Dynamic pricing with multiple products and partially specified demand distribution

Mathematics of Operations Research

Volumn 39, Issue 3, 2014, Pages 863-888

  Den Boer, Arnoud V ^a  

^a UNIVERSITY OF TWENTE   (Netherlands)

Author keywords

Marketing: Estimation statistical techniques; Pricing; Statistics: Estimation

Indexed keywords

DISPERSIONS; ECONOMICS; EIGENVALUES AND EIGENFUNCTIONS; OPTIMIZATION; PARAMETER ESTIMATION; SALES;

CURRENT PARAMETERS; DEMAND DISTRIBUTION; EXPLOITATION AND EXPLORATIONS; PARAMETER ESTIMATE; PARAMETERS UNKNOWN; QUADRATIC CONSTRAINT; SMALLEST EIGENVALUE; STATISTICAL ESTIMATION;

COSTS;

EID: 84906674322     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.2013.0636     Document Type: Article

References (70)

1. Abbasi-Yadkori Y, Szepesvári C(2011) Regret bounds for the adaptive control of linear quadratic systems. Kakade S, von Luxburg U, eds. 24th Annual Conf. Learn. Theory 4COLT5, Vol. 19, 1-26.
2. Abbasi-Yadkori Y, Pal D, Szepesvári C(2011) Improved algorithms for linear stochastic bandits. Shawe-Taylor J, Zemel RS, Bartlett P, Pereira FCN, Weinberger KQ, eds. Advances in Neural Information Processing Systems 24, 2312-2320.
3. Araman VF, Caldentey R (2009) Dynamic pricing for nonperishable products with demand learning. Oper. Res. 57(5): 1169-1188.
4. Arora S, Hazan E, Kale S (2012) The multiplicative weights update method: A meta-algorithm and applications. Theory Comput. 8: 121-164.
5. Auer P, Cesa-Bianchi N, Fischer P (2002) Finite-time analysis of the multi-armed bandit problem. Machine Learn. 47(2): 235-256.
6. Auer P, Ortner R, Szepesvári C(2007) Improved rates for the stochastic continuum-armed bandit problem. Bshouty N, Gentile C, eds. Learning Theory, Lecture Notes in Computer Science, Vol. 4539 (Springer, Berlin), 454-468.
7. Aviv Y, Pazgal A (2005) A partially observed Markov decision process for dynamic pricing. Management Sci. 51(9): 1400-1416.
8. Bartlett MS (1951) An inverse matrix adjustment arising in discriminant analysis. Ann. Math. Statist. 22(1): 107-111.
9. Bartlett PL, Tewari A (2009) REGAL: A regularization based algorithm for reinforcement learning in weakly communicating MDPs. Proc. Twenty-Fifth Conf. Uncertainty in Artificial Intelligence UAI '09 (AUAI Press, Arlington, VA), 35-42.
10. Baum LE, Katz M (1965) Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120(1): 108-123.
11. Bertsimas D, Perakis G (2006) Dynamic pricing: A learning approach. Mathematical and Computational Models for Congestion Charging (Springer, New York), 45-79.
12. Besbes O, Zeevi A (2009) Dynamic pricing without knowing the demand function: Risk bounds and near-optimal algorithms. Oper. Res. 57(6): 1407-1420.
13. Besbes O, Zeevi A (2011) On the minimax complexity of pricing in a changing environment. Oper. Res. 59(1): 66-79.
14. Boyd S, Vandenberghe L (2004) Convex Optimization (Cambridge University Press, New York).
15. Broder J, Rusmevichientong P (2012) Dynamic pricing under a general parametric choice model. Oper. Res. 60(4): 965-980.
16. Bubeck S, Cesa-Bianchi N (2012) Regret analysis of stochastic and non-stochastic multi-armed bandit problems. Foundations Trends Machine Learn. 58(1): 1-122.
17. Carvalho AX, Puterman ML (2005) Dynamic optimization and learning: How should a manager set prices when the demand function is unknown? Discussion Papers 1117, Instituto de Pesquisa Economica Aplicada-IPEA, Brasilia.
18. Carvalho AX, Puterman ML (2005) Learning and pricing in an internet environment with binomial demand. J. Revenue Pricing Management 3(4): 320-336.
19. Chang YI (1999) Strong consistency of maximum quasi-likelihood estimate in generalized linear models via a last time. Statist. Probab. Lett. 45(3): 237-246.
20. Clausen J (1999) Branch and bound algorithms-Principles and examples, http://janders.eecg.toronto.edu/1387/readings/b-and-b.pdf.
21. Cope E (2007) Bayesian strategies for dynamic pricing in e-commerce. Naval Res. Logist. 54(3): 265-281.
22. Cope EW (2009) Regret and convergence bounds for a class of continuum-armed bandit problems. IEEE Trans. Automatic Control 54(6): 1243-1253.
23. Dani V, Hayes TP, Kakade SM (2008) Stochastic linear optimization under bandit feedback. 21st Annual Conf. Learn. Theory (COLT) (Omnipress, Madison, WI), 355-366.
24. Davis JA (1968) Convergence rates for probabilities of moderate deviations. Ann. Math. Statist. 39(6): 2016-2028.
25. den Boer AV (2013) Dynamic pricing and learning: Historical origins, current research, and new directions. Working paper, http://ssrn.com/ abstractD2334429.
26. den Boer AV, Zwart B (2011) Mean square convergence rates for maximum quasi-likelihood estimators. Submitted for publication, https://www. researchgate.net/publication/257985753-Mean-square-convergence-rates-for- maximum-quasi-likelihood-estimators.
27. den Boer AV, Zwart B (2013) Simultaneously learning and optimizing using controlled variance pricing. Management Sci., ePub ahead of print December 10, http://dx.doi.org/10.1287/mnsc.2013.1788.
28. Duistermaat JJ, Kolk JAC (2004) Multidimensional Real Analysis: Differentiation, Cambridge Studies in Advanced Mathematics (86) (Cambridge University Press, Cambridge, UK).
29. Erdǒs P (1949) On a theorem of Hsu and Robbins. Ann. Math. Statist. 20(2): 286-291.
30. Erdǒs P (1950) Remark on my paper "On a theorem of Hsu and Robbins." Ann. Math. Statist. 21(1): 138.
31. Eren SS, Maglaras C(2010) Monopoly pricing with limited demand information. J. Revenue Pricing Management 9: 23-48.
32. Farias VF, van Roy B (2010) Dynamic pricing with a prior on market response. Oper. Res. 58(1): 16-29.
33. Filippi S, Cappe O, Garivier A, Szepesvari C(2010) Parametric bandits: The generalized linear case. Lafferty J, Williams CKI, Shawe-Taylor J, Zemel RS, Culotta A, eds. Advances in Neural Information Processing Systems 23 (Curran Associates, Red Hook, NY), 586-594.
34. Gill J (2001) Generalized Linear Models: A Unified Approach (Sage Publications, Thousand Oaks, CA).
35. Godambe VP, Heyde CC (1987) Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55(3): 231-244.
36. Hager WW (1989) Updating the inverse of a matrix. SIAM Rev. 31(2): 221-239.
37. Hall P, Heyde CC (1980) Martingale Limit Theory and Its Application, Probability and Mathematical Statistics (Academic Press, New York).
38. Harrison JM, Keskin NB, Zeevi A (2012) Bayesian dynamic pricing policies: Learning and earning under a binary prior distribution. Management Sci. 58(3): 570-586.
39. Heyde CC (1997) Quasi-Likelihood and Its Application, Springer Series in Statistics (Springer Verlag, New York).
40. Heyde CC, Morton R (1996) Quasi-likelihood and generalizing the EM algorithm. J. Roy. Statist. Soc. Ser. B 4Methodological5 58(2): 317-327.
41. Hsu PL, Robbins H (1947) Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33(2): 25-31.
42. Jaksch T, Ortner R, Auer P (2010) Near-optimal regret bounds for reinforcement learning. J. Machine Learn. Res. 11: 1563-1600.
43. Katz ML (1963) The probability in the tail of a distribution. Ann. Math. Statist. 34(1): 312-318.
44. Keskin NB, Zeevi A (2013) Dynamic pricing with an unknown linear demand model: Asymptotically optimal semi-myopic policies. Working paper, University of Chicago, Chicago, http://faculty.chicagobooth.edu/bora.keskin/pdfs/ DynamicPricingUnknownDemandModel.pdf.
45. Kleinberg R (2005) Nearly tight bounds for the continuum-armed bandit problem. Saul LK, Weiss Y, Bottou L, eds. Advances in Neural Information Processing Systems 17 (MIT Press, Cambridge, MA), 697-704.
46. Kleinberg R, Leighton T (2003) The value of knowing a demand curve: Bounds on regret for online posted-price auctions. Proc. 44th IEEE Sympos. Foundations of Comput. Sci. (IEEE Computer Society, Washington, DC), 594-605.
47. Lai TL (1974) Limit theorems for delayed sums. Ann. Probab. 2(3): 432-440.
48. Lai TL, Wei CZ (1982) Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist. 10(1): 154-166.
49. Le Guen T (2008) Data-driven pricing. Master's thesis, Massachusetts Institute of Technology, Cambridge, MA. Retrieved April 4, 2011, http://hdl.handle.net/1721.1/45627.
50. Lim AEB, Shanthikumar JG (2007) Relative entropy, exponential utility, and robust dynamic pricing. Oper. Res. 55(2): 198-214.
51. Lim AEB, Shanthikumar JG, Watewai T (2008) Robust multi-product pricing. Working paper, University of California, Berkeley, http://dx.doi.org/10.2139/ ssrn.1078012.
52. Lobo MS, Boyd S (2003) Pricing and learning with uncertain demand. Working paper, University of Stanford, Stanford, http://www.stanford.edu/~boyd/ papers/pdf/pric-learn-unc-dem.pdf.
53. Loève M (1977) Probability Theory I, 4th ed. (Springer Verlag, New York).
54. McCullagh P (1983) Quasi-likelihood functions. Ann. Statist. 11(1): 59-67.
55. McCullagh P, Nelder JA (1983) Generalized Linear Models (Chapman & Hall, London).
56. Nassiri-Toussi K, Ren W (1994) On the convergence of least squares estimates in white noise. IEEE Trans. Automatic Control 39(2): 364-368.
57. Osborne MR (1992) Fisher's method of scoring. Internat. Statist. Rev./Revue Internationale de Statistique 60(1): 99-117.
58. Pronzato L (2008) Optimal experimental design and some related control problems. Automatica 44(2): 303-325.
59. Pronzato L (2009) Asymptotic properties of nonlinear estimates in stochastic models with finite design space. Statist. Probab. Lett. 79(21): 2307-2313.
60. Rothschild M (1974) A two-armed bandit theory of market pricing. J. Econom. Theory 9(2): 185-202.
61. Rusmevichientong P, Tsitsiklis JN (2010) Linearly parameterized bandits. Math. Oper. Res. 35(2): 395-411.
62. Spataru A (2004) Precise asymptotics for a series of T.L. Lai. Proc. Amer. Math. Soc. 132(11): 3387-3395.
63. Spataru A (2009) Improved convergence rates for tail probabilities. Bull. Transilvania University of Brasov-Ser. III: Math., Informatics, Phys. 2(51): 137-142.
64. Stoica G (2007) Baum-Katz-Nagaev type results for martingales. J. Math. Anal. Appl. 336(2): 1489-1492.
65. Stout WF (1970) A martingale analogue of Kolmogorov's law of the iterated logarithm. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 15(4): 279-290.
66. Wang Y, Audibert JY, Munos R (2009) Algorithms for infinitely many-armed bandits. Koller D, Schuurmans D, Bengio Y, Bottou L, eds. Advances in Neural Information Processing Systems 21 (Curran Associates, Red Hook, NY), 1729-1736.
67. Wedderburn RWM (1974) Quasi-likelihood functions, generalized, linear models, and the Gauss-Newton method. Biometrika 61(3): 439-447.
68. Xia CH, Dube P (2007) Dynamic pricing in e-services under demand uncertainty. Production Oper. Management 16(6): 701-712.
69. Yin C, Zhang H, Zhao L (2008) Rate of strong consistency of maximum quasi-likelihood estimator in multivariate generalized linear models. Comm. Statist.-Theory Methods 37(19): 3115-3123.
70. Yu JY, Mannor S (2011) Unimodal bandits. Getoor L, Scheffer T, eds. Proc. 28th Internat. Conf. Machine Learn., Bellevue, Washington, (Omnipress, Madison, WI), 41-48.