Finite Markov Chain Analysis

Finite, discrete-time Markov chain models of genetic algorithms have been used successfully in the past to understand the complex dynamics of a simple GA. Markov chains can exactly model the GA by accounting for all of the stochasticity introduced by various GA operators, such as initialization, selection, crossover, and mutation. Although such models quickly become unwieldy with increasing population size or genome length, they provide initial insights that guide our development of approximate, scalable models. In this study, we use Markov chains to analyze the stochastic eeects of the \niching operator" of a niched GA. Speciically, we model the eeect of tness sharing on a single-locus genome. Without niching, our model is an absorbing Markov chain. With niching, we are dealing with a \quasi-ergodic" Markov chain. Rather than calculating expected times to absorption, we are interested in steady-state probabilities for positive recurrent states. Established techniques for analyzing ergodic Markov chains give us new insights into the dynamic nature of a niched GA. We explore the stability of the expected steady state distribution achieved by the niched GA. We demonstrate the \niching pressure" as a force separate from the forces of selection, drift, and mutation. Through visualization, we gain intuitions of the relationships among these separate forces. These results generalize beyond the tness sharing algorithm to all types of GA optimization of context dependent functions. In any such function, the GA must nd and maintain a diverse population of cooperative individuals rather than converging to the truly steady state of a uniform population.