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 a of buyers u(1), u(2), ... , u(sigma) arriving online and he needs to decide, for each u(i), the amount of product to be sold to ui 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 is an element of, we have an algorithm A(h,is an element of) that has competitive ratio O(log r*(log((2)) r*) ... (lod((h-1)) r*)(log((h)) r*)(1+is an element of)) if the value of r* = p*/p(1), the ratio of the highest market price p* = max(i) p(i) and the first price pi, is large and satisfies log((h)) r* > 1, where log((i)) x denotes the application of the logarithm function i times to x; otherwise, Ah,, has a constant competitive ratio Gamma(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 a with log((h)) r* > 1 such that the ratio between the optimal revenue and the revenue obtained by A is Omega(logr*(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 are 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 one-way 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 vi at the then-prevailing market price pi. Her objective is to minimize the cost. The optimal competitive algorithms