Stochastic and PDE Models of Multilevel Selection with Pairwise Between-Group Competition #

Konstantinos Alexiou, Dan Cooney, Yoichiro Mori

11:50 Monday in 3Q16.

Part of the Game theory and agent-based models session.

Abstract #

In this talk, we revisit a model of the evolution of altruistic punishment using a PDE framework for describing the dynamics of multilevel selection. The individuals within the group compete each other based on their payoff functions which affect their corresponding birth rates. In addition, there is also a between-group competition based on the average payoff of the group members. This is expressed as a pairwise conflict between the groups, where the outcome is affected by some probability $\rho$. We focus our attention and we analyze the evolutionary dynamics of altruistic cooperation and punishment. In particular, we see that there is an asymmetry between altruistic cooperation and altruistic punishment, where this process allows both of them to be maintained even when groups are large enough. We also discuss the scenarios where there is absence of punishment, and when the altruistic punishers are common within the group. Furthermore, we prove the weak convergence of the underlying stochastic particle process to a hyperbolic PDE on the interval $[0,1]$ with dependence on the single parameter $\lambda$, as well as the probability $\rho$. Under a different scaling, we also prove the weak convergence of the stochastic process to a parabolic-type PDE that includes also a diffusive term. We perform analytical and numerical approaches for our model in order to explore the infinite population limit under the different scalings and various functions in probability, considering small and large-scale societies. This is part of my PhD project and a joint work with Dr. Dan Cooney (University of Pennsylvania) and Professor Yoichiro Mori (University of Pennsylvania)