Abstract. 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 A h,ϵ that has competitive ratio O(log r, the ratio of the highest market price p * = maxi pi and the first price p1, 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(log r * (log (2) r * ) . . . (log (h−1) r * )(log (h) r * )).
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