On-Line Difference Maximization

TL;DR

This paper develops optimal online algorithms for maximizing rank gains in sequentially arriving data, achieving near-optimal expected gains compared to offline strategies in financial and game-theoretic contexts.

Contribution

It introduces and analyzes the first optimal online algorithms for both single and multiple low/high value selection problems with provable performance bounds.

Findings

Expected gain for single pair is n-O(1), close to offline gain

Multiple pair strategy achieves expected gain of n^2/8- heta(n log n)

Online algorithms are within 3/4 of offline optimal performance

Abstract

In this paper we examine problems motivated by on-line financial problems and stochastic games. In particular, we consider a sequence of entirely arbitrary distinct values arriving in random order, and must devise strategies for selecting low values followed by high values in such a way as to maximize the expected gain in rank from low values to high values. First, we consider a scenario in which only one low value and one high value may be selected. We give an optimal on-line algorithm for this scenario, and analyze it to show that, surprisingly, the expected gain is n-O(1), and so differs from the best possible off-line gain by only a constant additive term (which is, in fact, fairly small -- at most 15). In a second scenario, we allow multiple nonoverlapping low/high selections, where the total gain for our algorithm is the sum of the individual pair gains. We also give an optimal on-line algorithm for this problem, where the expected gain is n^2/8-\Theta(n\log n). An analysis shows that the optimal expected off-line gain is n^2/6+\Theta(1), so the performance of our on-line algorithm is within a factor of 3/4 of the best off-line strategy.