Models based on agents has been widely used to study the paradigmatic emergence of cooperation in social systems. In the Spatial Prisoners' Dilemma game introduced by Martin Nowak and Robert May, the agents are greedy and imitate indiscriminately the action of the wealthiest neighbor. Other strategies, for example stochastic or pavlovian, have also been considered showing similar results as the greedy rule. That is, in general matter, the asymptotic emergence or maintenance of cooperation. For these spatial models, it can be proved, and that is the dilemma, that cooperation extinguishes when agents exhibit completely rational behavior. In this work, we explore the behavior of a system with bounded rational agents. For that, we consider a modified Spatial Prisoners' Dilemma on an adaptive network where each agent can play different actions with different neighbors. The co-evolutive dynamic obeys a scheme of rational imitation of the wealthiest agent of each neighborhood. We show the existence of a phase transition (absence of cooperation - presence of cooperation) that depends on the incentive to defect. We compute the critical value and report a simulation study that evidences that the emergence or survival of cooperation at the steady state is a critical phenomenon. These results provide a fascinating point of view to understand the trade-off between cooperation and rationality in wealthy societies. Our results also include the emergence of a rich social structure living in asymptotic regimen. Throughout a simulation study, we analyze the distribution of wealth and other complex aspects of the social network at the steady state
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