  Two mechanisms that have been used to study the evolution of cooperative
behavior are altruistic punishment, in which cooperative individuals pay
additional costs to punish defection, and multilevel selection, in which
competition between groups can help to counteract individual-level incentives
to cheat. Boyd, Gintis, Bowles, and Richerson have used simulation models of
cultural evolution to suggest that altruistic punishment and pairwise
group-level competition can work in concert to promote cooperation, even when
neither mechanism can do so on its own. In this paper, we formulate a PDE model
for multilevel selection motivated by the approach of Boyd and coauthors,
modeling individual-level birth-death competition with a replicator equation
based on individual payoffs and describing group-level competition with
pairwise conflicts based on differences in the average payoffs of the competing
groups. Building off of existing PDE models for multilevel selection with
frequency-independent group-level competition, we use analytical and numerical
techniques to understand how the forms of individual and average payoffs can
impact the long-time ability to sustain altruistic punishment in
group-structured populations. We find several interesting differences between
the behavior of our new PDE model with pairwise group-level competition and
existing multilevel PDE models, including the observation that our new model
can feature a non-monotonic dependence of the long-time collective payoff on
the strength of altruistic punishment. Going forward, our PDE framework can
serve as a way to connect and compare disparate approaches for understanding
multilevel selection across the literature in evolutionary biology and
anthropology.
