What is your strategy?

[kut2k2]

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:

	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?

Oy! Sorry, I have to do it my way.

Let x == fraction of unit bet on Red ,
Let y == fraction of unit bet on R-14 ,
Let z == fraction of unit bet on R-16.

x + y + z == 1

E == (4/37)(35z+x-y) + (3/37)(35y+x-z) + (16/37)(x-y-z) + (14/37)(-1)

Now comes the hard part:

0 == 4(35z+x-y)/(1+k*(35z+x-y)) + 3(35y+x-z)/(1+k*(35y+x-z)) + 16(x-y-z)/(1+k*(x-y-z)) - 14/(1-k)

Solve for k eek

The quick-and-dirty solution for k (and probably a serious underestimate in this case) is

k1 == (4(35z+x-y) + 3(35y+x-z) + 16(x-y-z) - 14) / (4(35z+x-y)^2 + 3(35y+x-z)^2 + 16(x-y-z)^2 + 14)

Instead I strongly recommend a numerical equation cruncher like Excel's Solver routine.

 

[SplawnDarts]

Instead I strongly recommend a numerical equation cruncher like Excel's Solver routine.

Good call on Excel solver - I'd never used that before.
I get

8.1% on 16
5.4% on 14
10.8% on red

 

[nonlinear5]

That makes sense...

I managed to crash my machine twice, because I was doing so many simulations, but I think I've resolved the discrepancy.

Your top result (maximizing the F(), numerically):
{R-16: 8%, R-14: 5%, Red: 11%}

My top result (Monte-Carlo, utility function is the average log(endingBankroll)):
{R-16: 8%, R-14: 5%, Red: 11%} (same as yours)

My top result (Monte-Carlo, utility function is the median log(endingBankroll)):
{R-16: 8%, R-14: 8%, Red: 6%}

Kut2k2, you appear to be optimizing for the absolute profit, so your result would be different.

 

[nonlinear5]

Oy! Sorry, I have to do it my way.

Let x == fraction of unit bet on Red ,
Let y == fraction of unit bet on R-14 ,
Let z == fraction of unit bet on R-16.

x + y + z == 1

This should be:
x + y + z <= 1

 

[SplawnDarts]

Here's an amusing thing I found: if you add the option to bet on black and/or green, the solver gives almost the same solution, but puts 0.6% on green and ups red to 11.4%. It appears that green, even with a slightly negative expectation, is such a nice hedge that you're better off including a tiny bit and betting more. This isn't supper surprising I suppose - there are analogous results in portfolio theory where including negative expectation, negatively correlated "assets" actually improves things.

Black is so horribly negative expectation you could never use it that way though.

 

[nonlinear5]

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.

Did you arrive at 1/5 to 1/10 Kelly analytically, or is it just a gut feel? Do you use discrete or continuous Kelly?

 

[kut2k2]

I managed to crash my machine twice, because I was doing so many simulations, but I think I've resolved the discrepancy.

Your top result (maximizing the F(), numerically):
{R-16: 8%, R-14: 5%, Red: 11%}

My top result (Monte-Carlo, utility function is the average log(endingBankroll)):
{R-16: 8%, R-14: 5%, Red: 11%} (same as yours)

My top result (Monte-Carlo, utility function is the median log(endingBankroll)):
{R-16: 8%, R-14: 8%, Red: 6%}

Kut2k2, you appear to be optimizing for the absolute profit, so your result would be different.

The Kelly equation I posted indicates the total betting fraction (k) that grows a given asset allocation {x,y,z} the fastest in this roulette game. So it doesn't find the optimal allocation {x_opt, y_opt, z_opt}.

x_opt, y_opt and z_opt are the values that maximize k*E.

 

[SplawnDarts]

Did you arrive at 1/5 to 1/10 Kelly analytically, or is it just a gut feel? Do you use discrete or continuous Kelly?

It's based on monte carlo equity curves with various kelly fraction bets. 1/5 to 1/10 was the range where they started looking "acceptable" to my eye - no stomach churning drawdowns.

Most of the stuff I do has discrete outcomes, so I use discrete Kelly (or the general log/maximization form thereof when there are weird correlations like in this case).

 

[kut2k2]

This should be:
x + y + z <= 1

No. Reread the definitions. They're based on a unit bet.

To put it another way, x == R/(R+F+S), and so on.

 

[kut2k2]

I managed to crash my machine twice, because I was doing so many simulations, but I think I've resolved the discrepancy....

My top result (Monte-Carlo, utility function is the median log(endingBankroll)):
{R-16: 8%, R-14: 8%, Red: 6%}

Kelly == 0.1963808
Expectation == 1.81572481
k*E == 0.35657349



There is overbetting here (betting fraction == 22% > k) but the proportions are very good.