Criteria of optimality and existence of $\bar{p}$-optimal plans

Abstract

Theorem 3.8 is a necessary and sufficient criterion for $\bar{p}$-optimality. But it seems to be of rather limited value since its application does not only require the functions G_n (which in principle may be obtained by solving the OE) but also the functions G_nf which in general are not easy to obtain. The criterion usually used in the literature reads, generalized to our model, as follows: if f_n(y) is a maximum point of L_nG_n+1(h_nf(y),·) for all n ∈ N and all y∈Sⁿ then f is $\bar{p}$-optimal. It is easy to construct counterexamples which show that this criterion is in general not necessary. It becomes necessary if we consider only those y∈Sⁿ that occur under the use of f with positive probability. This leads to