Optimalf 137
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
2.00 1.82 1.67 1.54 1.43 1.33 1.25 1.18 1.11 1.05 1.00
3.50 3.32 3.17 3.04 2.93 2.83 2.75 2.68 2.61 2.55 2.50
1.75 1.49 1.27 1.06 0.88 0.71 0.55 0.40 0.26 0.13 0.00
3.06 2.73 2.41 2.10 l.80 1.51 1.21 0.91 0.61 0.31 0.00
2.681.53 2.241.23 1.830.96 1.450.74 1.110.54 0.800.38 0.530.24 0.140.31 0.060.14 0.020.04 0.000.00
1.34 1.01 0.73 0.51 0.33 0.20 0.11 0.05 0.01 .00 0.00
2.34 1.84 1.39 1.00 0.68 0.42 0.23 0.11 0.03 .00 0.00
2.051.17 1.510.83 1.060.56 0.690.35 0.420.20 0.230.11 0.100.05 0.020.04 0.01.00 .00.00 0.000.00
1.03 0.68 0.42 0.24 0.13 0.06 0.02 0.01 .00 .00 0.00
1.80 1.24 0.80 0.48 0.26 0.12 0.05 0.01 .00 .00 0.00
0.90 0.56 0.32 0.17 0.08 0 03 0.01 .00 .00 .00 0.00
0.791.57 0.461.02 0.240.61 0.120.33 0.050.16 0.020.06 0.02.00 .00.00 .00.00 .00.00 0.000.00
0.69 0.38 0.19 0.08 0.03 0.01 .00 .00 .00 .00 0.00
1.20 0.69 0.35 0.16 0.06 0.02 .00 .00 .00 .00 0.00
1.050.60 0.310.56 0.270.14 0.060.11 0.040.02 0.01.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.53 0.25 0.11 0.04 0 01 .00 .00 .00 .00 .00 0.00
0.92 0.46 0.20 0.08 0.02 0.01 .00 .00 .00 .00 0.00
0.46 0.21 0.08 0.03 0.01 .00 .00 .00 .00 .00 0.00
0.400.81 0.170.38 0.060.15 0.020.05 0.01.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.70 0.31 0.12 0.04 0.01 .00 .00 .00 .00 .00 0.00
0.35 0.14 0.05 0.01 .00 .00 .00 .00 .00 .00 0.00
0.310.62 0.120.26 0.040.09 0.010.02 0.01.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.54 0.21 0.07 0.02 .00 .00 .00 .00 .00 .00 0.00
0.27 0.09 0.03 0.01 .00 .00 .00 .00 .00 .00 0.00
0 47 0.17 0.05 0.01 .00 .00 .00 .00 .00 .00 0.00
0.410.24 0.140.08 0.040.02 0.01.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0 21 0.06 0.02 .00 .00 .00 .00 .00 .00 .00 0.00
0.36 0.12 0.03 0.01 .00 .00 .00 .00 .00 .00 0.00
0.320.18 0.100.05 0.020.01 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 .00.00 0.000.00
0.16 0.04 0.01 .00 .00 .00 .00 .00 .00 .00 0.00
0.28 0.08 0.02 .00 .00 .00 .00 .00 .00 .00 0.00
0.14 0.04 0.01 .00 .00 .00 .00 .00 .00 .00 0.00
0.07 0.02 .00 .00 .00 .00 .00 .00 .00 .00 0.00
of .1 or .4 your TWR is 4.66. This is not even half of what it is at .25, yet you
are only 15% away from the optimal and only 40 bets have elapsed! What
does this mean in terms of dollars? At f=.1, you would be making one
bet for every $10 in your stake. Atf=.4 you would be making one bet for
every $2.50 in your stake. Both make the same amount, with a TWR of 4.66.
Atf=.25, you are making one bet for every $4 in your stake. Notice that if
you make one bet for every $4 in your stake, you will make more than twice
as much as you would if you were making one bet for every $2.50 in your
stake! Clearly, it does not pay to overbet. At one bet for every $10 in your
stake you make the same amount as if you had bet four times that amount,
one bet for every $2.50 in your stake! Notice that in a 50/50 game where
you win twice the amount that you lose, at anfof .5 you are only breaking
even! That means you are only breaking even if you made one bet for every
$2 in your stake. At anfgreater than .5 you are losing in this game, and it
is simply a matter of time until you are completely tapped out!
Now let’s increase the winning payout from two units to five units, as
is demonstrated in the data in Figure 4.6. Here your optimalfis .4, or bet
$1 for every $2.50 in your stake. After 20 sequences of+5,−1, 40 bets, your