Abstract
Cooperation is a central mechanism for evolution. It consists of an individual paying a cost in order to benefit another individual. However, natural selection describes individuals as being selfish and in competition among themselves. Therefore explaining the origin of cooperation within the context of natural selection is a problem that has been puzzling researchers for a long time. In the paradigmatic case of the Prisoner’s Dilemma (PD), several schemes for the evolution of cooperation have been proposed. Here we introduce an extension of the Replicator Equation (RE), called the Optimal Replicator Equation (ORE), motivated by the fact that evolution acts not only at the level of individuals of a population, but also among competing populations, and we show that this new model for natural selection directly leads to a simple and natural rule for the emergence of cooperation in the most basic version of the PD. Contrary to common belief, our results reveal that cooperation can emerge among selfish individuals because of selfishness itself: if the final reward for being part of a society is sufficiently appealing, players spontaneously decide to cooperate.
Highlights
Cooperation is a central mechanism for evolution
This condition is not trivial in this case, at least for two reasons: firstly, it is obtained as the result of an optimal strategy; secondly, and most importantly, because the two variables pC and pD are dynamical, with dynamical equations (7) and terminal conditions determined by the choice of the final average fitness according to (8)
We have proposed a modified version of the Replicator Equation (RE), the equation governing natural selection, called the Optimal Replicator Equation (ORE), which stems from the assumption that evolution is an optimization process that on the one side selects at any given time those individuals with higher fitness and on the other side favours those populations with higher average fitness
Summary
Cooperation is a central mechanism for evolution. It consists of an individual paying a cost in order to benefit another individual. Dynamical models for the coevolution of the existing species and their fitness landscape have been proposed[33,34,35,36] This subject is fundamentally linked to the fact that evolution is similar to an optimization process: a population evolves in order to adapt as much as possible to some external conditions (which in turn may change in time). In[38] it has been argued that living systems adapt to the environment by performing an optimal control
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