Fundamental clusters in spatial 2×2 games

Abstract

The notion of fundamental clusters is introduced, serving as a rule of thumb to characterize the statistical properties of the complex behaviour of cellular automata such as spatial 2 Â 2 games. They represent the smallest cluster size determining the fate of the entire system. Checking simple growth criteria allows us to decide whether the cluster-individuals, e.g. some mutant family, are capable of surviving and invading a resident population. In biology, spatial 2 Â 2 games have a broad spectrum of applications ranging from the evolution of cooperation and intraspecies competition to disease spread. This methodological study allows simple classi¢cations and long-term predictions in various biological and social models to be made.For minimal neighbourhood types, we show that the intuitive candidate, a 3 Â 3 cluster, turns out to be fundamental with certain weak limitations for the Moore neighbourhood but not for the Von Neumann neighbourhood. However, in the latter case, 2 Â 2 clusters generally serve as reliable indicators to whether a strategy survives.Stochasticity is added to investigate the e¡ects of varying fractions of one strategy present at initialization time and to discuss the rich dynamic properties in greater detail. Finally, we derive Liapunov exponents for the system and show that chaos reigns in a small region where the two strategies coexist in dynamical equilibrium.

Spatial Ultimatum Games, collaborations and the evolution of fairness

“…For n 2 we can deduce analytically, by an extension of the methods used by Killingback et al (1999) and Hauert (2001) to study the behaviour of spatial games, that an estimate of the asymptotic value of the mean o¡er made by the dominant individuals in the population is p 1/3. We see from ¢gure 2a that this estimate is in extremely good agreement with the results of our evolutionary simulations.…”

Effects of punishment in a mobile population playing the prisoner's dilemma game

“…They found cooperators and defectors both persist indefinitely (in shifting clusters), without the need to assume the use of complicated strategies, not even that any individual remembers previous interactions (each player was either C or D, and after each round each lattice site was occupied by the player with the highest payoff among the previous owner and its neighbors). A huge amount of work has been undertaken ever since concerning evolutionary games on graphs [3], exploring many diverse combinations of realistic network topologies (link dynamics) and strategy update rules that lead to the survival of cooperation [12][13][14][15][16][17][18][19][20].…”

Evolutionary stability on graphs

“…Spatial games can lead to very different evolutionary dynamics than games in well-mixed populations (Nowak & May 1992, 1993, Wilson et al 1992, Ellison 1993, Herz 1994, Lindgren & Nordahl 1994, Nowak et al 1994, Killingback & Doebeli 1996, Nakamaru et al 1997, Eshel et al 1998, 1999, Szabó & Tőke 1998, van Baalen & Rand 1998, Szabó et al 2000, Szabó et al 2005, Hauert 2001, Irwin & Taylor 2001, Szabó & Hauert 2002, Le Galliard et al 2003, Hauert & Doebeli 2004, Ifti et al 2004, Santos & Pacheco 2005, Santos et al 2006, Szabó & Fáth 2007. Spatial models have also been studied in ecology (Levin 1974, Levin & Paine 1974, Durrett & Levin 1994, Hassell et al 1994, Tainaka 1994, Durrett & Levin 1997, Tilman & Karieva 1997, Haraguchi & Sasaki 2000, Neuhauser 2001, Pastor-Satorras & Vespignani 2001, Wootton 2001, May 2006) and population genetics (Wright 1943, Kimura 1953, Kimura & Weiss 1964…”