# Competitive Algorithms for Unbounded One-Way Trading

• Francis Y. L. Chin
• Bin Fu
• Minghui Jiang
• Hing-Fung Ting
• Yong Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8546)

## Abstract

In the one-way trading problem, a seller has some product to be sold to a sequence σ of buyers u 1, u 2, …, u σ arriving online and he needs to decide, for each u i , the amount of product to be sold to u i at the then-prevailing market price p i . The objective is to maximize the seller’s revenue. We note that most previous algorithms for the problem need to impose some artificial upper bound M and lower bound m on the market prices, and the seller needs to know either the values of M and m, or their ratio M/m, at the outset. Moreover, the performance guarantees provided by these algorithms depend only on M and m, and are often too loose; for example, given a one-way trading algorithm with competitive ratio Θ(log(M/m)), its actual performance can be significantly better when the actual highest to actual lowest price ratio is significantly smaller than M/m.

This paper gives a one-way trading algorithm that does not impose any bounds on market prices and whose performance guarantee depends directly on the input. In particular, we give a class of one-way trading algorithms such that for any positive integer h and any positive number ε, we have an algorithm A h,ε that has competitive ratio O (logr  ∗ (log(2) r  ∗ ) … (log(h − 1) r  ∗ )(log(h) r  ∗ )1 + ε ) if the value of r  ∗  = p  ∗ /p 1, the ratio of the highest market price p  ∗  =  max i p i and the first price p 1, is large and satisfy log(h) r  ∗  > 1, where log(i) x denotes the application of the logarithm function i times to x; otherwise, A h,ε has a constant competitive ratio Γ h . We also show that our algorithms are near optimal by showing that given any positive integer h and any one-way trading algorithm A, we can construct a sequence of buyers σ with log(h) r  ∗  > 1 such that the ratio between the optimal revenue and the revenue obtained by A is at least Ω(logr  ∗ (log(2) r  ∗ ) …(log(h − 1) r  ∗ ) (log(h) r  ∗ )).

## References

1. 1.
Babaioff, M., Dughmi, S., Kleinberg, R., Slivkins, A.: Dynamic Pricing with Limited Supply. In: Proceedings of the 13th ACM Conference on Electronic Commerce, pp. 74–91 (2012)Google Scholar
2. 2.
Badanidiyuru, A., Kleinberg, R., Singer, Y.: Learning on a budget: posted price mechanisms for online procurement. In: Proc. of the 13th ACM Conference on Electronic Commerce, pp. 128–145 (2012)Google Scholar
3. 3.
Balcan, M.-F., Blum, A., Mansour, Y.: Item pricing for revenue maximization. In: Proceedings of the 9th ACM Conference on Electronic Commerce, pp. 50–59 (2008)Google Scholar
4. 4.
Blum, A., Hartline, J.D.: Near-optimal online auctions. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1156–1163 (2005)Google Scholar
5. 5.
Blum, A., Gupta, A., Mansour, Y., Sharma, A.: Welfare and Profit Maximization with Production Costs. In: Proceedings of 52th Annual IEEE Symposium on Foundations of Computer Science, pp. 77–86 (2011)Google Scholar
6. 6.
Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press (1998)Google Scholar
7. 7.
Chakraborty, T., Even-Dar, E., Guha, S., Mansour, Y., Muthukrishnan, S.: Approximation schemes for sequential posted pricing in multi-unit auctions. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 158–169. Springer, Heidelberg (2010)
8. 8.
Chakraborty, T., Huang, Z., Khanna, S.: Dynamic and non-uniform pricing strategies for revenue maximization. In: Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 495–504 (2009)Google Scholar
9. 9.
Chen, G.-H., Kao, M.-Y., Lyuu, Y.-D., Wong, H.-K.: Optimal buy-and-hold strategies for financial markets with bounded daily returns. SIAM J. Compt. 31(2), 447–459 (2001), A preliminary version appeared in STOC 1999, pp. 119–128Google Scholar
10. 10.
El-Yaniv, R., Fiat, A., Karp, R.M., Turpin, G.: Competitive analysis of financial games. In: Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science, pp. 372–333 (1992)Google Scholar
11. 11.
El-Yaniv, R., Fiat, A., Karp, R.M., Turpin, G.: Optimal search and one-way trading online algorithms. Algorithmica 30(1), 101–139 (2001)
12. 12.
Fujiwara, H., Iwama, K., Sekiguchi, Y.: Average-case competitive analyses for one-way trading. Journal of Combinatorial Optimization 21(1), 83–107 (2011)
13. 13.
Koutsoupias, E., Pierrakos, G.: On the Competitive Ratio of Online Sampling Auctions. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 327–338. Springer, Heidelberg (2010)
14. 14.
Lorenz, J., Panagiotou, K., Steger, A.: Optimal algorithms for k-search with application in option pricing. Algorithmica 55, 311–328 (2009); A preliminary version appeared in Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 275–286. Springer, Heidelberg (2007)Google Scholar
15. 15.
Myerson, R.B.: Optimal auction design. Mathematics of Operations Research 6, 58–73 (1981)
16. 16.
Zhang, Y., Chin, F.Y.L., Ting, H.-F.: Competitive Algorithms for Online Pricing. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 391–401. Springer, Heidelberg (2011)
17. 17.
Zhang, Y., Chin, F.Y.L., Ting, H.-F.: Online pricing for bundles of multiple items. Journal of Global Optimization 58(2), 377–387Google Scholar

© Springer International Publishing Switzerland 2014

## Authors and Affiliations

• Francis Y. L. Chin
• 1
• Bin Fu
• 2
• Minghui Jiang
• 3
• Hing-Fung Ting
• 1
• Yong Zhang
• 1
• 4
1. 1.Department of Computer ScienceThe University of Hong KongHong Kong
2. 2.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA
3. 3.Department of Computer ScienceUtah State UniversityLoganUSA