Abstract: Natural selection in complex biological and social systems can simultaneously operate across multiple levels of organization, ranging from genes and cells to animal groups and complex human societies. Such scenarios typically present an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. In this talk, we will explore this conflict by considering a game-theoretic model of multilevel selection in a group-structured population featuring two types of individuals, cooperators and defectors. Assuming that individuals compete based on their payoff and groups also compete based on the average payoff of group members, we consider how the distribution of cooperators within groups changes over time depending on the relative strength of within-group and between-group competition. In the limit of infinitely many groups and of infinite group size, we can describe the state of the population through the probability density of the fraction of cooperators within groups, and derive a hyperbolic partial differential equation for the changing composition of the population. We show that there exists a threshold relative selection strength such that defectors will take over the population for sufficiently weak between-group competition, while cooperation persists in the long-time population when the strength of between-group competition exceeds the threshold. Surprisingly, when groups are best off with an intermediate level of within-group cooperation, individual-level selection casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. This is joint work with Yoichiro Mori.