  In this work we give a tight lower bound on makespan approximations for
envy-free allocation mechanism dedicated to scheduling tasks on unrelated
machines. Specifically, we show that no mechanism exists that can guarantee an
envy-free allocation of jobs to $m$ machines with a makespan of less than a
factor of $O(\log m)$ of the minimal makespan. Combined with previous results,
this paper definitively proves that the optimal algorithm for obtaining a
minimal makespan for any envy-free division can at best approximate the
makespan to a factor of $O(\log m)$.
