Effects of neighbourhood size and connectivity on the spatial Continuous Prisoner's Dilemma

Abstract

The Prisoner's Dilemma, a two-person game in which the players can either cooperate or defect, is a common paradigm for studying the evolution of cooperation. In real situations cooperation is almost never all or nothing. This observation is the motivation for the Continuous Prisoner's Dilemma, in which individuals exhibit variable degrees of cooperation. It is known that in the presence of spatial structure, when individuals "play against" (i.e. interact with) their neighbours, and "compare to" ("learn from") them, cooperative investments can evolve to considerable levels. Here, we examine the effect of increasing the neighbourhood size: we find that the mean-field limit of no cooperation is reached for a critical neighbourhood size of about five neighbours on each side in a Moore neighbourhood, which does not depend on the size of the spatial lattice. We also find the related result that in a network of players, the critical average degree (number of neighbours) of nodes for which defection is the final state does not depend on network size, but only on the network topology. This critical average degree is considerably (about 10 times) higher for clustered (social) networks, than for distributed random networks. This result strengthens the argument that clustering is the mechanism which makes the development and maintenance of the cooperation possible. In the lattice topology, it is observed that when the neighbourhood sizes for "interacting" and "learning" differ by more than 0.5, cooperation is not sustainable, even for neighbourhood sizes that are below the mean-field limit of defection. We also study the evolution of neighbourhood sizes, as well as investment level. Here, we observe that the series of the interaction and learning neighbourhoods converge, and a final cooperative state with considerable levels of average investment is achieved.

Adaptive Dynamics with Interaction Structure

“…Our work also unifies previous findings that spatial structure can promote cooperation in continuous-strategy social dilemmas (van Baalen and Rand, 1998;Killingback et al, 1999;Le Galliard et al, 2003, 2005Ifti et al, 2004). Our work shows that, across models and games, the strength of this spatial benefit to cooperation is quantified by the structure coefficient σ.…”

Cooperation among cancer cells as public goods games on Voronoi networks

“…In other words, in order to study diffusible public goods, we must decouple the interaction neighborhood (the group playing the public goods game) and the update neighborhood (the one-step neighbors). This approach has been used to study a two-person game with a linear benefit function (the Prisoner's Dilemma) on a regular lattice [Ifti et al 2004, Ohtsuki et al 2007 and for non-linear public goods games on regular lattices with a fixed diffusion range [Archetti 2013b]. …”

Active linking in evolutionary games

“…For detailed reviews of the replicator equation and other approaches to evolutionary game dynamics, see Fudenberg and Tirole (1991), Weibull (1995), Samuelson (1997), Cressman (2003), Sigmund (1998, 2003), Gintis (2000) and Nowak and Sigmund (2004). in finite populations Imhof and Nowak, 2006;Taylor et al, 2004;Fudenberg et al, 2006;Traulsen et al, 2006a,b), in spatially extended systems (Nowak and May, 1992;Nakamaru et al, 1998;Killingback and Doebeli, 1996;van Baalen and Rand, 1998;Irwin and Taylor, 2001;Hauert and Doebeli, 2004;Ifti et al, 2004;Nakamaru and Iwasa, 2005;Jansen and van Baalen, 2006) or on graphs (Lieberman et al, 2005;Santos et al, , 2006a. Taylor and Nowak (2006) analyze a scenario where the interaction rate does depend on the strategies.…”