2 tion tendency, and was robust against misjudgments of the worst interaction. The possibility of opponent differ- entiation introduces a mechanism of punishing without costs and without any kind of internal preferences except the desire of maximize own payoff. We call this punish- ment “implicit punishment”. But in that work [18], the possibility of adoption of different strategies was intro- duced in the context of network reciprocity. What hap- pens if network reciprocity is excluded? Here we analyze this model in well mixed populations, what means that we are excluding network reciprocity effects. The other important feature of the model is the synchronous up- date assumption. It is well known that results may be striking different if asynchronous update is used [19, 20]. Here we analyze the model with both synchronous and asynchronous updates. We show that cooperation still re- mains alive, although for asynchronous update it achieves its lower bound level. We analyze the model using com- puter simulations and a mean-field technique. We made our simulations using networks of size N = 40 and N = 100 and evaluate the fraction of cooper- ation ( f$_{c}$ ) adopted by all of the players in all of their interactions. If n$_{c}$ is the quantity of C strategies used in all of the interactions by all of the players, we have 0 ≤ n$_{c}$ ≤ N ( N − 1) and f$_{c}$ = n$_{c}$/N ( N − 1). The ran- dom initial configuration consists of 50% of cooperation and the averages are made over 100 different initial con- ditions. We use ϵ = 0 . 001 and we show here only the case b = 2, although we simulated the model also for other values of b . In fact, the b value has no effect in the simulations and in the mean-field results. Before going on, let us state one fundamental feature of the model that is independent if the update is syn- chronous or not. Suppose a focal player imitates a defec- tion strategy. We state that any interaction of type (C,C) will never be replaced by (D,C). If some opponent adopts defection, the focal player must have at least one (C,D) or (D,D) interaction. But these interactions give payoffs worst than (C,C). So (C,C) will never be replaced. This proves the existence of a lower bound for the fraction of cooperation given by the initial fraction of mutual coop- eration. On the other hand, if the focal player imitates a D strategy, he/she will first seek for (C,D) interactions. If they are present, (C,D) will be replaced by (D,D). One can see that mutual cooperation is never destroyed and every exploitation is punished. Let us state the model formally. We study the pris- oner dilemma in a population of size N as the scenario for the cooperation problem. We consider a well-mixed population, what means that each player can interact with everybody. The strategy vector of an individual is ⃗ S = ( S 1 , . . . , S$_{j}$, . . . , S$_{N}$$_{−}$ 1 ), where S$_{j}$ can be C (coop- eration) or D (defection). So individuals are merged in N − 1 interactions. If in one of these interactions an individual plays C against an opponent who is playing D, we denote this interaction as (C,D) (the first entry is the strategy of the focal player and the second en- try is the opponent strategy). The payoff of a D strat- egy against a C strategy is P ( D, C ) = b , where b > 1 is the defection tendency. Using the same notation, we have that P ( D, D ) = ϵ , with ϵ << 1, P ( C, C ) = 1 and P ( C, D ) = 0. For synchronous update, each player in- teracts with the other N − 1 players, plays a round of one game against each opponent, and earns a cumulative payoff. After that, each player chooses randomly one neighbor and compare their cumulative payoffs. If the opponent cumulative payoff is bigger than its own one, it imitates the strategy the opponent is using against him/her with probability proportional to the difference of cumulative payoffs, Π = | Δ P$_{cum}$ | / (( N − 1) b ) [21]. On the other hand, if the opponent cumulative payoff is lower than its own one, the focal player remains with the same strategy. If imitation takes place, the new strategy re- places the strategy used in the interaction that gives the worst payoff. The worst payoff of the focal player is given by the interaction (C,D), followed by (D,D), (C,C) and (D,C). For asynchronous update, a random individual is chosen so that it can imitate and possibly replace one of its strategies like in the synchronous case. After this individual update, the entire population play a round of the game, and each player earns new cumulative payoffs, and another random individual is chosen to update its strategies. A time step consists of N of such processes. For the usual game, where each player adopts a sin- gle strategy against all of its opponents, a single defector can invade a population of cooperators in infinity well- mixed population [2]. The first remarkable feature of “implicit punishment” is that cooperation is evolutionary stable in well-mixed population for both synchronous and asynchronous update. If a mutant that adopts defection against everybody appears in a population where every- body is cooperating, the mutant initially earns a huge payoff. But as soon as others imitate it, the exploited (C,D) interactions will be replaced by (D,D), neutralizing the exploitations. What is simple, but remarkable, is that the interactions that are changing are just those in which the exploiter is involved and all of the other mutual coop- erations are maintained. The “implicit punishment” will take place until the mutant cumulative payoff is equal to cooperator cumulative payoff. If we call n$_{exp}$ the quan- tity of defection adopted by the mutant exploiter, the payoff of the mutant exploiter and the payoff of the co- operators are P$_{exp}$ = n$_{exp}$ ( b + ϵ ) and P$_{coop}$ = N − 2, respectively. By equating both expression we have the equilibrium fraction Let us now discuss the results obtained by numerical simulations with both synchronous and asynchronous up- dates. In the initial conditions, we set the players to co- operate with probability 0 . 5 against each one of its oppo- nents. This gives an initial cooperation fraction of 50%. 2. The model 3. Evolutionary analysis n$_{exp}$ = N − 2 b + ϵ .