Unbounded One-Way Trading on Distributions with Monotone Hazard Rate

Abstract

One-way trading is a fundamental problem in the online algorithms. A seller has some product to be sold to a sequence of buyers \(\{u_1, u_2, \dots \}\) in an online fashion and each buyer \(u_i\) is associated with his accepted unit price \(p_i\), which is known to the seller on the arrival of \(u_i\). The seller needs to decide the amount of products to be sold to \(u_i\) at the then-prevailing price \(p_i\). The objective is to maximize the total revenue of the seller. In this paper, we study the unbounded one-way trading, i.e., the highest unit price among all buyers is unbounded. We also assume that the highest prices of buyers follow some distribution with monotone hazard rate, which is well-adopted in Economics. We investigate two variants, (1) the distribution is on the highest price among all buyers, and (2) a general variant that the prices of buyers is independent and identically distributed. To measure the performance of the algorithms, the expected competitive ratios, \(\mathrm {E}[OPT]/\mathrm {E}[ALG]\) and \(\mathrm {E}[OPT/ALG]\), are considered and constant-competitive algorithms are given if the distributions satisfy the monotone hazard rate.