5 W − CC = N ∑ k $_{2}$=1 N − k 1 ∑ k $_{3}$=0 k 3 N [ PF$_{CC}$ − PF ]Θ( PF$_{CC}$ − PF ) bN P$_{k}$$_{2}$$_{,k}$$_{3}$ (1 − δ$_{k}$$_{3}$$_{,}$ 0 δ$_{k}$$_{2}$$_{,N}$ ) , and W − DC = N ∑ k $_{2}$=1 N − k 1 ∑ k $_{3}$=0 N − k 2 − k 3 N [ PF$_{DC}$ − PF ]Θ( PF$_{DC}$ − PF ) bN P$_{k}$$_{2}$$_{,k}$$_{3}$ (1 − δ$_{k}$$_{3}$$_{,}$ 0 δ$_{k}$$_{2}$$_{,N}$ ) . x ∞ DD = x 0 DD + 2 x 0 $_{CD}$, x ∞ CC = x 0 $_{CC}$, and x ∞ CD = x ∞ DC = 0 . x$_{CC}$ = x 0 $_{CC}$, dx$_{DD}$ dt = 2 1 N ( W + CD + W + DD − W − CC − W − $_{DC}$) . dx$_{CD}$ dt = − 1 2 ( dx$_{DD}$ dt + dx$_{CC}$ dt ) . dx$_{DD}$ dt = 2 1 N $_{2}${ x$_{CD}$ + x$_{DD}$ ϵ b [1 − (1 − x$_{CD}$ ) N − $^{1}$] − x$_{CC}$ b [(1 − x$_{CD}$ ) N − 1 − ( x$_{CC}$ + x$_{DC}$ ) N − $^{1}$] } dx$_{CC}$ dt = 1 Nx$_{DD}$ ( W − CC + W − $_{DC}$) $^{2}$. dx$_{CC}$ dt = 2 1 N $_{4}${ x$_{CC}$ b [(1 − x$_{CD}$ ) N − $^{1}$− ( x$_{CC}$ + x$_{DC}$ ) N − $^{1}$] } $^{2}$, dx$_{DD}$ dt = 2 1 N $_{2}${ x$_{CD}$ + x$_{DD}$ ϵ b [1 − (1 − x$_{CD}$ ) N − $^{1}$] − x$_{CC}$ b [(1 − x$_{CD}$ ) N − 1 − ( x$_{CC}$ + x$_{DC}$ ) N − $^{1}$] } x$_{DD}$ = x 0 $_{DD}$+$^{ϵ}$ $_{b}$x 0 DD + x 0 CD 1 2 + ϵ b [1 − exp( − 2 N $_{2}$( 1 2 + ϵ b ) t )] , dx$_{DD}$ dt = 2 1 N $_{2}$( x$_{CD}$ + ϵ b x$_{DD}$ ) , and dx$_{CC}$ dt = 0 . As all of the nodes have the same typical behavior, be- cause the population is well-mixed, we can approximate the global concentrations by the local ones. The above expressions determine the rate of (D,D) variation by one unit. If we want the time derivative of x$_{DD}$ , we need do divide the expressions by N and multiply by a factor of two, because there is the contribution of the opponents update . So we have Following the same reasoning, one can see that x$_{CC}$ does not change in time. Finally, as all of the mean-field vari- ables are normalized to one, we obtain that where the index 0 refers to the initial conditions. If we set ϵ = 0, just for simplicity, the solution reaches the fixed point One can see that only the initial mutual cooperation can be maintained and all of the other interactions are mutual defections. Note that all of the exploitations are neutral- ized. Note that this approximation gives good results if compared to simulation data. We can simplify further these expressions if we replace k 1 , k 2 and k 3 inside the payoff expressions of the focal player by the expected value of such quantities given by the configuration probability: Nx$_{CD}$ , Nx$_{DD}$ , and Nx$_{CC}$ , respectively. With this extra approximation the Θ function can be easily evaluated in the limit of large N and we have the following equation: This equation can be solved numerically. Fig.1 shows the numerical solution and the simulation results. Note that this approximation furnishes good results when com- pared with in silico evolution. For the initial condition used here, we have that the terms inside the parentheses are 0 . 75 powered to N − 1 and 0 . 5 powered to N − 1. If N is large, the terms that are powered to N are very small and they can be neglected, at least for short times. This gives the following simplified equations, The solution of these equation are straightforward, Let us explain the term in the denominator. If the focal player makes a (D,D) to (C,D) transition on an specific interaction, the mean-field player associated to this spe- cific interaction should choose exactly this interaction, what happens with probability 1 /Nx$_{DD}$ . If we perform the same simplifications that was already done for the asynchronous case, we have Let us treat the synchronous case. Now (C,C) can in- crease, because it is possible to have a (D,D) to (C,C) transition whenever two players make a (D,D) to (C,D) transition in their shared (D,D) interaction. This is an essential feature of the synchronous model. This kind of transition does not take place in asynchronous update and that is the reason why cooperation assumes the lower bound value in the asynchronous case. We can approxi- mate the rate of this transition by and mean-field approximation for the synchronous update