CHAPTER 4 Optimalf
Optimal Fixed Fraction
We have seen that in order to consider betting/trading a given situation/
system you must first determine if a positive mathematical expectation ex-
ists. We have seen in the previous chapter that what is seemingly a “good
bet” on a mathematical expectation basis (i.e., the mathematical expecta-
tion is positive) may in fact not be such a good bet when you consider
reinvestment of returns.^1 Reinvesting returns never raises the mathemati-
cal expectation (as a percentage—although it can raise the mathematical
expectation in terms of dollars, which it does geometrically, which is why
we want to reinvest). If there is in fact a positive mathematical expectation,
however small, the next step is to exploit this positive expectation to its
fullest potential. This has been shown, for an independent trials process,
to be by reinvesting a fixed fraction of your total stake,^2 which leads to the
following axiom:For any given independent trials situation where you
(^1) If you are reinvesting too high a percentage of your winnings relative to the disper-
sion of outcomes of the system.
(^2) For a dependent trials process the idea of betting a proportion of your total stake
also yields the greatest exploitation of a positive mathematical expectation, just like
an independent trials process. However, in a dependent trials process you optimally
bet a variable fraction of your total stake, the exact fraction for each individual bet
determined by the probabilities and payoffs involved for each individual bet.
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