Prisoner's Dilemma in One-Dimensional Cellular Automata: Visualization of Evolutionary Patterns

Abstract

The spatial Prisoner's Dilemma is a prototype model to show the emergence of cooperation in very competitive environments. It considers players, at site of lattices, that can either cooperate or defect when playing the Prisoner's Dilemma with other z players. This model presents a rich phase diagram. Here we consider players in cells of one-dimensional cellular automata. Each player interacts with other z players. This geometry allows us to vary, in a simple manner, the number of neighbors ranging from one up to the lattice size, including self-interaction. This approach has multiple advantages. It is simple to implement numerically and we are able to retrieve all the previous results found in the previously considered lattices, with a faster convergence to stationary values. More remarkable, it permits us to keep track of the spatio-temporal evolution of each player of the automaton. Giving rise to interesting patterns. These patterns allow the interpretation of cooperation/defection clusters as particles, which can be absorbed and collided among themselves. The presented approach represents a new paradigm to study the emergence and maintenance of cooperation in the spatial Prisoner's Dilemma.

Exhaustive exploration of Prisoner's Dilemma Parameter space in one-dimensional cellular automata

“…However, the total payoff of each player depends on the neighborhood, then, if the player belongs to a cooperative cluster he/she has a higher payoff than the player from a defective one. In the cooperative/defective clusters border, the differences among payoffs are essential to determine the system dynamics [15].…”

“…If z is even, there are α = z/2 adjacent interacting players to the right and to the left hand side of this player. If z is odd, each side has α = (z − 1)/2 players and player i interacts with his/her own state (self-interaction) [15,19,20]. In addition to ρ 0 and z the other free parameter in this model is the temptation T in the conflict range 1 ≤ T ≤ 2.…”

“…We have used the simplest lattice topology, i.e. onedimensional lattice to represent regular lattices at any dimensionality [15]. The computational implementation of PD in the one-dimensional case is simpler than in other topologies, and it requires less computational time to run the numerical codes.…”

“…The computational implementation of PD in the one-dimensional case is simpler than in other topologies, and it requires less computational time to run the numerical codes. In one-dimensional cellular automata, it is simpler to understand the way that oscillations in the cooperator proportion take place [15]. Beyond the topologies, it is also possible to consider the mobility of players [16].…”

Examining the role of individual movement in promoting coexistence in a spatially explicit prisoner's dilemma

“…Several models have been considered within this theoretical framework since the original works proposed by Nowak and May [vid., for instance, Alonso-Sanz (2014); Fogarty et al (2012); Fu et al (2010); Gianetto and Heydari (2015); Grim (1997); Grujić et al (2014); Kirchkamp (2000); Qin et al (2008); Lindgren and Nordahl (1994); Nakamaru et al (1997); Oliphant (1994); Pereira et al (2008); Roca et al (2009) ;Schweitzer et al (2002); Szabó and Fath (2007); Vainstein and Arenzon (2001); Xia et al (2015)]. Traditionally, explicit motion is not included in these models, with strategies moving via imitation or colonisation of neighbouring sites.…”