An improved bound for the price of anarchy for related machine scheduling

TL;DR

This paper improves the theoretical upper bounds on the price of anarchy in related machine scheduling games, showing tighter efficiency guarantees for Nash equilibria under specific machine speed conditions.

Contribution

It introduces a new upper bound for the price of anarchy in related machine scheduling games, refining previous bounds for both two machines and multiple divisible-speed machines.

Findings

Upper bound of 2 - 1/(2(2m-1)) for general m machines

Bound of 3/2 for two machines

Bound of 2 - 1/(2m) for divisible-speed m machines

Abstract

In this paper, we introduce an improved upper bound for the efficiency of Nash equilibria in utilitarian scheduling games on related machines. The machines have varying speeds and adhere to the Shortest Processing Time (SPT) policy as the global order for job processing. The goal of each job is to minimize its completion time, while the social objective is to minimize the sum of completion times. Our main result provides an upper bound of $2-\frac{1}{2\cdot(2m-1)}$ on the price of anarchy for the general case of $m$ machines. We improve this bound to 3/2 for the case of two machines, and to $2-\frac{1}{2\cdot m}$ for the general case of $m$ machines when the machines have divisible speeds.