Competitive algorithms for unbounded One-Way Trading

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 Ah,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*/p1, 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, 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(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*)). © 2014 Springer International Publishing.