We consider the problem of maximizing the minimum load for

machines that are controlled by selfish agents, who are only

interested in maximizing their own profit. Unlike the classical

load balancing problem, this problem

has not been considered for selfish agents until now.

For a constant number of machines, $m$, we show a

monotone polynomial time approximation scheme (PTAS) with running

time that is linear in the number of jobs. It uses a new

technique for reducing the number of jobs while remaining close

to the optimal solution. We also present an FPTAS for the classical

machine covering problem, i.e., where no selfish agents are involved

(the previous best result for this case was a PTAS)

and use this to give a monotone FPTAS.

Additionally, we give a monotone approximation algorithm with

approximation ratio $min(m,(2+eps)s_1/s_m)$ where $eps>0$ can

be chosen arbitrarily small and $s_i$ is the (real) speed of

machine $i$. Finally we give improved results for two machines.