Notes on two elementary evolutionary games

Thus spoke the wise Queen Delores, saying, “I have studied mathematics. I will question this young man, in my tent tonight, and in the morning I will report the truth as to his pretensions”. Jurgen, James Branch Cabell (1919), Ch. XXXII: Jurgen proves it by mathematics Evolutionary game theory [3] defines something like a functor from the classical theory of games to dynamical systems, imagining biological entities whose rate of reproduction is proportional to their success at playing the game in question. It is a beautiful and accessible subject; these notes on two interesting examples grew out of JHU’s 2007 BioCalc I course, and I’d like to thank the students there for their interest and forebearance. This work was also suggested by DARPA’s Fundamental Problems of Biology initiative. 1. The Battle of the Sexes ” C’est magnifique, mais ce n’est pas la guerre . . . ” Marshal Pierre Bosquet, Balaclava 1854 1.1 In the evolutionary version of Richard Dawkins’ toy model for marriage markets as asymmetric two-player games, the rate of reproduction of a population type is proportional to its success at beating the mean expectation for the game. Following Hofbauer and Sigmund [3 §10.2], this is expressed by a system ẋk = xk · [(Ay)k − x · Ay] , ẏk = yk · [(Bx)k − y ·Bx] . of ‘replicator’ equations, in which there are two types of players: those of type I (males) characterized by a state vector x = (x1, . . . , xn) in a unit simplex in some space of strategies, and type II (females) defined similarly by a vector y = (y1, . . . , ym). Date: 31 October 2010. 1991 Mathematics Subject Classification. draft. 1