X1,t1 +∆t ≤ q|X1,t1 = x, V(1) = v1 , Z∆t = 2, M∆t = m} 0 = q<x m = 0, 1 1 x≤q 0 = q<x (G(q))m + m(G(q))m−1 (1 − (G(q))) (G(q))m + m(G(q))m−1 (1 − (G(q))) + x ≤ q < v1 m 2 (G(q))m−2 (1 − (G(q)))2 m≥2 v1 ≤ q For auction 2, F22 (q|0) = = ∞ m=0 P x Pr{X2,t2 +∆t ≤ q|X2,t2 = 0, V(1) = v1 , Z∆t = 2}g(v1 |x)dv1 ∞ m=0 P x Pr{X2,t2 +∆t ≤ q|X2,t2 = 0, Z∆t = 2, M∆t = m}ρM (m|∆t) q<x x≤q Pr{X2,t2 +∆t ≤ q|X2,t2 = 0, Z∆t = 2, M∆t = m}ρM (m|∆t) ∞ m=0 q<x Pr{X2,t2 +∆t ≤ q|X2,t2 = 0, X1,t1 = x, V(1) = v1 , Z∆t = 2, M∆t = m}ρM (m|∆t)g(v1 |x)dv1 where Pr{X2,t2 +∆t ≤ q|X2,t2 = 0, Z∆t = 2, M∆t = m} 0 = q<0 m=0 1 0≤q<x 0 = q<p (G(q))m + m(G(q))m−1 (1 − (G(q))) 120 m≥1 p≤q<x x≤q and Pr{X2,t2 +∆t ≤ q|X2,t2 = 0, V(1) = v1 , Z∆t = 2, M∆t = m} 0 q<x = 1 x≤q (G(q))m + m(G(q))m−1 (1 − (G(q))) = (G(q))m + m(G(q))m−1 (1 − (G(q))) + m = 0, 1 x ≤ q < v1 m 2 (G(q))m−2 (1 − (G(q)))2 m≥2 v1 ≤ q Proof Lemma 3.7 - See Appendix B Lemma 3.8. If the system state is ([x1 , x1 + k, t1 ; x2 , x2 + k, t2 ], 2), where p ≤ x2 < x1 and t2 < t1 , and if neither auction will expire in the time-interval [t, t + ∆t], then the CDF of the within period price-transition of auction i, i = 1, 2, is given by, Fi2 (q|xi ) = Pr{V(2) < x1 }Fi2 q|xi , V(2) < x1 + Pr{V(2) ≥ x1 }Fi2 q|xi , V(2) ≥ x1 = G(x1 ) − G(x2 ) 2 1 − G(x1 ) 2 Fi q|xi , V(2) < x1 + F q|xi , V(2) ≥ x1 1 − G(x2 ) 1 − G(x2 ) i where, 1) For V(2) < x1 , F12 q|x1 , V(2) < x1 P = x1 P Pr{X1,t1 +∆t ≤ q|X1,t1 = x1 , V(1) = v1 , V(2) < x1 , Z∆t = 2}g(v1 |x1 )dv1 ∞ = x1 m=0 Pr{X1,t1 +∆t ≤ q|X1,t1 = x1 , V(1) = v1 , V(2) < x1 , Z∆t = 2, M∆t = m}ρM (m|∆t)g(v1 |x1 )dv1 where 121 Pr{X1,t1 +∆t ≤ q|X1,t1 = x1 , V(1) = v1 , V(2) < x1 , Z∆t = 2, M∆t = m} 0 = q < x1 1 m = 0, 1 x1 ≤ q 0 = q < x1 (G(q))m + m(G(q))m−1 (1 − (G(q))) (G(q))m + m(G(q))m−1 (1 − (G(q))) + x1 ≤ q < v1 m 2 (G(q))m−2 (1 − (G(q)))2 m≥2 v1 ≤ q and F22 q|x2 , V(2) < x1 P = x1 P = x1 x1 x2 x1 x2 Pr{X2,y2 +∆t ≤ q|X2,y2 = x2 , V(1) = v1 , V(2) = v2 < x2 , Z∆t = 2}φ(v2 |x2 , x1 )g(v1 |x1 )dv2 dv1 ∞ Pr{X2,y2 +∆t ≤ q|X2,y2 = x2 , V(1) = v1 , V(2) = v2 < x1 , Z∆t = 2, M∆t = m}ρM (m|∆t)φ(v2 |x2 , x1 )g(v1 |x1 )dv2 dv1 m=0 where 122 Pr{X2,t2 +∆t ≤ q|X2,t2 = x2 , V(1) = v1 , V(2) = v2 < x1 , Z∆t = 2, M∆t = m} 0 = 1 q < x2 x2 ≤ q 0 = m=0 G(q) 1 q < x2 v2 ≤ q 0 (G(q))m = (G(q))m + m(G(q))m−1 (1 − (G(q))) (G(q))m + m(G(q))m−1 (1 − (G(q))) + m=1 x2 ≤ q < v2 q < x2 x2 ≤ q < v2 v2 ≤ q < v1 m 2 and 123 (G(q))m−2 (1 − (G(q)))2 v1 ≤ q m≥2 2) For V(2) ≥ x1 , and i = 1, 2 Fi2 q|xi , V(2) ≥ x1 P = x1 P = x1 v1 x1 Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , V(2) = v2 ≥ x2 , Z∆t = 2}φ(v2 |x1 , v1 )g(v1 |x1 )dv2 dv1 v1 ∞ x1 m=0 Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , V(2) = v2 ≥ x1 , Z∆t = 2, M∆t = m}ρM (m|∆t)φ(v2 |x1 , v1 )g(v1 |x1 )dv2 dv1 where Pr{Xi,ti +∆t ≤ q|Xi,ti = xi , V(1) = v1 , V(2) = v2 ≥ x1 , Z∆t = 2, M∆t = m} 0 q < xi = 1 x ≤q m=0 i 0 = G(q) 1 q < xi v2 ≤ q 0 (G(q))m = (G(q))m + m(G(q))m−1 (1 − (G(q))) (G(q))m + m(G(q))m−1 (1 − (G(q))) + m=1 xi ≤ q < v2 q < xi xi ≤ q < v2 v2 ≤ q < v1 m 2 Proof Lemma 3.8 - See Appendix B 124 (G(q))m−2 (1 − (G(q)))2 v1 ≤ q m≥2 3.3. Numerical Example. We illustrate the above calculations with a numerical example. Let bidders valuations be uniformly distributed on [.01, 1], and the number of arriving bidders in the interval [t, t + ∆t], be uniform distributed over 0, 1, 2, 3, and 4, i.e. ρM (m|∆t) = .2, for m = 0, 1, 2, 3, 4, and ρM (m|∆t) = 0, for m ≥ 5. The minimal bid-increment is assumed to be very small (< .00001), and does not affect the derivations. Similarly, the value of ∆t is irrelevant, but can be thought of as, for instance, 24 hours. The resulting within period price-transition CDF for different starting states are shown in Figure 3.4. The upper left graph shows the CDF when only one auction (defined as auction 1), is underway with current price X1,t1 = 0, .01, .2, .4, .6, .8, as labeled in the graph. For instance, we see that if X1,t1 = $0 (the top line), then the probability that X1,t1 +∆t ≤ $.4, is slightly greater than .6. Whereas if X1,t1 = $.2 (the third line from the top), then the probability that X1,t1 +∆t ≤ $.4, has decreased to slightly below .4. The upper right graph shows the CDF for two simultaneously released auctions when the price in the auctions (X1 , X2 ) = (0,0), (.1,0), (.1,.1), (.2,.2), (.4,.4), (.6,.6), (.8,.8), as labeled. The second case when (X1 , X2 ) = (.1, 0), is slightly different from the others and therefore represented by a dotted line. The difference is that F12 (q|.01) = 0, while F22 (q|0) > 0, for q ≤ $.01. However, for q > $.01, F12 (q|.01) = F22 (q|0). We see that if the price in both auctions is $0 (the top line), then the probability that Xi,ti +∆t ≤ $.4, is about .85, i = 1, 2. While if both auctions are priced at $.2, the probability that Xi,ti +∆t ≤ $.4, is slightly less than .6, i = 1, 2. The two graphs in the bottom represent the CDF when the two auctions were started sequentially such that t2 < t1 . The bottom left graph shows the CDF for auction 1 and 125 2 when the prices at time t, (X1 , X2 ) = (.2,0), (.4,0), (.6,0), (.8,0). The CDF for auction 1 is represented by the dashed line, and is of course 0 for X1,t1 +∆t < $x1 . The CDF for auction 2 is represented by the dotted line for X2,t1 +∆t < $x1 , and coincide with auction 1’s dashed line for X2,t1 +∆t ≥ $x1 . That is, the CDF for both auctions are identical for Xi,ti +∆t > $x1 , but differ for Xi,ti +∆t < $x1 , i = 1, 2. For instance, if (X1 , X2 ) = (.2, 0), then the probability that either auction is priced ≤ $.4, at time t + ∆t is about .7. While the probability that Xi,ti +∆t ≤ $.1, is 0 for auction 1, and slightly above .4 for auction 2. The bottom right graph shows the CDF for auction 1 and 2 when (X1 , X2 ) = (.2,.1), (.4,.3), (.6,.5), (.8,.7). The features are similar to the previous case in that for q ≥ x1 , the CDF for both auctions are identical, F12 (q|x1 ) = F22 (q|x2 ). While for x2 ≤ q < x1 , the CDF for auction 1 is 0 while the CDF for auction 2 is positive. For q < x2 both auctions’ CDF is of course 0. For instance, if (X1 , X2 ) = (.2, .1), then the probability that Xi,ti +∆t ≤ $.4 is about .55, i = 1, 2, and the probability that Xi,ti +∆t ≤ $.15, is 0 for auction 1 (i = 1) and about .25 for auction 2 (i = 2). From the graphs we can visually assess and confirm that Assumptions 2.1, 2.2, and 2.3 from Chapter 2 holds. Assumption 2.1 was that the CDF of the within period pricetransitions is decreasing in price. That is, at a higher current price the CDF decreases for a given threshold. This can immediately be confirmed since in each of the four graphs the CDF for higher initial prices are always beneath the CDF for the lower initial prices. For instance, in the top two graphs, the top line represents when current price is $0, and the bottom most line when current price is $.8. Assumption 2.2 stated that the CDF of the within period price-transitions increases when there are two rather than one ongoing auctions. In order to confirm this assumption, 126 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 X=0 0.3 X1=.01 0.3 0.2 X = .2 0 0.1 Xi=.01 0.2 X = .01 0.1 0 Xi=0 0.2 X = .4 0.3 0.4 X = .6 0.5 Xy+∆ t 0.6 X = .8 0.7 0.8 0.9 1 0 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 Xi=.8 0 0.1 0.2 0.3 0.4 0.5 Xi,y+∆ t 0.6 0.7 0.8 0.9 1 0.4 X2=0 0.3 0.3 0.2 0.2 X1=.2 0.1 0 Xi=.6 CDF for Two Simultaneous Auctions 1 0 Xi=.4 0.1 CDF for One Auction 0.4 Xi=.2 0.1 0.2 X1=.4 0.3 0.4 X1=.6 0.5 Xi,y +∆ t 0.6 X1=.8 0.7 0.8 0.1 0.9 1 i 0 0 0.1 X1=.2 X1=.4 X1=.6 X1=.8 X2=.1 X2=.2 X2=.5 X2=.7 0.2 0.3 0.4 0.5 Xi,y +∆ t 0.6 0.7 0.8 0.9 i CDF for Two Sequential Auctions (X2 = 0) CDF for Two Sequential Auctions Figure 3.4. The conditional CDF of the within period price-transitions when bidders valuation is uniform on [.01, 1], the number of arriving bidders is uniform between 0 and 4, and each bidder follows the minimal bid increment strategy. The top left graph is when there is only one auction, the top right when there are two simultaneously started auctions, and the bottom when there are two overlapping auctions. In each graph the lines represent the distribution function given the initial prices (X1 , X2 ) as labeled. for a given current price Xi,ti = xi and future price Xi,ti +∆t , i = 1, 2, we compare the graphs when there are two auctions with the graph when there is only one auction. For instance, let Xi = $.4, and suppose we are interested to evaluate the probability that Xi,ti +∆t ≤ $.5. In the top left graph we see that when there is only one ongoing auction then the probability is slightly above .4. If there are two simultaneously released auctions (top right graph), 127 1 then the probability has increased to about .6. While if the two auctions were started sequentially, we see that if (X1 , X2 ) = (.4, 0) (bottom left) or (X1 , X2 ) = (.4, .2) (bottom right), then the probability is about .75 and about .6 respectively. Since the probability increased for all cases when there were two ongoing auctions, this ‘confirms’ that there is a cannibalization effect and that the CDF is increasing in number of ongoing auctions. The final Assumption 2.3, which states that the cannibalization effect is diminishing in price, is a bit more difficult to visually confirm. Again we compare, for a given initial price Xi,ti = xi and future price Xi,ti +∆t , i = 1, 2, the value of the CDF in the top left graph with the corresponding value in one of the other three graphs. For instance, suppose we are interested to compare the cannibalization effect when Xi,ti = $.2 and Xi,ti = $.4. If there is only one auction then the probability that Xi,ti +∆t ≤ $.6, is about .62 and .59 when Xi,ti = $.2 and $.4 respectively. When there are two simultaneously released auctions the probabilities have increased to about .82 and .74 for Xi,ti = $.2 and $.4 respectively. We note that the difference in CDF at Xi,ti = $.4, about .15 (=.74-.59), is less than the difference in CDF at Xi,ti = $.2, about .2 (=.82-.62). A similar comparison can be made with the bottom graphs to the top left graph as well. This indicates that the cannibalization effect is diminishing in price, and provides support for Assumption 2.3 Next we discuss the bidding strategy when bidders bid truthfully in the lowest priced auction. 128 4. Truthful Bidding Strategy A potential implication of the previous bidding strategy is that each bidder may end up placing several small bids and eventually still be outbid. Though we do not consider the effort or time it may take to submit a bid, in reality this might be a very tiresome and frustrating strategy. As an alternative to the previous strategy we consider when bidders simply bid their valuation in the lowest priced auction, i.e. truthfully bid their valuation. That is, the bidding behavior is as follows, (1) A bidder with valuation V = v arrives. (2) He observes the price in all ongoing auctions. (3) If the lowest priced auction is below his valuation (v > min{X1 , X2 }), then he bids in the lowest priced auction his valuation (B = v). If the auctions are priced equally, then with probability .5 he chooses one. (a) If he is successful and becomes the high-bidder, then he continuously observe how the auction progress and if he is ever outbid returns to step (2). (b) If he is not successful and his bid does not make him the high-bidder, then he returns to step (2). (4) If the lowest priced auction is above his valuation (v ≤ min{X1 , X2 }), then he leaves the auction site and never returns. Similar to Section 3 we assume placing a bid takes no time. That is, no time elapses from a bidder arriving until he either is registered as a high-bidder or leaves the auction site. When there is only one ongoing auction, and no speculation regarding additional auctions released, this leads to an equilibrium outcome as originally shown by Vickrey (1961). Furthermore, this bidding strategy is consistent with what eBay promotes bidders to do, as seen in the following quote, 129 ”When you place a bid, we suggest that you enter the maximum amount that you’re willing to pay for the item. (You won’t necessarily pay the amount of your maximum bid.) eBay compares your bid to those of other bidders and increases your bid on your behalf using only as much of your bid as is necessary to maintain your high bid position.”2 However, it is fairly clear why this is not an optimal bidding strategy if there are more than two auctions underway. The issue that can arise, as described below and unlike the previous bidding strategy, is that the bidders with the two highest valuations bid against each other. This results in that the bidder with the highest valuation has to pay a price equal to the second highest valuation, while the bidder with the second highest valuation only has to pay a price equal to the third highest valuation. Therefore, a bidder with a high valuation, who considers the chance that he has the highest valuation to be sufficient, will have an incentive not to bid truthfully. Instead, he should attempt to keep at par with the bidder with the second highest valuation, and make sure he does not end up in a bidding war with him. That is, he would be better of by following the minimal bid-increment strategy. Numerical examples will illustrate possible sequence of events. Suppose that at time t, two auctions are underway, the system state is [.25, .5, 2; .1, .25, 1], and a bidder with valuation V = v arrives. Recall that the high-bids are censored and that the bidder will always participate in the lowest priced auction. There are then three possible transitions depending on the value of v. [.25, .5, 2; .1, .25, 1] v ≤ .1 [.25, .5, 2; .1, .25, 1] =⇒ [.25, .5, 2; v, .25, 1] [.25, .5, 2; .25, v, 1] 2http://pages.ebay.com/help/buy/buying-ov.html 130 .1 < v ≤ .25 .25 ≤ v In the first case, when v ≤ .1, the bidder simply leaves the auctions and the state of the auctions remain the same. In the second case, when .1 < v ≤ .25, the bidder places a bid v in auction 2 and is immediately outbid and leaves the auction site. And in the third case, when .25 < v, the bidder places a bid v in auction 2 and becomes the high-bidder, while the previous high-bidder whose previous bid was .25 leaves the auctions. Suppose instead the system state is [.1, .5, 2; .1, .25, 1], and again a bidder with valuation V = v arrives. There are now six possible transitions depending on v. [.1, .5, 2; .1, .25, 1] [v, .5, 2; v, .25, 1] [v, .5, 2; .25, v, 1] [.1, .5, 2; .1, .25, 2] =⇒ v ≤ .1 .1 ≤ v ≤ .25 w. P r. = .5 [.25, .5, 2; .25, v, 1] w. P r. = .5 [.5, v, 2; .25, .5, 1] w. P r. = .5 [.25, .5, 2; .25, v, 1] w. P r. = .5 .25 ≤ v ≤ .5 .5 ≤ v In the first case the bidder simply leaves the auction and the state of the auctions remain the same. In the second case the bidder places a bid v first in one auction, and upon being outbid tries the other auction and then leaves (the order in which he bids in the auctions does not matter). In the third and fourth case, when .25 ≤ v ≤ .5, the bidder will with probability .5 choose one auction. If he chooses auction 1 first then he is immediately outbid and places a bid in auction 2 where he becomes the high-bidder. If he chooses auction 2 then he outbids the current high-bidder, who then tries to place a bid in auction 1. And in the fifth and sixth case, when .5 ≤ v, the same logic applies. The bidder will choose one auction with equal probability and therefore either end up as the high-bidder 131 in auction 1 or auction 2, and the high-bidder he replaces will try to bid in the other auction. In order to derive the CDF of Xi,ti +∆t , i = 1, 2, at the end of the time interval [t, t+∆t], the same approach as in the previous bid strategy is employed. That is, first we condition on the valuation of the high-bidder in each auction, and then on the number of bidders that arrive in [t, t + ∆t]. Based on this conditional information we derive the probability that Xi,ti +∆t ≤ q, for a given threshold q, i = 1, 2. However, due to that it is possible the bidder with the second highest valuation bids against the bidder with the highest valuation, there are situations in which the price is different in the two auctions. Therefore the calculations of the CDF are a bit more complicated. In Appendix C we outline how upper and lower bounds on the CDF can be derived for the case when there is one respectively two simultaneously released auctions. The case for sequentially released auctions further complicate the price-transitions as described in Appendix C. The underlying idea with the calculations in Appendix C is that given an initial state [X1 , H1 , t1 ; X2 , H2 , t2 ], by conditioning on the value of H1 , H2 , and number of arriving bidders, for a given threshold q, upper and lower bounds on the probability that Xi,ti +∆t ≤ q can be given. We illustrate the resulting CDF for the one and two simultaneously started auctions with a numerical example. 4.1. Numerical example. Similar to the numerical example in Section 3.3, let bidders valuation be uniformly distributed on [.01, 1], and the number of arriving bidders in the interval [t, t + ∆t], be uniform distributed over 0, 1, 2, 3, and 4, i.e. ρM (m|∆t) = .2 for m = 0, 1, 2, 3, 4, and ρM (m|∆t) = 0 for m ≥ 5. The value of ∆t is irrelevant, but can be thought of as, for instance, 24 hours. The resulting within period price-transition CDF, for different starting states of one respectively two simultaneously started auctions, are shown in Figure 3.5. The upper left graph shows the CDF when there is only one ongoing auction and is identical to the graph for the minimum bid-increment strategy. See Section 3.3 for 132 more comments. The upper right graph shows the CDF when the auctions were started simultaneously, and the prices are identical as labeled, i.e. when t1 = t2 and (X1 , X2 ) = (.01,.01), (.2,.2), (.4,.4), (.6,.6), (.8,.8). The upper solid line represents the upper bound, and the lower dashed line represents the lower bounds. For instance, if (X1 , X2 )=(.2,.2), then the probability that Xi,ti +∆t ≤ $.4, has an upper bound of about .55, and lower bound of about .48. A property with the auctions when they are simultaneously released and priced equally, is that the price-transitions are Markovian. Specifically, no additional information regarding which auction (auction 1 or 2) has the high-bid corresponding to the highest valuation V(1) , is gained by keeping track of the auctions for each arriving bidder. This is summarized in Lemma C.1 of Appendix C. The two bottom graphs are the CDF for each individual auction when the auctions were started simultaneously but X2 < X1 . The bottom left graph is for the CDF of auction 1, F12 (q|x1 ), when X1,t = 0, .01, .2, .4, .6, .8. Note that the bounds do not depend on the price of auction 2, only that it is priced below auction 1. For instance, suppose X1,t1 = .2, then the probability that Xi,ti +∆t ≤ $.4, is minimum .45 (lower bound), and maximum .65. A distinctive feature of the graphs is that for X1,t1 > .1, the lower and upper bounds start apart. This is to be expected, and can be confirmed by the calculations shown in Appendix C. The bottom right graph is for the CDF of auction 2, F12 (q|x2 ), when (X1 , X2 ) = (0,0), (.1,0), (.2,.3), (.5,.4), (.7,.6), (.9,.8). Note that in this case the price of auction 1 does effect the CDF, and that the lower and upper bounds for the probability that X2,t2 +∆t ≤ x1 , coincide. Another interesting feature is that, unlike the case of auction 1, the bounds for 133 X2,t2 +∆t ≥ x1 , are much closer. That is, there is less difference between the upper and lower bounds for the CDF of auction 2, than there is for the CDF of auction 1. The reason for this is that the only difference in between the upper and lower bounds are for probabilities of X2,t2 +∆t ≥ H1,t1 . Which can be seen in the calculations shown in Appendix C. To illustrate, suppose (X1 , X2 )=(.3,.2), then the upper and lower bound for the probability that X2,t2 +∆t ≤ .4, is about .64 and .59 respectively. Although the figures only display the bounds, similar to Figure 3.4, Assumptions 2.1, 2.2, and 2.3 from Chapter 2 can be visually assessed. For instance, we see that in all graphs and for a given bound, the CDF associated with a higher initial price is always beneath the CDF for a lower initial price. The assumption regarding the cannibalization also seem to hold. For instance, suppose X1,t1 = .2, and we are interested in the probability that X1,t1 +∆ ≤ .4. When there is only one ongoing auction (top left), the probability is about .4. If the two auctions were started simultaneously and are priced equally (top right), then the upper bound is about .55, and lower bound about .48. While if the two auctions were started simultaneously but auction 2 is priced below auction 1 (bottom left), then the upper and lower bounds are about .65 and .45 respectively. Though this does not formally establish Assumption 2.2, it does provide some support that it would hold. The final Assumption 2.3 is visually assessed same as with the previous bidding strategy. Suppose we are interested in comparing the cannibalization effect between X1,t1 = $.2 and $.4. If there is only one ongoing auction, then the probability that X1,t1 +∆t ≤ $.6, is about .62 and .59, for X1,t1 = $.2 and $.4 respectively. If the two auctions were started simultaneously and are priced equally, then the probability that X1,t1 +∆t ≤ $.6, has an upper/lower bound of about .79/.69 and .74/.64 for X1,t1 = $.2 respectively $.4. In other words, the upper/lower difference between having one or two ongoing auctions is about .17/.07 when 134 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 X=0 0.3 0.3 0.2 X = .2 0.1 0 0 0.1 0.2 X = .4 0.3 0.4 X = .6 0.5 Xy+∆ t 0.6 X = .8 0.7 0.8 0.9 1 0 0 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 X1=0 0.3 0.3 0.2 0.2 X1=.2 0.1 0.4 0.5 Xi,y+∆ t 0.6 0.7 0.8 0.9 1 X1=0 0.4 X1=.01 0 0.3 Xi=.8 X =0 0.5 0.1 0.2 Xi=.6 2 0.5 0.4 0.1 Xi=.4 CDF for Two Simultaneous Auctions, X1 = X2 1 0 Xi=.2 0.1 CDF for One Auction 0.6 Xi=.01 0.2 X = .01 0.2 X1=.4 0.3 0.4 0.5 X1,y+∆ t X1=.6 0.6 X1=.8 0.7 0.8 X2=0 X1=.01 0.1 0.9 1 CDF for Auction 1, t2 = t1 , X2 < X1 0 0 0.1 X2=.2 X2=.4 X2=.6 X2=.8 X1=.3 X1=.5 X1=.7 X1=.9 0.2 0.3 0.4 0.5 X2,y+∆ t 0.6 0.7 0.8 0.9 CDF for Auction 2, t2 = t1 , X2 < X1 Figure 3.5. The conditional CDF of the within period price-transitions when bidders valuation is uniform on [.01, 1], the number of arriving bidders is uniform between 0 and 4, and each bidder follows the truthful bidding strategy. The top left graph is when there is only one auction (Z∆t = 1), and the top right when there are two simultaneously started auctions and the prices are the same. The bottom graphs are for auction 1 (left) and auction 2 (right), when the auctions were started simultaneously but the prices are different. In each of the graph when there are two ongoing auctions, the solid lines indicates the upper bounds and the dashed lines the lower bounds. X1,t1 = $.2, and about .15/.05 when X1,t1 = $.4. That is, the cannibalization seems to be diminishing in the price of an auction. 135 1 5. Discussion This chapter has provided a discussion on how the conditional distribution function for the within period price-transitions can be computed. Although the focus has been with regard to two specific bidding strategies, the same methodology could be applied to other bidding strategies. Depending on the complexity of the auction rules, bidder attributes, and other assumptions, the within period price transitions might be challenging to evaluate. In some instance, like the minimal bid increment strategy, closed form solutions can be derived. In other cases, like the truthful bidding strategy, bounds can be evaluated. If the price transitions are much more complicated then a possible solutions is to estimate the conditional distribution function using simulation. A comment regarding the truthful bidding strategy, is that it would have been possible to derive the exact CDF, by for instance evaluating the possible sample paths. That is, given an initial starting price Xi,ti , number of bidders that arrive m, and threshold q, one could generate the paths for which Xi,ti +∆t ends above or below q. From this one could first calculate the probability of each path, and then determine the conditional distribution function given all possible paths. Though this might be a time consuming procedure (since it is exponentially growing in m), it is feasible and provides the exact conditional distribution function. The reason we did not show this was to illustrate how bounds can be evaluated. The main objective of this chapter has been to show how the conditional distribution function for the within period price transitions can be derived based on the individual bidding behavior. The benefit with this is that it provides a mean for a seller to derive the CDF, based on particular bidding behavior, and verify if the assumptions from Chapter 2 holds. In other words, a seller that, rather than assume a particular distribution function 136 of the within period price-transitions, prefers to assume certain aspects of the individual bidding behavior, can still verify if a threshold type policy is optimal based on the framework discussed in this chapter. In addition, a seller can use the methodology presented to derive the distribution function of the final price. That is, if the length of the time-interval ∆t coincides with the remaining time, then the conditional CDF of the final price is derived. Therefore, the chapter has also provided a method for deriving the conditional distribution function of the final price for an auction that has elapsed for some time t and is currently at a price level Xt . This information can then be used by a seller or buyer to make a better informed decision. Although the two bidding strategies may, under certain conditions, result in an equilibrium outcome, the purpose has not been to analyze the equilibrium outcome. The bid strategic equilibrium analysis for auctions that are not started simultaneously is, to the best of our knowledge, still an open research topic and source for possible extensions to this chapter. In Chapter 5 we conduct an empirical analysis on how bidders actually behave. 137 CHAPTER 4 Empirical Analysis of Within Period Price-Increments 1. Introduction In Chapter 2 a model for analyzing the optimal release of inventory for online auctions was discussed. The problem was formulated as a discrete time Markov Decision Process, where each period the seller has to decide whether or not to start a new auction. The framework of the analysis was that auctions evolve according to a stochastic process, and given certain properties the optimal release policy is of a threshold type. It might be natural to ask how a seller could go about and determine the process by which auctions evolve, and how to determine if the conditions in Chapter 2 holds. The objective of this chapter is three-fold. First, to present a statistical model that describes how auction prices progress. Second, to provide some structural properties on the statistical model such that the results of Chapter 2 hold. And thirdly, to empirically test and validate the proposed model based on the auction data from Dell Financial Services (DFS). Overview of Chapter 4. This chapter is organized as follows. In Section 2 a statistical model for the within period price-increments is discussed. Section 3 provides some sufficient conditions under which the structural results of Chapter 2 hold. Section 4 gives an overview of Generalized Linear Models. Section 5 and 6 discuss the specific model formulation and the results of the empirical analysis pertaining to the DFS data. Section 7 concludes the chapter with an overall discussion. 138 2. Zero-Inflated Gamma Distributed Price-Increments Let XY define the price of an auction that has elapsed Y periods, where XY is assumed continuous and non-negative and Y discrete and finite; XY ≥ 0, Y = {0, 1, . . . , τ }. That is, each auction is divided into τ periods of equal length, and at the end of each period the price of the auction is observed. Note that there is a direct relationship between Y , the number of elapsed periods, and the period number. The period between Y and Y + 1 is defined as period Y , i.e. the first period is defined as period 0, the second period as period 1, and so on. Unlike the previous chapter, were we kept track of both items waiting to be released for auctions as well as ongoing auctions, we now only keep track of ongoing auctions. Which means that Y = 0 implies the auction in question has been released. Define ZY to be the discrete number of ongoing auctions after Y periods, ZY = {0, 1, 2, . . . }. More specifically, an auction that has elapsed Y < τ periods will ‘compete’ with ZY − 1 additional auctions in the upcoming period Y . Since an auction is over after τ periods, and hence the number of ongoing auctions irrelevant, we define Zτ ≡ 0. The state of an auction is defined by the three variables, S ≡ (X, Y, Z). A time-line for a three-period auction is provided in Figure 4.1. We define CY as the price-increment in period Y , CY = XY +1 − XY , which we assume follows a ‘zero-inflated gamma distribution’. That is, the within period price-increments X0 , Z0 Y =0 C0 X1 , Z1 period 0 Y = 1 C1 X2 , Z2 period 1 Y = 2 C2 X3 , Z3 period 2 Y = 3 Figure 4.1. Time-line for an auction with 3 periods. The auction starts at Y = 0 and ends three periods later at Y = 3. 139 have the following density function, (4.1) fCY |S (c|s) = 1 − πs πs 1 Γ(νs ) c=0 νs µs νs cνs −1 e − cνs µs c>0 The above density indicates that with probability 1 − πs , the within period priceincrement is zero, i.e. no bids arrived and the price did not change. While if a positive price-transition occurs, then the price-increment is gamma distributed with mean µs and shape parameter νs . Note that the gamma distribution is a member of the exponential family, and that πs , µs , and νs are functions of the auction state. More specifically we assume that transformations of πs and µs are linear functions of the auction state as follows, for s = (x, y, z), (4.2) ln πs 1 − πs βb0 + βb2 z = (4.3) y=0 βm0 + βm1 x + βm2 z + βm3 x × z + βm4 1{y=2} + · · · + βmτ +1 1{y=τ −2} βe0 + βe1 x + βe2 z + βe3 x × z γb0 + γb2 z ln(µs ) = y = 0, τ − 1 y =τ −1 y=0 γm0 + γm1 x + γm2 z + γm3 x × z + γm4 1{y=2} + · · · + γmτ +1 1{y=τ −2} γe0 + γe1 x + γe2 z + γe3 x × z y = 0, τ − 1 s=τ −1 The symbol 1{·} represents the indicator function, and is 1 if the argument in the brackets is true and 0 otherwise. The gamma shape parameter νs is assumed to only depend on the period of the auction as follows, (4.4) νs = νb 1{y=0} + νm 1{y=0,τ −1} + νe 1{y=τ −1} 140 The three cases and subscripts on the coefficients refer to the beginning (Y = 0), middle (Y = 1, 2, . . . , τ − 2), and end (Y = τ − 1) of an auction. The logit function (4.2) indicates that the log-odds of observing a positive price-increment is linear with respect to XY , ZY , and the interaction between XY and ZY . For the middle periods we also assume there is an additive effect depending on the period Y . Similarly, equation (4.3) means that the log of the average positive price-increment is linear with respect to the listed covariates. The reason we have separated three cases for each function is that we anticipate different dynamics for the three stages. For the first period this should be evident as all auctions start with X0 = 0. That the final period might be different is due to the well-established observations that there is dramatically more bidding activity toward the end of an auction than during [24, 28]. 2.1. Some Comments Regarding The Model. An immediate question one may have is why use the density function (4.1), and why base the parameters on the functions (4.2) and (4.3). The main attribute of the within period price-increments that we seek to model is that there is a ‘high’ chance of observing a ‘low’ price-increment, including a positive probability of a zero price-increment, and a ‘low’ chance of observing a high price-increment. This would seem a plausible assumption and supported by data shown in Figure 4.7, 4.8, and 4.9. In the graphs, with perhaps an exception for the final periods, most increments tend to be ‘small’ but there are also some observations with ‘large ’increments. Therefore, the general characteristics of (4.1) seem to be supported by the data. However, the formal statistical analysis will reveal if (4.1) indeed is appropriate. Note that, if ν=1 then the positive price-increments are exponentially distributed and the model simplifies considerably. 141 The second part of the model regarding the linear functions, called link functions, defines how the parameters relate to the covariates of interest. The logit function for πs and log of µs are standard choices in Generalized Linear Models (GLM) and are readily available in most statistics packages. The main benefits of using the two functions, are ease of interpretation and the restriction of the range of πs and µs to the interval [0, 1], respectively the non-negative real numbers. Some more details including a general overview of GLMs are discussed in Section 4. The conditional expected within period price-increment, for s = (x, y, z), is given by, (4.5) E[CY |S = s] = πs µs From equations (4.2) and (4.3), we can solve for πs and µs . To simplify the notation, we assume a given elapsed auction time Y = y, and write βj (γj ) instead of βbj , βmj , or βej (γbj , γmj , γej ), j > 0. In addition, we ignore the indicator functions 1{y=2} , . . . , 1{y=τ −2} , and define β0 (γ0 ) as a generic intercept, e.g. β0 = βm0 + βm4 . Consequently, πs and µs can be written as follows, for s = (x, y, z), eβ0 +β1 x+β2 z+β3 x×z eβx = 1 + eβ0 +β1 x+β2 z+β3 x×z 1 + eβx (4.6) πs = (4.7) µs = eγ0 +γ1 x+γ2 z+γ3 x×z = eγx Note that for y = 0, β1 , β3 , γ1 , γ3 = 0. Stern and Coe, 1984, fit a similar model to the analysis of daily rainfall. In their model they divide the analysis into two parts. In the first part they model the likelihood of a day having rain as a Bernoulli variable, i.e. a day is either wet or dry. While in the second part they model the amount of rainfall on wet days as being gamma distributed. Though their linear predictors are based on a harmonic series depending on previous days’ rainfall the 142 general link functions are the same. They use the logit function for the probability of observing rain/no-rain, and the log of the expected rainfall for the rate of rainfall. The paper includes a numerical illustration based on daily rainfall data from Morogoro, Tanzania. A summary of the paper appears in Section 8.4 of McCullagh and Nelder, 1989. 2.2. Maximum Likelihood Estimation. We use maximum likelihood to estimate the coefficients of the two link functions (4.2) and (4.3). The likelihood of (4.1) given n independent price-transitions, ordered such that the first k had a positive price-increment (c > 0), and the remaining n − k had a zero price-increment (c = 0) is, k L= πsi i=1 1 Γ(νsi ) νsi µsi νsi νsi −1 − ci e n ci νsi µsi (1 − πsj ) j=k+1 The log-likelihood is then, (4.8) k ln(L) = i=1 ln(1 − πsj ) + νsi µsi νsi − ln(Γ(νsi )) + νsi ln ln(1 − πsj ) + j=k+1 1 Γ(νsi ) k n ln(πsi ) + i=1 ln i=1 j=k+1 k = k n ln(πsi ) + i=1 νsi −1 − ci νsi µsi e ci νsi µsi + (νsi − 1) ln(ci ) − Note that maximizing (4.8) is identical to separately maximizing a likelihood corresponding to n independent Bernoulli distributed random variables with Pr{success} = πsi and Pr{failure} = 1 − πsi , with k ‘success’ (n − k ‘failure’), and k independent gamma distributed random variables with mean µ and shape parameter ν. We formally summarize this in the next lemma. Lemma 4.1. The maximization of the log-likelihood function (4.8) is identical to separately maximizing the log-likelihood of n independent Bernoulli distributed random variables with Pr{success} = πsi , with k ‘success’ and n − k ‘failure’, and k independent gamma distributed random variables with parameters µsi and νsi . 143 ci νsi µsi Proof Lemma 4.1 - The likelihood of n independent Bernoulli distributed random variables, Oi = 0, 1, i = 1, . . . , n, with Pr{success} = πsi , ordered such that the first k observations were ‘success’ and the remaining n − k observations were ‘failure’, is given by, k i=1 πsi LB = n j=k+1 ln(1 n j=k+1 (1 − πsj ). The log-likelihood is therefore, ln(LB ) = k i=1 ln(πsi ) + − πsj ). The likelihood of k independent gamma distributed random variables, Ci > 0, i = 1, . . . , k, with mean µsi and shape parameter νsi is given by, LE = k νsi νsi −1 exp(−ci νsi /µsi ). i=1 (1/Γ(νsi )(νsi /µsi ) ci k i=1 ln(1/Γ(νsi )) The log-likelihood is therefore ln(LE ) = + νsi ln(νsi /µsi ) + (νsi − 1) ln(ci ) − ci νsi /µsi . If all n observations were taken during the same stage, i.e. beginning, middle, or end, then the same instance of (4.2), (4.3), and gamma shape parameter νs applies to all observations, and (4.8) becomes, ln(L) k = n k ln(πsi ) + i=1 ln i=1 eβxi 1 + eβxi = βxi − i=1 βxj − ln(1 + e ) + k(ν ln(ν) − ln(Γ(ν))) − ν k ln(1 + e γxi − i=1 j=k+1 βxj ν−1 ln(ci ) + ci /µsi ν k n n k ln(µsi ) − i=1 j=k+1 k = ln(1 − πsj ) + k(ν ln(ν) − ln(Γ(ν))) − ν ) + k(ν ln(ν) − ln(Γ(ν))) − ν j=1 βxi − i=1 ν−1 ln(ci ) + ci e−γxi ν ν−1 ln(ci ) + ci e−βxi ν where βxi and γxi are the linear predictors of observation i’s vector of covariates. The objective therefore becomes to find the β and γ coefficients that maximizes the above expression. As a consequence of Lemma 4.1, we can estimate the β and γ coefficients using the framework of GLM in two steps. In the first step we create a new random variable as 144 follows, Oi = 0 ci = 0 i = 1, 2, . . . , n 1 ci > 0 and then use a standard GLM algorithm to fit the β coefficients such that 1 − πs o = 0 (4.9) Pr{O = o|S = s} = π o=1 s and (4.2) holds. In the second step we limit the analysis to the auctions for which a positive price-increment occurred. In other words, we only consider the k auctions for which a positive price-increment occurred and use a GLM algorithm to fit the γ coefficients to the density function gC|S (c|s) = (1/Γ(νs ))(νs /µs )νs cνs −1 exp(−cνs /µs ) such that (4.3) holds. Section 4 provides a discussion and references for the most common algorithm to estimate β and γ. Note that, due to Lemma 4.1, the proposed model with zero-inflated gamma distributed within period price-increments, is almost identical to the rainfall model proposed by Stern and Coe, 1984. Next we discuss what properties on the β and γ coefficients will support the main results from the Chapter 2. 3. Structural Properties For Optimal Auction Release Policy In Chapter 2 we presented a model for the optimal release of inventory for online auctions. The main results, that the value function is increasing and that a threshold-type release policy is optimal, relied on three assumptions regarding the cumulative distribution function of the within period price-transitions. The assumptions were: 1) ‘Monotonicity in price ’, meaning that the probability of observing jumps to high prices is increasing in the current price of an auction, 2) ‘Cannibalization effect’, defined to be that for a specific auction, the more ongoing auctions there are, the less likely you are to observe price-transitions 145 to high prices, and 3) ‘Diminishing cannibalization’, meaning that the gain for a specific auction, of having fewer ongoing auctions, is decreasing in the current price of the auction. The objective of this section is to provide structural properties on the β and γ coefficients under which the assumptions and consequently the results of Chapter 2 holds. If the within period price-increment follow (4.1), then the cumulative distribution function of the within period price-transition is given by, for s = (x, z, y), c ≥ 0, c (4.10) FXY +1 |S (x + c|s) = 1 − πs + πs = 1 − πs 1− 0 1 Γ(ν) 1 Γ(ν) ν µs cν/µs ν uν−1 e−(uν/µs ) du uν−1 e−u du 0 = 1 − πs (1 − GC|S (c|s)) where, ln πs 1 − πs = β0 + β1 x + β2 z + β3 x × z ln(µs ) = γ0 + γ1 x + γ2 z + γ3 x × z and GC|S (c|s) is the cumulative distribution of positive price-increments. The integral term in the second equality of (4.10) is the lower incomplete gamma function.1 For simplicity of notation, the above equations are, similar to (4.6) and (4.7) in Section 2.1, based on a given period Y = y, and written with respect to generic βj and γj coefficients. Furthermore, the gamma shape parameter ν in (4.10), has for simplicity of notation, been written without a subscript s. Implicitly, however, we assume that ν depends on the state of the auction s according to (4.4). 1cf. http://mathworld.wolfram.com/IncompleteGammaFunction.html 146 Though the assumptions in Chapter 2 were stated with respect to the distribution function of the within period price-transitions, in this section we focus on the conditional expected within period price-transition E[XY +1 |S], and provide structural conditions on the β and γ coefficients to ensure various monotone properties. The reason for this is two fold. First, it provides a more intuitive discussion by focusing on the effect on the expected or average price-transition, rather than the effect of the distribution function. Second, it simplifies the analysis, as might be evident by (4.10). The conditional expected within period price-transition is given by, for s = (x, y, z), (4.11) E[XY +1 |S = s] = x + πs µs = x + eβx γx e 1 + eβx where, ln πs 1 − πs = β0 + β1 x + β2 z + β3 x × z ln(µs ) = γ0 + γ1 x + γ2 z + γ3 x × z To summarize, the objective with this section is to provide sufficient conditions such that (4.11) is increasing in x and decreasing in z. Next we discuss each of the three main assumptions in detail. 3.1. Monotonicity in price. The first assumption was that the probability of observing price-transitions to the ‘high’ prices is increasing in the current price. That is, we assumed, for s = (x, y, z) and c ≥ 0, Pr{XY +1 ≤ c + x|S = s} is decreasing in x. An implication of this assumption is that, for s = (x, y, z), E[XY +1 |S = s] is increasing in x. As mentioned earlier, rather than providing structural properties such that (4.10) is decreasing in x, we focus on the latter consequence and provide sufficient conditions such that E[XY +1 |S = s] is increasing in x. One approach to characterize the conditions under 147 which this holds, is to analyze the derivative of (4.11) with respect to x. A sufficient condition for E[XY +1 |S = s] to be increasing in x, is ∂ ∂x E[XY +1 |S = s] ≥ 0. The derivative of E[XY +1 |S = s] with respect to x is given by, (4.12) ∂ ∂ E[XY +1 |S = s] = x+ ∂x ∂x =1+ ∂ πs µs + πs ∂x ∂ µs ∂x (β1 + β3 z)eβx (1 + eβx − eβx ) µs + πs (γ1 + γ3 z)eγx (1 + eβx )2 = 1 + (β1 + β3 z)πs (1 − πs )µs + (γ1 + γ3 z)πs µs where the last equality holds because 1 − πs = 1/(1 + eβx ). Since µs , πs , (1 − πs ) ≥ 0, it is clear that if either β1 or γ1 are negative then (4.12) is not necessarily non-negative. Note that β1 < 0 implies at higher price you are less likely to see positive price-transition, and that γ1 < 0 implies that the expected price-increment is smaller the higher the current price. Since it is reasonable to assume that β1 and γ1 are negative, some structural properties to ensure that (4.12) remains non-negative will be provided. This is sufficient for the value function in Chapter 2 to be increasing in x, and consequently for Proposition 2.11 to hold. Note that β1 , γ1 < 0 or convexity in x is neither sufficient nor necessary for E[XY +1 |S] to be increasing in x as illustrated in the left graph of Figure 4.2. In Figure 4.2, the left graph provides four examples of E[XY +1 |S] as a function of x for various combinations of negative β1 and γ1 . The solid lines ‘A’ and ‘B’ are examples for which E[XY +1 |S] is increasing in x. The dashed lines ‘C’ and ‘D’ are examples for which E[XY +1 |S] is not monotone in x. All four examples are based on β and γ parameter settings which are close to the values observed in the empirical analysis later on. Specifically we used, πs = e2+β1 x−.01z 1 + e2+β1 x−.01z µs = e5+γ1 x−.01z 148 0.000 200 γ1 C D 0 0 50 100 150 −0.025 −0.020 −0.015 −0.010 −0.005 150 100 B 50 E[XY+1 | XY=x] A C B D −0.06 200 A −0.05 −0.04 −0.03 −0.02 −0.01 β1 x Figure 4.2. The graph to the left illustrates E[XY +1 |S = s] as a function of x. Each example is based on β0 = 2, β2 = −.01, β3 = 0, γ0 = 5, γ2 = −.01, γ3 = 0, and z = 1, but different β1 , γ1 as follows: (A) β1 = −.025, γ1 = −.001, (B) β1 = −.01, γ1 = −.006, (C) β1 = −.05, γ1 = −.001, (D) β1 = −.05, γ1 = −.02. The solid lines A and B are increasing in x, while the dashed lines C and D are not monotone in x. Lines B and D are convex in x, while A and C are neither convex nor concave. The graph to the right displays the coordinates of (β1 , γ1 ) for the four examples. The triangular region, enclosed by the dashed lines, represents the sufficient conditions for E[XY +1 |S = s] to be increasing in x, as specified by the second set of conditions in Lemma 4.2 and the values of β0 , β2 , γ0 , γ2 . The coordinates for A and B are slightly outside the region, which shows that the conditions are not necessary. The coordinates for C and D are, however, too far from the bounded region. with z = 1, and β1 and γ1 as follows, A β1 -.025 B C D -.01 -.05 -.05 γ1 -.001 -.006 -.001 -.02 The following lemma gives some sufficient conditions for which consequently E[XY +1 |S] is increasing in x. 149 ∂ ∂x E[XY +1 |S] ≥ 0, and 0.00 Lemma 4.2. Define π0 ≡ exp(β0 + β2 )/(1 + exp(β0 + β2 )) and µ0 ≡ exp(γ0 + γ2 ). For s = (x, y, z), sufficient conditions for E[XY +1 |S] given by (4.11) to be increasing in x include, (1) β1 , β3 , γ1 , γ3 ≥ 0 (2) (a) β2 , γ2 ≤ 0 (b) β3 , γ3 ≥ 0 (c) − µ40 − β3 z ≤ β1 < 0 (d) − π01µ0 − β1 +β3 z 4π0 − γ3 z ≤ γ1 < 0 1 −γ1 (e) z ≤ min{ −β β3 , γ3 } (3) (a) β2 , γ2 ≤ 0 (b) β3 , γ3 < 0 (c) − µ40 − β3 z ≤ β1 < 0 (d) − π01µ0 − β1 +β3 z 4π0 − γ3 z ≤ γ1 < 0 (e) z ≤ min{ ββ13 , γγ13 } Proof Lemma 4.2 - Since πs , µs ≥ 0, if β1 , β3 , γ1 , γ3 ≥ 0, then β1 + β3 z ≥ 0 and γ1 + γ3 z ≥ 0, and consequently, 1 + (β1 + β3 z)πs (1 − πs )µs + (γ1 + γ3 z)πs µs > 0. Therefore, if the first set of conditions holds then ∂ ∂x E[XY +1 |S] ≥ 0, where the derivative us given by (4.12), holds. The second set of conditions is the case when β1 , γ1 < 0 and β3 , γ3 ≥ 0. Condition (a) ensures that πs and µs are decreasing in z. Condition (e) ensures that β1 + β3 z, γ1 + γ3 z ≤ 0, and hence πs and µs are decreasing in x. Note that if 1+(β1 +β3 z).25µ0 +(γ1 +γ3 z)π0 µ0 ≥ 0, then ∂ ∂x E[XY +1 |S = s] ≥ 0 for all s. By substituting the lower bounds for β1 and γ1 , 1 + (β1 + β3 z).25µ0 + (γ1 + γ3 z)π0 µ0 = 0, and ∂ ∂x E[XY +1 |S] ≥ 0 holds. Note that if β3 , γ3 = 0 then the bounds on β1 and γ1 simplify and condition (e) holds vacuously (define min{−β1 /0, −γ1 /0} ≡ ∞). The third set of conditions is the case when β1 , β3 , γ1 , γ3 < 0. The last condition (e) ensures 150 that β1 + β3 z ≥ 2β1 and γ1 + γ3 z ≥ 2γ1 . In other words, the effect of an increase in x is at most two times β1 for πs and two times γ1 for µ2 . No changes to the previous proof is necessary, since the bound on β1 and γ1 adjusts according to β3 and γ3 . A few comments follow. • The first condition in Lemma 4.2 does not have much intuitive appeal, since it implies auction price-increments are increasing in x, i.e. the higher the price, the more likely to see a larger positive price-increment. • The lower bound on γ1 in the last two set of conditions states that, for given values of β0 , β2 , β3 , γ0 , γ2 , γ3 , the space defined by γ1 and β1 defines a ‘sufficient’ region such that E[XY +1 |S] is increasing in x. See the right graph of Figure 4.2. Recall that the stated conditions are only sufficient, and hence outside the defined area, E[XY +1 |S] may or may not be increasing in x. Furthermore, the bounds on γ1 are conservative, as it guarantees ∂ ∂x E[XY +1 |S] ≥ 0 in the worst case scenario when πs (1 − πs )µs = .25µ0 and πs µs = π0 µ0 (which only happens if π0 = .5). • Alternatively the lower bound on β1 can be stated in terms of γ1 , for β3 ≥ 0, β1 ≥ − 1 γ 1 π0 − − β3 z .25µ0 .25 • The intuition behind the bound of γ1 is as follows. A $1 increase in x cannot result in a decrease of µs by more than the maximum average price-increment π0 µ0 . That is, if γ1 is ‘small’ (and γ3 = 0) then exp(γ1 x) ≈ 1 + γ1 x and, E[XY +1 |S = s] = x + πs µs = x + πs eγ0 +γ2 z eγ1 x ≈ x + πs eγ0 +γ2 z (1 + γ1 x) ≤ x + π0 µ0 (1 + γ1 x) = x + π0 µ0 + π0 µ0 γ1 x Therefore, since γ1 < 0, unless γ1 > −1/π0 µ0 an increase in x will result in a decrease of E[XY +1 |S]. The lower bound on γ1 is then adjusted up by a factor of 151 .25/π0 given β1 . A similar though more convoluted argument leads to the lower bound of β1 , where .25 represents the maximum variance, πs (1 − πs ), of a positive price-increment. • If πs = 1 the condition on γ1 simplifies to the following, for γ3 ≥ 0, γ1 ≥ − 1 µ0 • The upper bound on z in the second condition ensures that πs and µs are decreasing in x. While the upper bound on z in the third condition limits the negative effect an increase in x has on πs and µs . • The condition β2 , γ2 ≤ 0 ensures that πs and µs are decreasing in z, but are not crucial for establishing lower bounds on β1 and γ1 . However, if β2 , γ2 > 0 then it would not seem reasonable that β3 , γ3 > 0, since it would imply that πs and µs are increasing in z. See Section 3.2. Therefore, if β2 , γ2 > 0 then β3 , γ3 ≤ 0, and a simple amendment is to replace π0 , µ0 with π0 , µ0 , where π0 ≡ exp(β0 + β2 z )/(1 + exp(β0 + β2 z )), µ0 ≡ exp(γ0 + γ2 z ), and z = min{β1 /β3 , γ1 /γ3 } • Since each coefficient can be positive, negative or zero, there are in total 81 possible combinations of β1 , β3 , γ1 , γ3 to consider. However, the main combinations of interest are covered in the three cases listed in Lemma 4.2. In addition, most combinations are not of interest or acceptable, e.g. cases where β1 , γ1 = 0 and β3 , γ3 = 0. • The conditions stated in Lemma 4.2 can be directly verified by data. See Section 5. 3.2. Cannibalization Effect. The second assumption imposed in Chapter 2 was that, for a given price, the seller is less likely to see transitions to the higher prices the more auctions are underway. That is, for s = (x, y, z) and c ≥ 0, Pr{XY +1 ≤ c + x|S = s} is increasing in z. This would imply that, for s = (x, y, z), E[XY +1 |S] is decreasing in z. 152 Similar to the discussion with regard to price, rather than analyzing (4.10), we focus on the conditional expected price-transition E[XY +1 |S] given by (4.11). Since z is discrete the equivalent to analyzing the derivative is to look at differences due to increments in z. The difference in E[XY +1 |S] due to an unit increase in z is given by, for s = (x, y, z), s+ = (x, y, z + 1), (4.13) E[XY +1 |S = (x, y, z)] − E[XY +1 |S = (x, y, z + 1)] = (x + πs µs ) − (x + πs+ µs+ ) = πs µs − πs+ µs+ where πs , πs+ are given by (4.6), and µs , µs+ are given by (4.7). Sufficient conditions for (4.13) to be positive, and hence E[XY +1 |S] decreasing in z, is β2 , γ2 < 0. In other words, the more ongoing auctions there are, the less likely a positive price-increment occurs and the lower the average positive price-increment. This is formally summarized in the next lemma. Lemma 4.3. Sufficient conditions for E[XY +1 |S = (x, y, z)] − E[XY +1 |S = (x, y, z + 1)] ≥ 0 to hold include, (1) β2 , γ2 < 0, β3 , γ3 ≤ 0 (2) β2 , γ2 < 0, β3 , γ3 > 0, and x ≤ min{−β2 /β3 , −γ2 /γ3 } Proof Lemma 4.3 - E[XY +1 |S = (x, y, z)] − E[XY +1 |S = (x, y, z + 1)] = πs µs − πs+ µs+ = µs (πs − πs+ eγ2 +γ3 x ) = µs eγ2 +γ3 x 1 − 1 + e−βx 1 + e−βx e−β2 −β3 x where exp(−βx) = exp(−(β0 + β1 x + β2 z + β3 x × z)). Therefore, if β2 , γ2 < 0, β3 , γ3 ≤ 0 then µs ≥ 0, eγ2 +γ3 x < 1, and e−βx e−β2 −β3 x > e−βx . Consequently, E[XY +1 |S = (x, y, z)] − E[XY +1 |S = (x, y, z + 1)] ≥ 0. 153 For the second set of conditions, the upper bound on x ensures that eγ2 +γ3 x < 1, and e−βx e−β2 −β3 x > e−βx , and the same proof as in the previous case holds. An alternative to the ‘strict’ cannibalization effect, would be that the cannibalization effect effects the auctions at ‘high prices’. That is E[XY +1 |S] is increasing in z for auctions below some price-level, and decreasing for auctions above the price-level. The following lemma summarizes this result. Lemma 4.4. If β2 , γ2 > 0, β3 , γ3 ≤ 0 and x ≤ min{−β2 /β3 , −γ2 /γ3 }, then E[XY +1 |S] is increasing in z, and if x ≥ max{−β2 /β3 , −γ2 /γ3 } then E[XY +1 |S] is decreasing in z. Proof Lemma 4.4 - The proof is based on the same logic as in the previous proof. If x is below the lower bound then eγ2 +γ3 x > 1, and e−βx e−β2 −β3 x < e−βx . And consequently, E[XY +1 |S = s] − E[XY +1 |S = s+ ] ≤ 0. While if x is above the upper bound then the same conditions as in the proof of Lemma 4.3 holds. A few comments follow. • The result in Lemma 4.4 can be extended to the following. There exists a pricethreshold pc ∈ [min{−β2 /β3 , −γ2 /γ3 }, max{−β2 /β3 , −γ2 /γ3 }], such that if β2 , γ2 < 0, β3 , γ3 > 0, and x ≤ pc , then E[XY +1 |S] is increasing in z, while for x > pc , E[XY +1 |S] is decreasing in z. • If β2 , γ2 , β3 , γ3 < 0, then it implies that the effect of cannibalization is increasing in both the number of auctions as well as the current price, i.e. having more auctions underway is much worse at a higher current price than at a lower current price. • Though there are other possible combinations of β2 , β3 , γ2 , γ3 , the three listed are the ones of most interest. Note that scenarios where β2 , γ2 = 0 (non-significant) and β3 , γ3 = 0 (significant) will not be considered in the ensuing data analysis. 154 • Recall that Z is the discrete number of ongoing auctions. However, in the ensuing data analysis we define Z as the average number of ongoing auctions. Which implies that Z is continuous. Consequently we may analyze the partial derivative with respect to z, ∂ E[XY +1 |S = (x, y, z)] = (β2 + β3 x)πs (1 − πs )µs + (γ2 + γ3 x)πs µs ∂z Note that Lemma 4.3 and 4.4 holds for z continuous. 3.3. Diminishing Cannibalization Effect in the Current Price. The third assumption in Chapter 2, which lead to the threshold policy of Theorem 2.12, is that the cannibalization effect is diminishing in the current price. That is the higher the current price, the less impact of cannibalization. This holds under the second set of conditions listed in Lemma 4.2 and the second set of conditions of Lemma 4.3. Note, that these conditions impose an upper bound on the current price. If, for instance, prices are not restricted, then there exist a price-level pu such that, if x < pu then E[XY +1 |S = (x, y, z)]−E[XY +1 |S = (x, y, z +1)] ≥ 0 and decreasing in x, while if x > pu then E[XY +1 |S = (x, y, z)]−E[XY +1 |S = (x, y, z+1)] < 0 and decreasing in x. That is higher priced auctions are better off with more ongoing auctions. Though this might seem a bit strange and counterintuitive, due to the decaying exponential shape of (4.6) and (4.7), this does not imply that high priced auctions are likely to see high price-increments. Contrariwise, the expected price-increments are still decreasing in price. See Figure 4.6 in Section ?? for examples. Another interesting scenario is the third set of conditions of Lemma 4.2 and the first set of conditions in Lemma 4.3, where β2 , β3 , γ2 , γ3 < 0. In this scenario the cannibalization effect is increasing. That is the exponential decline is steeper the more ongoing auctions there are. However, due to the shape of a negative exponential curve this cannot hold forever. And at some point the exponential curve flattens out and the cannibalization effect 155 becomes diminishing. In other words, the cannibalization effect is increasing in x up to a price-level pn , after which the cannibalization effect is decreasing in x. See Figure 4.6 on page 188 for an example. Another possible scenario is when the interaction terms are countering the main effects, i.e. the conditions specified in Lemma 4.4. This scenario does have intuitive appeal as it displays that the ‘cannibalization effect’ is working against the ‘high’ priced auctions and for the ‘low’ priced auctions. In other words, rather than a cannibalization effect, there is a price competition effect favoring the low priced auctions. However, this would presumably reflect that the lower priced auctions benefit from competition only in the presence of higher priced auctions. And not categorically from the fact that there are more auctions underway. In other words, a low price auction would presumably not benefit from having more ongoing auctions if they also were at low prices. In this scenario the cannibalization effect would be negative for x < pi , and positive for x > pi , i.e. the expected price-increments are higher (lower) with more auctions underway if x < pi (x > pi ). In addition, there would exist a price-level pii > pi such that for x < pii (x > pii ) the cannibalization effect is increasing (diminishing) in x. In Section 6 illustrations of the different cannibalization effects are provided in Figure 4.6. Next we provide an overview of generalized linear models (GLM). 4. Generalized Linear Models The statistical models for analyzing the within period price-transitions of online auctions, is based on the theory of Generalized Linear Models (GLM). As the name implies GLM is a generalization of the normal (Gaussian) linear regression models. Linear models, 156 which date back to the works by Gauss and Legendre, has been extensively developed over the last century, including the individual work by, for instance, Tukey, Fisher, and Cox [17, Ch.1],[9, p.56]. The term ‘generalized linear model’ was first introduced by Nelder and Wedderburn in 1972, who provided a unified theory and a general algorithm for computing the maximum likelihood estimates for a class of generalized linear models [17, p.19], [9, p.56]. A multivariate extension to GLM, labeled Exponential Dispersion Models (EDM), is proposed and discussed in Jørgensen (1987). Some of the underlying ideas of GLM and EDM, can be found in Tweedie (1947) [13, p.128, 145, 148]. One aspect of the ‘generalization’ is that instead of assuming a response variable Y to be normally distributed, we may assume it simply belongs to a distribution from the exponential family. That is, Y is derived from a distribution with a density that can be written as, fY (y|θ, φ) = exp yθ − b(θ) + c(y, φ) a(φ) The parameter θ is known as the canonical parameter and represents location, while φ is called the dispersion parameter and represents the shape. The functions a(·), b(·) and c(·) are specific to the distribution in question. Illustrations based on the binomial and gamma distributions are provided in Section 4.2 and 4.3 below. The exponential family distributions have mean µ = E[Y ] = ∂ ∂θ b(θ) 2 ∂ = b (θ), and variance var(Y ) = a(φ) ∂θ 2 b(θ) = a(φ)b (θ). It is interesting to note that the exponential family includes both continuous as well as discrete distributions. The most common are the normal, Poisson, binomial, gamma, and inverse Gaussian distribution. In addition to specifying the distribution of the response variable, GLM requires a link function, g(µ), that describes how the mean response variable, µ, relates to the vector of 157 known explanatory variables (covariates), x, through a linear predictor, η = β0 + β1 x1 + · · · + βp xp = βx The most common algorithm to fit the β coefficients for a GLM is based on the NewtonRaphson method with Fisher scoring. This numerical iterative optimization method is equivalent to iteratively weighted least squares [17, p.42], [30, p.185], [9, p.64], [8, p.117], [15, p.200]. The reason why a numerical approach is used is that in general there is no exact analytical expression that maximizes the log-likelihood for the members of the exponential family. One member that does have an exact solution is the Gaussian [8, p.117]. We omit the details but note that the estimated variance of the β coefficients estimates are given by, (4.14) ˆ = a(φ) ˆ XT W ˆX var( ˆ β) β −1 where X is the matrix of explanatory variables (covariates), Wβˆ is a diagonal matrix with the weights used in the final iteration, and φˆ is the estimate of the dispersion parameter ˆ se(β), ˆ are the square-roots of the diagonal entries of var( ˆ φ. The standard errors of β, ˆ β). To evaluate the statistical significance of a particular βˆ coefficient, we compare the ratio ˆ ˆ to the standard normal distribution (N (0, 1)) or use a likelihood ratio test. β/se( β) More details regarding the algorithm can be found in McCullagh and Nelder (1991, p.40), Firth (1991, p.62), Venables and Ripley (2002a, p.185), and Faraway (2006, p.117). Note that other methods of fitting a GLM are available. For instance, Venables and Ripley (2002a, p.445) provide a method for direct maximization of the likelihood for a binomial logistic regression. For the case of binary response variables see also McCullagh and Nelder (1991, p.115). 158 A second aspect of the ‘generalization’ is that the variance of Y , var(Y ) = a(φ)b (θ), may vary with respect to the covariates. For instance, var(Y ) might be increasing in the mean µ. How the variance relates to the mean is captured by the variance function V (µ) ≡ b (θ). Recall from above that µ = b (θ). Examples of the variance function include: V (µ) = 1 (normal), V (µ) = µ (Poisson), V (µ) = µ2 (gamma), and V (µ) = µ(1−µ) (binomial). Note that var(Y ) is also affected by the dispersion parameter φ through a(φ). Therefore in order to fully characterize the variance, φ either has to be given (fixed) or estimated. Examples for which φ = 1 include the binomial, Poisson, and exponential distribution. Next follows a discussion how to estimate φ, two GLM examples based on the binomial and gamma distributions, and a discussion regarding model validation and residual analysis. 4.1. Estimation of Dispersion Parameter. The two most common estimates of φ are the maximum likelihood estimate (MLE) and the moment estimator. For distributions with a constant coefficient of variation,2 such as the gamma distribution, McCullagh and Nelder (1991) argue for the use of the moment estimator. The two arguments they provide are: 1) the maximum likelihood estimate is very sensitive to rounding errors in observations with values close to zero, and 2) if the assumption regarding the distribution is false then the estimate does not consistently estimate the coefficient of variation [17, p.295]. Though the first comment most likely does not apply to our data set, we will nevertheless follow their recommendation and base our estimate on the moment estimator, X2 φˆ = n−p where X 2 is the square-sum of Pearson residuals, n the number of observations, and p the number of fitted parameters. For the gamma distribution, X 2 = 2Coefficient of variation = standard deviation/mean 159 n i=1 ((yi −µ ˆi )/ˆ µi )2 [17, p.296]. Note that φˆ is derived for a given set of βˆ coefficients, whose estimates are not based ˆ In other words, the maximum likelihood estimate and GLM fitting algorithm on φ or φ. ˆ However, as a consequence of (4.14) the of the β coefficients do not depend on φ or φ. ˆ Therefore, even small changes in φˆ may lead to inference regarding βˆ does depend on φ. different implications regarding the significance of the βˆ coefficients. The other method of estimating φ is to solve for the maximum log-likelihood function of the distribution. That is, for given β coefficients, φˆ is determined by computing the MLE of φ. Note that the two estimates may result in different values and therefore reach different conclusions regarding the significance of the βˆ coefficients. For more details regarding φˆ refer to McCullagh and Nelder (1991, p.295), Firth (1991, p.64), and Venables and Ripley (2002a, p.186; 2002b, p.9). Firth (1991, p.65) also alludes to a third possible estimator. Though the moment estimator of φ is the default from the output summary in ‘R’, both estimates are available. For all estimates of φ in this chapter, the MLE estimates were always larger than the moment estimates. 4.2. Binomial Distribution as a GLM. To transform a binomial distribution, with π Pr[success] = π, to a member of the exponential family, let θ = ln( 1−π ), b(θ) = n ln(1 + n y eθ ) = −n ln(1 − π), and c(y, φ) = ln . Note that for the binomial distribution the dispersion parameter φ = 1 and a(φ) = 1, and hence the issue of which estimator of φ to use does not apply. Though there are many options for the link function, the most common and the one we use, is the logit function, (4.15) g(π) = ln π 1−π = β0 + β1 x1 + · · · + βp xp = βx The main benefit of using the logit function is that it ensures the range of π to be within the interval [0, 1]. Another benefit is the direct interpretation of the β coefficients. Namely, an 160 unit increase in a covariate, while keeping the other variables fixed, results in a β increase of the log-odds. Alternatively, a unit increase in a covariate results in exp(β) increase of the odds. 4.3. Gamma Distribution as a GLM. The gamma distribution, G(λ, ν), has two parameters, a scale parameter λ and a shape parameter ν. The general density is, f (y) = 1 ν ν−1 −λy λ y e Γ(ν) y>0 where Γ(·) is the gamma function.3 The gamma distribution has mean ν/λ, and variance ν/λ2 . Note that the exponential distribution is the special case of the gamma distribution for which ν = 1, and that the gamma distribution with ν < 1 is steeper than an exponential. While a gamma distribution with ν > 1 has the shape of a right skewed bell-shaped curve, where the skew becomes less dramatic the larger the value of ν. See Figure 4.3. In order to write the gamma density as a member of the exponential family, reparametrize by defining µ = ν/λ and re-write the gamma density as follows, f (y) = 1 Γ(ν) ν µ ν − y ν−1 e yν µ y>0 Let the canonical parameter θ = −1/µ, the dispersion parameter φ = 1/ν, b(θ) = − ln(−θ) = − ln(1/µ), a(φ) = φ = 1/ν, and c(φ, y) = φ−1 ln(φ−1 ) + (φ−1 − 1) ln(y) + ln(1/Γ(φ−1 )) = ν ln(ν) + (ν − 1) ln(y) + ln(1/Γ(ν)). Similar to the binomial distribution there are many possible link functions. However, in order to ensure the range of λ, the main parameter of interest, is positive we use the log-link function, (4.16) g(µ) = ln(µ) = γ0 + γ1 x1 + · · · + γp xp = γx 3For a complex number z with a positive real part Γ(z) = Γ(z) = (z − 1)!. 161 ∞ z−1 −t t e dt. 0 If z is a positive integer then 1.0 0.8 0.6 0.4 0.0 0.2 Gamma density 0 2 4 6 8 10 y Figure 4.3. Illustration of the gamma distribution G(1, ν) for various values of ν. The solid line represents ν = 1, the dashed line represents ν = 1/2, and the dotted line represents ν = 2. Note that the solid line depicts the exponential distribution with λ = 1. Since exp(γx) > 0, this ensures the range of µ, and hence λ, is positive. As mentioned above, the GLM estimation algorithm of the γ coefficients is independent of the dispersion parameter and its estimate. In other words, estimating the γ coefficients for a G(λ, 1) or G(λ, 2) results in exactly the same γˆ . However, the standard errors, and resulting p-values, will be different depending on the estimated value of φ. Since a(φ) = φ, we see from (4.14) that the standard errors are increasing in φ. Therefore, if φˆ < 1 then, compared to an exponential distribution, the standard errors and associate p-values will be smaller. Consequently, by fitting an exponential distribution with fixed φ = 1, a γˆ coefficient might be statistically insignificant, while by fitting the gamma distribution and deriving an estimate φˆ < 1 the γˆ coefficient might be significant. 162 4.4. Model Validation and Residual Analysis. With a statistical or probabilistic model, the question arises whether the model accurately reflects reality as represented by the data. That is, does the data support the proposed model. The test of model validation is referred to as goodness-of-fit test, and tests how probable it would be to observe the data given that the model is correct. A model ‘fails’ the test when the probability of observing the given data is very low. While a ‘pass’ implies that there is not enough evidence to refute the model, i.e. the probability of observing the data, as predicted by the model, is ‘good’ or ‘large’. The advantage, or curse, of data validation is one aspect that separates statistical models from, for instance, normative economics or management science models. The most common bases for GLM goodness-of-fit test are the scaled deviance, D(y; µ ˆ)/φ, and the generalized Pearson statistics, X 2 . Both measures give a value of how close a model M fits the data, and are functions of the vector of observed values y, and the vector of estimated means µ ˆ. The scaled deviance is defined as twice the difference in maximum log-likelihood between the saturated model and the model in question. Let S denote the saturated model, which has one parameter per observation and thus fits the data perfectly. Let lS (y, φ|y) and lM (ˆ µ, φ|y) represent the maximum log-likelihoods for the two models respectively, then D(y; µ ˆ)/a(φ) = 2[lS (y, φ|y) − lM (ˆ µ, φ|y)], which for GLM can be rewritten as, for a(φ) = φ, n [yi (θ˜S,i − θ˜M,i ) − b(θ˜S,i ) + b(θ˜M,i )]/φ D(y; µ ˆ)/φ = 2 i=1 where θ˜S,i and θ˜M,i are the estimate of θi under S and M respectively. The generalized Pearson statistics is defined as, n X2 = i=1 (yi − µ ˆi )2 V (ˆ µi ) 163 If model M has p parameters and n observations, then under the normal-theory linear models D(y; µ ˆ) and X 2 are χ2n−p distributed. For other distributions, though some asymptotic results are available, this is often not even approximately correct (even for large n). And it seems the following comment from McCullagh and Nelder (1991) still applies, ”Further work on the asymptotic distribution of D(Y; µ ˆ) remains to be done.” [17, p.36] The main asymptotic results available seem to pertain to distributions that, in the limit of certain parameters, resemble the normal distribution. Examples of these include Poisson with ‘large’ means (µi ), binomial with ‘large’ number of trials (mi ), and gamma with a ‘large’ shape parameter ν or alternatively a ‘small’ dispersion parameter φ. These results are referred to as small-dispersion asymptotics, and imply that the scaled deviance can be approximated by a χ2n−p distribution. Though there are no general asymptotic results for the deviance or Pearson X 2 , there are results for the analysis of nested models. Similar to the normal-theory linear models, the difference in scaled deviance between nested GLM models is approximately χ2 distributed. That is, if model 1 and model 2 have p1 and p2 parameters respectively, and the models are nested with p2 < p1 , then (D2 (y; µ ˆ)−D1 (y; µ ˆ))/φ is approximately χ2p1 −p2 distributed. It should be noted that the difference in Pearson X 2 between nested models is not monotone and need not have a χ2 distribution. For more details regarding goodness-of-fit and comparison of nested models refer to McCullagh and Nelder (1991, p.33), Firth (1991, p.68), Venables and Ripley (2002a, p.186), Faraway (2006, p.120), and Jørgensen (1987, p.134). In addition to the goodness-of-fit test another form of model validation is residual analysis. Residual analysis can be used both to detect individual or clusters of observations that do not fit the overall pattern of the data, as well as the overall validation of model assumptions. The latter part, which will be the main focus of our analysis, can further be 164 divided into the aspects that validate the structural part, and the stochastic part of the model. For example, for the positive price-increments the validation is with regards to that the log-link should be linear (structural part) and that the price-increments are gamma distributed (stochastic part). Though there are many different types of residuals, the most common include deviance residuals, rd , and Pearson residuals, rP . Deviance residuals are defined such that rd2 = D(y; µ ˆ), and Pearson residuals such that rP2 = X 2 . However, due to that the Pearson residuals tend to be skewed for non-normal distributions we limit our analysis to the deviance residuals [17, p.38], [8, p.123]. In addition, we will analyze the response residuals y − µ ˆ. Both the deviance residuals and response residuals will be depicted against the fitted linear predictor ηˆ. Two comments regarding the residual analysis follow. First, we will not investigate the residual plots for the Bernoulli distributed price-increments, as their residual plot are bound to be rather uninformative. The reason for this is that binary data will only generate two bands, one for y = 0 and one for y = 1. Second, when analyzing the residuals, note that we are assuming the positive price-increments are gamma distributed with a non-constant variance function. Specifically, we are assuming the variance is decreasing in the mean, e.g. the higher the current price, and thus smaller expected price-increments, the smaller the variance for the price-increments. As a result, the residual plots will look different than, for instance, the residual plots from a normal linear regression model. The patterns we should expect are as follows. For the deviance residuals, due to that the variance function has been scaled out, if the model is correct then there should not be any pattern in the data. That is we should expect to have the deviance residuals randomly distributed around zero with a constant range. If a pattern is observed it could indicate that an inappropriate link function and/or set of covariates was chosen. For the residual plot, on the other hand, due to that the variance is non-constant, we would expect to see a pattern. In particularly, we 165 would expect the response residuals to be more spread for values of ηˆ for which the variance is larger. For more details, see Chapter 12.6 in McCullagh and Nelder (1991), and Chapter 6.4 in Faraway (2006). 4.4.1. Goodness-of-fit for Bernoulli distributed observations. In the case of Bernoulli distribution, a special case of the binomial, the goodness-of-fit test regarding the distributional assumption of the observed values becomes degenerate and vacuous. With only two possible outcomes and independent observations, there are no other plausible alternative distributions. Consider, for instance, a data set with n observations, out of which k were ‘success’ (1) and n − k ‘failure’ (0). Since µ ˆi = π ˆ = k/n and V (ˆ µi ) = π ˆ (1 − π ˆ ), n X2 = i=1 (yi − µ ˆi )2 (1 − π ˆ )2 π ˆ2 1−π ˆ π ˆ =k + (n − k) =k + (n − k) V (ˆ µi ) π ˆ (1 − π ˆ) π ˆ (1 − π ˆ) π ˆ 1−π ˆ =k (n − k)/n k/n + (n − k) = (n − k) + k = n k/n (n − k)/n For the binomial distribution the deviance can be written as follows, n D(y; µ ˆ) = 2 yi ln(yi /ˆ µi ) + (mi − yi ) ln((mi − yi )/(mi − µ ˆi )) i=1 Therefore, for the special case of Bernoulli distribution, where mi = 1 and again assume there were k ‘success’and n − k ‘failure’, we have, n D(y; µ ˆ) = 2 yi ln(yi /ˆ µi ) + (1 − yi ) ln((1 − yi )/(1 − µ ˆi )) i=1 = 2k ln(1/ˆ π ) + (n − k) ln(1/(1 − π ˆ )) = 2n[−ˆ π ln(ˆ π ) − (1 − π ˆ ) ln(1 − π ˆ )] which does not depend on the observed vector y and therefore useless as a measure of goodness-of-fit [17, p.119], [9, p.69], [8, p.121]. Though the issue of testing the Bernoulli distribution might not be of concern, the choice of link function may be. In our analysis we 166 use the logit or log-odds function, mainly due to ease of interpretation and wide use of applicability. Consequently for the Bernoulli distribution we only use D(y; µ ˆ) for comparison of nested models. 4.4.2. Goodness-of-fit for gamma distributed observations. The deviance for the gamma distribution can be written as follows, n D(y; µ ˆ) = 2 − ln(yi /ˆ µi ) + (yi − µ ˆi )/ˆ µi i=1 Despite that the goodness-of-fit analysis based on the deviance also has certain limitations, there are some asymptotic results. As mentioned above, though there are no general asymptotic results, there are asymptotic results pertaining to the case when the dispersion parameter φ is small [13, p.134], [9, p.69]. In these instances, labeled small-dispersion asymptotics, D(y; µ ˆ)/φ has approximately a χ2n−p distribution. However, a working definition of what defines a ‘small’ φ does not seem documented, which makes it hard to judge when the approximation might be good or OK. It appears that the small-dispersion asymptotics apply to when the shape of the gamma distributions resembles the normal distribution. Therefore, it seems a reasonable working definition would be that at a minimum φ < .5, i.e. ν > 2. See also discussion in Section 8.3.6 of McCullagh and Nelder (1991). 5. Model Formulation and Selection for DFS Data Statistical analysis is based on the six data subsets described in Chapter 1, and for convenience summarized in Table 4.1 below. All auctions lasted for three days (τ = 72 hours), and for the purpose of analysis divided into 12 hour intervals. Therefore, there are in total seven observations per auction, and six periods for the three stages for which we estimate πs and λs . To provide a more intuitive index of the elapsed time we count elapsed 167 Product D1 L1 D3 D4 L4 L5 PC Model Processor Model Processor Speed Memory Hard Drive Count (GHz) (MB) (GB) Desktop Intel Pentium 4 1.7-2.8 256, 512, 1024 20, 40 2,072 Laptop Intel Pentium 4, M 1.8-2.4 256, 512 20, 30, 40 2,046 Desktop Intel Pentium 4 2.4 256 40 274 Desktop Intel Pentium 4 2.0 512 40 167 Laptop Intel Pentium 4 1.8 512 30 172 Laptop Intel Pentium M 1.4 512 40 163 Table 4.1. Characteristics of the six products analyzed. periods in increments of 12, Y = 0, 12, 24, 36, 48, 60, 72. In Figure 4.4 and Table 4.2 below, boxplots of price and descriptive statistics, for the 12 hour intervals are shown. Note that the information regards the distribution of price at the specific intervals, and not the distribution of the price-increments. The leftmost columns in Table 4.2 represents descriptive statistics for the average number of ongoing auctions in each 12 hour period. In other words, in each period an auction is underway the average number of ongoing auctions is recorded. For instance, for product D1 in the first period (0h), the smallest and largest average number of ongoing auctions observed were 1.00 and 83.88 respectively. More details are given below. An immediate observation for the boxplots in Figure 4.4, is the steady increase of the median price over time (the line inside each box). Informal statistical support to the observation that the median price is increasing, is that the notches of the boxes do not overlap. In Table 4.2 we see that for each product line, the median price-increments are almost identical for the first four periods, while in the final period the increase is almost twice as large as the previous periods’ price-increments. For desktops, the median increase for the first period is almost zero, while in the next four periods the price increases between $25-30 per period. And in the last period the increase is about $75-80. For laptops, the median increase for the first five periods is about $50, while for the last period the increase is over $100. For the mean number of ongoing auctions, since there is no reason the numbers should fluctuate systematically between time-periods, 168 500 400 500 0 0 100 100 200 200 300 300 400 500 400 300 200 100 0 12h 24h 36h 48h 12h 60h 72h (Final) 24h 36h 48h 60h 72h (Final) 60h 72h (Final) 600 700 500 400 500 300 400 200 300 100 200 0 100 0 48h 48h D4 600 700 600 500 400 300 200 100 36h Current Price after __ hours 36h Current Price after __ hours D3 0 24h 24h 700 D1 12h 12h 60h 72h (Final) Current Price after __ hours Current Price after __ hours 12h 24h 36h 48h 60h 72h (Final) Current Price after __ hours L1 L4 12h 24h 36h 48h 60h 72h (Final) Current Price after __ hours L5 Figure 4.4. Distribution of the current price at the 12hour periods. The upper and lower edge of each box represents the 75th respectively 25th percentile of the observations. The line inside each box represents the median. Non-overlapping notches indicate significant difference in median. the consistency of the values is to be expected. Besides the observation that the price-increment in the last period is larger than the previous period, there is an additional interesting aspect to the dynamics in the last period. Namely that, almost all auctions have a positive price-increment. See Table 4.3 below. With the exception of a handful of auctions for the two aggregated categories D1 and L1, all auctions have a positive price-increment in the final 12 hours. Therefore, we assume πe = 1 and that price-increments in the final 12 hours are strictly gamma distributed. The 169 Product Time Period Price, X Min Median Mean Max D1 Average # of Min Median 1.00 20.73 1.00 20.97 1.00 21.23 1.00 21.36 1.00 21.04 1.00 20.95 Auctions, Z Mean Max 24.14 83.88 24.69 83.88 25.07 83.88 25.07 83.88 24.69 83.82 24.14 83.82 0h 12h 0.99 1.04 29.70 255.60 24h 0.99 50.00 59.20 455.00 36h 0.99 75.00 83.46 455.00 0.99 100.00 104.60 455.00 48h 60h 0.99 129.00 136.10 455.00 72h (τ ) 93.92 200.00 208.20 501.00 D3 0h 1.00 5.000 5.933 14.700 12h 0.99 0.99 16.18 187.40 1.00 5.051 6.009 14.590 24h 0.99 25.62 42.10 187.40 1.00 5.240 6.041 14.700 0.99 66.60 68.85 210.00 1.00 5.154 6.041 14.510 36h 48h 0.99 97.00 92.72 210.00 1.00 5.033 6.009 14.700 60h 38.00 125.00 123.20 226.00 1.00 5.018 5.933 14.700 72h (τ ) 127.50 190.70 193.60 338.50 D4 0h 1.00 4.157 5.764 19.150 12h 0.99 28.00 41.05 150.00 1.00 4.249 6.220 19.510 24h 0.99 60.10 75.77 202.00 1.00 4.660 6.505 19.270 36h 0.99 90.05 100.50 251.00 1.00 4.666 6.505 19.520 48h 0.99 120.00 122.00 256.10 1.00 4.389 6.220 19.540 55.00 150.00 154.50 256.10 1.00 4.071 5.764 19.560 60h 72h (τ ) 129.20 230.00 230.60 348.30 L1 0h 1.00 19.09 19.17 53.91 0.99 58.76 94.51 600.00 1.027 19.57 19.52 53.91 12h 24h 0.99 125.00 154.40 643.00 1.00 19.55 19.77 53.90 36h 0.99 186.50 200.60 643.00 1.00 19.57 19.77 53.90 48h 0.99 207.00 236.40 675.00 1.00 19.39 19.52 53.91 10.00 288.90 289.80 675.00 1.00 19.16 19.17 53.91 60h 72h (τ ) 10.50 405.00 412.20 700.00 L4 0h 1.00 4.279 5.549 17.650 12h 0.99 50.01 84.51 510.00 1.00 4.002 5.632 17.570 24h 0.99 100.00 144.10 550.00 1.00 4.183 5.742 17.580 36h 0.99 165.00 187.50 550.00 1.00 4.350 5.742 17.610 48h 23.00 200.00 219.00 550.00 1.00 4.232 5.632 17.620 60h 61.44 255.00 276.50 550.00 1.00 4.236 5.549 17.650 72h (τ ) 266.00 424.40 428.00 613.00 L5 0h 1.00 3.00 3.148 8.000 12h 0.99 59.00 100.10 481.00 1.00 3.00 3.239 7.884 24h 0.99 150.00 163.40 481.00 1.00 3.00 3.268 8.000 36h 0.99 185.00 208.30 481.00 1.00 3.00 3.268 7.871 48h 0.99 222.00 241.70 490.00 1.00 3.00 3.239 8.000 60h 68.30 301.00 302.50 490.00 1.00 3.00 3.148 8.000 72h (τ ) 281.00 405.00 409.60 597.00 Table 4.2. Descriptive statistics for price X and mean number of ongoing auctions Z. proposed model is therefore modified as follows, for s = (x, y, z) and y = 0, 12, 24, 36, 48, fCY |S (c|s) = 1 − πs πs 1 Γ(νs ) c=0 νs µs 170 νs cνs −1 e − cνs µs c>0 Time Period 0h → 12h 12h → 24h 24h → 36h 36h → 48h 48h → 60h 60h → 72h D1 D3 D4 L1 L4 L5 Zero Pos. Zero Pos. Zero Pos. Zero Pos. Zero Pos. Zero Pos. 330 1,742 63 211 8 159 93 1,953 6 166 4 159 887 1,185 120 154 54 113 685 1,361 62 110 62 100 939 1,133 117 157 78 89 832 1,214 72 100 80 83 1,022 1,050 125 149 82 85 948 1,098 80 92 89 74 689 1,383 95 179 59 108 670 1,376 46 126 55 108 1 2,071 0 274 0 167 6 2,040 0 172 0 163 Table 4.3. Number of zero vs positive price-increment transitions. and for y = 60, fCY |S (c|s) = 1 Γ(νs ) νs µs νs − cνs −1 e cνs µs c>0 where ln πs 1 − πs β0 = y=0 β0 + β1 x + β2 z + β3 x × z + β4 1{y=24} + β5 1{y=36} + β6 1{y=48} n.a. γ0 ln(µs ) = + β2 z + γ2 z y = 12, 24, 36, 48 y = 60 y=0 γ0 + γ1 x + γ2 z + γ3 x × z + γ4 1{y=24} + γ5 1{y=36} + γ6 1{y=48} γ0 + γ1 x + γ2 z + γ3 x × z y = 12, 24, 36, 48 y = 60 The above models are referred to as the unabridged models, to reflect that all parameters are incorporated into the analysis. The final models are then derived by iteratively eliminating non-significant parameters as follows. First the β (γ) coefficients for the unabridged models are estimated by maximum likelihood. The parameter with the most non-significant β (γ) coefficient, i.e. largest p-value, is then removed and the coefficients for the reduced model are re-estimated. The process of iteratively eliminating one parameter is repeated until all coefficients are found to be significant. However, if the coefficient for X or Z is found non-significant but the coefficient for the interaction X × Z is significant, then the interaction term is also removed. For instance, if β2 is non-significant while β3 is significant, 171 then in the next iteration both Z and X × Z are removed. Note that, due to Lemma 4.1 the β and γ coefficients are estimated independently, which implies that a parameter might, for instance, be found non-significant with respect to πs and significant with respect to λs . The term unabridged is to contrast with the general statistics terminology of the null and full (or saturated) models. The null model refers to the model with intercept only, i.e. no dependence on the covariates. And the full model usually refers to the model with one parameter per observation, i.e. by fitting a function that perfectly matches all observations. The model that remains after all non-significant coefficients have been eliminated is referred to as the final model. The analysis was run using the ‘glm’ function in the statistical software R [22]. Though the data for estimating the β and γ coefficients came from the same products, there was one difference between the two estimation procedures. In estimating the γ coefficients only the observations with positive price-increments were used. That is, for each product, when estimating the γ coefficients for λb , λm , and λe , all zero price-increments were filtered out. To illustrate, suppose a product consists of 100 auctions, which means that for each of the six periods (0 → 12h, 12 → 24h, 24 → 36h, 36 → 48h, 48 → 60h, 60 → 72h,) there are 100 observations. Suppose further that for each of the periods there were 50, 60, 70, 80, 90, and 100 auctions with positive price-increments, i.e. out of the 100 auctions, 50 observed a positive price-increment in the first period, 60 in the second period, and so on. Note that it is not necessarily the same set of auctions with positive price-increments in each period. Then when estimating the γb coefficients in the first period only the 50 observations were used, when estimating γm for the four middle periods only the 300 (=60+70+80+90) observations were used, and when estimating γe for the final period the 100 observations were used. In contrast, for estimating πb and πm all the within period 172 observations were used. Recall that we assume πe = 1. This means that when estimating the βb coefficients for the first period all 100 observations were used, and when estimating βm for the four middle periods 400 (= 100 × 4) observations were used. The implicit assumption behind the above set-up is that auctions’ within period pricetransitions are independent, both with respect to competing auctions as well as with respect to the previous price-transitions. In other words, besides the cannibalization effect, auctions are assumed to be independent of each other, and the price-transitions are memoryless or Markovian. Since the data was not collected using a designed study, the number of ongoing auctions in each 12-hour interval varies. That is, the auctions analyzed were not released in synchronized 12-hour intervals, but instead with various time overlap. Consequently, for almost all auctions the number of ongoing auctions in a given 12-hour period fluctuates. Therefore, the arithmetic mean number of ongoing auctions in a time-period is used as a proxy for Z. There are of course other possible variables that could have been included. In particularly, for the aggregated products D1 and L1, information regarding the product specifications such as processor speed, hard drive memory, and RAM, could have been included as covariates. We will leave these and other variables for possible future extensions to the above model. Before discussing the results we examine the individual products a bit closer. 5.1. Hypothesis Testing of β and γ Coefficients. In Figures 4.7, 4.8 and 4.9 at the end of this chapter, the within period price-increments for the six products are displayed. Each individual graph has price at the beginning of a period on the x-axis, and the 12 hour within period price-increment on the y-axis. Each observation represents an 173 individual auction. Observations along the y-axis are auctions that did not change price during the period, i.e. received no bids in the 12 hour period. The line in each graph represents the fitted values of the conditional expected price-increment for the final model, and is discussed in Section 6. For the products and periods where z or x × z are significant, the median value of z was used in evaluating the fitted values. Note the following distinction. Z is the average number of ongoing auctions in a period. In the figures, the median of the observed values of Z was used to fit the line. in A few comments follow. First, there is clear variability in the price-increments. For a given price, or a small interval around a given price, the 12 hour within period price-increment varies. Most auctions tend to have ‘small’ price-increments and a few auctions have ‘large’ price-increments. The main motivation of the proposed model is to model this feature. Second, one can see that the price-increments are decreasing in the price. In other words, as the price increases the within period price-increments tend to decrease. Though this would seem intuitive, the objective of the analysis is to capture the rate at which the mean within period priceincrement decreases with respect to price, and assess if the structural properties discussed in Section 3.1 hold. We summarize our observations regarding price in the following testable hypothesis. Hypothesis 4.5. The within period log-odds of observing a positive price-increment are on average decreasing in price. Hypothesis 4.6. The within period log-mean of the positive price-increments are on average decreasing in price. A third comment is, it seems that the middle four 12 hour periods, i.e. 12 → 24h, 24 → 36h, 36 → 48h, and 48 → 60h, all exhibit similar though distinct dynamics. As 174 noted earlier, the last period is different in that almost all auctions exhibit a positive priceincrement. Furthermore, though price-increments are still clearly decreasing in price, for auctions at ‘low’ prices there are no ‘small’ price-increments. That is, there are no observations in the lower left quadrant of the graphs for the final period. The main take-away is that elapsed time seems to have an effect on the auctions dynamics, which we do not state in terms of formal hypotheses. However, an important implication regarding the price-dynamics in the final period, is that the price-transitions for the first five periods might not affect the final price. In other words, if the expected price-increment in the final period decreases linearly at a rate of $1 per unit increase in price, then the expected final price would be constant. And consequently the previous periods’ dynamics may be irrelevant. Note that the proposed model with (4.3), is not based on the mean price-increment being linearly related to price. It assumes that the log mean price-increment is linearly related to price. If a normal linear regression relationship was used there would be a price, beyond which, all within period price-increments are negative, which would seem as a rather restrictive implication. A benefit of (4.3) is that it ensures price-increments remain positive. However, since running an additional regression analysis does not entail much effort, in Appendix E a normal linear regression model for the last period will be discussed. The results of which, can be used to support or refute that the mean price-increment is decreasing at a rate of $1, and consequently the previous periods dynamics irrelevant. In Figure 4.10, 4.11 and 4.12 at the end of this chapter, the six within period priceincrements as a function of the average number of ongoing auctions is displayed. On the x-axis is the average number of ongoing auctions for a particular period, and on the y-axis is the within period price-increment. Each observation represents an individual auction. 175 Similar to the previous set of graphs the price-increments again display variability with regard to the average number of ongoing auctions. However, unlike the previous graphs there does not seem to be any clear or obvious trend. Intuitively, it would seem reasonable that the within period price-increments are negatively correlated with the number of ongoing auctions. The formal statistical analysis and hypothesis testing will reveal if this is the case. Hypothesis 4.7. The within period log-odds of observing a positive price-increment are on average decreasing in the number of ongoing auctions. Hypothesis 4.8. The within period log-mean of the positive price-increments are on average decreasing in the number of ongoing auctions. The final hypothesis testing is with regard to the interaction term ‘x × z’. From the discussion in Section 3 we would expect to see the interaction term counter the main effects. That is, for the structural properties to hold we expect β3 and γ3 to be positive. Therefore, the final set of hypothesis are as follows. Hypothesis 4.9. The expected decrease of the within period log-odds of observing a positive price-increment due to an increase in the number of ongoing auctions, is diminishing in price. Hypothesis 4.10. The expected decrease of the within period log-mean of the positive price-increments due to an increase in the number of ongoing auctions, is diminishing in price. A few comments follow. • The stated hypothesis are meant to represent the interpretation of the actual hypothesis testing. That is, the actual null hypothesis is, of course, βk = 0 or γk = 0, 176 and the alternative hypothesis is βk = 0 or γk = 0, k = 1, 2, . . . , 6. In order to provide a more interesting and intuitive discussion the hypothesis have been stated to be consistent with our observations. • Though our goal is to derive general conclusions regarding the various variables, the hypothesis testing will be conducted for each product individually. The reason for this is that the final model for each product is likely to differ, and therefore we will not conduct a simultaneous hypothesis testing for all products. Recall that the model selection is such that non-significant variables are sequentially removed for each product. However, based on the individual products’ test results we attempt to derive some general conclusions. A different statistical approach would have been to conduct a simultaneous hypothesis test for all products. • Finally, we report the exact p-value (a posteriori probability) as the significance level of the hypothesis test. All p-values less than .0001 will be reported as ‘< .0001’. Note that the reported p-values from the hypothesis tests that β = 0 and γ = 0 are used for the stated hypotheses. 6. Results for DFS Data To assess the proposed model we first look at the fitted means and qualitatively evaluate the fit. In Figures 4.7, 4.8 and 4.9 at the end of this chapter, the line in each graph represents the fitted values for the conditional expected price-increments for the final model. Note that the line is the conditional expected price-increment as specified by (4.5), and not the conditional expected price-increment for the positive price-increments, i.e. the line represents πs µs and not µs . For products and periods where either z or x × z were found to be significant, the median value of z was used to determine the fitted values. Recall that in the last period πs = 1. For the first period the fitted value of the expected price-increment 177 is displayed by the dashed line and is of course independent of price. Overall the lines fit the data and are in accord with the trends exhibited by the graphs. In particular, for the final period, with perhaps the exception of D1, we see that the line follows the trends very well and seems to represent the mean price-increments perfectly. A possible problem with D1 in the final period might be the outlier observed around $450. For the middle four periods the fit is also good and seems to follow the overall trends very well. This is perhaps best seen in the four product specific products D3, D4, L4, and L5, for which the fitted values provide a very good fit. For the first period it is hard to comment, even qualitatively, regarding the proposed model. Table 4.4 shows the resulting equations for the fitted means. The first, second and third line for each equation represents the first, middle and last periods respectively. A ‘n.s.’ entry represents not significant, meaning that none of the variables were found to be significant, while ‘n.a.’ represents not applicable. Detail ‘R’ output for each product is provided in Appendix D. Next we discuss the quantitative results and formal measures of goodness-of-fit. 6.1. Effect of Price. The first thing to note is that price, x, is always significant and always represented by a negative coefficient. In other words, the higher the price, the less likely it is for a positive price-transition to occur. And if there is a positive price-transition then the expected price-increment is smaller. Though this might be expected, and consistent with the trends observed in Figures 4.7, 4.8 and 4.9, the issue arises if the β1 and γ1 coefficients are ‘too negative’. That is, are the decreasing rates such that E[XY +1 |S] for each product is not monotone in the current price. Before we discuss more details we summarize the formal hypotheses results. 178 (D1) (D3) (D4) (L1) (L4) (L5) n.s. πs ln 1−π = .841 − .010x − .009z − .00005x × z + (.228)1{y=24} + (.327)1{y=36} + (1.307)1{y=48} s n.a. 4.100 − .004z ln(µs ) = 4.029 − .002x − (.131)1{y=24} − (.149)1{y=36} 4.699 − .004x + .003z .759 − .180z πs ln 1−πs = .931 − .021x − .059z + (.603)1{y=24} + (1.056)1{y=36} + (2.116)1{y=48} n.a. 3.824 − .064z ln(µs ) = 4.000 − .007x + (.305)1{y=48} 4.681 − .005x + .108z − .0008x × z n.s. πs = ln 1−π 1.375 − .012x − .027z + (.948)1{y=48} s n.a. n.s. ln(µs ) = 4.241 − .005x − .023z + (.294)1{y=48} 5.442 − .008x 2.198 − .044z πs = 1.867 − .006x − .020z − .00008x × z + (.144)1{y=24} + (.250)1{y=36} + (1.197)1{y=48} ln 1−πs n.a. 5.183 − .023z ln(µs ) = 4.838 − .002x − .011z − (.057)1{y=24} − (.119)1{y=36} + (.107)1{y=48} 6.000 − .004x 1.481 − .090z πs = 1.331 − .007x + (1.320)1{y=48} ln 1−πs n.a. 5.057 − .065z ln(µs ) = 4.620 − .001x − (.170)1{y=24} − (.328)1{y=36} 5.736 − .003x + .079z − .0002x × z 2.421 − .295z πs = 1.671 − .007x − .140z + (1.299)1{y=48} ln 1−πs n.a. n.s. ln(µs ) = 5.155 − .003x − .123z + .0004x × z − (.186)1{y=36} 5.959 − .005x Table 4.4. The resulting equations for the final model. In each set of equations, the first corresponds to the first period, the second to the middle four periods, and the third to the final period. All coefficients shown are significant, ‘n.a.’ represents not-applicable, and ‘n.s.’ represents cases with all covariates non-significant. 179 Test of Hypothesis 4.5 - For all products during the middle periods we reject the hypothesis that β1 ≥ 0 (p < .0001 for all products), and conclude that the within period log-odds of observing a positive price-increment are on average decreasing in price. Test of Hypothesis 4.6 - For all products during the middle and final periods we reject the hypothesis that γ1 ≥ 0 (p < .005 for all products), and conclude that the within period price-increments are on average decreasing in price. In Lemma 4.2 of Section 3.1 sufficient conditions to ensure that E[XY +1 |S] is increasing were provided. The illustration of these conditions are provided in Figure 4.5. The top six graphs of Figure 4.5 represents the coordinates for β1 (x-axis) and γ1 (y-axis) for each product. The first row are for D1, D3 and D4, while the second row are for L1, L4 and L5 (from left to right). The triangular area enclosed by the dashed lines, display the region for which the sufficient conditions of Lemma 4.2 hold for the four middle periods. The ‘x’ in each graph is the coordinate for the point estimate of (β1 , γ1 ). For all six products we see that the ‘x’ falls inside the ‘sufficient’ region and hence support that E[XY +1 |S] is increasing in the current price. For the final 12 hour period, since πs = 1, there is no estimate for β1 . Consequently the condition on γ1 simplifies and results in a lower bound. See the discussion at the end of Section 3.1. The line on the y-axis labeled ‘60h’ indicates the lower bound. The circle on the y-axis is the point estimate for γ1 for the final period. We see that, with the exception of product D1, the circle is below the line. This does not, however, necessarily mean that E[XY +1 |S] is non-monotone in price as the conditions are only sufficient. Therefore to further assess if E[XY +1 |S] is increasing in the final period, the graph of E[XY +1 |S] for the final period is analyzed. The bottom six graphs of Figure 4.5 display the conditional expected price transition for the final period. On the x-axis is X60 , and on the y-axis is the final price Xτ . The 180 first row of graphs are for D1, D3 and D4, while the second row are for L1, L4 and L5 (left to right). The 45 degree line represents the line at which auctions remain unchanged in the final 12 hours. The bold curve represents E[Xτ |S] = x + µs . The main feature of interest is if this curve is monotone. We see that for the desktop products E[Xτ |S] is indeed increasing, although for D3 and D4 there seems to be a slight downward trend at the lower prices. However, for the laptop products the curve is clearly non-monotone. The implication is that, for laptop products, there exist a price p such that E[Xτ |S] is decreasing in price for x < p and increasing in price for x > p . Which in turn implies that the result of Proposition 2.11 of Chapter 2 does not hold. In other words, it is not necessarily true that the seller is always better off the higher the price. However, since for all the previous periods, the expected price transitions are increasing in price, overall the seller is better off the higher the price. Meaning that, though at the start of the final period, there exist a ‘low’ price which has the same expected final price as a ‘high’ price, the seller would be even better off above the ‘high’ price. And at the extreme end, the seller would want to be above the ‘high’ price that has an equivalent expected final price as the ‘lowest’ priced auction. Therefore, we conclude that, in general, the seller is always better of the higher the price of an auction. 6.2. Effect of Number of Ongoing Auctions and Interaction Term. The second and perhaps more interesting results regards the cannibalization effect. In the equations in Table 4.4 this is represented by the coefficients for z and x × z. We observe that the probability of observing a positive price-transition is decreasing in the average number of ongoing auctions, z. This is seen by observing that the coefficients for z in the equations pertaining to ln(πs /1 − πs ) are either negative or non-significant. A third observation is that, though the interaction term x × z is rarely significant, it does exhibit some influence. This means that, with the exception of D4, the dynamics between price and number of 181 −0.005 −0.010 12h −0.06 −0.04 −0.02 0.00 −0.10 −0.08 −0.06 −0.04 −0.015 12h 24h 36h −0.02 0.00 −0.10 −0.08 −0.06 −0.04 X −−−−−− 60h −0.005 −−−−−− 60h O 48h γ1 γ1 −0.010 0.00 X −0.005 48h 12h 24h 36h −−−−−− 60h O O −0.010 −0.005 X −0.02 β1 0.000 β1 0.000 β1 γ1 γ1 36h 24h 0.000 −0.08 48h 48h 12h 24h −0.010 −0.10 60h −−−−−− −0.020 −0.010 γ1 −0.015 −0.020 48h −0.025 −0.025 12h X O −0.030 −0.020 48h O X −0.030 −0.005 −0.010 −0.015 60h −−−−−− −0.030 γ1 O −0.005 −−−−−− 60h X −0.025 24h 0.000 0.000 0.000 36h 36h −0.015 −0.015 −0.015 12h 24h −0.03 −0.02 −0.01 0.00 −0.06 −0.05 −0.04 −0.03 −0.01 0.00 −0.06 200 300 400 500 600 600 400 500 600 0 200 500 600 700 300 400 500 600 700 500 600 700 500 Final Price 400 0 100 200 Final Price 300 200 Figure 4.5. 100 Current Price After 60hours 100 400 0.00 400 Final Price 300 0 300 Current Price After 60hours −0.01 200 200 600 700 600 500 400 300 200 100 200 −0.02 100 100 Current Price After 60hours 0 100 −0.03 0 0 Current Price After 60hours 0 −0.04 500 600 500 300 Final Price 200 100 0 100 −0.05 β1 400 500 400 300 Final Price 200 100 0 0 Final Price −0.02 β1 600 β1 300 −0.04 400 −0.05 300 −0.06 −0.020 −0.020 −0.020 36h 0 100 200 300 400 Current Price After 60hours 500 600 700 0 100 200 300 400 500 600 Current Price After 60hours The top six graphs display the feasible region for which the sufficient conditions of Lemma 4.2 holds (first row: D1, D3, D4, second row: L1, L4, L5; left to right). Each graph has β1 on the x-axis and γ1 on the y-axis. The triangular area enclosed by the dashed lines provides the region such that Lemma 4.2 holds for the middle four periods. The ‘x’ represents the point estimate of (β1 , γ1 ). The line labeled ‘60h’ on the y-axis is the lower bound for γ1 , and the circle represents the point estimate of γ1 for the final period. If the ‘x’ or circle is outside the region respectively lower bound it implies Lemma 4.2 does not hold. The bottom six graphs show the conditional expected price transitions for the final period (first row: D1, D3, D4, second row: L1, L4, L5; left to right). The x-axis represents price after 60 hours, and the y-axis represent the final price. The 45 degree line indicates auctions that received no bids. The bold curve line shows E[Xτ |S] = x + µs . The main feature of interest is to see if E[Xτ |S] is increasing. 182 700 ongoing auctions will at some point during an auction come into effect. More details follow but first we summarize the formal hypothesis results regarding the number of competing auctions and the interaction term. Test of Hypothesis 4.7 - For D3, L1, L4, L5 in the first period we reject the hypothesis that β2 ≥ 0 (p < .02 for all products), while for D1 and D4 in the first period we fail to reject the hypothesis that β2 ≤ 0 (p > .42 for D1 and p > .75 for D4). For D1, D3, D4, L1, L5 during the middle periods we reject the hypothesis that β2 ≥ 0 (p < .005 for D1, D3, L1, L5, and p < .091 for D4), while for L4 we fail to reject the hypothesis that β2 ≤ 0 (p > .66). We therefore conclude that the within period log-odds of observing a positive price-increment are on average decreasing in the number of competing auctions. Test of Hypothesis 4.8 - For D1, D3, L1, L4 in the first period we reject the hypothesis that γ2 ≥ 0 (p < .04 for all products), while for D4 and L5 we fail to reject the hypothesis that γ2 ≤ 0 (p > .18 for D4 and p > .33 for L5). For D4, L1, L5 during the middle periods we reject the hypothesis that γ2 ≥ 0 (p < .002 for all products), while for D1, D3, L4 we fail to reject the hypothesis that γ2 ≤ 0 (p > .53 for D1, p > .22 for D3, and p > .83 for L4). For D1, D3, L4 in the final period we reject the hypothesis that γ2 ≤ 0 (p < .002 for all products), while for D4, L1, L5 we fail to reject the hypothesis that γ2 ≥ 0 (p > .17 for D4, p > .98 for L1, p > .78 for L5). Therefore, for the first and middle periods we conclude that the within period priceincrements are on average decreasing in the number of competing auctions, while for the final periods we conclude that the within period price-increments are on average not decreasing in the number of competing auctions. In other words, Hypothesis 4.8 only holds 183 for the first and middle periods, and not for the final period. Test of Hypothesis 4.9 - For D1 and L1 during the middle periods we reject the hypothesis that β3 ≥ 0 (p < .075 for both), while for D3, D4, L4, L5 we fail to reject the hypothesis that β3 = 0 (p > .77 for D3, p > .94 for D4, p > .66 for L4, and p > .18 for L5). We therefore conclude that the decreasing rate of the within period log-odds of observing a positive price-increment due to an increase in the number of ongoing auctions, is not diminishing in price. That is, Hypothesis 4.9 does not hold. Test of Hypothesis 4.10 - For L5 during the middle periods we reject γ3 ≤ 0 (p < .03), while for D1, D3, D4, L1, L4 we fail to reject γ3 = 0 (p > .64 for D1, p > .22 for D3, p > .27 for D4, p > .50 for L1, and p > .83 for L4). For D3 and L4 during the final period we reject γ3 ≥ 0 (p < .004 for both products), while for D1, D4, L1, L5 we fail to reject γ3 = 0 (p > .39 for D1, p > .43 for D4, p > .36 for L1, and p > .51 for L5). We therefore conclude that during both the middle and final periods the expected decrease of the within period price-increments due to an increase in the number of ongoing auctions, is diminishing in price. In other words, Hypothesis 4.10 holds. A comment regarding the test of Hypothesis 4.10 is that although the sign of the γ3 coefficient for the final period is different than during the middle periods, it is also different from the sign of the γ2 coefficient for the final period. In other words, we note that for both the middle and final periods, the main effect of the number of ongoing auctions is ‘diminished’ by the interaction effect from price. 184 Similar to the effect of price, there seems to be a different dynamics between the first five periods and the final period. In the first five periods, though z is not always significant and x × z hardly ever is, the coefficients are always such that Lemma 4.3 holds. For example, for D4 we see that an unit increase of z, decreases the log-odds of observing a positive price-increment with .027, and decreases the log-mean of the positive price-increment with .023. Note, however, that for the middle periods of L5, there is a slight violation regarding the condition on price, as −γ2 /γ3 = .123/.0004 = 308, and there are several L5 auctions during the middle periods at prices higher than $308. Consequently, we conclude that during the first five periods, the more ongoing auctions the lower the conditional expected price-increment E[CY +1 |S]. Though this might not clearly be visible in Figures 4.10, 4.11, and 4.12, it has intuitive appeal and should not be too surprising. In the final period, for the three instances where z is significant, namely D1, D3, and L4, γ2 is positive. This would imply that the main effect of having more ongoing auctions, is that it increases the conditional expected price-increment. Though this might seem a bit counter-intuitive and paradoxical, it ignores the interaction effect x × z. For both D3 and L4, since γ3 is negative the conditions for Lemma 4.4 are satisfied. Specifically, the pricelevel pi = −γ2 /γ3 , is the point such that if x > pi (x < pi ) then the conditional expected price-increment decreases (increases) the more ongoing auctions there are. As mentioned in Section 3.2, the ‘cannibalization effect’, or rather price competition effect, benefits the lower priced auctions and works against the higher priced auctions. With the interpretation that this only happens in the presence of both low and high priced auctions, this has intuitive appeal and seem natural to expect. The specific price levels pi are as follows. For D3, pi = $135, which is slightly above the median of X60 and about $50 below the median Xτ . And for L4, pi = $395, which though it is almost $100 above the median of X60 , is just slightly below the median of Xτ . Therefore, the loose interpretation would be that auctions 185 for D3 or L4 products, which are priced below the median of Xτ and only has 12 hours remaining, are considered ‘good deals’, and consequently will either attract more bidders and/or higher bids. In the bottom two graphs of Figure 4.6 the ‘cannibalization effect’ in the last period for D3 and L4 is shown. On the x-axis is the price after five periods (X60 ) and on the y-axis is the conditional expected within period price-increment. The solid, dashed and dotted lines, represents the scenario with 1, 5 and 10 ongoing auctions. We see that for D3 and L4, if prices are below $135 respectively $395, then the more ongoing auctions, the expected within period price-increment increases. And for prices above the thresholds the expected price-increment decreases. The only anomaly regarding a positive γ2 is the aggregated product D1. There is no intuitive reason why this should be the case, and the graph in Figure 4.10 does not indicate any strong positive trend. One possibility might be that, since the expected price transitions are decreasing in the number of ongoing auctions for the first five periods, in the final period there is some compensation. Regardless, we conclude that, in general, each auction will experience a lower price transition the more ongoing auctions there are. The third result of interest is regarding the diminishing cannibalization effect, which would imply that a threshold type policy is optimal. For the middle four periods only β3 for D1 and L1, and γ3 for L5 were found to be significant. In the former cases, β3 is negative, which implies that the cannibalization effect is increasing in price. That is, the higher the price and the more ongoing auctions the less likely there will be a positive within period price-increment. While in the latter case, γ3 is positive, which implies that the cannibalization effect is diminishing in price. Illustration of the two different cannibalization effects are shown in the top three graphs of Figure 4.6. Each graph has the expected within period price-increment for the second period (12 → 24h) on the y-axis, and price after 12 hours on the x-axis. Each line represents the expected within period price-increments for 186 a given number of ongoing auctions as labeled. For L5 we see that as price increases the lines get closer to each other, i.e. the cannibalization effect is diminishing in price. While for D1 and L1 we see that if there are more ongoing auctions then the exponential curve becomes steeper, and hence the cannibalization effect is increasing in price. However, we also see that, as discussed in Section 3.3, due to the shape of the negative exponential curve, there is a price-level at which the difference between the lines is maximized and after which the difference diminishes. That is, the cannibalization effect is first increasing and then decreasing in price. For D1 and L1 the cannibalization effect seems to be increasing up to about $125 respectively $250, and thereafter decreasing or diminishing. The other noteworthy observation to point out is the difference between D1 and L1 regarding the cannibalization effect. For D1 there is a much smaller shift in going from 1 to 20, or from 20 to 40, ongoing auctions, as compared with L1. In other words, the laptop auctions are much more sensitive to an increase in the number of ongoing auctions. One explanation for this might be that laptops are approximately twice as expensive, and hence there is more room for price variability (in particularly downward). While for desktops the prices might in general already be so low that there is not much room for them to decrease to. For the final period, as mentioned above and illustrated in Figure 4.6, only D3 and L4 have significant interaction terms. Although, as discussed in Section 3.3, there exist a price-level pii , such that if x > pii (x < pii ) then the cannibalization effect is decreasing (increasing) in price, it is hard to see if this price-level is within range of values displayed. The graphs in Figure 4.6 are limited to the range of values displayed by the data. Consequently, we conclude that the cannibalization effect is not diminishing but rather increasing in price (note that at first the cannibalization effect is negative). The implication on the optimal release policy for all six products will be discussed in Section 7. 187 40 z=1 z=20 30 20 10 0 Expected Periodic Price−Increment z=40 0 50 100 150 200 250 Current Price After 12hours 80 100 120 z=1 60 z=4 z=8 0 0 20 40 80 z=40 40 z=20 60 100 Expected Periodic Price−Increment z=1 20 Expected Periodic Price−Increment 120 140 140 D1 - Y=12h (β2 , β3 , γ2 < 0, γ3 = 0) 0 100 200 300 400 500 600 0 100 Current Price After 12hours 500 600 600 200 400 L5 - Y=12h (β2 < 0, β3 = 0, γ2 < 0, γ3 > 0) 400 z=5 z=1 0 0 100 50 z=1 300 z=5 z=10 200 Expected Periodic Price−Increment 150 500 z=10 100 300 Current Price After 12hours L1 - Y=12h (β2 , β3 , γ2 < 0, γ3 = 0) Expected Periodic Price−Increment 200 0 50 100 150 200 0 Current Price After 60hours 100 200 300 400 500 600 Current Price After 60hours D3 - Y=60h (γ2 > 0, γ3 < 0) L4 - Y=60h (γ2 > 0, γ3 < 0) Figure 4.6. Each graph shows the expected within period price-increment (yaxis) as a function of price (x-axis). The top three graphs are for D1, L1 and L5 in the second period (12 to 24h), while the bottom two graphs are for D3 and L4 in the final period (60 to 72h). Each line represents the conditional expected priceincrement given the number of ongoing auctions as labeled. The solid line is the base case when there is only one ongoing auction. 188 6.3. Effect of Elapsed Auction Time. Another observation to note is that the individual periods exhibit different dynamics. Clearly there is a difference between the first, middle and final periods, since they all have different coefficient values. In addition, for the middle periods we note that, for all products, at least one of the elapsed period, y, indicator functions is significant. This is true both with regard to ln(πs /1−πs ) as well as ln(µs ). 6.4. Estimation of Gamma Shape Parameter. The other result of interest concerns the shape parameter of the gamma distribution for the positive price-increments. Recall from Figure 4.3 in Section 4.3, that depending on the shape parameter ν, the gamma distribution will look and behave differently. If ν > 1 then the gamma distribution resembles a right skewed unimodal curve, and if ν < 1 then it resembles a steeper exponential curve. In the last two columns of Tables 4.5, 4.6, and 4.7 below, the resulting estimates of φ and ν for the different periods of the auctions are shown (recall φ = 1/ν). First thing to note is that all products have very similar parameters and hence similar shapes. The only striking exception is for the first period of D3, where the distribution resembles a steeper exponential distribution due to that νˆ < 1. In all other instances, the distribution behaves like a heavily skewed unimodal curve. That is, there is a long tail to the right (large price-increments) and almost no tail to the left (small price-increments). This is expected and natural since price-increments cannot be negative. Moreover, we see that as the auctions progress the unimodal curve shifts more and more to the right, i.e. to higher price-increments. In particular, in the final period νˆ is approximately twice as large as during the middle periods. This further reflects the difference in auction dynamics in the final period, where the low priced auctions do not exhibit small price-increments. In other word
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Analytical and empirical models of online auctions Ødegaard, Fredrik 2008
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Title | Analytical and empirical models of online auctions |
Creator |
Ødegaard, Fredrik |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | This thesis provides a discussion on some analytical and empirical models of online auctions. The objective is to provide an alternative framework for analyzing online auctions, and to characterize the distribution of intermediate prices. Chapter 1 provides a mathematical formulation of the eBay auction format and background to the data used in the empirical analysis. Chapter 2 analyzes policies for optimally disposing inventory using online auctions. It is assumed a seller has a fixed number of items to sell using a sequence of, possibly overlapping, single-item auctions. The decision the seller must make is when to start each auction. The decision involves a trade-off between a holding cost for each period an item remains unsold, and a cannibalization effect among competing auctions. Consequently the seller must trade-off the expected marginal gain for the ongoing auctions with the expected marginal cost of the unreleased items by further deferring their release. The problem is formulated as a discrete time Markov Decision Problem. Conditions are derived to ensure that the optimal release policy is a control limit policy in the current price of the ongoing auctions. Chapter 2 focuses on the two item case which has sufficient complexity to raise challenging questions. An underlying assumption in Chapter 2 is that the auction dynamics can be captured by a set of transition probabilities. Chapter 3 shows with two fixed bidding strategies how the transition probabilities can be derived for a given auction format and bidder arrival process. The two specific bidding strategies analyzed are when bidders bid: 1) a minimal increment, and 2) their true valuation. Chapters 4 and 5 provides empirical analyzes of 4,000 eBay auctions conducted by Dell. Chapter 4 provides a statistical model where over discrete time periods, prices of online auctions follow a zero-inflated gamma distribution. Chapter 5 provides an analysis of the 44,000 bids placed in the auctions, based on bids following a gamma distribution. Both models presented in Chapters 4 and 5 are based on conditional probabilities given the price and elapsed time of an auction, and certain parameters of the competing auctions. Chapter 6 concludes the thesis with a discussion of the main results and possible extensions. |
Extent | 27980677 bytes |
Subject |
Online auctions Markov decision processes Fixed bidding strategies Empirical analysis |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0066588 |
URI | http://hdl.handle.net/2429/1615 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Graduation Date | 2008-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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