A Simulation-Based Model for Final Price Prediction in Online Auctions

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1 經濟與管理論叢 (Journal of Economics and Management), 2007, Vol. 3, No. 1, 1-16 A Simulation-Based Model for Final Price Prediction in Online Auctions Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho, and Yu-Chen Hsieh * Online auctions, a profitable, exciting, and dynamic part of e-commerce, have enjoyed increasing public interest. However, there is still a paucity of literature on final price prediction for online auctions. Although Markov process models provide a mathematical approach to predicting online auction prices, estimating parameters of a Markov process model in practice is a challenging task. In this paper we propose a simulation-based model as an alternative approach to predicting the final price in online auctions. The simulation results show that the proposed model can predict the final price more accurately than a Markov process model. Additionally, the consistent lower predictions of the Markov process model suggest a direction for future research into improving performance in both models. Keywords: Markov process, simulation, e-commerce, auction, final price prediction JEL classification: C15, D44, D61 1 Introduction An auction has been one of the main exchange mechanisms for thousands of years. Modeling the relevant issues surrounding auctions is increasingly important today, especially as auctions have been brought to the internet and evolved into online auctions. Online auctions are a new type of e-commerce that started in America and have become one of the most successful business models on the internet. For example, the most well-known online auction website ebay has more than 12 million Received March 22, 2006, revised October 13, 2006, accepted October 18, * Authors are respectively at: Department of Marketing, Department of Business Administration, and Institute of Electronic Commerce, National Chung Hsing University, Taiwan, and Educational Measurement and Statistics, National Taichung Teachers College, Taiwan.

2 2 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh products across 18,000 categories and made as much as 18.3 billion US dollars in The company claims that their annual gross sales exceeded 30 billion US dollars in Unlike traditional auctions, online auctions bend the rules of time and space. Participants do not have to be in the same place at the same time to participate. People bid and win items halfway around the world in the middle of the night. Auction websites are now equally or more popular than typical shopping websites. More and more people are changing their shopping habits by turning to online auctions due to its convenience and compatibility with their lifestyles, needs, and shopping preferences. With the great success of online auctions, there has been a vast body of research in economics, finance, and e-commerce proposing different models to explain the online auction process (McAfee and McMillan, 1987). However, most recent research focuses on the design of the mechanisms of an auction site from the seller s or buyer s point of view or as welfare (allocation efficiency) analysis. Few studies forecast the final price of an online auction. Knowing the possible final price of an online auction in advance is obviously important to both bidders and auctioneers. Bidders can set a price limit to avoid being overcharged; auctioneers can set an appropriate reserve price to ensure a minimum return. Motivated by the need for a useful online auction final price prediction model and to fill a gap in the existing literature, the main purpose of this study is to develop a model to predict the final price of online English auctions. Unlike the posted price in regular trade, the final price for an online auction is determined by competing bids submitted by bidders. These bids are placed over time randomly. The distribution of the resulting inter-arrival time intervals can have a great impact on the final price. Instead of assuming some predetermined distribution and estimating its parameters, we propose a simulation-based model that uses bootstrapping to derive the sampling distribution and to predict the final price. Another common approach relies on Markov process modeling to evaluate the outcome of an online English auction. Since this technique can be used to predict the price trajectory and the final selling price of an online auction, we also construct a Markov process final price prediction model as a benchmark for comparisons and to examine the validity of the proposed model.

3 A Simulation-Based Model for Final Price Prediction in Online Auctions 3 The remainder of this paper is organized as follows. In section 2, we briefly review various types of online auctions. A Markov process model and the proposed simulation-based model are described in sections 3 and 4, respectively. The simulation results of the proposed model are compared to those obtained from the Markov process model based on real digital camera online auction data in section 5. Concluding remarks and future research directions are given in section 6. 2 Online auctions Due to the popularity of the Internet, online auctions have grown at a very fast rate since their birth in Auctioneers (sellers) describe their items on the auction website, and bidders (buyers) bid for these items. As in other markets, auctioneers want to sell goods at prices as high as possible while bidders intend to buy goods at prices as low as possible (Vakrat and Seidmann, 2000). If we can predict the final prices correctly, bidders can avoid being overcharged and auctioneers can set reserve prices properly to ensure minimum returns. As a result, the online auction process can be made more efficient and effective in resource allocation and price determination. Today, there are hundreds of websites dedicated to online auctions. An incredible variety of goods and services are auctioned on the internet. According to Vulkan (2003), online auctions can be divided into three major types: consumer-to-consumer, business-to-consumer, and business-to-business auctions. Consumer-to-consumer (C2C) auctions, such as ebay, involve an individual as the auctioneer and many other individuals as bidders. Business-to-consumer (B2C) auctions, such as onsale, are characterized by a firm as the auctioneer and many individual consumers as bidders. Business-to-business (B2B) auctions, like Moai, include a firm as the auctioneer (often a material or components supplier) and many other firms as bidders. We focus our discussion on C2C type online auctions due to its popularity on the internet. In terms of the auction mechanism, online auctions can be classified into four types: English, Dutch, first-price sealed-bid, and second-price sealed-bid auctions. In an English auction, the price of a good is continuously raised until only one bidder remains. After the auctioneer announces the starting price for an online

4 4 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh English auction, bidders show their willingness to buy the good by electronically submitting bids at or above the current price (McAfee and McMillan, 1987). Each bidder knows the price of the current bid at any point in time during the auction process. Most auction sites send outbid notification messages to inform a bidder that another bidder has submitted a higher bid. Usually online English auctions last longer than their offline counterparts; a typical online English auction can last from a few hours to a couple days. English auctions can be further divided into one-unit auctions and multi-unit auctions. For multi-unit English auctions, each winning bidder either pays the amount of his own bid or the amount of the lowest accepted bid, depending on the rule adopted by the auctioneers. In a Dutch auction, the auctioneer sets a high initial price and then lowers the price until a bidder accepts the current price (McAfee and McMillan, 1987). In a first-price sealed-bid auction, bidders submit sealed bids and the winning bidder pays the amount submitted. The difference between a first-price sealed-bid auction and an English auction is that, in an English auction, bidders are able to observe rival bids and adjust their own bids accordingly while in a sealed-bid auction they can t. Typically, in sealed-bid auctions each bidder is allowed to submit only one bid (McAfee and McMillan, 1987). In a second-price sealed-bid auction, bidders submit sealed bids. They are told that the highest bidder wins the item but pays a price equal to the second-highest bid. This type of auction is also called a Vickery auction. Bidding prices are very close to market valuations of items in this type of auction. In addition to these basic types of auctions, there are other innovative types of online auctions. For example, in a name your price auction, a buyer lists his product requirement and the maximum price he is willing to pay, then potential sellers post their bids for this buyer. Since they are one of the most commonly used formats on the internet, this paper proposes a final price prediction model for online English auctions. 3 The Markov process model Markov process theory was developed by the Russian mathematician A. A. Markov. The theory provides a foundation for modeling a stochastic process whose one-period-ahead state depends solely on its current state and is completely

5 A Simulation-Based Model for Final Price Prediction in Online Auctions 5 independent of its state in previous periods. 3.1 Some Markov process concepts A Markov process can adequately model uncertainty in many real word situations that evolve dynamically over time. In practical applications, we seek an adequate state description such that the associated stochastic process has the Markovian property that the knowledge of the present state is sufficient to predict the next-period state of the process. Segev et al. (2001) model an online auction in terms of a Markov process on a state space defined by the current price of the auctioned item and the number of bidders that were previously bumped. With some approximations, their approach first converts an online auction into a small- to medium-sized theoretical auction. Then the transition probability matrix of system states is derived and the price trajectory of the small-scale Markov process is obtained. Finally, the final price prediction can be determined. Table 1 presents notation for the Markov process. Variable Definition Table 1: Definitions of variables A R c p L t The auction. Reserve price of the auction. Minimum bid increment of the auction. Going price of the auction. This is the lowest bid needed to become the new winner. Number of bidders in the orbit queue. Time index. L max Maximum probable number of bidders in the orbit queue. This variable changes with p. λ μ Arrival rate of new bidders. Arrival rate of bidders who awakens from the orbit queue and revisit the auction site. v i The valuation of the ith bidder. All v i are drawn from the same CDF. F( v i ) The CDF of bidder valuations for the item. F( v i ) The survivor function of bidder valuations for the item, or F( vi) = 1 F( vi).

6 6 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh The sequence of auction events in the Markov process model is as follows: (1) New bidders arrive at the auction site. (2) New bidders view the auction and see the current price p. Each bidder has two choices: submit a bid for p and become the new current winner or drop out of the auction. (3) If a bidder is unwilling to pay the current price (i.e., if v i < p ), he or she drops out of the auction. (4) If a bidder is willing to pay at least this price (i.e., if v i p ), he or she submits a bid for p. (5) If a new bidder submits a bid, the auction records the new current winner and updates the current price to p+ c. The previous winner is bumped into an orbit queue containing all previous bidders who were bumped. (6) A previous winner from the orbit queue awakens and revisits the auction site and sees the new current price, restarting the cycle. The state of an auction is denoted A t ( p, L), where t represents the time index, p the current price, and L the number of bidders in the orbit queue. Assume that the arrival process of new bidders follows a Poisson process with parameter λ and the arrival process of bidders awakening from the orbit queue follows a non-stationary Poisson process with parameter μ L. Transition probabilities of different states are given in Table 2. Table 2: Transition probabilities of different states Transition Event Probability A ( p, L) A ( p, L) 1 t t+ A( p, L) A ( p+ c, L+ 1) 1 t t + A new bidder arrives at the site, sees the auction price p, and decides to drop out of the auction. A new bidder arrives at the site, sees the auction price p, and decides to submit a bid p. λ ( F( p)) L λ + μ λ ( F ( p)) L λ + μ A ( p, L) A + 1( p, L 1) t t An old bidder awakens from the orbit queue, sees the auction price p, and decides to drop out of the auction. μ L * ( F ( p)) λ + μl At ( p, L) At + 1( p c, L) auction price p, and decides to submit a bid p. An old bidder awakens from the orbit queue, sees the + μ L * ( F ( p)) λ + μl

7 A Simulation-Based Model for Final Price Prediction in Online Auctions 7 In Table 2, F ( p) is the cumulative distribution function (CDF) of new bidder valuations and is assumed to follow a uniform distribution with lower and upper limits a and b. In Table 2, F ( p) Q is the conditional CDF used for the posterior CDF of the bidder in the orbit queue. For convenience, we replace the * exact expression F ( p) Q with F ( p) because we can t keep track of each * bidder in the orbit queue individually. The calculation of F ( p) is as follows: F ( p) = ( L 1 L ) F ( p 1) + (1 L ) F( v v p 1) * * max max max i i This equation is a weighted average of the CDFs of all previous occupants in the queue and the posterior CDF of the latest arrival to the queue. Using the transition probabilities formulas in Table 2, we can construct the one-step transition probability matrix P. The simulation results of the online auction Markov process model can be obtained according to the equation P n = P P, where n represents the number of transitions. In other words, the n 1 price trajectory can be obtained through the transition matrix. However, applying the Markov process model to predict the final price in practice, we need to specify some parameters beforehand. These parameters include the lower and upper limits of what bidders are willing to pay, the arrival rate for new bidders, and the arrival rate for bidders awakening from the orbit queue. Next we describe how the parameters are jointly estimated. 3.2 Estimating the parameters Lower and upper limits of what bidders are willing to pay Bidder willingness to pay is assumed to follow a uniform distribution on the interval [ ab., ] The lower and upper limits, a and b, are assumed to be the starting bid and the market value of the item in each auction. In our empirical study, the market value of each auctioned digital camera is obtained from prices posted in professional digital camera retail websites. We choose the market value of an auctioned item as the upper limit because the highest bid for a rational bidder will not exceed the market value.

8 8 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh Arrival rate of new bidders Recall that the new bidders are assumed to arrive at the auction site following a Poisson process with parameter λ. To estimate this parameter, we observe intervals between bids and find the average number of arrivals in an hour from the bidding history records of Yahoo! Auction Arrival rate from the orbit queue The arrival process awakening from the orbit queue is assumed to follow a non-stationary Poisson process with parameter μ L. From the Yahoo! bidding history records, we find those bidders who submit bids more than once. These bidders are viewed as the bidders awakening from the orbit queue. As for λ, we use observed intervals between bids awakening from the orbit queue to estimate μ. 4 The simulation-based model Although the Markov process model provides a theoretical framework for estimating the final price of an online auction, there are several difficulties in implementing this model and in estimating the parameters in particular. To avoid the difficulties in estimating the Markov process model, we propose a different approach to attack this problem. We utilize the real bidding history of past auctions to build an empirical probability distribution for arrival rates and bid increments and simulate the auction process by sampling from the constructed distribution many times to get the final price distribution. This process is developed based on the logic of bootstrapping (Mooney and Duval, 1993), which is different from the traditional parametric approach. Bootstrapping uses a large number of samples to form the shape of a statistic s sampling distribution rather than strong distributional assumptions and analytic formulas. 4.1 Experimental setting The experimental setting of the simulation-based model in this study tries to

9 A Simulation-Based Model for Final Price Prediction in Online Auctions 9 simulate the real world scenario as faithfully as possible. We assume that there is only one item to be auctioned to n bidders who will submit their bids to the auctioneer. The auctioneer will award the item to the highest bidder at closing time. Each bidder can repeatedly submit new bids until the auction is closed. Let R denote the latest bid, T the time elapsed since the auction started, t j the time interval between bid j 1 and bid j, c the bid increment of bidder j, d the j duration of the auction, and P m the market price. The arrival time between two bids and the bid increment can be collected from the real process to form an empirical joint probability distribution F ( t i, c ) as j shown in Figure 1. It can be seen that most of the time intervals fall within one hour and the bid increments fall in the range 0 to 90. Figure 1: The joint distribution of time intervals and bid increments The stopping criterion of an auction simulation in many studies is set at a fixed number of bidding events (Lee et al., 2001; Mizuta and Steiglitz, 2000; Kim, 2004). Instead of following the same criterion, we adopt two different stopping rules to better match reality. The first criterion is when the sum of sequential interval times exceeds the pre-determined duration ( d ) of the auction. The second criterion is when the latest bidding price ( R ) exceeds the market price ( P m ). A brief description

10 10 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh of the simulation process is presented below. 4.2 Simulation process The simulation process proceeds as follows. Step 1: An auction begins with some parameters set by the auctioneer, such as the description of the item, starting price, bid increment, reserve price, and starting and closing times. Step 2: Two joint empirical distributions of inter-arrival times and bid increments are constructed for newly arriving bidders and for bidders awakening from the orbit queue from historical data. Variables R and T are initialized at the starting price and zero. Step 3: The simulation proceeds by drawing the first bidder randomly from the distribution obtained from the new arrivals and setting T = T + t. j R = R + c and Step 4: A random draw simulates a new arrivals or a bidder awakening from the orbit queue, updating R + = R c and j T T + t j =. Step 5: The simulation terminates when either stopping criteria is reached (i.e., T d or R Pm ); otherwise the process returns to Step 4. Following the simulation outline, a final price for an online auction is obtained. The process is repeated 1000 times for each auction. The average predicted final price is then calculated to serve as the predicted final price for the auction. Simulated 95% confidence intervals can be constructed at the same time. j 5 Simulation results We collect data from 100 online auctions for digital cameras from the Yahoo! Auction website. All auctions occurred between December 3, 2004, and January 20, For each auction, the collected data includes all intervals between bids, price increments, starting price, reserve price, product description, and number of bids. For each auction, the Markov process model and the simulation-based model are employed to predict the final price. The simulation results are shown in Table 3. Among the 100 auctions, 8 cases are ties and the proposed simulation-based model

11 A Simulation-Based Model for Final Price Prediction in Online Auctions 11 outperforms than the Markov process model in 62 auctions (i.e., the percentage error is smaller) and underperforms in 30 auctions. The average percentage error is 16.89% for the Markov model and 5.0% for proposed simulation model. A paired t-test for the percentage error for these two models is significant with p-value less than It can be seen that the proposed simulation approach outperforms the Markov model in terms of prediction accuracy. We believe that the accuracy can be further improved if the number of simulations is increased. Table 3: Simulation results Actual Markov model Simulation model Auction a b λ μ price (% error) (% error) A % 17.38% A % 7.69% A % 14.76% A % 0.00% A % 0.00% A % 6.75% A % 2.55% A % 6.83% A % 0.28% A % 0.61% A % 0.48% A % 8.19% A % 6.26% A % 0.17% A % 3.14% A % 6.70% A % 0.27% A % 0.96% A % 9.85% A % 1.22% A % 3.37% A % 0.00% A % 0.59% A % 0.36% A % 35.51% A % 31.81%

12 12 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh Table 3: Simulation results (continued) Actual Markov model Simulation model Auction a b λ μ price (% error) (% error) A % 1.88% A % 11.78% A % 4.60% A % 49.60% A % 1.79% A % 0.00% A % 11.36% A % 1.54% A % 25.52% A % 28.95% A % 0.00% A % 6.31% A % 2.09% A % 0.05% A % 0.16% A % 0.00% A % 0.00% A % 16.22% A % 7.82% A % 16.51% A % 7.24% A % 3.10% A % 18.29% A % 22.44% A % 1.17% A % 30.43% A % 3.67% A % 3.04% A % 7.63% A % 1.44% A % 0.00% A % 1.29% A % 5.75% A % 19.20%

13 A Simulation-Based Model for Final Price Prediction in Online Auctions 13 Table 3: Simulation results (continued) Actual Markov model Simulation model Auction a b λ μ price (% error) (% error) A % 9.34% A % 28.34% A % 7.96% A % 2.06% A % 4.08% A % 14.25% A % 5.97% A % 3.11% A % 6.71% A % 7.00% A % 7.79% A % 3.27% A % 12.91% A % 47.06% A % 7.06% A % 1.19% A % 9.72% A % 4.00% A % 0.00% A % 17.65% A % 5.43% A % 0.69% A % 18.37% A % 22.60% A % 1.45% A % 44.43% A % 0.98% A % 0.56% A % 6.09% A % 12.70% A % 1.53% A % 2.47% A % 28.38% A % 0.66%

14 14 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh Table 3: Simulation results (continued) Actual Markov model Simulation model Auction a b λ μ price (% error) (% error) A % 2.78% A % 11.23% A % 18.50% A % 5.72% A % 52.64% A % 12.69% Average 16.89% 5.00% The inferior performance of the Markov process model and its consistent lower prediction compared to the actual final price (see Figure 2) may be due to the inappropriate estimate of the arrival rate λ. Since the true value of λ should include two parts, one for new bidders who place bids and another for new arrivals who drop out of the auction (which cannot be observed in the data), λ is underestimated. Percentage of error 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% % % % % % Markov No. of Auction Simulation Figure 2: Comparison of percent error: Markov and simulation-based models

15 A Simulation-Based Model for Final Price Prediction in Online Auctions 15 6 Conclusion The final price in an online auction results from a complex decision making process determined by interactions between the auctioneer and many bidders. A Markov process model provides a mathematical approach to final price prediction. However, the parameters in the Markov process model are difficult to estimate in practice. We propose a simulation-based model as an alternative to the Markov process model. Simulation results demonstrate that the proposed simulation-based model can predict the final price more accurately than the Markov model does. Since the real arrival process does not show a decreasing trend as assumed in the Markov process model and since we do not consider this phenomenon in our simulation-based model, the accuracy of both models can probably be improved if the arrival rate assumption is appropriately modified. In particular, a different joint distribution can be constructed for different stages in the auction in the two models. In addition, current econometric research is investigating relating website mechanisms to arrival rate and other parameter estimation. References Lee, A., M. Lyu and I. King, (2001), Agent-Based Multimedia Data Sharing Platform, Proceedings of the International Symposium on Information Systems and Engineering, Las Vegas. Kim, Y., (2004), An Optimal Auction Infrastructure Design: An Agent-Based Simulation Approach, Proceedings of the Tenth Americas Conference on Information System, McAfee, R. P. and J. McMillan, (1987), Auctions and Bidding, Journal of Economic Literature, 25, Mizuta, H. and K. Steiglitz, (2000), Agent-Based Simulation of Dynamic Online Auctions, Proceedings of Winter Simulation Conference. Mooney, C. Z. and R. D. Duval, (1993), Bootstrapping: A Nonparametric Approach to Statistical Inference, Newbury Park: Sage. Segev, A., C. Beam and J. Shanthikumar, (2001), Optimal Design of Internet-Based Auctions, Information Technology and Management, 2,

16 16 Shihyu Chou, Chin-Shien Lin, Chi-hong Chen, Tai-Ru Ho and Yu-Chen Hsieh Vakrat, Y. and A. Seidmann, (2000), Implications of the Bidders Arrival Process on the Design of Online Auctions, Proceedings of the 33rd Hawaii International Conference on System Sciences. Vulkan, N., (2003), The Economics of E-Commerce: A Strategic Guide to Understanding and Designing the Online Marketplace, Princeton University Press.