=+Show that Q_(x) ≤ Q. (x) for all x and all policies w. Such a Tro is optimal.

Theorem 7.3 is the special case of this result for p = {, bold play in the role of Tro, and u(x) = 1 or u(x) = 0 according as x =1 or x < 1.

The condition (7.34) says that gambling with policy To is at least as good as not gambling at all; (7.35) says that, although the prospects even under To become on the average less sanguine as time passes, it is better to use To now than to use some other policy for one step and then change to " o.