Kelly criterion for high variance games

Hi all, I am starting to get a bit of a handle/understanding of how the Kelly criterion works. One question I had relates to something I was discussing the other day.

Does the Kelly risk-management approach work equally well for games with high variance? BJ has a very low variance on a per hand basis (1.33 per hand) - how about if your game of choice involved a straight-up bet on roulette (at 35-1) or some other game where, though you know that you have an edge, the payoff might be 100-1, 200-1 or 500-1 when that winning bet does come in.

I was told that in those cases using Kelly may not be appropriate. Could anyone shed light on this question for me?

Many thanks in advance,
Matt21

Kelly criterion maximizes the growth of your bankroll, no matter what the game or the variance.

Normally: Max(EV)
Kelly: Max( log(Bankroll +/- Result) )

The equation that people use to estimate "Result" (which is donned "CE" or "Certainty Equivalent") is EV - Var / (2 * KellyCriteria * Bankroll), but that's only appropriate for blackjack or other low EV games (EV < 10%)

To maximize log(Bankroll + CE) you take into account variance, and hence it is applicable to anything with variance, such as stocks, loans, hi-variance casino games, etc.

Now, you don't actually have to maximize log, but that is pretty standard. See wikipedia for a discussion of other things to maximize.

Here's an example of a 2-pay system (either win or lose).

Fraction of Bankroll to Bet = (P_win * Payout - P_lose) / Payout.

And the CE:
log(Bankroll + CE) = P_win * log(Bankroll + Payout) - P_lose * log(Bankroll - 1.0) which assumes you lose 100% of your original bet if you lose.

Then just rearrange and solve for CE.

Now Kelly betting, however, has some downsides. Yes, it will, in the long run, grow your bankroll the fastest. At the end of Infinite years a person who used Kelly to size his/her bets will have a larger bankroll than anybody else under any other system. That is a mathematical fact.

However, Kelly criteria says that at any point you have 50% chance of losing 50% of your bankroll, and 30% chance of losing 30% of your bankroll. Those swings may be way too much for you to stomach or even desire. Let's say $1 million is worth a lot to you. You might not want a 50% chance of losing $500k. But Warren Buffet wouldn't mind taking that 50% chance of losing $500k because that money doesn't mean the same to him. And what if you die before reaching the "long run"? Wouldn't you want to keep some of your money safe instead of risking it for the "maximum" growth? Kelly betting maximizes CE.

If you accidentally miscalculated your EV or variance (from the P_lose) and accidentally bet twice your Kelly fraction, then your CE is $0! You don't want to risk $1 million when your CE could possibly be $0 instead of the maximum.

That's why people advocate 1/2 kelly or 1/4 kelly. If you bet half of the kelly fraction, then you have way less chance of losing 50% of your bankroll. For example, a full Kelly better has 1/3 chance of losing 50% of his bankroll before doubling, while a 1/2 Kelly better has 1/9 chance of losing 50% of his bankroll before doubling.

Finally, I'll leave you with a quote from Ed Thorp:
"Those individuals or institutions who are long term compounders should consider the possibility of using the Kelly criterion to asymptotically maximize the expected compound growth rate of their wealth. Investors with less tolerance for intermediate term risk may prefer to use a lesser fraction. Longterm compounders ought to avoid using a greater fraction ("overbetting"). Therefore, to the extent that future probabilities are uncertain, long term compounders should further limit their investment fraction enough to prevent a significant risk of overbetting."

Bet size and risk

Optimal bet size is a function of bankroll times edge divided by risk. Risk is a function of variance so you can substitute it in the relationship above. As you can see this function considers variance into bet size.

This is why risk averse play in blackjack raises your optimal return. By lowering risk more than the decrease in EV for any hand matchup the adjustment allows a higher optimal bet to be made which makes more money.

It is a function of it, but not exactly. If you are forced to play -EV hands, or whatever, you actually have to multiply what you stated by a factor "k" to compensate from the lost winrate from -EV hands.

I'm considering playing a video poker machine on a local joint, which according to my calculations is favourable. I also have a corresponding simulation which supports those +EV results.
Now I turned my interest towards risk management, so I used extensive simulation for recording total payout probability estimates, and feed those results into the Kelly log-utility "<log(bankroll + betsize*outcome)>", where "outcome" are the payouts on a unit bet (positive for a win, negative for a loss). Since variance is comparable high (compared to edge, hence N0 is large) I will most surely stick to minimum betsize. Large N0 (as compared to Blackjack) is not much of a problem, since VP can be played reasonable fast.

I'm now trying to figure out the kind of bankroll I would need for that game, before I pursue the actual training part. As the betsize increases only with increments, my approach is to keep the betsize (flatbetting) at minimum, and calculate the bankroll for which my utility increment is largest (i.e. maximizing "<log(bankroll + betsize*outcome)> - log(bankroll)". For this bankroll the minimum bet would be a (generalized) Kelly bet, and hence I will call that bankroll the "Kelly bankroll".

It turns out I'm willing to invest 3 times the Kelly bankroll, so the minimum bet would be 1/3 Kelly bet. My question is, what would be my RoR if flat-betting, and starting with 1/3 Kelly bet ?

There is a second question: From the relation <log(bankroll + betsize*outcome)> = log(bankroll + CEV) it turns out that betting more than twice the Kelly bet has a CEV < 0, and it would be of more utility not to play at all. If I would follow log-utility, I should hence stop playing if my bankroll drops below half of Kelly bank.

Now I'm stuck here: if I would be forced to play on minimum bet, a strict log-utiltiy player would play from 3*Kelly bank until he reaches 1/2 * Kelly bank (in other words, play as long as CEV is positive). But then he is risking only 5/6 of his bankroll, which means his "true" bankroll (subject to risk of total loss) is only 5/6*3 =2.5 * Kelly bank. Now with the same argument starting from a 2.5 * Kelly bank he would only risk the loss of 2*Kelly bank (since he will stop at 0.5*Kelly bank)....

This doesn't make much sense to me. Actually the third question is: What is "bankroll" ? Is this the amount at risk ? Then I run into the dilemma at a "stop-loss" of 0.5 Kelly bank, which reduces the actual amount of risk.
I would guess "bankroll" in the log-utility is the entire wealth, and that 2.5*Kelly bankroll would be my real "risk" (i.e. the bankroll used in RoR calculations).

As I said, I'm really new to bankroll management.. Any help ?

Let's take this one step at a time.

Okay.

Well let's hold on a minute here. You need to figure out if Utility = log(X) is a good utility function for you. Utility(0) = -inf and Utility(1) = 0 might not be good for you. Let's quantify this. Firstly, let's say we want Utility(1/2 * Bankroll) = 0, and Utility(2 * Bankroll) = 1000. The units of utility don't matter, but rather the shape. What you need to do is plot out Utility = A * log(X + B) and decide on a good value for A and B which is in line with your view of risk. Should losing half your bankroll decrease your utility as much as doubling your bankroll? You need to remember that perhaps losing half your bankroll means you won't be able to play anymore (because of the min bet)! Or perhaps Utility(1/3 * Bankroll) = 0 would be better. I do recommend keeping the log function in there, but it's up to you.

Then, you do:

argmax <A * log(Bankroll + Bet * Payout + B) >

to solve for whatever you want to solve for.

Okay so I will need to brush up on my math for RoR when it comes to resizing. But for flat betting until either infinite or bust, you just do RoR = ((1 + EV/Std)/(1 - EV/Std))^(Bankroll/Std) which assumes a Gaussian.

Yes that is very true! Don't overbet! log-utility assumes, however, that you are constantly resizing your bets to account for your fluctuating bankroll.

No, a strict log-utility player would change from $0.25 machines to theoretical $0.125 machines when he reaches 1/2 * Bankroll, to build his bankroll back up.

If you are playing at a certain machine, and won't or can't resize, then anytime your bankroll dips below Kelly Bank, you are essentially overbetting and not maximizing CE anymore. The whole point of Kelly betting assumes you are able to resize. Honestly, if you are playing 1/3 kelly, and resizing as you go up to maintain 1/3 Kelly and maximize CE, you have a very low chance of busting out.

Bankroll is whatever you want to maximize for your personal utility. If you own a house valued at $500k, and you're staking $20k on this video poker machine, you have to ask yourself how much is that $20k worth it terms of utility. Do you want to maximize perhaps < 1/10 * log($520k + Payout * Odds + $30k)>? It's up to you. Plot out different Utility functions

When it comes to RoR, your bankroll should be whatever you're willing to put into the machine until your last penny loses. That's why a true kelly better would always be able to resize and essentially have a 0% RoR, but we have to deal with minimums and such.

See here for some other ways people maximize their utility.

Do you have more questions? I do enjoy discussing this stuff.

Yes it is an interesting topic, since it's totally new for me (and it freaks me out). You can do much on paper and simulations, but the computer don't care about "bad luck".

I have no experience how risky a 1/2, 1/3, 1/4 Kelly bet is, as I have no experience on that. After reading the Wikipedia article, if someone would ask me what amount I would accept in cash upfront, instead of playing it out - I would probably answer with half my EV. I guess that would define my risk tolerance, I will now need to translate into a Kelly fraction if I decide to use a log-like utility function.

I must think/read more about this topic though... I hope we can discuss this further!

sorry for jumping in here as i'm a complete lost puppy far as the maths of Kelly stuff.
but one thing that's always been a question (in the back of my confused mind, lol) has to do with bet resizing and Kelly betting and computer simulations (such as CVCX) for blackjack games.
the seeming paradox to me is that this Kelly betting theory stuff seems to call for resizing how you bet with respect to bankroll......... but like for CVCX i don't see where bets are re sized at all with respect to a given bankroll..... errh, i mean bets are sized according to true counts (ie. advantage) but i don't see where units utilized are resized, doesn't Kelly theory require the ability to re-size units utilized?.... yet CVCX refers to risk of ruin and kelly factors........
i mean heck, say to your mind the only acceptable unit for you risk utility wise is the table minimum, errhh well then for you, you could never re size your unit bet downward, so how the heck could CVCX associate such a unit and table minimum situation with Kelly betting theory?
:whip:

I'm not sure if cvcx actually takes into account resizing when computing RoR when the "Kelly Factor" option is turned on. When you click "Manually Adjust Min Bet" I see that it removes the box for Kelly betting. So honestly Norm would have to answer that question. Not sure if cvcx does it, and I'm actually not sure how to calculate RoR when you are resizing. Maybe it's in blackjack attack somewhere that I haven't seen yet...

Sid Browne

http://www2.gsb.columbia.edu/divisions/dro/browne_research.html (Archive copy)

CAN YOU DO BETTER THAN KELLY IN THE SHORT RUN ?
Sid Browne 
Columbia University
Published in Finding the Edge: Mathematical Analysis of Casino Games, Pages 215 - 231,
Vancura, O., Cornelius, J., Eadington, W.R. Eds. Institute for the Study of Gambling and
Commercial Gaming, University of Nevada, 2000.

OMG!! - raise a thread and then forget you raised it! checkk back a month later and see all the dicsussion!! Thanks Assume_R for your helpful replies......cz