  Here we consider a game theoretic model of multilevel selection in which
individuals compete based on their payoff and groups also compete based on the
average payoff of group members. Our focus is on multilevel social dilemmas:
games in which individuals are best off cheating, while groups of individuals
do best when composed of many cooperators. We analyze the dynamics of the
two-level replicator dynamics, a nonlocal hyperbolic PDE describing
deterministic birth-death dynamics for both individuals and groups. While past
work on such multilevel dynamics has restricted attention to scenarios with
exactly solvable within-group dynamics, we use comparison principles and an
invariant property of the tail of the population distribution to extend our
analysis to all possible two-player, two-strategy social dilemmas. We
characterize the threshold level of between-group selection dividing a regime
in which the population converges to a delta function at the equilibrium of the
within-group dynamics from a regime in which between-group competition
facilitates the existence of steady-state densities supporting greater levels
of cooperation. In particular, we see that the threshold selection strength and
average payoff at steady state depend on a tug-of-war between the
individual-level incentive to be a defector in a many-cooperator group and the
group-level incentive to have many cooperators over many defectors. We also
find that lower-level selection casts a long shadow: if groups are best off
with a mix of cooperators and defectors, then there will always be fewer
cooperators than optimal at steady state, even in the limit of infinitely
strong competition between groups.
