The price of anarchy for a berth allocation game

In this work, we investigate a berth allocation game involving m identical berths and n vessels. Each vessel selfishly selects a consecutive set of berths for unloading, with the objective of minimizing its own cost represented by the maximum load among the chosen berths. The social cost is defined as the makespan, i.e., the maximum load over all berths. Our game generalizes classical machine scheduling games where jobs (vessels) may require multiple consecutive machines (berths). We analyze the price of anarchy (PoA) of the berth allocation game, which quantifies the impact of selfish behaviors of vessels. Specifically, we first consider a special case where each job can occupy at most two consecutive machines, and derive exact upper and lower bounds for the PoA based on the number of machines m. We show that the PoA asymptotically approaches 94\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{9}{4}$$\end{document}. For the general case where each job can occupy an arbitrary number of consecutive machines, we obtain a tight bound for the PoA, which is Theta logmloglogm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta \left( \frac{\log m}{\log \log m}\right) $$\end{document}.