Loughborough University Institutional Repository Competitive algorithms for unbounded one-way trading

In the one-way trading problem, a seller has L units of 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 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 Ah,ε that has competitive ratio O(log r∗(log r∗) . . . (log(h−1) r∗)(log 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 satisfies log (h) r∗ > 1, where log 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 r∗ > 1 such that the ratio between the optimal revenue and the revenue obtained by A is Ω(log r∗(log r∗) . . . (log(h−1) r∗)(log 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 v1, v2, . . . , vn arriving online, and she needs to decide the fraction to purchase from each vi at the then-prevailing market price pi. Her objective is to minimize the cost. The optimal competitive algorithms whose performance guarantees depend only on the lowest market price p∗ = mini pi, and one of M and φ, the price fluctuation ratio, are presented.