Phase diagrams for an evolutionary prisoner’s dilemma game on two-dimensional lattices

Abstract

The effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary Prisoner's Dilemma game with players located on the sites of different two-dimensional lattices. This system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive. Using systematic Monte Carlo simulations and different levels of the generalized mean-field approximations we have determined the phase boundaries (critical points) separating the two phases on the plane of the temperature (noise) and temptation to choose defection. In the zero temperature limit this analysis suggests that the cooperation can be sustained only for those connectivity structures where three-site clique percolation occurs. 05.50.+q, 02.50.+Le, 87.23.Ge In the original (two-player and one-shot) Prisoner's Dilemma (PD) game [1, 2] the players should simultaneously choose between two options, called defection and cooperation. The selfish players wish to maximize their own income in the knowledge of payoffs dependent on their choices. The curiosity of PD game is hidden in the fact that the choice of defection yields higher income independently of the partner's choice. However, if both players choose defection then their individual income is lower than those obtained for mutual cooperation when the maximum total payoff is shared equally. The rational (intelligent) players cannot resolve this dilemma and both of them choose defection (this is the so-called Nashequilibrium in the PD game). At the same time we find many examples in the nature where the mutual cooperation (altruism, ethical norms, etc.) emerges spontaneously among the selfish individuals [3]. In the last decades several mechanisms (e.g., kin selection [4], application of retaliating strategies [5], and voluntary participation [6]) are reported which enforce the appearance of cooperation in the societies.The spatial versions [7,8] of the evolutionary PD games can explain the maintenance of cooperation for the iterated games with a limited range of interaction if the players follow one of the two simplest strategies. For the two simplest strategies, denoted shortly as D and C, the player choose always defection and cooperation, respectively. In the evolutionary games the players wish to maximize their total payoff, coming from PD games with the neighbors, by adopting one of the more successful strategies available in their neighborhood. This type of dynamics describes the behavior of the ecological systems controlled by the Darwinian selection [9,10].Following the pioneering work of Nowak et al. [7,8] the two-strategy spatial evolutionary PD games have already been studied by several authors using different evolutionary rules on a large class of backgrounds including social networks [11,12,13,14] (for a survey of lattice models see the papers [15,16,17] and further references therein). In the present paper our attention is focused on the effect of noise built into the dynamical rule. It is tur...

The prisoner’s dilemma on co-evolving networks under perfect rationality

“…Effects of entries in the payoff matrix and addition of noise have been examined on different types of two-dimensional lattices [19]. Other topology related topics, such as the role of an 'influential node' [20] and optional participation [21] have been examined as well.…”

Spatial prisoner’s dilemma games with increasing size of the interaction neighborhood on regular lattices

“…After each time step, player x can reassess and imitate one of the more successful neighbor's strategies by comparing his/her payoff with that of a randomly selected neighbor y. Following previous studies [11][12][13][14]22,[33][34][35][36][37][38][39][40], player x can follow the strategy of a randomly selected neighbor y with the probability depending on the payoff difference (P x P y ):…”

“…Notably, increasing the number of neighbors n has a favorable effect on defection [35][36][37][38][39][40]. Our goal is to explore whether the effect of the size of the neighborhood n is beneficial for the maintenance of cooperation.…”

Promotion of cooperation induced by the interplay between structure and game dynamics

“…Furthermore, interactions in real-world network of contacts are heterogeneous, often associated with scale-free (power-law) dependence on the degree distribution, P (k) ∼ k −γ with 2 < γ < 3. Accordingly, the evolution of cooperation on model networks with features such as lattices [6,7,8,9], smallworld [10,11,12], scale-free [13,14,15], and community structure [16] has been scrutinized. Interestingly, Santos et al found that scale-free networks provide a unifying framework for the emergency of cooperation [13].…”