Emergence of Cooperation as a Non-equilibrium Transition in Noisy Spatial Games

The emergence of cooperation among selfish agents that have no incentive to cooperate is a non-trivial phenomenon that has long intrigued biologists, social scientists and physicists. The iterated Prisoner's Dilemma (IPD) game provides a natural framework for investigating this phenomenon. Here, agents repeatedly interact with their opponents, and their choice to either cooperate or defect is determined at each round by knowledge of the previous outcomes. The spatial version of IPD, where each agent interacts only with their nearest neighbors on a specified connection topology, has been used to study the evolution of cooperation under conditions of bounded rationality. In this paper we study how the collective behavior that arises from the simultaneous actions of the agents (implemented by synchronous update) is affected by (i) uncertainty, measured as noise intensity K, (ii) the payoff b, quantifying the temptation to defect and (iii) the nature of the underlying connection topology. In particular, we study the phase transitions between states characterized by distinct collective dynamics as the connection topology is gradually altered from a two-dimensional lattice to a random network. This is achieved by rewiring links between agents with a probability p following the small-world network construction paradigm. On crossing a specified threshold value of b, the game switches from being Prisoner's Dilemma, characterized by a unique equilibrium, to Stag Hunt, a well-known coordination game having multiple equilibria. We observe that the system can exhibit three collective states corresponding to a pair of absorbing states (viz., all agents cooperating or defecting) and a fluctuating state characterized by agents switching intermittently between cooperation and defection. As noise and temptation can be interpreted as temperature and an external field respectively, a strong analogy can be drawn between the phase diagrams of such games with that of interacting spin systems. Considering the 3-dimensional p-K-b parameter space allows us to investigate the different phase transitions that occur between these collective states and characterize them using finite-size scaling. We find that the values of the critical exponents depend on the connection topology and are different from the Directed Percolation (DP) universality class.