  In a refreshing mathematical investigation, Mark (2018) shows that status
hierarchy may facilitate the emergence of cooperation in groups. Despite the
contribution, the present paper notes that there are limitations in Mark's
model that makes it less realistic than it could in explaining real-world
experiences. Consequently, we present a more generalized modified framework in
which his model is a special case, by developing and introducing a new
hierarchy measure into the model to estimate the cooperation level in a set of
hierarchical structures omitted in Mark's work yet common in everyday
life--those with multiple leaders. We derived the conditions under which
cooperation can emerge in these groups, and verified our analytical predictions
in agent-based computer simulations. In so doing, not only does our model
elaborate on its predecessor and support Mark's general prediction. For theory,
our work further reveals two novel phenomena of group cooperation: Both the
relative number of cooperators to defectors in groups and the assortativity
among these different roles can backfire; they are not always the higher, the
better for cooperation to thrive. For methodology, the hierarchy measure
developed and our model using the measure may also be applied in future
research on a wide range of related topics.
