  We consider the minimum makespan problem for $n$ tasks and two unrelated
parallel selfish machines. Let $R_n$ be the best approximation ratio of
randomized monotone scale-free algorithms. This class contains the most
efficient algorithms known for truthful scheduling on two machines. We propose
a new $Min-Max$ formulation for $R_n$, as well as upper and lower bounds on
$R_n$ based on this formulation. For the lower bound, we exploit pointwise
approximations of cumulative distribution functions (CDFs). For the upper
bound, we construct randomized algorithms using distributions with piecewise
rational CDFs. Our method improves upon the existing bounds on $R_n$ for small
$n$. In particular, we obtain almost tight bounds for $n=2$ showing that
$|R_2-1.505996|<10^{-6}$.
