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The problem of cooperation 1−8 is that defection is evolutionarily stable. If everybody in a population defects and one individual cooperates then this individual has a lower payoff and will be opposed by selection. Thus, the emergence of cooperation is thought to require specific mechanisms: for example, several cooperators have to arise simultaneously to overcome an invasion barrier 9 or arise as spatial clusters 10,11 . This understanding is based on traditional concepts of evolutionary stability and dynamics of infinite populations 12−16 . Here we study evolutionary game dynamics in finite populations 17−20 and show that a single cooperator using a reciprocal strategy 3,21 can invade a population of defectors with a probability that corresponds to a net selective advantage. We specify the conditions for natural selection to favor the emergence of cooperation and derive conditions for evolutionary stability in finite populations.Explaining the evolution of cooperation by natural selection has been a major theme of evolutionary biology since Darwin. The standard game dynamical formulation, which captures the essence of the problem, is the Prisoner's Dilemma. In the non-repeated game, defection dominates cooperation. In the repeated game, stratetegies like tit-for-tat (TFT) or win-stay, lose-shift allow cooperation, but the question is how do they arise in the first place? Always defect (AllD) is evolutionarily stable against invasion by TFT in traditional game dynamics of infinite populations.Let us investigate a game between two strategies, A and B, with payoff matrix If the initial frequency of TFT is less than this value, then it will be eliminated by natural selection.TFT can only replace AllD if its initial frequency exceeds this invasion barrier.Let us now study a stochastic process describing a finite population of size N . At each time step, one individuals is chosen for reproduction proportional to fitness. The offspring replaces a randomly chosen individual. The population size is strictly constant 22 .The fitness of each player depends on the number of TFT or AllD players. In addition,we introduce a parameter w, which determines the contribution of the game's payoff to fitness. This parameter, quantifying the intensity of selection, cancels out in deterministic replicator dynamics of infinite populations, but plays a crucial role in finite populations, as we shall see.We can calculate the probability, ρ, that starting from a single individual strategy A will invade and take over a population of B players (Methods). in very large populations, the selection against TFT at low frequencies is too strong.Thus, neither small nor large but intermediate population sizes are optimum for initiating cooperation.Can we derive the underlying prinicple that determines whether a particular payoff matrix (1) allows selection for TFT replacing AllD? The exact expression for ρ is complicated. The condition ρ > 1/N requires the solution of N -th order polynomials, and a diffusion approximation yields transcendental e...

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A key aspect of human behaviour is cooperation. We tend to help others even if costs are involved. We are more likely to help when the costs are small and the benefits for the other person significant. Cooperation leads to a tension between what is best for the individual and what is best for the group. A group does better if everyone cooperates, but each individual is tempted to defect. Recently there has been much interest in exploring the effect of costly punishment on human cooperation. Costly punishment means paying a cost for another individual to incur a cost. It has been suggested that costly punishment promotes cooperation even in non-repeated games and without any possibility of reputation effects. But most of our interactions are repeated and reputation is always at stake. Thus, if costly punishment is important in promoting cooperation, it must do so in a repeated setting. We have performed experiments in which, in each round of a repeated game, people choose between cooperation, defection and costly punishment. In control experiments, people could only cooperate or defect. Here we show that the option of costly punishment increases the amount of cooperation but not the average payoff of the group. Furthermore, there is a strong negative correlation between total payoff and use of costly punishment. Those people who gain the highest total payoff tend not to use costly punishment: winners don't punish. This suggests that costly punishment behaviour is maladaptive in cooperation games and might have evolved for other reasons.