What is your strategy?

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Incidentally, assuming no cheating the ratio of what you should bet is

S = fraction of bankroll bet on 16
F = fraction of bankroll bet on 14
R = fraction of bankroll bet on red

maximize f(S,F,R) over the range 0 to S+F+R < 1 using gradient=0 method where
f(S,F,R) = ln(4/37(35S - F + R) + 3/37(-S+35F+R)+16/37(-S-F+R)+14/37(-S-F-R))

The issue is then what bankroll to bet against $1000 or life roll or some combination.

And as always with Kelly math, you may want to derate some to give a smoother equity curve. I generally size most of my bets at 1/5 to 1/10th Kelly on life bankroll.
 
And here is what I think is the answer to the original question. The "best" strategy is a close tie between these top 10 strategies:

Code: 
	R16	R14	Red	MedianProfit
	8	8	5	2473.37
	7	7	5	2472.91
	8	8	4	2472.69
	7	7	6	2472.38
	8	7	4	2472.12
	8	8	6	2470.02
	7	7	4	2469.58
	8	8	3	2468.00
	7	7	7	2467.99
	8	7	3	2466.19

The simulation effectively maximizes the following, as suggested by SplawnDarts:
maximize f(S,F,R) over the range 0 to S+F+R < 1 using gradient=0 method where
f(S,F,R) = ln(4/37(35S - F + R) + 3/37(-S+35F+R)+16/37(-S-F+R)+14/37(-S-F-R))

kut2k2, can you calculate the E*K for these, to see f the E*K score agrees with the "median profit" score?
 
SplawnDarts said:
Incidentally, assuming no cheating the ratio of what you should bet is

S = fraction of bankroll bet on 16
F = fraction of bankroll bet on 14
R = fraction of bankroll bet on red

maximize f(S,F,R) over the range 0 to S+F+R < 1 using gradient=0 method where
f(S,F,R) = ln(4/37(35S - F + R) + 3/37(-S+35F+R)+16/37(-S-F+R)+14/37(-S-F-R))
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Ok, what's the solution then? And why are you taking a log? Trying to find the first derivative?
 
Visaria said:
Ok, what's the solution then? And why are you taking a log? Trying to find the first derivative?
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The underlying math of Kelly criterion is that you maximize the expected log of your bankroll. So I actually typed slightly the wrong thing. I should have typed:

f(S,F,R) = 4/37*log(35S - F + R + 1) + 3/37*log(-S+35F+R+1)+16/37*log(-S-F+R+1)+14/37*log(-S-F-R+1)

Oops - I pulled out the log when I should have distributed it and forgot to include the starting bankroll in the ending bankroll. I rarely do Kelly math. As to the numeric solution, I'm not taking that gradient by hand and don't have Mathematica handy to do it.
 
SplawnDarts said:
The underlying math of Kelly criterion is that you maximize the expected log of your bankroll. So I actually typed slightly the wrong thing. I should have typed:

f(S,F,R) = 4/37*log(35S - F + R + 1) + 3/37*log(-S+35F+R+1)+16/37*log(-S-F+R+1)+14/37*log(-S-F-R+1)

Oops - I pulled out the log when I should have distributed it and forgot to include the starting bankroll in the ending bankroll. I rarely do Kelly math. As to the numeric solution, I'm not taking that gradient by hand and don't have Mathematica handy to do it.
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Your solution is much more elegant than mine, since it does not require any random number generation or Monte Carlo simulations. So, it's much faster and more accurate. Here is what I get with your proposed solution (top 20 strategies):

Code: 
	R16	R14	Red	F(R16, R14, Red)
	8	5	11	0.11634
	8	5	12	0.11629
	8	5	10	0.11628
	8	6	11	0.11622
	8	6	10	0.11621
	8	5	13	0.11612
	8	6	12	0.11612
	8	5	9	0.11610
	8	6	9	0.11609
	9	5	11	0.11598
	9	5	10	0.11596
	8	6	13	0.11590
	9	5	12	0.11589
	8	6	8	0.11585
	8	5	14	0.11584
	9	6	10	0.11584
	9	5	9	0.11583
	8	5	8	0.11582
	9	6	11	0.11581
 
nonlinear5 said:
What's the meaning of "+1" in each log, such as in
log(35S - F + R + 1) ?
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That's your ending bankroll assuming that case occurred. So you start out with a bankroll of 1 (arbitrary unit). If that case (16) hits, you win 35x whatever fraction of the bankroll you bet on sixteen, lose whatever fraction you bet on fourteen, and win whatever fraction you bet on red. The +1 is just the observation you still have the original bankroll.
 
SplawnDarts said:
That's your ending bankroll assuming that case occurred. So you start out with a bankroll of 1 (arbitrary unit). If that case (16) hits, you win 35x whatever fraction of the bankroll you bet on sixteen, lose whatever fraction you bet on fourteen, and win whatever fraction you bet on red. The +1 is just the observation you still have the original bankroll.
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Ok, thanks. My Monte-Carlo disagrees somewhat with your more analytical solution (specifically with respect to the bet on red), so I am looking for he explanation of the discrepancy.
 
nonlinear5 said:
Ok, thanks. My Monte-Carlo disagrees somewhat with your more analytical solution (specifically with respect to the bet on red), so I am looking for he explanation of the discrepancy.
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Actually, I know why. I am ranking the strategies based on the mean profit, while you rank them based on the compound rate of return.
 
nonlinear5 said:
Ok, thanks. My Monte-Carlo disagrees somewhat with your more analytical solution (specifically with respect to the bet on red), so I am looking for he explanation of the discrepancy.
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One thing I will note is that Monte Carlo simulation tends to converge slowly near Kelly bet because the volatility is pretty high (hence the desirability of betting half Kelly or less).

Incidentally, if you're seeing disagreement did you manage to take the max of that function? If so I'd be curious what it was.

Edit: nevermind - looks like you managed to take the max numerically.
 
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