Unbiased coevolutionary solution concepts

The primary concern of this paper is the question: Is the goal of an optimal well behaved coevolutionary algorithm attainable? We approach this question from the point of view of the No Free Lunch (NFL) theorem. The NFL theorem has been shown to, in general, not hold in coevolution and as such we can hope for optimal (in the NFL sense) coevolutionary algorithms. We attempt to shed light on this question by investigating the relationship between Ficici's notion of monotonicity and algorithm performance by introducing the notion of solution concept bias. Informally, the unbiased solution concepts are those for which the NFL theorem holds. We show that the notion of solution concept bias and Ficici's notion of monotonicity are orthogonal in the sense that all possible combinations of bias and monotonicity are possible. We also explore some possible consequences and trade-offs which might arise in coevolutionary algorithm design due to different combinations of bias and monotonicity. For example, in biased monotonic solution concepts there may be a trade-off between the guarantee of good algorithmic behavior and optimality. An algorithm which improves monotonically may be suboptimal. The results presented in this paper raise the possibility that the goals of monotonicity and an optimality may be in conflict and bring to light the question: Which is more important, the quality of the solution produced by a coevolutionary algorithm, or the dynamics by which that solution was obtained?