4 We first consider a typical player, that we call focal player. We are going to study the dynamics of the local concentration of (C,D), (D,D), (C,C), and (D,C) interac- tions around the focal player. Let k 1 , k 2 , k 3 , and k 4 be the quantity of such interactions around the focal player. Note that k 1 + k 2 + k 3 + k 4 = N . Let us assume that the probability of having k 1 , k 2 , k 3 , and k 4 interactions are given by the respective global concentrations. The probability of a focal player configuration ( k 1 ,k 2 ,k 3 ) is given by Let us first define a local and a global interaction con- centrations for a population of size N + 1. If in an in- teraction a player i adopts strategy A and its opponent adopts B, where A,B ∈ { C,D } , we say that player i has a (A,B) interaction. Player i can have N$_{AB}$ ( i ) (A,B) in- teractions. We define the local concentration of (A,B) as the fraction of (A,B) interactions around player i , namely x$_{AB}$ ( i ) = N$_{AB}$ ( i ) /N . For the global concentration of (A,B) we define x$_{AB}$ = ∑ $_{i}$N$_{AB}$ ( i ) / (( N + 1) N ). We consider the other nodes as mean-field nodes. So the the local concentration of (C,D), (D,D), and (C,C) inter- actions around those nodes is given by the configuration vector N ( x$_{CD}$,x$_{DD}$,x$_{CC}$ ). Now we are going to derive the rate of variation of the local concentration around the focal player. Let us derive the rate of increasing of the ( D,D ) lo- cal concentration. Suppose that the focal player is on a ( k 1 ,k 2 ,k 3 ) configuration. Note that the payoff of this configuration of the focal player is the focal player can imitate the D strategy from (C,D) or (D,D) interactions. Let us focus on the first alterna- tive, that happens with probability k 1 /N . The opponent associated with the (C,D) interaction has a payoff of The probability of imitating the D strategy from (C,D) is given by where Θ( x ) = 1 if x > 0, and Θ( x ) = 0 otherwise. The mean rate of increasing of (D,D) by one unit due to the imitation from (C,D) is There is just one transition that increases this quantity: (C,D) to (D,D). In this replacement, the focal player adopts C and imitates an opponent that adopts a D. So at least one interaction of type (C,D) must be present. But Following the same lines we obtain the possibility of imitating from (D,D), that is given by The same analysis can be done to calculate the rate of decreasing of ( D,D ). There is just one transition in- volved, namely, (D,D) to (C,D). At least one (D,D) must be present. The C strategy can be imitated from (C,C) or (D,C). But note that the focal player cannot be currently adopting a (C,D), because in that case (C,D) would give the worst payoff. So imitating a C strategy would not change the quantity of (D,D). Following the previous steps, we can define and we obtain that P$_{k}$$_{1}$$_{,k}$$_{2}$$_{,k}$$_{3}$ = N ! k 1 ! k 2 ! k 3 !( N − k 1 − k 2 − k 3 )! x k 1 $_{CD}$x k 2 $_{DD}$x k 3 $_{CC}$x N − k 1 − k 2 − k 3 DC PF$_{CD}$ = b + ( N − 1)( x$_{DC}$b + x$_{CC}$ + x$_{DD}$ϵ ) . PF = ( N − k 1 − k 2 − k 3 ) b + k 3 + k 2 ϵ. [ PF$_{CD}$ − PF ]Θ( PF$_{CD}$ − PF$_{k}$ ) bN , W + CD = N ∑ k $_{1}$=1 N − k 1 ∑ k $_{2}$=0 N − k 1 − k 2 ∑ k $_{3}$=0 k 1 N [ PF$_{CD}$ − PF ]Θ( PF$_{CD}$ − PF ) bN P$_{k}$$_{1}$$_{,k}$$_{2}$$_{,k}$$_{3}$ W + DD = N ∑ k $_{1}$=1 N − k 1 ∑ k $_{2}$=0 N − k 1 − k 2 ∑ k $_{3}$=0 k 2 N [ PF$_{DD}$ − PF ]Θ( PF$_{DD}$ − PF ) bN P$_{k}$$_{1}$$_{,k}$$_{2}$$_{,k}$$_{3}$. P$_{k}$$_{2}$$_{,k}$$_{3}$ = N ! k 2 ! k 3 !( N − k 2 − k 3 )! x k 2 $_{DD}$x k 3 $_{CC}$x N − k 2 − k 3 DC , mean-field approximation for the asynchronous update