On-line Difference Maximization

Title: On-line Difference Maximization

Abstract: This research investigates the problem of selecting values from an online source to maximize the difference in their ranks. The values can be arbitrary distinct real numbers, and the goal is to determine the optimal strategy for selecting low values followed by high values to achieve the maximum expected gain. The study presents two scenarios: one allowing only one low/high selection and another allowing multiple nonoverlapping selections. The paper provides optimal on-line algorithms for both scenarios and analyzes their performance. The results show that the expected gain for the first scenario is n-O(1), and for the second scenario, it is n2/8-Θ(nlogn), which is within a factor of 3/4 of the best off-line strategy. The research has implications for on-line financial problems and stochastic games, contributing to the understanding and application of on-line algorithms in realistic scenarios.

Main Research Question: How can we devise optimal strategies for selecting values from an online source to maximize the expected difference in their ranks?

Methodology: The study uses mathematical modeling and analysis to investigate the problem. It considers two scenarios: one allowing only one low/high selection and another allowing multiple nonoverlapping selections. The research employs the concept of on-line algorithms, which means that the selection must be made without knowledge of future values. The analysis involves determining the optimal strategy and the expected gain for each scenario.

Results: The research presents optimal on-line algorithms for both scenarios. The expected gain for the first scenario is n-O(1), and for the second scenario, it is n2/8-Θ(nlogn). These results show that the performance of the on-line algorithms is within a factor of 3/4 of the best off-line strategy.

Implications: The research has implications for on-line financial problems and stochastic games. It contributes to the understanding and application of on-line algorithms in realistic scenarios. The study also relates to the secretary problem, a standard textbook example in mathematical analysis, and has led to interesting extensions in the field.

Link to Article: https://arxiv.org/abs/0101024v2 Authors: arXiv ID: 0101024v2