What is your strategy?

nonlinear5 · Apr 1, 2014

kut2k2 said:
OK let's try a different question.

Which of the three strategies below is best according to MCS?

A) bet 10% on Red ;
B) bet 5% on Red and 5% on R-16 ;
C) bet 5% on Red, 3% on R-16 and 2% on R-14.

Now we can make an apples-to-apples comparison looking only at asset allocation and with no preordained performance metric.

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?

kut2k2 · Apr 1, 2014

nonlinear5 said:
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.

nonlinear5 · Apr 1, 2014

kut2k2 said:
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.

nonlinear5 · Apr 1, 2014

kut2k2 said:
OK let's try a different question.

Which of the three strategies below is best according to MCS?

A) bet 10% on Red ;
B) bet 5% on Red and 5% on R-16 ;
C) bet 5% on Red, 3% on R-16 and 2% on R-14.

Now we can make an apples-to-apples comparison looking only at asset allocation and with no preordained performance metric.

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.

kut2k2 · Apr 1, 2014

nonlinear5 said:
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.

nonlinear5 · Apr 1, 2014

kut2k2 said:
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 specify 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.

nonlinear5 · Apr 1, 2014

nonlinear5 said:
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.

nonlinear5 · Apr 1, 2014

nonlinear5 said:
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?

kut2k2 · Apr 1, 2014

nonlinear5 said:
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.

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!

nonlinear5 · Apr 1, 2014

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.