  Hassett and Keel predicted that there is a descending sequence of critical
$\alpha$ values where the log canonical model for the moduli space of stable
curves with respect to $\alpha \delta$ changes. We derive a conjectural formula
for the critical values in two different ways, by working out the intersection
theory of the moduli space of hyperelliptic curves and by computing the GIT
stability of certain curves with tails and bridges. The results give a rough
outline of how the log minimal model program would proceed, telling us when the
log canonical model changes and which curves are to be discarded and acquired
at the critical steps.
