the column labeled "to contribute" we multiply
each of the expectations by its probability; the total
of this column, or "bottom line," is our
expectation if we draw a card. Since a loss of 48
cents by drawing is preferable to one of 54 cents
from standing, basic strategy is to draw to (T,6)
v 9. Note that it was assumed that we would not draw
a card to (T,6,A), (T,6,2), etc. This decision would
rest on a previous and similar demonstration that
it was not in our interest to do so.
our best strategy and consequent expectation with
a smaller total of possibly more than two cards, such
as (5,4,3), would be based on a sort of recursive
reference to previous calculations of our optimal
expectation and strategy with (5,4,3,A), (5,4,3,2)....
(5,4,3,9), (5,4,3,A,A), etc. All this is very tedious
and time consuming, but necessary if the exact player
expectation is sought. This, of course, is what computers
ire designed for; limitations on the human life span
and supply of paper preclude an individual doing the
calculations by hand.
is it time to move to a new limit? If you double your
bankroll, it is time to consider playing higher. Just
recognize that the competition generally is better
the higher you go, so you shouldn't double the stakes
you're playing just because you've doubled your bankroll,
at least not as a long-term plan.