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Optimal
Kelly wagering requires precise bet sizing (which can result
in wagers in fractions of a chip). Though powerful in theory,
in practice, perfect Kelly betting is not realistic. A somewhat
more practical approach to betting is a method referred
to as "modified proportional betting." The wagering
is still quasi-Kelly, but there are important differences.
First, it's assumed that any win or loss for a playing session
is small compared to the entire bankroll. This way, the
bankroll need not be re-evaluated prior to every wager.
Before a session begins, the bankroll can be evaluated once
and bet sizes predetermined for the entire session.
Furthermore, bets are capped at a maximum level we'll call
the "ceiling." Ideally, we'd like to bet proportionally
(to our advantage) no matter how high the expectation rises.
In practice, however, we cannot get away with spreading
from, say, 1 to 100 units, even though the count may merit
it. In the casino, we must have a ceiling at some point,
if only because of the table limit. The range of the floor
to ceiling levels is commonly called the "bet spread."
It's always somewhat problematic to develop a benchmark
for determining the performance of a card-counting system
(for purposes of comparing with other systems). For this
book, we've chosen to use a modified proportional betting
comparison, which places the respective systems on a similar
scale in terms of risk of ruin.
Assuming a modified proportional betting scheme is the appropriate
vehicle of comparison, two important variables must be determined.
First, what is the bet spread? Second, how quickly does
the wagering traverse the bet spread?
We've already touched on some of the interesting ramifications
of choosing and attempting to implement a bet spread. Ideally,
we'd like to use an infinite spread. But again, in practice
this isn't possible. We believe a reasonable bet spread
for which counters should strive is 1 to 5 in a single-
or double-deck game, and 1 to 10 in 6- and 8-deck games.
Traversing the bet spread is a concept that also merits
further attention. The issue here is how fast we change
our wagering from the minimum (at a disadvantage) to the
maximum (with the advantage). With a finite bankroll, we'll
move up and down with our bets in direct proportion to our
prevailing expectation (a la Kelly). The slope of the ramp
is proportional to our starting bankroll, with a greater
bankroll implying a steeper ramp (we'll make more maximum
bets).
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