Agent incentives of a proportional sharing mechanism in resource sharing

In a resource sharing system, resources are shared among multiple interconnected peers. Peers act as both suppliers and customers of resources by making a certain amount of their resources directly available to other network participants. Their utilities are determined by the total of resources received from all neighbors. The allocation of the shared resources is determined by a preset mechanism that depends on the information submitted from the agents. The participating agents, however, may try to strategically manipulate its submitted information to improve its utility. In this paper, we consider a tit-for-tat popular proportional sharing mechanism and discuss the incentives an agent may lie, by a so-called vertex splitting strategy, for personal gains. We use the concept of incentive ratio to characterize the extent to which utilities can be increased. For the resource sharing system where the underlying network is a cycle, we prove that the incentive ratio is bounded by 2≤ζ≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\le \zeta \le 4$$\end{document}. Furthermore, the incentive ratio on a cycle with even number of vertices is proved to be exactly 2.