  Given a sequence of random variables ${\bf X}=X_1,X_2,\ldots$ suppose the aim
is to maximize one's return by picking a `favorable' $X_i$. Obviously, the
expected payoff crucially depends on the information at hand. An optimally
informed person knows all the values $X_i=x_i$ and thus receives $E (\sup
X_i)$. We will compare this return to the expected payoffs of a number of
observers having less information, in particular $\sup_i (EX_i)$, the value of
the sequence to a person who only knows the first moments of the random
variables.
  In general, there is a stochastic environment (i.e. a class of random
variables $\cal C$), and several levels of information. Given some ${\bf X} \in
{\cal C}$, an observer possessing information $j$ obtains $r_j({\bf X})$. We
are going to study `information sets' of the form $$ R_{\cal C}^{j,k} = \{
(x,y) | x = r_j({\bf X}), y=r_k({\bf X}), {\bf X} \in {\cal C} \}, $$
characterizing the advantage of $k$ relative to $j$. Since such a set measures
the additional payoff by virtue of increased information, its analysis yields a
number of interesting results, in particular `prophet-type' inequalities.
