Abstract
We present a mathematical analysis of the long-run behavior of genetic algorithms (GAs) that are used for modeling social phenomena. Our analysis relies on commonly used mathematical techniques in the field of evolutionary game theory. We make a number of assumptions in our analysis, the most important one being that the mutation rate is positive but infinitely small. Given our assumptions, we derive results that can be used to calculate the exact long-run behavior of a GA. 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 GA. To demonstrate the usefulness of our mathematical analysis, we replicate a well-known study by Axelrod in which a GA is used to model the evolution of strategies in iterated prisoner’s dilemmas. The theoretically predicted long-run behavior of the GA 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 GAs in the prisoner’s dilemma context are provided as well.
Highlights
The field of evolutionary computation is concerned with the study of all kinds of evolutionary algorithms
We present a mathematical analysis of the long-run behavior of genetic algorithms (GAs) that are used for modeling social phenomena
We want to show how evolutionary algorithms that are used for modeling social phenomena can be analyzed mathematically using techniques that are popular in evolutionary game theory
Summary
The field of evolutionary computation is concerned with the study of all kinds of evolutionary algorithms. It is not our aim to argue in favor of either the agent-based computational economics approach, which emphasizes algorithms and computer simulations, or the evolutionary game-theoretic approach, which emphasizes models and mathematical analysis. This is a quite remarkable result that, to the best of our knowledge, has not been reported before in the theoretical literature on GAs. The result implies that when GAs are used for modeling social phenomena the crossover rate is likely to be a rather insignificant parameter, at least when one is mainly interested in the behavior of GAs in the long run (for the short run, see Thibert-Plante and Charbonneau 2007). Proofs of our mathematical results are provided in the Appendix
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