Competitive Algorithms for Unbounded One-Way Trading

In the one-way trading problem, a seller has some product to be sold to a sequence σ of buyers u1, u2, . . . , uσ arriving online and he needs to decide, for each ui, the amount of product to be sold to ui at the then-prevailing market price pi. 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 Ah,ε that has competitive ratio O(log r ∗(log(2) r∗) . . . (log(h−1) r∗)(log(h) r∗)1+ε) if the value of r∗ = p/p1, the ratio of the highest market price p∗ = maxi pi and the first price p1, is large and satisfy log r∗ > 1, where log x denotes the application of the logarithm function i times to x ; otherwise, Ah,ε has a constant competitive ratio Γh. 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 r∗ > 1 such that the ratio between the optimal revenue and the revenue obtained by A is at least Ω(log r∗(log(2) r∗) . . . (log(h−1) r∗)(log(h) r∗)). ⋆ Research supported by HK RGC grant HKU-711709E and Shenzhen basic research project (NO.JCYJ20120618143038947). ⋆⋆ Research supported by HK RGC grant HKU-716412E. ⋆ ⋆ ⋆ Research supported by NSFC 11171086 and Natural Science Foundation of Hebei Province A2013201218. 2 Francis Y.L. Chin et al.