This study investigates a robust monopoly pricing problem, where a seller seeks to maximize expected revenue when selling a product to a buyer with limited information, specifically knowing only the mean, variance, and asymmetric information (semivariance) of the buyer’s valuation distribution. We explore both deterministic and randomized pricing strategies and evaluate their performance under these constraints. We formulate two maximin problems aimed at maximizing the worst-case expected revenues. By employing the primal-dual approach in infinite linear programming, we derive the closed-form for the optimal pricing strategies for both maximin problems. Our results show that incorporating semivariance significantly enhances the performance of pricing strategies by better capturing distributional asymmetry. Additionally, we compare the performance of deterministic and randomized pricing in various scenarios, offering valuable insights into how risk and reward can be effectively balanced. Moreover, our results on both pricing strategies provide new bounds for the value of personalized pricing. This comprehensive analysis enables a deeper understanding of semivariance information to improve decision-making and offer practical insights for managing complex pricing decisions.
BandiCBertsimasD (2014) Optimal design for multi-item auctions: A robust optimization approach. Mathematics of Operations Research39(4): 1012–1038.
2.
BarlowREMarshallAWProschanF (1963) Properties of probability distributions with monotone hazard rate. The Annals of Mathematical Statistics34(2): 375–389.
3.
BeiXGravinNLuP, et al. (2019) Correlation-robust analysis of single item auction. In: Proceedings of the thirtieth annual ACM-SIAM symposium on discrete algorithms, San Diego, California, USA, 6–9 January 2019, pp.193–208. Philadelphia, Pennsylvania, USA: SIAM.
4.
BergemannDSchlagK (2011) Robust monopoly pricing. Journal of Economic Theory146(6): 2527–2543.
5.
BergemannDSchlagKH (2008) Pricing without priors. Journal of the European Economic Association6(2–3): 560–569.
6.
BertsimasDPopescuI (2005) Optimal inequalities in probability theory: A convex optimization approach. SIAM Journal on Optimization15(3): 780–804.
7.
CarrascoVLuzVFKosN, et al. (2018) Optimal selling mechanisms under moment conditions. Journal of Economic Theory177: 245–279.
8.
CarrollG (2017) Robustness and separation in multidimensional screening. Econometrica85(2): 453–488.
9.
ChampernowneDG (1953) A model of income distribution. The Economic Journal63(250): 318–351.
10.
ChenHHeYQiJ, et al. (2023) Distributionally robust pricing with asymmetric information. Available at SSRN 4365395.
11.
ChenHHuMPerakisG (2022) Distribution-free pricing. Manufacturing & Service Operations Management24(4): 1939–1958.
12.
ChenZHuZWangR (2024) Screening with limited information: A dual perspective. Operations Research72(4): 1487–1504.
13.
ChenZSimMXiongP (2020) Robust stochastic optimization made easy with rsome. Management Science66(8): 3329–3339.
14.
ChoobinehFBrantingD (1986) A simple approximation for semivariance. European Journal of Operational Research27(3): 364–370.
15.
CondorelliDSzentesB (2020) Information design in the holdup problem. Journal of Political Economy128(2): 681–709.
16.
DelageEYeY (2010) Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research58(3): 595–612.
17.
De VanyAWallsWD (2002) Does hollywood make too many r-rated movies? risk, stochastic dominance, and the illusion of expectation. The Journal of Business75(3): 425–451.
18.
DuS (2018) Robust mechanisms under common valuation. Econometrica86(5): 1569–1588.
19.
EliashbergJElberseALeendersMA (2006) The motion picture industry: Critical issues in practice, current research, and new research directions. Marketing Science25(6): 638–661.
20.
ElmachtoubANGuptaVHamiltonML (2021) The value of personalized pricing. Management Science67(10): 6055–6070.
21.
ErenSSMaglarasC (2010) Monopoly pricing with limited demand information. Journal of Revenue and Pricing Management9: 23–48.
22.
GallegoGKatirciogluKRamachandranB (2007) Inventory management under highly uncertain demand. Operations Research Letters35(3): 281–289.
23.
GrootveldHHallerbachW (1999) Variance vs downside risk: Is there really that much difference?European Journal of Operational Research114(2): 304–319.
24.
GuoJHeSJiangB, et al. (2022) A unified framework for generalized momentproblems: a novel primal-dual approach. arXiv preprint arXiv:2201.01445.
25.
HanSGuptaSLehmannDR (2001) Consumer price sensitivity and price thresholds. Journal of Retailing77(4): 435–456.
26.
HuangX (2008) Portfolio selection with a new definition of risk. European Journal of Operational Research186(1): 351–357.
27.
HuangYWenQ (2019) Auction–lottery hybrid mechanisms: Structural model and empirical analysis. International Economic Review60(1): 355–385.
KosNMessnerM (2015) Selling to the mean. Available at SSRN 2632014.
33.
LiYLuPYeH (2019) Revenue maximization with imprecise distribution. In: Proceedings of the 18th international conference on autonomous agents and multiAgent aystems, Montreal, Quebec, Canada, 13–17 May, pp.1582–1590. Richland, SC: ACM.
MarkowitzHToddPXuG, et al. (1993) Computation of mean-semivariance efficient sets by the critical line algorithm. Annals of Operations Research45(1): 307–317.
36.
Mohajerin EsfahaniPKuhnD (2018) Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming171(1): 115–166.
37.
MunozAVassilvitskiiS (2017) Revenue optimization with approximate bid predictions. In: Advances in neural information processing systems 30: Annual conference on neural information processing systems 2017, Long Beach, CA, USA, 4–9 December 2017, pp. 1856–1864.
38.
MyersonRB (1981) Optimal auction design. Mathematics of Operations Research6(1): 58–73.
39.
NatarajanKSimMUichancoJ (2018) Asymmetry and ambiguity in newsvendor models. Management Science64(7): 3146–3167.
40.
PınarMKızılkaleC (2017) Robust screening under ambiguity. Mathematical Programming163(1): 273–299.
41.
RahimianHMehrotraS (2019) Distributionally robust optimization: A review. arXiv preprint arXiv:1908.05659.
42.
RileyJZeckhauserR (1983) Optimal selling strategies: When to haggle, when to hold firm. The Quarterly Journal of Economics98(2): 267–289.
43.
RoeslerA-KSzentesB (2017) Buyer-optimal learning and monopoly pricing. American Economic Review107(7): 2072–2080.
44.
RoosEBrekelmansRVan EekelenW, et al. (2022) Tight tail probability bounds for distribution-free decision making. European Journal of Operational Research299(3): 931–944.
45.
SaezEZucmanG (2016) Wealth inequality in the united states since 1913: Evidence from capitalized income tax data. The Quarterly Journal of Economics131(2): 519–578.
46.
SalemABZMountTD (1974) A convenient descriptive model of income distribution: The gamma density. Econometrica42(6): 1115–1127.
47.
ShapiroA (2001) On Duality Theory of Conic Linear Problems. New York: Springer, pp.135–165.
48.
ShapiroCVarianHR (1999) Information Rules: A Strategic Guide to the Network Economy. Boston, MA: Harvard Business Press.
49.
SuzdaltsevA (2022) Distributionally robust pricing in independent private value auctions. Journal of Economic Theory206: 105555.
50.
TamuzO (2013) A lower bound on seller revenue in single buyer monopoly auctions. Operations Research Letters41(5): 474–476.
51.
Van ZandenJLBatenJFoldvariP, et al. (2014) The changing shape of global inequality 1820–2000; exploring a new dataset. Review of Income and Wealth60(2): 279–297.
52.
WangS (2024) The power of simple menus in robust selling mechanisms. Management Science71(6): 5268–5287.
53.
WangSLiuSZhangJ (2024) Minimax regret robust screening with moment information. Manufacturing & Service Operations Management26(3): 992–1012.
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