On the expected number and distribution of equilibria in multi-player evolutionary games

Evolutionary game theory (EGT) has become a powerfulmathematical framework for the modelling and analysis ofcomplex biological/economical systems whenever there isfrequency dependent selection – the fitness of an individualdoes not only depend on its strategy, but also on the com-position of the population in relation with (multiple) otherstrategies (Maynard Smith and Price, 1973; Hofbauer andSigmund, 1998). The payoff from the games is interpretedas individual fitness, naturally leading to a dynamical ap-proach. Random evolutionary games in which the payoffentries are random variables form an important subclass ofEGT. They are necessary to model social and biological sys-tems in which very limited information is available, or wherethe environment changes so rapidly and frequently that onecannot describe the payoffs of their inhabitants’ interactions(Fudenberg and Harris, 1992; Gross et al., 2009). As inclassical game theory with the Nash equilibrium, see e.g.(McLennan, 2005), the analysis of properties of equilibriumpoints in EGT has been of special interest, see e.g. (Gokhaleand Traulsen, 2010).