The Annals of Applied Statistics

Optimal pricing using online auction experiments: A Pólya tree approach

Edward I. George and Sam K. Hui

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We show how a retailer can estimate the optimal price of a new product using observed transaction prices from online second-price auction experiments. For this purpose we propose a Bayesian Pólya tree approach which, given the limited nature of the data, requires a specially tailored implementation. Avoiding the need for a priori parametric assumptions, the Pólya tree approach allows for flexible inference of the valuation distribution, leading to more robust estimation of optimal price than competing parametric approaches. In collaboration with an online jewelry retailer, we illustrate how our methodology can be combined with managerial prior knowledge to estimate the profit maximizing price of a new jewelry product.

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Ann. Appl. Stat., Volume 6, Number 1 (2012), 55-82.

First available in Project Euclid: 6 March 2012

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Bayesian nonparametrics Pólya tree distribution second-price auctions internet auctions optimal pricing


George, Edward I.; Hui, Sam K. Optimal pricing using online auction experiments: A Pólya tree approach. Ann. Appl. Stat. 6 (2012), no. 1, 55--82. doi:10.1214/11-AOAS503.

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  • Adams, C. P. (2007). Estimating Demand from eBay prices. International Journal of Industrial Organization 25 1213–1232.
  • Baldwin, L. H., Marshall, R. C. and Richard, J.-F. (1997). Bidder collusion at forest timber sales. Journal of Political Economy 105 657–699.
  • Bapna, R., Chang, S. A., Goes, P. and Gupta, A. (2009). Overlapping online auctions: Empirical characterization of bidder strategies and auction prices. MIS Quarterly 33 763–783.
  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
  • Boatwright, P., Borle, S. and Kadane, J. B. (2010). Common value vs. private value categories in online auctions: A distinction without a difference? Decision Analysis 7 86–98.
  • Bradlow, E. T. and Park, Y. H. (2007). Bayesian estimation of bid sequences in internet auctions using a generalized record-breaking model. Marketing Science 26 218–229.
  • Braun, M., Fader, P. S., Bradlow, E. T. and Kunreuther, H. (2006). Modeling the “Pseudodeductible” in insurance claims decision. Management Science 52 1258–1272.
  • Breidert, C. (2006). Estimation of willingness-to-pay: Theory, measurement, application, DUV.
  • Brezger, A. and Steiner, W. J. (2008). Monotonic regression based on Bayesian P-splines: An application to estimating price response functions from store-level scanner data. J. Bus. Econom. Statist. 26 90–104.
  • Canals-Cerda, J. and Pearcy, J. (2010). Arriving in time: Estimation of english auctions with a stochastic number of bidders. Working paper. Available at
  • Casella, G. and Berger, R. L. (2001). Statistical Inference, 2nd ed. Duxbury, Pacific Grove, CA.
  • Chakravarti, D., Greenleaf, E., Sinha, A., Cheema, A., Box, J. C., Friedman, D., Ho, T. H., Issac, R. M., Mitchell, A. A., Rapoport, A., Rothkopf, M. H., Srivastava, J. and Zwick, R. (2002). Auctions: Research opportunities in marketing. Marketing Letters 13 281–296.
  • Chan, T. Y., Kadiyali, V. and Park, Y.-H. (2007). Willingness to pay and competition in online auctions. Journal of Marketing Research 44 324–333.
  • Dey, D., Müller, P. and Sinha, D., eds. (1998). Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statistics 133. Springer, New York.
  • Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2 615–629.
  • George, E. and Hui, S. (2011). Supplement to “Optimal pricing using online auction experiments: A Pólya tree approach.” DOI:10.1214/11-AOAS503SUPP.
  • Green, P. E. and Srinivasan, V. (1978). Conjoint analysis in consumer research: Issues and outlook. Journal of Consumer Research 5 103–123.
  • Haruvy, E., Leszczyc, P., Carare, O., Cox, J. C., Greenleaf, E. A., Jank, W., Jap, S., Park, Y.-H. and Rothkopf, M. H. (2008). Competition between auctions. Marketing Letters 19 431–448.
  • Hou, J. and Rego, C. (2007). A classification of online bidders in a private value auction: Evidence from eBay. International Journal of Electronic Marketing and Retailing 1 322–338.
  • Houser, D. and Wooders, J. (2006). Reputation in auctions: Theory, and evidence from eBay. Journal of Economics and Management Strategy 15 353–369.
  • Jank, W. and Zhang, S. (2011). An automated and data-driven bidding strategy for online auctions. Informs J. Comput. 23 238–253.
  • Kim, J. G., Menzefricke, U. and Feinberg, F. (2004). Assessing heterogeneity in discrete choice models using a Dirichlet process prior. Review of Marketing Science 2 1–39.
  • Kim, J. G., Menzefricke, U. and Feinberg, F. (2007). Capturing flexible heterogeneous utility curves: A Bayesian spline approach. Management Science 53 340–354.
  • Klemperer, P. (1999). Auction theory: A guide to the literature. Journal of Economic Surveys 13 227–286.
  • Laffont, J.-J. and Vuong, Q. (1996). Structural analysis of auction data. American Economic Review 86 414–420.
  • Lavine, M. (1992). Some aspects of Pólya tree distributions for statistical modelling. Ann. Statist. 20 1222–1235.
  • Lavine, M. (1994). More aspects of Pólya tree distributions for statistical modelling. Ann. Statist. 22 1161–1176.
  • Mauldin, R. D., Sudderth, W. D. and Williams, S. C. (1992). Pólya trees and random distributions. Ann. Statist. 20 1203–1221.
  • Mitchell, R. C. and Carson, R. T. (1989). Using Surveys to Value Public Goods: The Contingent Valuation Method. RFF Press, Washington, DC.
  • Muliere, P. and Walker, S. (1997). A Bayesian non-parametric approach to survival analysis using Polya trees. Scand. J. Statist. 24 331–340.
  • Ockenfels, A. and Roth, A. E. (2006). Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction. Games Econom. Behav. 55 297–320.
  • Paddock, S. M. (2002). Bayesian nonparametric multiple imputation of partially observed data with ignorable nonresponse. Biometrika 89 529–538.
  • Paddock, S. M., Ruggeri, F., Lavine, M. and West, M. (2003). Randomized Pólya tree models for nonparametric Bayesian inference. Statist. Sinica 13 443–460.
  • Park, Y.-H. and Bradlow, E. T. (2005). An integrated model for bidding behavior in internet auctions: Whether, who, when, and how much. Journal of Marketing Research 42 470–482.
  • Rasmusen, E. B. (2006). Strategic implications of uncertainty over one’s own private value in auctions. Adv. Theor. Econ. 6 Art. 7, 24 pp. (electronic).
  • Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd ed. Springer, New York.
  • Roth, A. E. and Ockenfels, A. (2002). Last-minute bidding and the rules for ending second-price auctions: Evidence from eBay and amazon auctions on the internet. American Economic Review 92 1093–1103.
  • Song, U. (2004). Nonparametric estimation of an eBay auction model with an unknown number of bidders. Working paper, Univ. British Columbia, Vancouver.
  • Sood, A., James, G. and Tellis, G. (2009). Functional regression: A new model for predicting market penetration of new products. Marketing Science 28 36–51.
  • Tellis, G. J. (1986). Beyond the many faces of price: An integration of pricing strategies. Journal of Marketing 50 146–160.
  • Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance 16 8–37.
  • Walker, S. G., Damien, P., Laud, P. W. and Smith, A. F. M. (1999). Bayesian nonparametric inference for random distributions and related functions. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 485–527.
  • Wong, W. H. and Ma, L. (2010). Optional Pólya tree and Bayesian inference. Ann. Statist. 38 1433–1459.
  • Yao, S. and Mela, C. F. (2008). Online auction demand. Marketing Science 27 861–885.
  • Zeithammer, R. (2006). Forward-looking bidding in online auctions. Journal of Marketing Research 43 462–476.

Supplemental materials

  • Supplementary material: Web Appendix for “Optimal pricing using online auction experiments: A Pólya tree approach”. Robustness checks for the left telescoping hierarchy and the IPV assumption can be found in the supplemental article.