J. Brustle, J. Correa, Paul Dütting, Victor Verdugo
{"title":"动态定价的竞争复杂性","authors":"J. Brustle, J. Correa, Paul Dütting, Victor Verdugo","doi":"10.1145/3490486.3538366","DOIUrl":null,"url":null,"abstract":"We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward Am(F) achievable by the optimal online policy on m i.i.d. random variables drawn from F to the expected maximum Mn(F) of n i.i.d. draws from the same distribution. We ask how big does m have to be to ensure that (1+ε) Am(F) ≥ Mn(F) for all F. We resolve this question and exhibit a stark phase transition: When ε = 0 the competition complexity is unbounded. That is, for any n and any m there is a distribution F such that Am(F) > Mn(F). In contrast, for any ε < 0, it is sufficient and necessary to have $m = φ(ε)n where φ(ε) = Θ(log log 1/ε). Therefore, the competition complexity not only drops from being unbounded to being linear, it is actually linear with a very small constant. The technical core of our analysis is a loss-less reduction to an infinite dimensional and non-linear optimization problem that we solve optimally. A corollary of this reduction, which may be of independent interest, is a novel proof of the factor ~0.745 i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds.","PeriodicalId":209859,"journal":{"name":"Proceedings of the 23rd ACM Conference on Economics and Computation","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"The Competition Complexity of Dynamic Pricing\",\"authors\":\"J. Brustle, J. Correa, Paul Dütting, Victor Verdugo\",\"doi\":\"10.1145/3490486.3538366\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward Am(F) achievable by the optimal online policy on m i.i.d. random variables drawn from F to the expected maximum Mn(F) of n i.i.d. draws from the same distribution. We ask how big does m have to be to ensure that (1+ε) Am(F) ≥ Mn(F) for all F. We resolve this question and exhibit a stark phase transition: When ε = 0 the competition complexity is unbounded. That is, for any n and any m there is a distribution F such that Am(F) > Mn(F). In contrast, for any ε < 0, it is sufficient and necessary to have $m = φ(ε)n where φ(ε) = Θ(log log 1/ε). Therefore, the competition complexity not only drops from being unbounded to being linear, it is actually linear with a very small constant. The technical core of our analysis is a loss-less reduction to an infinite dimensional and non-linear optimization problem that we solve optimally. A corollary of this reduction, which may be of independent interest, is a novel proof of the factor ~0.745 i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds.\",\"PeriodicalId\":209859,\"journal\":{\"name\":\"Proceedings of the 23rd ACM Conference on Economics and Computation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-07-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 23rd ACM Conference on Economics and Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3490486.3538366\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the 23rd ACM Conference on Economics and Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3490486.3538366","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward Am(F) achievable by the optimal online policy on m i.i.d. random variables drawn from F to the expected maximum Mn(F) of n i.i.d. draws from the same distribution. We ask how big does m have to be to ensure that (1+ε) Am(F) ≥ Mn(F) for all F. We resolve this question and exhibit a stark phase transition: When ε = 0 the competition complexity is unbounded. That is, for any n and any m there is a distribution F such that Am(F) > Mn(F). In contrast, for any ε < 0, it is sufficient and necessary to have $m = φ(ε)n where φ(ε) = Θ(log log 1/ε). Therefore, the competition complexity not only drops from being unbounded to being linear, it is actually linear with a very small constant. The technical core of our analysis is a loss-less reduction to an infinite dimensional and non-linear optimization problem that we solve optimally. A corollary of this reduction, which may be of independent interest, is a novel proof of the factor ~0.745 i.i.d. prophet inequality, which simultaneously establishes matching upper and lower bounds.