Competitive Algorithms for Unbounded One-Way Trading

Abstract

In the one-way trading problem, a seller has L units of 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 all 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.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 * (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 satisfies 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 Ω(log r * (log (2) r * ) . . . (log (h−1) r * )(log (h) r * )). A special case of the one-way trading is also studied, in which the L units of product is comprised of L items, each of which must be sold atomically (or equivalently, the amount of product sold to each buyer must be an integer).Furthermore, a complementary problem to the one-way trading problem, say, the oneway buying problem, is studied in this paper. In the one-way buying problem, a buyer wants to purchase one unit of product through a sequence of n sellers v 1 , v 2 , . . . , v n arriving online, and she needs to decide the fraction to purchase from each v i at the then-prevailing market price p i . Her objective is to minimize the cost. The optimal competitive algorithms whose performance guarantees depend only on the lowest market price p * = min i p i , and one of M and φ, the price fluctuation ratio, are presented.