2 9 Ja n 20 07 epl draft Influence of initial distributions on robust cooperation in evolu - tionary Prisoner ’ s Dilemma

We study the evolutionary Prisoner's Dilemma game on scale-free networks for different initial distributions. We consider three types of initial distributions for cooperators and defectors: initially random distribution with different frequencies of defectors; intentional organization with defectors initially occupying the most connected nodes with different fractions of defectors; intentional assignment for cooperators occupying the most connected nodes with different proportions of defectors at the beginning. It is shown that initial configurations for co-operators and defectors can influence the stationary level of cooperation and the evolution speed of cooperation. Organizations with the vertices with highest connectivity representing individuals cooperators could exhibit the most robust cooperation and drive evolutionary process to converge fastest to the high steady cooperation in the three situations of initial distributions. Otherwise, we determine the critical initial frequencies of defectors above which the extinction of cooperators occurs for the respective initial distributions, and find that the presence of network loops and clusters for cooperators can favor the emergence of cooperation. Introduction. – Evolutionary game theory has become an important tool for investigating cooperative behavior of biological, ecological, social and economic systems [1, 2]. The Prisoner' Dilemma game (PDG) is one of the most commonly employed games for this purpose. Originally, in the PDG, two individuals adopt one of the two available strategies, cooperate or defect; both receive R under mutual cooperation and P under mutual defec-tion, while a cooperator receives S when confronted to a defector, which in turn receives T , where T > R > P > S and T +S < 2R. Under these conditions it is best to defect for rational individuals in a single round of the PDG, regardless of the opponent strategy. However, mutual cooperation would be preferable for both of individuals. Thus, the dilemma is caused by the selfishness of the individuals. However, the unstable cooperative behavior is opposite to the observations in the real world. This disagreement thus motivates to find under what conditions the cooperation can emerge on the PDG. Graph theory provides a nat