What is your strategy?

[dom993]

The Kelly fraction is the optimal geometric growth fraction. It by definition is the betting strategy that will grow your bankroll the fastest for a given trading strategy. If you had only a single spin, which better betting strategy would you use instead of Kelly?

Remember: we have yet to determine what the best trading strategy is for this roulette example, which boils down to which bets to take. It appears that incorporating Kelly is a necessary component to deciding what will be the best trading strategy.

On a single-spin opportunity, I would not even bother looking at Red16 as the odds clearly favor a net negative P&L, in case you didn't notice.

The law of large numbers requires really large sample-set to apply.

Back on betting strategy for Red16, betting fixed $50 until the 10th spin or a single win, yields an ending bankroll of +2500 at the 50th percentile, with a low stdev of 997.

I'd say that beats your Kelly-based bets by a large margin.

 

[kut2k2]

Here are some MC results:

8%-bankroll Red-16 : ending bankroll 50th percentile +1823, stdev 68626
4%-bankroll Red-16 : ending bankroll 50th percentile +1682, stdev 7463
$50 fixed bet Red-16: ending bankroll 50th percentile +2350, stdev 1821

Those 3 strategies guarantee a minimum ending bankroll of +438, +669, +500 respectively.

6.7%-bankroll Red-16 is the one fractional strategy that guarantees a minimum ending bankroll of +503, ie. similar to fixed bet $50. Its results are:
ending bankroll 50th percentile +1821, stdev 22539


Now do you still think Kelly applies to small sample-sets?

You should not use the word 'guarantee' in the context of MCS. MCS is a computer program, and as with all computer programs, it's Garbage In, Garbage Out. And even assuming you've made no bad inputs, how good is your random-number generator? I'll bet it's not the world's best. And even the world's best RNG program won't be as random as a well-built and well-operated roulette wheel.

MCS is what you do when you don't have access to the high-quality information we do have access to in this theoretical situation. I trust my probability calculations here over your MCS results.

 

[dom993]

You should not use the word 'guarantee' in the context of MCS. MCS is a computer program, and as with all computer programs, it's Garbage In, Garbage Out. And even assuming you've made no bad inputs, how good is your random-number generator? I'll bet it's not the world's best. And even the world's best RNG program won't be as random as a well-built and well-operated roulette wheel.

MCS is what you do when you don't have access to the high-quality information we do have access to in this theoretical situation. I trust my probability calculations here over your MCS results.

I used "guarantee minimum ending bankroll" as I computed it assuming 10 losses in a row. It is your true guaranteed minimum ending bankroll using those betting strategies.

 

[dom993]

My bad, all of the MC results I posted were using 36-1 for a single number win, instead of 35-1 as mentioned in post #1.

Here are the corrected results:

8%-bankroll Red-16 : ending bankroll 50th percentile +1786, stdev 26483
4%-bankroll Red-16 : ending bankroll 50th percentile +1656, stdev 5208
$50 fixed bet Red-16: ending bankroll 50th percentile +2300, stdev 1763

Fixed $50, stop on 1st win: ending bankroll 50th percentile +2450, stdev 970

 

[dom993]

Now for Red betting:

24%-bankroll Red : ending bankroll 50th percentile +1213, stdev 1342
$93-fixed bet: ending bankroll 50th percentile +1186, stdev 286

Both betting strategies have the same minimum guaranteed ending bankroll of $66, assuming 10 losses in a row.

Do you still think Kelly is optimal for small sample-sets?

 

[dom993]

Anyway, thank you for posting that interesting problem

 

[kut2k2]

Now for Red betting:

24%-bankroll Red : ending bankroll 50th percentile +1213, stdev 1342
$93-fixed bet: ending bankroll 50th percentile +1186, stdev 286

Both betting strategies have the same minimum guaranteed ending bankroll of $66, assuming 10 losses in a row.

Do you still think Kelly is optimal for small sample-sets?

You just proved that it is. Kelly has the higher expected outcome. As to the greater std dev, so what? You've swallowed the economystic kool-aid that standard deviation is a measure of risk. In fact, standard deviation is a measure of uncertainty, and uncertainty can work in your favor as often as it works against you. Things that work in your favor are not 'risk', logically speaking (I.e., outside of ivory-tower economics).

Edit: you're still practicing GIGO. Your standard deviation for Kelly is too big. You must be taking a constant 24% of the initial bankroll ($1000) rather than 24% of the current bankroll. Kelly never takes you into negative bankroll territory.

 

[nonlinear5]

Here are the top 10 strategies, if the goal is to maximize the risk-adjusted return, rather than the absolute return. The score is the average return divided by the standard deviation of returns. The top strategy is to bet 1% of the bankroll on Red. This strategy would probably resonate with most traders, and with the conventional wisdom of "bet a small percentage on the most probable outcome".

	R16	R14	Red	Score
	0	0	1	11.27
	0	1	0	9.01
	1	0	0	7.13
	0	0	2	5.79
	0	2	0	4.73
	0	0	3	3.96
	2	0	0	3.80
	0	1	1	3.72
	0	3	0	3.15
	1	0	1	3.11

Dom993: can you verify this on your end?

 

[kut2k2]

Here are the top 10 strategies, if the goal is to maximize the risk-adjusted return, rather than the absolute return. The score is the average return divided by the standard deviation of returns. The top strategy is to bet 1% of the bankroll on Red. This strategy would probably resonate with most traders, and the conventional wisdom of "bet a small percentage on the most probable outcome".

	R16	R14	Red	Score
	0	0	1	11.27
	0	1	0	9.01
	1	0	0	7.13
	0	0	2	5.79
	0	2	0	4.73
	0	0	3	3.96
	2	0	0	3.80
	0	1	1	3.72
	0	3	0	3.15
	1	0	1	3.11

Dom993: can you verify this on your end?

1% on Red? Sheesh! Why bother? (A) I'll stick with Kelly. (B) You might get better results with the downside deviation rather than the standard deviation. PMPT uses the downside deviation.

 

[Visaria]

I'm pretty much with kut2k2 . You guys don't understand the meaning of the Kelly fraction. To say you would bet the whole $1k is absurd. Maybe because $1k is not much. I think you guys might understand it more if instead of $1k, the bankroll was defined as your entire net worth.

Let's say your entire net worth was $1 million and so the bankroll for this casino offer is your $1 million. Are u now really going to bet the $1 million, your entire net worth, on red? I doubt it.