Maybe (k*E) does not exactly correspond to LN(bankroll), which is the rate of geometric growth.
What is your strategy?
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Maybe (k*E) does not exactly correspond to LN(bankroll), which is the rate of geometric growth.
You didn't post your result for Green = 0. What is it?
And the rate of geometric growth is
((1+35F-S-G)^(3/37))*((1-F+35S-G)^(4/37))*((1-F-S+35G)^(1/37))*((1-F-S-G)^(29/37)).
I don't know what the above equation is. We are looking to maximize the rate of growth g, which is the log of the terminal bankroll so I believe my version (well, SplawnDarts' version) is correct:
Code:
F(R16,R14,Red,Green) = (4/37) * log1p(35 * R16 - Green - R14 + Red)
+ (3/37) * log1p(-R16 + 35 * R14 - Green + Red)
+ (16/37) * log1p(-R16 - R14 - Green + Red)
+ (13/37) * log1p(-R16 - R14 - Green - Red)
+ (1/37) * log1p(-R16 - R14 + 35 * Green - Red)I am not surprised that the green could be part of the solution, but I am surprised that the red is not, which would indicate that its marginal effect is zero to negative. Did anyone give an intuitive explanation why that is the case?
Red is part of the solution, too. It's just that we are experimenting with various rules, such as excluding bet on red altogether.
You are right in that there is nothing special about Green-0. A bet on Green-0 is just as good as a bet on any other number, as long as it's not R16, R14, or Red.
The expression I posted is the actual rate of geometric growth. If you take the log of that expression you get Splawndarts' formula.
The whole reason for dealing with logs is solution facilitation. Logs turn a product to be maximized into a sum to be maximized, because the bankroll rate of growth becomes a maximum at the same fractions its logarithm becomes a maximum.
Mea culpa. Your results are correct.
I managed to finally get accurate findings myself.
To summarize:
Betting on R-14 alone:
k14 == 5.483%
k*E14 == 0.10520714
Betting on R-16 alone:
k16 == 8.263%
k*E16 == 0.23894396
Betting on Green alone:
k0 == 0% because k<0 is not allowed.
k*E0 == 0.
Betting on both R-14 and R-16:
F1 == 5.723% > k14
S1 == 8.426% > k16
k*E1 == 0.3535 > k*E14 + k*E16
Betting on all three {R-14, R-16, G-0}:
F2 == 5.733% > F1
S2 == 8.436% > S1
G == 0.328% > k0
k*E2 == 0.3539 > k*E1
I find this synergy from combining bets fascinating. Anybody else?
Looks like the (k*E) metric has its merit, right?
Very right. It's still the best performance metric here IMO. Just deucedly hard to calculate k for three or more bets is all.
But what about this crazy synergy? Even adding a negative expectation bet can increase performance! :eek:
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