For asynchronous update, cooperation cannot dominate population but it can coexist with defection and assumes values near the lower-bound (25% for the initial condition assumed here). Fig. 1 shows simulation and mean-field approximation results for the asynchronous update. For the synchronous update, cooperation dominates the en- tire population. Fig. 2 shows the simulation and mean- field approximation results for the synchronous update. One can see in Fig. 3 that, for short times, the syn- chronous update behaves like the asynchronous update. At the beginning, all of the exploitations are neutralized and only the initial mutual cooperation survives. After that, cooperation starts to increase very slowly until it dominates the population. So we can define a short-time regime and a long-time regime for synchronous dynam- ics. The same qualitative result holds for large popula- tions, but simulation time gets extremely huge for syn- chronous update. Fig. 4 shows a finite size analysis for the time spent to reach the minimum value of the co- operation fraction before cooperation dominates in the synchronous update. Note that as along as N increases, 1 /T goes to zero, implying that there is no long-time regime for N → ∞ . On the other hand, if we have N large but finite, the long-time regime is the asymptotic regime, which is characterized by domination of cooper- ation. proximation gives not only the stationary solutions, but fits reasonably the in silico time evolution, although for the synchronous update it fits well only in the short-time regime. Let us now derive the mean-field solution. We first analyze the asynchronous update followed by the synchronous one. mean-field solution provides a good equilibrium anal- ysis in well-mixed population if the usual game is con- sidered [22]. But in structured population, it is not a good approximation [23]. Although in the present work we deal with well-mixed population, the nature of the “implicit punishment” model is not so simple. It is not obvious that a mean-field approach would work. So it is a remarkable result the fact that our mean-field ap- FIG. 4: The plot shows the relation 1 /T × 1 /N , where N is the population size and T is the time to reach the minimum value of cooperation for synchronous update. FIG. 1: Fraction of cooperation for asynchronous update with N = 100 and b = 2 . 0. FIG. 3: Short-time regime of the cooperation fraction for syn- chronous update with N = 40 and b = 2 . 0. 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