A Mathematical Framework for the Study of Coevolution

Despite achieving compelling results in engineering and optimization problems, coevolutionary algorithms remain difficult to understand, with most knowledge to date coming from practical successes and failures, not from theoretical understanding. Thus, explaining why coevolution succeeds is still more art than science. In this paper, we present a theoretical framework for studying coevolution based on the mathematics of ordered sets. We use this framework to describe solutions for coevolutionary optimization problems, generalizing the notion of Pareto non-dominated front from the field of multi-objective optimization. Our framework focuses attention on the order structure of solution and test sets, which we argue is a key source of difficulty in coevolutionary optimization problems. As an application of the framework we show, in the special case of two-player games, that Pareto dominance is closely related to intransitivities in the game.