certainty equivalent

[rrwoods]

Originally Posted by assume_R

Firstly, the equation we use is WinRate - Std^(2/(2*KellyFraction*Bankroll))

[ i've added parentheses to this, i think it's equivalent but i want to check -- Std = standard deviation, yes? ]

What if I'm in a game where the Kelly Fraction is hard to define, but I know my risk of ruin based on my bets (and win rate and standard deviation)? Is there a formula that can derive the kelly fraction from that information?

[assume_R]

Originally Posted by rrwoods

i've added parentheses to this, i think it's equivalent but i want to check -- Std = standard deviation, yes?

No, adding parentheses makes the units not work out. It is WinRate - Variance / (2 * KellyFraction * Bankroll)

Originally Posted by rrwoods

What if I'm in a game where the Kelly Fraction is hard to define, but I know my risk of ruin based on my bets (and win rate and standard deviation)? Is there a formula that can derive the kelly fraction from that information?

I don't know personally, but here's a quote from the help file in cvcx which might help you. qfit could answer this better:

"Kelly Factor is an alternate method of specifying risk for those familiar with Kelly theory. A Kelly Factor of 1.0 equates to a risk of 13.5%. A factor of 0.5 means that you will be betting with double the bankroll required for a risk of 13.5%. In a perfect world, this would mean that yours bets would be half as much. But, this is not quite true because of bet simplification and the inability to bet fractions of a dollar. Some professionals play .33, .25 or even lower Kelly factors. These represent substantially less risk; and obviously less income."

[MangoJ]

Originally Posted by assume_R

So I'll try to explain as best as I understand it.

Firstly, the equation we use is WinRate - Std^2/(2*KellyFraction*Bankroll)

So your "Kelly Fraction" is essentially how we are going to define how "risky" you are as a person. The whole concept is, according to wikipedia, " the guaranteed amount of money that an individual would view as equally desirable as a risky asset."

So what it means, is based on your individual kelly fraction (which quantifies your risk, based on how much of a RoR you'd accept), this will tell you if it's in your better interest to take a risky bet (i.e. a gamble), or a risk free bet (i.e. variance = 0).

Put it simply, for a given win rate, etc. of a game, and a given kelly fraction you play at, it would be better to take a job (we are defining a job as risk free) in which you are paid higher than your CE for the game you play.

So let's say you have a win rate of $50, and the CE is $30 (given your bankroll, kelly fraction, etc.) If somebody offered you a full time job for $25/hour, don't take it, because it's better to spend your time with a semi-risky $50/hour. But if they offered you a full time job for $35/hour, take it.

In other words, since Kelly optimizes for log(bankroll), the CE would be

log(bankroll+CE) = <log(bankroll+outcome-stake)>

where outcome is a random variable with certain probability and odds, and <...> is the average over this random variable.

Is that correct ?

[halibut]

Originally Posted by MangoJ

In other words, since Kelly optimizes for log(bankroll), the CE would be

log(bankroll+CE) = <log(bankroll+outcome-stake)>

where outcome is a random variable with certain probability and odds, and <...> is the average over this random variable.

Is that correct ?

Yes, that's what A3 in Kelly FAQ (http://www.bjmath.com/bjmath/kelly/kellyfaq.htm) says.

[MangoJ]

Thanks man. I didn't know the FAQ yet. But then I would rather name it "utility equivalent certainty" to make the origin more clearly.

[blackjack avenger]

The probability of not losing 20% of bank with constant resizing kelly fractions:

kelly 20.0%
1/2 kelly 48.8%
1/3 kelly 67.2%
1/4 kelly 79.0%
1/5 kelly 86.6%
1/6 kelly 91.4%
1/8 kelly 96.4%

One can come close to eliminating ror or risk of drawdown at which point CE=WR almost, we can't eliminate all variance.


happy variance

[MJ1]

Originally Posted by blackjack avenger

The probability of not losing 20% of bank with constant resizing kelly fractions:

kelly 20.0%
1/2 kelly 48.8%
1/3 kelly 67.2%
1/4 kelly 79.0%
1/5 kelly 86.6%
1/6 kelly 91.4%
1/8 kelly 96.4%

One can come close to eliminating ror or risk of drawdown at which point CE=WR almost, we can't eliminate all variance.


happy variance

If CE nearly equals WR, is that bad? Wouldn't that suggest that you are severely under-betting your BR?

It is said that the ratio of CE to WR should be 0.5, when betting optimally.

OTOH, if CE nearly equals WR, wouldn't that mean that the counter should only forgo the opportunity if somebody offers them their EV (or close to it) upfront? This would seem to suggest that CE approaching WR means that the opportunity is quite favorable, thus the counter requires more $$$ to pass-up the opportunity whereas if CE = 0.5 x EV he would require less money to pass up the opportunity.

So to reiterate, is CE approaching EV a good thing or a bad thing? It would seem that one can make a case for either side. Many thanks to anybody that can clarify this point of confusion.

MJ

[paddywhack]

Originally Posted by MJ1

So to reiterate, is CE approaching EV a good thing or a bad thing? It would seem that one can make a case for either side. Many thanks to anybody that can clarify this point of confusion.

MJ

Depends how you look at it.

Say you are betting a very small Kelly Fraction, such that your RoR is miniscule. Therefore your CE is close to your Win Rate. This means that you are pretty much guaranteed that level of income (CE).

Now, as your Kelly Fraction rises, your Win Rate increases but so does your RoR. And your CE is diminished due to the variance involved and the probability of losing a significant portion of the bankroll.

So the question becomes, how much Certainty do you want in that income stream. Higher CE/WR ratio maximizes the certainty that you will make that money. Maximizing the WR by increasing the Kelly Fraction decreases the CE/WR ratio.

I've always heard to maximize the CE as much as possible. You may not want it higher than .75 or .8, otherwise you may only be making peanuts.

[blackjack avenger]

Well said paddywhack
I would add:
Most pros & teams probably play from 1/4 to 1/8 Kelly.
With small fractions of Kelly one does not have to resize bets on losses dramatically. Their long run is a lot shorter.

Probably at/beyond 1/8 Kelly one resizes so little that the long run is basically that of a fixed bettor, but with 0% ror.

It's hard to be conservative with small banks.

Most APs fail due to variance, so eliminate it!

[MJ1]

Originally Posted by paddywhack

Depends how you look at it.

Say you are betting a very small Kelly Fraction, such that your RoR is miniscule. Therefore your CE is close to your Win Rate. This means that you are pretty much guaranteed that level of income (CE).

Thanks for your response. I do have a couple of questions. Why would using a small K-F (kelly-factor) make CE approach WR?

CE = EV - [Var / (2 * Br * K-F)]

Now, as your Kelly Fraction rises, your Win Rate increases but so does your RoR. And your CE is diminished due to the variance involved and the probability of losing a significant portion of the bankroll.

As you increase K-F, wouldn't WR and Variance increase in direct proportion to each other, thereby leaving the ratio of CE/WR intact? I'm not trying to be argumentative, just unclear on how you arrive at these conclusions.

So the question becomes, how much Certainty do you want in that income stream. Higher CE/WR ratio maximizes the certainty that you will make that money.

The more certainty the better! So let me get this straight, the higher the ratio of CE/WR the more 'certain' I am of winning? I thought that ideally the ratio of CE/WR should be 0.5, thereby ensuring that optimal betting is being utilized.

MJ