6 Conclusion Here we analyzed the model that allows the individ- uals to choose different strategies against different op- ponents in well-mixed populations for both synchronous and asynchronous update. First we showed that coop- eration is evolutionary stable for both synchronous and asynchronous update, what means that a defector mu- tant cannot invade a population of cooperators. We also showed, for a initial condition of 50% of cooperation, that for synchronous update cooperation always dominates while for asynchronous update the cooperation fraction assumes the lower bound given by the initial mutual co- operation. For the synchronous update, population dy- namics exhibits a short-time behavior that is similar to the asynchronous update, but for suficient long times, co- operation spreads for large N but finite. The crucial dif- ference between synchronous and asynchronous is that in synchronous update it is possible to have a simultaneous update that allows a (D,D) to (C,C) transition. This does not happen in asynchronous update. In a preview work, the same model was analyse in a square lattice with syn- chronous update. Here we showed that the synchronous update is crucial for cooperation dominance while net- work reciprocity effects are not so important. Although the asynchronous update does not provide cooperation dominace, it allows cooperation to survive at its lower bound value. Note that this result in asynchronous up- date is still better than those of the usual game. Fig. 2 and Fig. 3 show the numerical solution of these equations. One can see from the above equations that x$_{CC}$ increases much slower than x$_{DD}$ . For the initial con- dition assumed here, x$_{CC}$ time derivative at the begin- ning is almost zero, because the values inside the brackets are equal to 0 . 5 powered to N . But when evolution starts, great part of (C,D) interactions are changed to (D,D) and x$_{CD}$ is reduced to some value near to zero making x$_{CC}$ to increase faster. So we have two regimes: short-time regime, when x$_{CC}$ is kept almost constant around its ini- tial values, and long-time regime, when x$_{CC}$ starts to increase faster. Fig. 2 and Fig. 3 show both coopera- tion evolution regimes. For short times, if we discard the terms that are powered to N − 1, we have the same solu- tion as the asynchronous case. This means that cooper- ation assumes a value near its lower bound value, given by the initial mutual cooperations. But for long-time regime, x$_{CD}$ is near zero and ˙ x$_{CC}$ cannot be neglected. So x$_{CC}$ starts to increase until it become equal to one. So for sufficient long times, the stationary solution is Note that for the short-time regime, shown in Fig. 3, the mean-field approximation fits well when compared to in silico evolution. For the long-time regime, the time evolution of the mean-field solution does not fit well, al- though it gives the right stationary solution. From the above expressions and simulation data, we see that the lower value of cooperation is reached very fast in the synchronous update. But as long as popu- lation size gets bigger, this value is reached very slowly (see Fig. 4). Besides that, if N is large, x$_{CC}$ increases very slowly. By these reasons, for large N , in the syn- chronous update the population seems to be wrapped in the lower value of cooperation, although what is actu- ally happening is that cooperation is slowly increasing, spreading until dominates the entire population. The authors thank to CNPq and FAPEMIG, Brazilian agencies. 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