Thoughts on solution concepts

This paper explores connections between Ficici's notion of solution concept and order theory. Ficici postulates that algorithms should ascend an order called weak preference; thus, understanding this order is important to questions of designing algorithms. We observe that the weak preference order is closely related to the pullback of the so-called lower ordering on subsets of an ordered set. The latter can, in turn, be represented as the pullback of the subset ordering of a certain powerset. Taken together, these two observations represent the weak preference ordering in a more simple and concrete form as a subset ordering. We utilize this representation to show that algorithms which ascend the weak preference ordering are vulnerable to a kind of bloating problem. Since this kind of bloat has been observed several times in practice, we hypothesize that ascending weak preference may be the cause. Finally, we show that monotonic solution concepts are convex in the order-theoretic sense. We conclude by speculating that monotonic solution concepts might be derivable from non-monotonic ones by taking convex hull. Since several intuitive solution concepts like average fitness are not monotonic, there is practical value in creating monotonic solution concepts from non-monotonic ones.