Evolutionary dynamics of cooperation Martin A. Nowak Abstract. Cooperation means a donor pays a cost, c, for a recipient to get a benefit b. In evolutionary biology, cost and benefit are measured in terms of fitness. While mutation and selection represent the main ‘forces’ of evolutionary dynamics, cooperation is a fundamental principle that is required for every level of biological organization. Individual cells rely on cooperation among their components. Multi-cellular organisms exist because of cooperation among their cells. Social insects are masters of cooperation. Most aspects of human society are based on mechanisms that promote cooperation. Whenever evolution constructs something entirely new (such as multi-cellularity or human language), cooperation is needed. Evolutionary construction is based on cooperation. I will present five basic principles for the evolution of cooperation, which arise in the theories of kin selection, direct reciprocity, indirect reciprocity, graph selection and group selection. Mathematics Subject Classification (2000). 92B05. Keywords. Mathematical biology, evolutionary dynamics, kin selection, evolutionary graph theory, indirect reciprocity, Prisoner’s Dilemma. 1. Kin selection In a pub conversation, J. B. S. Haldane, one of the founding fathers of a mathematical approach to biology, once remarked: ‘I will jump into the river to save two brothers or eight cousins.’ This insight was precisely formulated by William Hamilton many years later. He wrote a PhD thesis on this topic, submitted a long paper to the Journal of Theoretical Biology, disappeared into the Brazilian jungle and was world famous when he returned a few years later (Hamilton 1964a, b). The theory was termed ‘kin selection’ by John Maynard Smith (1964). The crucial equation is the following. Cooperation among relatives can be favored by natural selection if the coefficient of genetic relatedness, r , between the donor and the recipient exceeds the cost to benefit ratio of the altruistic act r > c/b. (1) Kin selection theory has been tested in numerous experimental studies. Many coop- erative acts among animals occur between close kin (Frank 1998, Hamilton 1998). The exact relationship between kin selection and other mechanisms such as group selection and spatial reciprocity, however, remains unclear. A recent study even sug- gests that much of cooperation in social insects is due to group selection rather than kin selection (Wilson & Hölldobler 2005). Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society