Genetic Algorithms with a State Dependent Fitness Function

In the last chapter we have mainly reviewed existing literature and models, which have proven to be of great importance for the theoretical analysis of genetic algorithms. In this chapter we will deal with a new problem, which has not been dealt with up to now. The analytical models presented in chapter 3 all assume that the genetic algorithm is used to solve an optimization problem for a given non changing fitness function. However, this is not the case if we think of an economic system, like a market, where the payoff of a single market member depends crucially on the actions of the rest of the market. The same argument holds also for two agents playing a normal form game, where the payoff of a strategy depends on the opponents’ strategy. This means that two major aspects of the genetic algorithm will change when it is used to model learning in economic systems. Firstly, the fitness of a single string depends on the state of the whole population. In an optimization problem the fitness values can be written in one r-dimensional vector f, but in economic systems the fitness is given by a r-dimensional function f : S → ℝr, where f _k (φ) is the fitness of string k when the whole population is in state φ ∈ S1. Secondly, we are no longer interested in the learning of optimal solutions, but we intend to learn equilibria. As the simplest concept of an economic equilibrium we say that the system is in equilibrium if the current action of every agent is optimal under the assumption, that all other agents behave according to the equilibrium. For later reference we give a formal definition of an economic equilibrium in definition 4.1.1. In what follows, we will often stress the term “economic” to distinguish it from dynamic equilibria which we will consider later on.