A new kind of turnpike theorem

Briefly and informally an (economic or adaptive) plan can be described in terms of its action with respect to two sets: Cfc, an effectively defined set of objects; M, a set of functions /*#: (P-*(R, £ £ 8 , where (P is the set of distributions over G, and (R is a ranking set of positive real numbers. Under the intended interpretation, each P £ ( P corresponds to a mix of goods, chromosomes, strategies, or programs, 8 is a set of possible conditions or environments under which the plan is expected to operate, and the ranking of P , IXE(P), specifies utility, expected offspring, payoff, or efficiency of the mix in condition or environment E £ 8 . The plan, then, is a procedure (cf. sequential sampling procedure, dynamic programming policy) for searching (P in an at tempt to locate mixes of high rank in any given £ £ 8 ; the object is to construct (if possible) a plan which is "robust with respect to M" in the sense that the search proceeds "efficiently" for any JJLEŒM. More formally, with any pair (r, E) where r is a plan and £ £ 8 , one can associate a trajectory through (P, ((P(r, £ ) ) = ((Pi(r, E), (P2(r, £ ) , • • • , (P*(r, £ ) , • • • ); coordinated with the trajectory is the sequence of rankings (IIE(T)) = (HE,I(T), /x^,2(r), • • • , iXE,t(j)> • • • ) where iiE,t{r) =At^((Pt(r, E)). A plan r0 will be called good in E relative to a set of plans 3 if