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Optimal digital product auctions with unlimited supply and rebidding behavior

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Abstract

We consider a digital product seller who needs to determine the number of items to sell and its price over an infinite horizon. The seller indeed owns unlimited supply of the digital products like music or software. Each buyer is risk-neutral and needs one unit of the product. The number of buyers in each period and their private valuations are random. The seller conducts an auction to allocate the items in each period. The buyers who fail to gain one item in the previous periods will rebid in the subsequent auctions. Regarding the formulated dynamic program, we prove that the optimal allocation solution is a variant of the base-stock policy. Based on the known Revenue Equivalence Principle, we also prove that the generalized second-price auction and the first-price auction will result in the same expected revenue for the seller. Finally, with mild technical modifications, the results of the infinite-horizon case can be extended to the finite-horizon case even if the demand is time-varying stochastic and independent.

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References

  • Akter, S., Michael, K., Uddin, M. R., McCarthy, G., & Rahman, M. (2021). Transforming business using digital innovations: The application of AI, blockchain, cloud and data analytics. Annals of Operations Research. https://doi.org/10.1007/s10479-020-03620-w

    Article  Google Scholar 

  • Andreoni, J., & Sprenger, C. (2012). Risk preferences are not time preferences. American Economic Review, 102(7), 3357–3376.

    Article  Google Scholar 

  • Avinadav, T., Chernonog, T., & Perlman, Y. (2014). Analysis of protection and pricing strategies for digital products under uncertain demand. International Journal of Production Economics, 158, 54–64.

    Article  Google Scholar 

  • Beil, D. R., & Wein, L. M. (2003). An inverse-optimization-based auction mechanism to support a multiattribute RFQ process. Management Science, 49(11), 1529–1545.

    Article  Google Scholar 

  • Bertsekas, D. P. (1987). Dynamic Programming: Deterministic and Stochastic Models. Prentice-Hall.

    Google Scholar 

  • Caldentey, R., & Vulcano, G. (2007). Online auction and list price revenue management. Management Science, 53(5), 795–813.

    Article  Google Scholar 

  • Cheng, M., Xu, S. X., & Huang, G. Q. (2016). Truthful multi-unit multi-attribute double auctions for perishable supply chain trading. Transportation Research Part E, 93, 21–37.

    Article  Google Scholar 

  • Chernonog, T., Avinadav, T., & Ben-Zvi, T. (2019). How to set price and quality in a supply chain of virtual products under bi-criteria and risk consideration. International Journal of Production Economics, 209, 156–163.

    Article  Google Scholar 

  • Clarke, E. H. (1971). Multipart pricing of public goods. Public Choice, 11(1), 17–33.

    Article  Google Scholar 

  • Conner, K. R., & Rumelt, R. P. (1991). Software Piracy: An analysis of protection strategies. Management Science, 37(2), 125–139.

    Article  Google Scholar 

  • Etzion, H., Pinker, E., & Seidmann, A. (2006). Analyzing the simultaneous use of auctions and posted prices for online selling. Manufacturing and Service Operations Management, 8(1), 68–91.

    Article  Google Scholar 

  • Goldberg, A. V., Hartline, J. D., Karlin, A. R., Saks, M., & Wright, A. (2006). Competitive auctions. Games and Economic Behavior, 55(2), 242–269.

    Article  Google Scholar 

  • Groves, T. (1973). Incentives in teams. Econometrica, 41(4), 617–631.

    Article  Google Scholar 

  • Herings, P. J. J., Peeters, R., & Yang, M. S. (2018). Piracy on the Internet: Accommodate it or fight it? A dynamic approach. European Journal of Operational Research, 266(1), 328–339.

    Article  Google Scholar 

  • Hotelling, H. (1929). Stability in competition. The Economic Journal, 39(153), 41–57.

    Article  Google Scholar 

  • Huang, Y. S., Lin, S. H., & Fang, C. C. (2017). Pricing and coordination with consideration of piracy for digital goods in supply chains. Journal of Business Research, 77, 30–40.

    Article  Google Scholar 

  • Khouja, M., & Rajagopalan, H. K. (2009). Can piracy lead to higher prices in the music and motion picture industries? Journal of the Operational Research Society, 60(3), 372–383.

    Article  Google Scholar 

  • Khouja, M., & Smith, M. A. (2007). Optimal pricing for information goods with piracy and saturation effect. European Journal of Operational Research, 176(1), 482–497.

    Article  Google Scholar 

  • Koçyiğit, Ç., Bayrak, H. I., & Pınar, M. Ç. (2018). Robust auction design under multiple priors by linear and integer programming. Annals of Operations Research, 260(1–2), 233–253.

    Article  Google Scholar 

  • Kogan, K., Ozinci, Y., & Perlman, Y. (2013). Containing piracy with product pricing, updating and protection investments. International Journal of Production Economics, 144(2), 468–478.

    Article  Google Scholar 

  • Liu, Y., Cheng, H., Tang, Q., & Eryarsoy, E. (2011). Optimal software pricing in the presence of piracy and word-of-mouth effect. Decision Support Systems, 51(2), 99–107.

    Article  Google Scholar 

  • Liu, Z., Li, M., & Kou, J. (2015). Selling information products: Sale channel selection and versioning strategy with network externality. International Journal of Production Economics, 166, 1–10.

    Article  Google Scholar 

  • Lucking-Reiley, D. (2000). Auctions on the Internet: What’s being auctioned, and how? Journal of Industrial Economics, 48(3), 227–252.

    Article  Google Scholar 

  • Lucking-Reiley, D., Bryan, D., Prasad, N., & Reeves, D. (2007). Pennies from eBay: The determinants of price in online auctions. Journal of Industrial Economics, 55(2), 223–233.

    Article  Google Scholar 

  • Maskin, E., & Riley, J. (1989). Optimal multi-unit auctions. In F. Hahn (Ed.), The Economics of Missing Markets, Information, and Games (pp. 312–335). Oxford University Press.

    Google Scholar 

  • Mata, R., Frey, R., Richter, D., Schupp, J., & Hertwig, R. (2018). Risk preference: A view from psychology. Journal of Economic Perspectives, 32(2), 155–172.

    Article  Google Scholar 

  • McAfee, R. P., & McMillan, J. (1987). Auctions with a stochastic number of bidders. Journal of Economic Theory, 43(1), 1–19.

    Article  Google Scholar 

  • Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations Research, 6(1), 58–73.

    Article  Google Scholar 

  • Ning, Y., Xu, S. X., Yan, M., & Huang, G. Q. (2018). Digital pricing with piracy and variety seeking. International Journal of Production Economics, 206, 184–195.

    Article  Google Scholar 

  • Peitz, M., & Waelbroeck, P. (2006). Why the music industry may gain from free downloading—the role of sampling. International Journal of Industrial Organization, 24(5), 907–913.

    Article  Google Scholar 

  • Pinker, E., Seidmann, A., & Vakrat, Y. (2010). Using bid data for the management of sequential, multi-unit, online auctions with uniformly distributed bidder valuations. European Journal of Operational Research, 202(2), 574–583.

    Article  Google Scholar 

  • Qian, X., Fang, S. C., Huang, M., An, Q., & Wang, X. (2018). Reverse auctions with regret-anticipated bidders. Annals of Operations Research, 268(1–2), 293–313.

    Article  Google Scholar 

  • Riley, J., & Samuelson, W. (1981). Optimal auctions. American Economics Review, 71(3), 381–392.

    Google Scholar 

  • Shaked, M., & Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press.

    Google Scholar 

  • Taleizadeh, A. A., Noori-daryan, M., Soltani, M. R., & Askari, R. (2021). Optimal pricing and ordering digitals goods under piracy using game theory. Annals of Operations Research. https://doi.org/10.1007/s10479-021-04036-w

    Article  Google Scholar 

  • UNCTAD. (2019). Digital economy report 2019. United Nations Conference on Trade and Development (UNCTAD), 2019. https://unctad.org/en/PublicationsLibrary/tdr2019_en.pdf

  • van Ryzin, G., & Vulcano, G. (2004). Optimal auctioning and ordering in an infinite horizon inventory-pricing system. Operations Research, 52(3), 346–367.

    Article  Google Scholar 

  • Varian, H. R., & Harris, C. (2014). The VCG auction in theory and practice. American Economic Review, 104(5), 442–445.

    Article  Google Scholar 

  • Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16(1), 8–37.

    Article  Google Scholar 

  • Vulcano, G., van Ryzin, G., & Maglaras, C. (2002). Optimal dynamic auctions for revenue management. Management Science, 48(11), 1388–1407.

    Article  Google Scholar 

  • Waters, J. (2015). Welfare implications of piracy with dynamic pricing and heterogeneous consumers. European Journal of Operational Research, 240(3), 904–911.

    Article  Google Scholar 

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Acknowledgements

The authors would like to express sincerest thanks to the editors and reviewers of this paper for their constructive comments and insightful suggestions. The study was conducted with the support of the National Natural Science Foundation of China under Grant 71991461 and 72071093, Guangdong Province Soft Science Research Project under Grant 2019A101002074, the Project for Philosophy and Social Sciences Research of Shenzhen City under Grant SZ2021B014, and the 2019 Guangdong Special Support Talent Program–Innovation and Entrepreneurship Leading Team (China) under Grant 2019BT02S593.

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Authors and Affiliations

  1. School of Business Administration, South China University of Technology, Guangzhou, China

    Yu Ning

  2. School of Intelligent Systems Science and Engineering, Jinan University, Guangzhou, China

    Su Xiu Xu

  3. Department of Industrial and Manufacturing Systems Engineering, The University of Hong Kong, Hong Kong, China

    George Q. Huang

  4. College of Management, Shenzhen University, Shenzhen, China

    Xudong Lin

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  1. Yu Ning
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  2. Su Xiu Xu
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Correspondence to Su Xiu Xu.

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Ning, Y., Xu, S.X., Huang, G.Q. et al. Optimal digital product auctions with unlimited supply and rebidding behavior. Ann Oper Res 307, 399–416 (2021). https://doi.org/10.1007/s10479-021-04245-3

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