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 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 ∗ )).
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