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## mean-field approximation for the asynchronous update 

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 

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 

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<img src='content_image/426818.jpg'>

<img src='content_image/426822.jpg'>

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 

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 

<img src='content_image/426819.jpg'>

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 

<img src='content_image/426821.jpg'>

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 

<img src='content_image/426823.jpg'>

change the quantity of (D,D). Following the previous steps, we can define 

<img src='content_image/426824.jpg'>

and we obtain that 

