56 J. Waketey and M. Nowak/Theoretical Population Biology 125 (2019)38-55 53 55832 from the John Templeton Foundation and W911 NF-18-2- 0265 from the US Defense Advanced Research Projects Agency to Martin Nowak. Appendix Analysis of the Markov chain Corresponding to the matrix in Eq. (3), we define the stochastic matrix a2 0 0 2a(1 — a)