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November  2018, 1(4): 383-392. doi: 10.3934/mfc.2018019

## Constant competitive algorithms for unbounded one-Way trading under monotone hazard rate

 1 Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, China 2 Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong 3 School of Computer Science and Technology, University of Science and Technology of China, China

Received  September 2018 Revised  September 2018 Published  December 2018

In the one-way trading problem, a seller has some product to be sold to a sequence of buyers in an online fashion, i.e., buyers come one after another. Each buyer has the accepted unit price which is known to the seller on his arrival. To maximize the total revenue, the seller has to carefully decide the amount of products to be sold to each buyer at the then-prevailing prices. In this paper, we study the unbounded one-way trading, i.e., the highest unit price among all buyers is positive and unbounded. We assume that the highest prices of buyers follow some distribution with monotone hazard rate, which is a well-adopted assumption. We investigate two variants, (1) the distribution is on the highest price among all buyers, and (2) the distribution of the highest price of each buyer is independent and identically distributed. To measure the performance of the algorithms, the expected competitive ratios, $\mathrm{E}[OPT]/\mathrm{E}[ALG]$ and $\mathrm{E}[OPT/ALG]$, are considered. If the distributions satisfy the monotone hazard rate, for both of the above two variants, constant-competitive algorithms can be achieved.

Citation: Yong Zhang, Francis Y. L. Chin, Francis C. M. Lau, Haisheng Tan, Hing-Fung Ting. Constant competitive algorithms for unbounded one-Way trading under monotone hazard rate. Mathematical Foundations of Computing, 2018, 1 (4) : 383-392. doi: 10.3934/mfc.2018019
##### References:
 [1] M. Babaioff, S. Dughmi, R. Kleinberg and A. Slivkins, Dynamic pricing with limited supply, In Proceedings of the 13th ACM Conference on Electronic Commerce (EC 2012), 3 (2015), 74–91. doi: 10.1145/2559152. [2] A. Badanidiyuru, R. Kleinberg and Y. Singer, Learning on a budget: posted price mechanisms for online procurement, In Proceedings of the 13th ACM Conference on Electronic Commerce (EC 2012), (2012), 128–145. doi: 10.1145/2229012.2229026. [3] M.-F. Balcan, A. Blum and Y. Mansour, Item pricing for revenue maximization, In Proceedings of the 9th ACM Conference on Electronic Commerce (EC 2008), (2018), 50–59. doi: 10.1145/1386790.1386802. [4] A. Blum and J. D. Hartline, Near-optimal online auctions, In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), 2005, 1156–1163. [5] A. Blum, A. Gupta, Y. Mansour and A. Sharma, Welfare and Profit Maximization with Production Costs, In Proceedings of 52th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011), (2011), 77–86. doi: 10.1109/FOCS.2011.68. [6] T. Chakraborty, E. Even-Dar, S. Guha, Y. Mansour and S. Muthukrishnan., Approximation schemes for sequential posted pricing in multi-unit auctions, In Proceedings of the 6th International Workshop on Internet and Network Economics (WINE 2010), 2010, 158–169. doi: 10.1007/978-3-642-17572-5_13. [7] T. Chakraborty, Z. Huang and S. Khanna, Dynamic and non-uniform pricing strategies for revenue maximization, In Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), 2009, 495–504. doi: 10.1109/FOCS.2009.43. [8] S. Chawla, J. D. Hartline and R. Kleinberg, Algorithmic pricing via virtual valuations, In Proceedings of the 8th ACM Conference on Electronic Commerce (EC 2007), 2007, 243–251. doi: 10.1145/1250910.1250946. [9] G.-H. Chen, M.-Y. Kao, Y.-D. Lyuu and H.-K. Wong, Optimal buy-and-hold strategies for financial markets with bounded daily returns, SIAM J. Compt., 31 (2001), 447-459. doi: 10.1137/S0097539799358847. [10] F. Y. L. Chin, B. Fu, J. Guo, S. Han, J. Hu, M. Jiang, G. Lin, H.-F. Ting, L. Zhang, Y. Zhang and D. Zhou, Competitive algorithms for unbounded one-way trading, Theor. Comput. Sci., 607 (2015), 35-48. doi: 10.1016/j.tcs.2015.05.034. [11] P. Dhangwatnotai, T. Roughgarden and Q. Yan, Revenue maximization with a single sample, Games and Economic Behavior, 91 (2015), 318-333. doi: 10.1016/j.geb.2014.03.011. [12] R. El-Yaniv, A. Fiat, R. M. Karp and G. Turpin, Competitive analysis of financial games, In Proceedings of 33th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1992), 1992, 372–333. doi: 10.1109/SFCS.1992.267758. [13] R. El-Yaniv, A. Fiat, R. M. Karp and G. Turpin, Optimal search and one-way trading online algorithms, Algorithmica, 30 (2001), 101-139. doi: 10.1007/s00453-001-0003-0. [14] H. Fujiwara, K. Iwama and Y. Sekiguchi, Average-case competitive analyses for one-way trading, Journal of Combinatorial Optimization, 21 (2011), 83-107. doi: 10.1007/s10878-009-9239-4. [15] E. Koutsoupias and G. Pierrakos., On the Competitive Ratio of Online Sampling Auctions., In Proceedings of the 6th International Workshop on Internet and Network Economics (WINE 2010), 2010, 327–338. doi: 10.1007/978-3-642-17572-5_27. [16] M. Mahdian, R. Preston McAfee and D. Pennock, The secretary problem with a hazard rate condition, In Proceedings of the 4th International Workshop on Internet and Network Economics (WINE 2008), 2008, 708–715. doi: 10.1007/978-3-540-92185-1_76. [17] R. B. Myerson, Optimal auction design, Mathematics of Operations Research, 6 (1981), 58-73. doi: 10.1287/moor.6.1.58. [18] Y. Zhang, F. Y. L. Chin and H.-F. Ting, Competitive algorithms for online pricing, In Proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011), 6842 (2011), 391–401. doi: 10.1007/978-3-642-22685-4_35. [19] Y. Zhang, F. Y. L. Chin and H.-F. Ting, Online pricing for bundles of multiple items, Journal of Global Optimization, 58 (2014), 377-387. doi: 10.1007/s10898-013-0043-4.

show all references

##### References:
 [1] M. Babaioff, S. Dughmi, R. Kleinberg and A. Slivkins, Dynamic pricing with limited supply, In Proceedings of the 13th ACM Conference on Electronic Commerce (EC 2012), 3 (2015), 74–91. doi: 10.1145/2559152. [2] A. Badanidiyuru, R. Kleinberg and Y. Singer, Learning on a budget: posted price mechanisms for online procurement, In Proceedings of the 13th ACM Conference on Electronic Commerce (EC 2012), (2012), 128–145. doi: 10.1145/2229012.2229026. [3] M.-F. Balcan, A. Blum and Y. Mansour, Item pricing for revenue maximization, In Proceedings of the 9th ACM Conference on Electronic Commerce (EC 2008), (2018), 50–59. doi: 10.1145/1386790.1386802. [4] A. Blum and J. D. Hartline, Near-optimal online auctions, In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), 2005, 1156–1163. [5] A. Blum, A. Gupta, Y. Mansour and A. Sharma, Welfare and Profit Maximization with Production Costs, In Proceedings of 52th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011), (2011), 77–86. doi: 10.1109/FOCS.2011.68. [6] T. Chakraborty, E. Even-Dar, S. Guha, Y. Mansour and S. Muthukrishnan., Approximation schemes for sequential posted pricing in multi-unit auctions, In Proceedings of the 6th International Workshop on Internet and Network Economics (WINE 2010), 2010, 158–169. doi: 10.1007/978-3-642-17572-5_13. [7] T. Chakraborty, Z. Huang and S. Khanna, Dynamic and non-uniform pricing strategies for revenue maximization, In Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2009), 2009, 495–504. doi: 10.1109/FOCS.2009.43. [8] S. Chawla, J. D. Hartline and R. Kleinberg, Algorithmic pricing via virtual valuations, In Proceedings of the 8th ACM Conference on Electronic Commerce (EC 2007), 2007, 243–251. doi: 10.1145/1250910.1250946. [9] G.-H. Chen, M.-Y. Kao, Y.-D. Lyuu and H.-K. Wong, Optimal buy-and-hold strategies for financial markets with bounded daily returns, SIAM J. Compt., 31 (2001), 447-459. doi: 10.1137/S0097539799358847. [10] F. Y. L. Chin, B. Fu, J. Guo, S. Han, J. Hu, M. Jiang, G. Lin, H.-F. Ting, L. Zhang, Y. Zhang and D. Zhou, Competitive algorithms for unbounded one-way trading, Theor. Comput. Sci., 607 (2015), 35-48. doi: 10.1016/j.tcs.2015.05.034. [11] P. Dhangwatnotai, T. Roughgarden and Q. Yan, Revenue maximization with a single sample, Games and Economic Behavior, 91 (2015), 318-333. doi: 10.1016/j.geb.2014.03.011. [12] R. El-Yaniv, A. Fiat, R. M. Karp and G. Turpin, Competitive analysis of financial games, In Proceedings of 33th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1992), 1992, 372–333. doi: 10.1109/SFCS.1992.267758. [13] R. El-Yaniv, A. Fiat, R. M. Karp and G. Turpin, Optimal search and one-way trading online algorithms, Algorithmica, 30 (2001), 101-139. doi: 10.1007/s00453-001-0003-0. [14] H. Fujiwara, K. Iwama and Y. Sekiguchi, Average-case competitive analyses for one-way trading, Journal of Combinatorial Optimization, 21 (2011), 83-107. doi: 10.1007/s10878-009-9239-4. [15] E. Koutsoupias and G. Pierrakos., On the Competitive Ratio of Online Sampling Auctions., In Proceedings of the 6th International Workshop on Internet and Network Economics (WINE 2010), 2010, 327–338. doi: 10.1007/978-3-642-17572-5_27. [16] M. Mahdian, R. Preston McAfee and D. Pennock, The secretary problem with a hazard rate condition, In Proceedings of the 4th International Workshop on Internet and Network Economics (WINE 2008), 2008, 708–715. doi: 10.1007/978-3-540-92185-1_76. [17] R. B. Myerson, Optimal auction design, Mathematics of Operations Research, 6 (1981), 58-73. doi: 10.1287/moor.6.1.58. [18] Y. Zhang, F. Y. L. Chin and H.-F. Ting, Competitive algorithms for online pricing, In Proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011), 6842 (2011), 391–401. doi: 10.1007/978-3-642-22685-4_35. [19] Y. Zhang, F. Y. L. Chin and H.-F. Ting, Online pricing for bundles of multiple items, Journal of Global Optimization, 58 (2014), 377-387. doi: 10.1007/s10898-013-0043-4.
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