A Markov Chain Analysis of Genetic Algorithms with a State Dependent Fitness Function

We an alyze the behavior of a Simple Geneti c Algorithm (GA) in syste ms where the fitn ess of a st ring is det ermined by a fun ct ion depending on the state of the whole populati on . The GA is mod eled by a Markov chain and we conclude that for small mutation probabiliti es the limi t distribution will put almost all t he weight to the homogeneous states. We derive condit ions under which a homogeneo us state will be stable for the dyn ami cs repr esenting the expected behavior of the GA. In terpret ing the state dep endent fitness fun ction as an economic syste m we prove that any strict economic equilibrium will be asymptotically st abl e with resp ect to the expect ed behavior of the GA.