A Mathematical Analysis of the Long-Run Behavior of Genetic Algorithms for Social Modeling

We present a mathematical analysis of the long-run behavior of genetic algorithms that are used for modeling social phenomena. The analysis relies on commonly used mathematical techniques in evolutionary game theory. Assuming a positive but infinitely small mutation rate, we derive results that can be used to calculate the exact long-run behavior of a genetic algorithm. Using these results, the need to rely on computer simulations can be avoided. We also show that if the mutation rate is infinitely small the crossover rate has no effect on the long-run behavior of a genetic algorithm. To demonstrate the usefulness of our mathematical analysis, we replicate a well-known study by Axelrod in which a genetic algorithm is used to model the evolution of strategies in iterated prisoner's dilemmas. The theoretically predicted long-run behavior of the genetic algorithm turns out to be in perfect agreement with the long-run behavior observed in computer simulations. Also, in line with our theoretically informed expectations, computer simulations indicate that the crossover rate has virtually no long-run effect. Some general new insights into the behavior of genetic algorithms in the prisoner's dilemma context are provided as well