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 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 ^∗)).