What is your strategy?

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nonlinear5 said:
My initial result of the Monte-Carlo came out nonsensical. I think that's because the quantity (R * r) / (s^2) does not make sense when r (average return) is negative. I need to figure out how to penalize it correctly, so that the average calculated metric over many trials makes sense.
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Okay, I think I am breaking new ground in computational finance.

Let
CK be the continuous Kelly,
R be the return over a given period,
s be the standard deviation of returns,
r be the average return.

First, I realized that my proposed performance metric:
Performance Metric = CK * R = (R * r) / (s^2)

should actually be this:
Performance Metric = CK * R
CK = R / (s^2)
Performance Metric = CK * R = [R / (s^2)] * R = (R^2) / (s^2)
Notice that (R^2) / (s^2) reduces to Sharpe squared:
(R^2) / (s^2) = (R / s)^2 = Sharpe^2


Next, I set up the Monte-Carlo simulation to maximize Log(Sharpe^2) with respect to the original problem as stated in the beginning of this thread. Here are the top 10 results which maximize Log(Sharpe^2):

Code: 
	R16	R14	Red
	6	5	11
	6	4	11
	6	5	10
	6	4	12
	6	5	12
	6	4	10
	6	4	13
	6	5	9
	6	5	13
	6	4	9

Notice that the best combo, {R16: 6, R14: 5, Red:11} is pretty close to what we collectively calculated as the best in a sense of maximizing the compounded rate of return, which is {R16: 8, R14: 5, Red:11}.

My calculated best, using Log(Sharpe^2), gives a slightly more conservative allocation, because it maximizes the compounded rate of risk-adjusted return.

Then I thought, what if I maximize Log(Sharpe), instead of Log(Sharpe^2)? The results came out identical between the two, predictably so.

So, my performance metric can be reduced to simply R/s, which is the Sharpe ratio.
 
Damn it, I had a bug in the calculation of standard deviation. After fixing it, here is what I get by maximizing Log(Sharpe):

Code: 
	R16	R14	Red
	7	5	11
	7	5	10
	7	5	12
	7	5	9
	7	5	13
	8	5	11
	8	5	10
	8	5	12
	8	5	9
	7	5	8

Based on the results, my point is still valid (I think).
 
One more adjustment needed, I think. In this particular problem, all betting strategies have the same number of betting opportunities. But what if we wanted to compare trading strategies with different number of trades? Since Sharpe increases with the square root of observations, I am adjusting my performance metric to this:

Performance Metric = (R / s) / sqrt(N),
where N is the number of trades.
 
kut2k2 said:
Interesting post found at another website:

" We asked the following question of interview candidates, and gave them twenty minutes to formulate their answer.

You decide to play poker in a casino. But on your way to the poker pit, you are lucky enough to stumble upon a promotional roulette wheel that has been set up differently from a normal wheel. A standard wheel has eighteen black, eighteen red, and one green space (though some have two green spaces). There are still thirty-seven spaces on the wheel, but
- There are four spaces marked Red-16 (the three extra spaces replacing three Black numbers) ,
- There are three spaces marked Red-14 (the two extra spaces replacing two Black numbers).

So there are
Twenty-three spaces marked Red = {Sixteen singles, four Red-16, three Red-14} ,
Thirteen spaces marked Black = {Thirteen singles} ,
One space marked Green = {One Green-0}.

- You have a starting bankroll of $1000, and are allowed to play for ten spins.
- You are only allowed to place one of these types of bets:
-- Any single number (including R-16, R-14, G-0), which pays off at 35-1 ,
-- Red or Black, which pays off at 1-1.
- You are allowed to place a bet for any amount (up to your current bankroll).
- You are allowed to place multiple bets on the same spin.
- You can use any winnings for future bets (e.g., if you bet $1000 on Red on spin one, and win, you now can place up to $2000 worth of bets on spin two.)

What is your strategy?"
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So if you just play Red, you will have a 24.32% advantage over the house, correct?
 
xelite777 said:
So if you just play Red, you will have a 24.32% advantage over the house, correct?
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yep.

if you just play r16 though, you have a 289% advantage over the house.

:D

We're looking for the optimal allocation. I think it is what i said in my previous post.
 
R-14 versus R-16

Or

A Tale of Two Horses

Suppose we had the same scenario as depicted in the OP with the exception that you can't bet on Red (casino management feels a winrate greater than 50% is too attractive).

So you can bet on any of the 34 numbers, you can bet on Black, or you can bet on any combination of those bets.

Q: Is there any combination of {R-14, R-16} that beats a bet on R-16 alone?

A: No. It turns out when you have two mutually exclusive bets and one is a standout, bet exclusively on the standout, either full Kelly or fractional Kelly if you are risk adverse.

The only time you should bet on both mutually exclusive bets is when you can "blend" them with a bet like Red.
 
kut2k2 said:
R-14 versus R-16

Or

A Tale of Two Horses

Suppose we had the same scenario as depicted in the OP with the exception that you can't bet on Red (casino management feels a winrate greater than 50% is too attractive).

So you can bet on any of the 34 numbers, you can bet on Black, or you can bet on any combination of those bets.

Q: Is there any combination of {R-14, R-16} that beats a bet on R-16 alone?

A: No. It turns out when you have two mutually exclusive bets and one is a standout, bet exclusively on the standout, either full Kelly or fractional Kelly if you are risk adverse.

The only time you should bet on both mutually exclusive bets is when you can "blend" them with a bet like Red.
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I don't think that is true.
 
It should be obvious. Let's say you go into the casino as described in the OP but there are two of these special wheels. You can only play either red 14 or red 16 on each. If you play r16 on table 1, you are only allowed to play r14 on the other wheel. Would you just play r16 on table 1 and forget about playing r14 on the other one? Of course not. You would play both wheels. This is scenario is identical to having one wheel where you can either play r14 or r16 or both. You simply allocate the respective kelly fraction to each bet per spin.
 
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