What is your strategy?

I can run the Monte-Carlo simulation, but to determine the "best" strategy, I have to use some performance metric. Which one would you like me to use?

 

Profit. Starting with $1000, what is your final bankroll after ten spins? On average, of course.

Which strategy is best and which is worst? Thanks.

 

Okay, I'll run it later today. Dom993, would you mind running it, too, just verify the validity? Let's run, say, 100000 simulations, each one for 10 spins of the wheel.

 

OK, I added 2 more strategies for reference, and ran the simulation 10 million times. I also added a condition that if the bankroll drops below $1, no bets can be made. The utility function is the average profit after 10 spins.

Here are the results:

Code:

Strategy	R16	R14	Red	AverageProfit
E		50	0	0	21481547
D		25	0	0	494454
B		5	0	5	4339
C		3	2	5	3544
A		0	0	10	272

The undisputed winner is Strategy E, which bets 50% of the bankroll on R16.

Insane? Yes. Are the results valid? Yes, the simulation simply shows how to maximize the utility function, which is the average profit, with no adjustment for risk.

 

Thanks for the thorough testing. Yikes! I didn't expect this.

I think the problem is the average. I should have specified the median, because 'average' usually is interpreted as the mean (even though, strictly speaking, 'average' can indicate the mean, the median or the mode).

And the mean here is meaningless (no pun intended). After all, if we put Bill Gates in a room with 99 assembly workers, the mean wealth in that room isn't representative of anybody there.

If it's not too much trouble, can you post the results based on the median bankrolls? Thanks.

 

Correct. The "average" result is skewed by just a few outliers which have an enormous profit, against the vast majority of the outcomes which result in a total loss of a bankroll. I'll re-run with the median, and post the results shortly.

 

Here is the same simulation, with both average and median profits:

Code:

Strategy	R16	R14	Red	AveProfit	MedianProfit
E		50	0	0	21481547	156
D		25	0	0	494454		2084
B		5	0	5	4339		1344
C		3	2	5	3544		1849
A		0	0	10	272		162

In deciding which strategy to use, the median profit certainly looks like an improvement, compared to the average profit. However, it still prescribes an insane bet of 25% on R16. Clearly, some risk-adjustment is needed.

 

And here is another one, with an additional metric, roughly equivalent to Sharpe ratio: average profit divided by the standard deviation of profits.

Code:

Strategy	R16	R14	Red	AveProfit	MedianProfit	MedianSharpe
E		50	0	0	21481547	156		0.04
D		25	0	0	494454		2084		0.20
B		5	0	5	4339		1344		0.85
C		3	2	5	3544		1849		0.88
A		0	0	10	272		162		1.18

Median Sharpe prescribes the most conservative of the 5 strategies, %10 bet on Red. So, three different metrics, and three different "best" strategies. Good stuff, eh?

 

Thanks, nonlinear5.

Code:

Strategy	MedianProfit  k*E
C		1849    0.35601508
B		1344    0.28061395
A		 162     0.05917

Just as I suspected, your original performance metric predicts exactly what the best strategy is, even when Kelly sizing itself isn't used.

Trust your first instinct, nonlinear!

 

As it turns out, there is a well-known solution to this problem. See the "many horses" section in this article. In our particular case, the Red-16, Red-14, and Red can be thought of as 3 different horses, with the well-defined odds and edges. So, the Kelly criterion can be applied to this "multiple horses" situation as a combination bet just as well as it can be applied to just one horse as a single bet.