Inefficiency of the Nash equilibrium for selfish machine covering on two hierarchical uniform machines

• We study the selfish machine covering game on two hierarchical uniform machines.
• We measure the inefficiency of the Nash equilibrium and strong Nash equilibrium.
• We obtain the exact (strong) price of anarchy and (strong) price of stability.

This paper studies the selfish scheduling game on two hierarchical uniform machines where the jobs are correspondingly classified into two hierarchical classes. The cost of a job is defined as the completion time of the machine to which it is assigned. Each selfish job is interested in minimizing its own cost, while the game seeks to meet the social objective of maximizing the machine cover. We obtain the (strong) price of anarchy and the (strong) price of stability as functions of the ratio between the speeds of the two machines s. We show that all the derived bounds are tight for any value of s, thus completely solving the problem of measuring the inefficiency of the Nash equilibrium on two hierarchical uniform machines.